Advances in Chemical Physics, Vol.133, Part B. Fractals, Diffusion, and Relaxation (Wiley 2006)

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

Edited By WILLIAM T. COFFEY AND YURI P. KALMYKOV Series Editor STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

AN INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS A SPECIAL VOLUME OF ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

EDITORIAL BOARD

BRUCE J. BERNE, Department of Chemistry, Columbia University, New York, New York, U.S.A. KURT BINDER, Institut fu¨r Physik, Johannes Gutenberg-Universita¨t Mainz, Mainz, Germany A. WELFORD CASTLEMAN, JR., Department of Chemistry, The Pennsylvania State University, University Park, Pennsylvania, U.S.A. DAVID CHANDLER, Department of Chemistry, University of California, Berkeley, California, U.S.A. M. S. CHILD, Department of Theoretical Chemistry, University of Oxford, Oxford, U.K. WILLIAM T. COFFEY, Department of Microelectronics and Electrical Engineering, Trinity College, University of Dublin, Dublin, Ireland F. FLEMING CRIM, Department of Chemistry, University of Wisconsin, Madison, Wisconsin, U.S.A. ERNEST R. DAVIDSON, Department of Chemistry, Indiana University, Bloomington, Indiana, U.S.A. GRAHAM R. FLEMING , Department of Chemistry, University of California, Berkeley, California, U.S.A. KARL F. FREED, The James Franck Institute, The University of Chicago, Chicago, Illinois, U.S.A. PIERRE GASPARD, Center for Nonlinear Phenomena and Complex Systems, Brussels, Belgium ERIC J. HELLER, Institute for Theoretical Atomic and Molecular Physics, HarvardSmithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A. ROBIN M. HOCHSTRASSER, Department of Chemistry, The University of Pennsylvania, Philadelphia, Pennsylvania, U.S.A. R. KOSLOFF, The Fritz Haber Research Center for Molecular Dynamics and Department of Physical Chemistry, The Hebrew University of Jerusalem, Jerusalem, Israel RUDOLPH A. MARCUS, Department of Chemistry, California Institute of Technology, Pasadena, California, U.S.A. G. NICOLIS, Center for Nonlinear Phenomena and Complex Systems, Universite´ Libre de Bruxelles, Brussels, Belgium THOMAS P. RUSSELL, Department of Polymer Science, University of Massachusetts, Amherst, Massachusetts, U.S.A. DONALD G. TRUHLAR , Department of Chemistry, University of Minnesota, Minneapolis, Minnesota, U.S.A. JOHN D. WEEKS, Institute for Physical Science and Technology and Department of Chemistry, University of Maryland, College Park, Maryland, U.S.A. PETER G. WOLYNES, Department of Chemistry, University of California, San Diego, California, U.S.A.

FRACTALS, DIFFUSION, AND RELAXATION IN DISORDERED COMPLEX SYSTEMS ADVANCES IN CHEMICAL PHYSICS VOLUME 133 PART B

Edited By WILLIAM T. COFFEY AND YURI P. KALMYKOV Series Editor STUART A. RICE Department of Chemistry and The James Franck Institute The University of Chicago Chicago, Illinois

AN INTERSCIENCE PUBLICATION

JOHN WILEY & SONS, INC.

Copyright # 2006 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, or online at http://www.wiley.com/go/permission. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic formats. For more information about Wiley products, visit our web site at www.wiley.com. Library of Congress Catalog Number: 58-9935 ISBN-13 ISBN-10 ISBN-13 ISBN-10 ISBN-13 ISBN-10

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CONTRIBUTORS TO VOLUME 133 ELI BARKAI, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel ALEXANDER BRODIN, Experimentalphysik II, Universita¨t Bayreuth, D 95440 Bayreuth, Germany THOMAS BLOCHOWICZ, Institute fur Festko¨rperphysik, Technische Universita¨t Darmstadt, D 64289 Darmstadt, Germany SIMONE CAPACCIOLI, Dipartimento di Fisica and INFM, Universita` di Pisa, I-56127, Pisa, Italy; and CNR-INFM Center ‘‘SOFT: Complex Dynamics in Structured Systems,’’ Universita` di Roma ‘‘La Sapienza,’’ I-00185 Roma, Italy RICCARDO CASALINI, Naval Research Laboratory, Washington, DC 20375, USA; and Chemistry Department, George Mason University, Fairfax, Virginia 20030, USA ALEKSEI V. CHECHKIN, Institute for Theoretical Physics, National Science Center, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine WILLIAM T. COFFEY, Department of Electronic and Electrical Engineering, School of Engineering, Trinity College, Dublin 2 Ireland YURI FELDMAN, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel VSEVOLD Y. GONCHAR, Institute for Theoretical Physics, National Science Center, Kharkov Institute of Physics and Technology, Kharkov 61108, Ukraine PAOLO GRIGOLINI, Department of Physics, University of North Texas, Denton, Texas, 76203 USA; and Department of Physics, University of Pisa, Pisa, Italy YURI P. KALMYKOV, Laboratoire de Mathe´matiques et Physique des Syste`mes, Universite de Perpignan, 66860 Perpignan Cedex, France JOSEPH KLAFTER, School of Chemistry, Tel Aviv University, Tel Aviv 69978, Israel FRIEDRICH KREMER, Universita¨t Leipzig, Fakultat fu¨r Physik und Geowissenschaften, 04103 Leipzig, Germany

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MASARU KUNO, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel GENNADY MARGOLIN, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46556, USA RALF METZLER, NORDITA–Nordic Institute for Theoretical Physics, DK-2100 Copenhagen Danish Denmark KIA L. NGAI Naval Research Laboratory, Washington, DC 20375, USA VITALY V. NOVIKOV, Odessa National Polytechnical University, 65044 Odessa, Ukraine MARIAN PALUCH, Institute of Physics, Silesian University, 40-007 Katowice, Poland NOE´LLE POTTIER, Matie`re et Syste`mes Complexes, UMR 7057 CNRS and Universite´ Paris 7—Denis Diderot, 75251 Paris Cedex 05, France VLADIMIR PROTASENKO, Department of Chemistry and Biochemistry, Notre Dame University, Notre Dame, Indiana 46566, USA; and Department of Physics, Bar Ilan University, Ramat Gan 52900, Israel ALEXANDER PUZENKO, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel C. M. ROLAND, Naval Research Laboratory, Washington, DC 20375, USA ERNST A. RO¨SSLER, Experimentalphysik II, Universita¨t Bayreuth, D 95440 Bayreuth, Germany YAROSLAV RYABOV, Department of Applied Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel. Maryland Center of Biomolecular Structure and Organization, University of Maryland, College Park, Maryland 20742-3360, USA ANATOLI SERGHEI, Fakultat fur Physik und Geowissenschaften, Universita¨t Leipzig, 04103 Leipzig, Germany SERGEY V. TITOV, Institute of Radio Engineering and Electronics of the Russian Academy of Seciences, Fryazino, Moscow Region, 141190, Russian Federation BRUCE J. WEST, Mathematical & Information Sciences Directorate, U.S. Army Research Office, Research Triangle Park, North Carolina 27709, USA

INTRODUCTION Few of us can any longer keep up with the flood of scientific literature, even in specialized subfields. Any attempt to do more and be broadly educated with respect to a large domain of science has the appearance of tilting at windmills. Yet the synthesis of ideas drawn from different subjects into new, powerful, general concepts is as valuable as ever, and the desire to remain educated persists in all scientists. This series, Advances in Chemical Physics, is devoted to helping the reader obtain general information about a wide variety of topics in chemical physics, a field that we interpret very broadly. Our intent is to have experts present comprehensive analyses of subjects of interest and to encourage the expression of individual points of view. We hope that this approach to the presentation of an overview of a subject will both stimulate new research and serve as a personalized learning text for beginners in a field. STUART A. RICE

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PREFACE Fractals, Diffusion, and Relaxation in Disordered Complex Systems, which is the subject of the present anthology, may be said to have evolved in two stages: (1) in the course of conversations with Stuart Rice during a remarkably pleasant lunch at the University of Chicago following the Indianapolis meeting of the American Physical Society in March 2002 and (2) following the Royal Irish Academy Conference on Diffusion and Relaxation in Disordered Fractal Systems held in Dublin in September 2002 [1]. During each of these meetings, the necessity of reviewing the progress both experimental and theoretical which has been made in our understanding of physical systems with relaxation differing substantially from exponential behavior was recognized. Furthermore, it was considered that the Advances in Chemical Physics, in line with its stated aspirations and with its wide circulation, would provide an ideal means of attaining this goal. For the best part of three centuries the fractional calculus constituted a subject area mainly of interest to mathematicians. Indeed many great mathematicians such as Leibniz, L’Hoˆpital, Euler, Fourier, Abel, Liouville, Weierstrass, Riemann, Letnikov, Wiener, Le´vy, and Hardy, to name but a few, have contributed to its development (for a historical survey see Ref. 2). In contrast, applications of fractional calculus in other branches of science have appeared only sporadically—for example, the application to the propagation of disturbances on transmission lines in the context of Heaviside’s operational calculus and Kohlrausch’s stretched exponential decay law [2,14]. However, the situation radically changed toward the end of the last century following the appearance of the famous books of Benoit Mandelbrot on fractals [3]. Thus, over the past few decades, the fractional calculus has no longer been restricted to the realm of pure mathematics and probability theory [2,4]. Indeed many scientists have discovered that the behavior of a variety of complex systems (such as glasses, liquid crystals, polymers, proteins, biopolymers, living organisms, or even ecosystems) may be successfully described by fractional calculus; thus it appears that complex systems governed by fractional differential equations play a dominant role in both the exact and life sciences [5]. In particular in the context of applications in physics and chemistry, the fractional calculus allows one to describe complex systems exhibiting anomalous relaxation behavior in much the same way as the normal relaxation of simple systems [6]. Examples include charge transport in amorphous semiconductors, the spread ix

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of contaminants in underground water, relaxation in polymer systems, and tracer dynamics in both polymer networks and arrays of convection rolls, and so on [6]. In general, the diffusion and relaxation processes in such complex systems no longer follow Gaussian statistics so that the temporal evolution of these systems deviates from the corresponding standard laws (where the mean-square displacement of a particle is proportional to the time between observations) for normal diffusion such as exhibited by classical Brownian particles. Furthermore, following the development for complex systems of higher-order experimental resolutions or via a combination of different probe techniques, the deviations from the classical diffusion and relaxation laws have become ever more apparent. Thus the ever larger data windows that are becoming accessible bring ever more refinement to the experimental data [5], with the result that fractional diffusion and kinetic equations have become extremely powerful tools for the description of anomalous relaxation and diffusion processes in such systems. In the present anthology we have tried to present a comprehensive account of the present state of the subject. It is obvious, however, that we cannot survey completely such an enormous area of modern research, and inevitably many important topics will have been omitted. In order to remedy this defect, we remark that the interested reader can find additional information concerning anomalous diffusion and relaxation and applications of fractional calculus in physics, chemistry, biology, radio engineering, and so on, in various review articles and books, a selection of which is given in Refs. 5–23. Roughly speaking, the contents of the two-volume anthology may be divided into four experimental and seven theoretical chapters that may be described as follows. Chapter 1, ‘‘Dielectric Relaxation Phenomena in Complex Materials,’’ by Y. Feldman, A. Puzenko, and Y. Ryabov, concerns dielectric spectroscopy studies of the structure, dynamics, and macroscopic behavior of materials, which may broadly be described by the generic term complex systems. Complex systems constitute an almost universal class of materials including associated liquids, polymers, biomolecules, colloids, porous materials, doped ferroelectric crystals, and so on. These systems are characterized by a new ‘‘mesoscopic’’ length scale, intermediate between molecular and macroscopic. The mesoscopic structures of complex systems typically arise from fluctuations or competing interactions and exhibit a rich variety of static and dynamic behavior. This growing field is interdisciplinary; it complements solid-state and statistical physics, and it overlaps considerably with chemistry, chemical engineering, materials science, and even biology. A common theme in complex systems is that while such materials are disordered on the molecular scale and homogeneous on the macroscopic scale, they usually possess a certain degree of order on a intermediate, or mesocopic, scale due to

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the delicate balance of interaction and thermal effects. The authors demonstrate how dielectric spectroscopy studies of complex systems can be applied to determine both their structures and dynamics, how they both arise, and how both may influence the macroscopic behavior. The glass transition is an unsolved problem of condensed mater physics. This question is addressed in chapter 2 by T. Blochowicz, A. Brodin, and E. Ro¨ssler, entitled ‘‘Evolution of the Dynamic Susceptibility in Supercooled Liquids and Glasses.’’ The emergence of the mode coupling theory of the glass transition has prompted the compilation of a large body of information on the glass transition phenomenon as well as on the glassy state that is reviewed in this contribution. Thus this chapter focuses on describing the evolution of the dynamic susceptibility; that is, its characteristic changes while supercooling a molecular liquid. The authors provide information on the relevant molecular dynamics, and a comparison between experiment and theory is given. The phenomenon is essentially addressed from an experimental point of view, by simultaneously discussing the results from three different probe techniques, namely quasi-elastic light scattering, dielectric spectroscopy, and nuclear magnetic resonance spectroscopy. The application of each of the three methods allows one to investigate the dynamics in the 0- to 1-THz frequency range. The crossover from liquid dynamics at the highest temperatures to glassy dynamics at moderate temperatures as well as the crossover to solid-state behavior at the lowest temperatures near the glass transition temperature, is described in detail. In addition, some remarks on the evolution of the susceptibility down to cryogenic temperatures are given. The lesson to be drawn from this contribution is that an understanding of the dynamics of disordered systems can only be achieved by joint application of the various techniques covering a large frequency range. In many complex systems such as glasses, polymers, and proteins, temporal evolutions differ as we have seen from the conventional exponential decay laws (and are often much slower). Very slowly relaxing systems remain out of equilibrium over very long times, and they display aging effects so that the time scale of response and correlation functions increases with the age of the system (i.e., the time elapsed since its preparation): Older systems relax more slowly than younger ones. Chapter 3 by N. Pottier, entitled ‘‘Slow Relaxation, Anomalous Diffusion, and Aging in Equilibrated or Nonequilibrated Environments,’’ describes recent developments in the physics of slowly relaxing out of equilibrium systems. Questions specifically related to out-of-equilibrium dynamics, such as (1) aging effects and (2) their description by means of an effective temperature, are discussed in the framework of a simple model. A system well adapted to the analysis of these concepts is a diffusing particle in contact with an

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environment, which is either itself in equilibrium (thermal bath) or out of equilibrium (aging medium). In an aging environment, the diffusing particle acts as a thermometer: Independent measurements, at the same age of the medium, of the particle mobility and mean-square displacement yield the effective temperature of the medium. Time-dependent fluctuations in the spectra of individual molecules appear in many single-molecule experiments. Since the dynamics of a single molecule is typically strongly coupled to the dynamics of the local environment of that molecule, it is not unusual that the time trace of the intensity of a single molecule should exhibit stochastic behavior. It is frequently assumed that the process of photon emission is stationary and ergodic. In contrast, the correlation function of single nanocrystals (or quantum dots) is nonstationary and nonergodic; thus these systems exhibit statistical behaviour very different from other single emitting objects. In this context, G. Margolin, V. Protasenko, M. Kuno, and E. Barkai in Chapter 4, entitted ‘‘Power-Law Blinking Quantum Dots: Stochastic and Physical Models.’’ discuss simple models that may explain the nonergodic behaviour of nanocrystals. The authors use a stochastic model to discuss statistical properties of blinking nanocrystals and to illustrate the concept of non ergodicity and aging. They study intensity correlation functions and discuss ensemble average correlation functions for both capped and uncapped nanocrystals. Different modes of aging appear; that is, a nonvanishing dependence of the correlation functions on the age of the system exists, and this dependence has different functional forms in each of the two cases. The authors also discuss nonergodicity of intensity fluctuations for capped nanocrystals, comparing trajectory (time) and ensemble intensity mean values and correlation functions. They analyze experimental data and show that due to weak ergodicity breaking, the time-averaged intensity of blinking dots is a random variable even for long measurement times. The distribution of the time-averaged intensity is not centered around the ensemble-averaged intensity; instead the authors find very large fluctuations, in good agreement with the predictions of stochastic theory. The main purpose of Chapter 5 by P. Grigolini, ‘‘The Continuous-Time Random Walk Versus the Generalized Master Equation,’’ is to show that the interpretation of certain experimental results concerning the spectroscopy of blinking quantum dots and single molecules requires new theoretical methods. He argues that traditional methods of statistical mechanics, based on either the quantum or the classical Liouville equation—and thus based on densities—must be replaced by the continuous-time random walk model introduced by Montroll and Weiss in 1965. To justify this change, the author reviews the recent work done in deriving Le´vy anomalous diffusion from a Liouville equation formalism. He demonstrates that this method, which is

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satisfactory for Poisson statistics, cannot reproduce the numerical and experimental results in the non-Poisson case. Using the continuous-time random walk formalism, the author determines the generalized master equations that should arise from the Liouville method and also proves that such equation are characterized by aging. He shows that, in spite of making a given generalized master equation totally equivalent to the continuoustime random walk picture, an external field perturbing the generalized master equations yields effects distinctly different from those obtained by applying the same external field in the continuous-time random walk picture. Here there is no need for the reader to know a priori the projection approach to the generalized master equations, and the fundamentals of continuous-time random walk calculus needed are included in the chapter. Thus this chapter aims at being an elementary introduction to these techniques and thus will be accessible to both researchers and graduate and undergraduate students with no special knowledge of the formalism. In Chapter 6, entitled ‘‘Fractal Physiology, Complexity, and the Fractional Calculus,’’ B. J. West concentrates on describing the new area of medicine called fractal physiology and focuses on the complexity of the human body and the characterization of that complexity through fractal measures. It is demonstrated that not only various anatomical structures within the human body—such as the convoluted surface of the brain, the lining of the bowel, neural networks, and placenta—are fractal, but also the output of many other dynamical physiological systems. For example, the time series for the interbeat intervals of the heart, interbreath intervals, and interstride intervals have all been shown to be fractal or multifractal statistical processes. Consequently, the fractal dimension turns out to be a significantly better indicator of health than more traditional measures, such as heart rate, breathing rate, and average gait. The observation that human physiology is fractal was first made by the author and his collaborators in the 1980s, based on the analysis of the data sets mentioned above. Subsequently, it was determined that the appropriate methodology for describing the dynamics of fractal time series is the fractional calculus, using either the fractional Langevin equation or the fractional diffusion equation, both of which are discussed in a biomedical context. The general goal of this chapter is to understand how complex phenomena in human physiology can be faithfully described using dynamical models involving fractional stochastic differential equations. Now various structures—for example, aggregates of particles in colloids, certain binary solutions, polymers, composites, and so on—can be conceived as fractal. Materials with a fractal structure belong to a wide class of inhomogeneous media and may exhibit properties differing from those of uniform matter, like crystals, ordinary composites, or homogeneous

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fluids. Thus in Chapter 7 by V. Novikov, entitled ‘‘Physical Properties of Fractal Structures,’’ hierarchical structure models are applied to study the dielectric, conductive, and elastic properties of inhomogeneous media with a chaotic, fractal structure. The power of the fractional calculus is demonstrated using as example the derivation of certain known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimensionality of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of inhomogeneous media exhibiting nonexponential relaxation behavior is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times shorter than the relaxation time at the lowest (primitive) self-similarity level, the relaxation should be of a classical, Debye-like type, irrespective of the pattern of nonclassical relaxation at longer times. The material described in this chapter can be used in the analysis of the frequency dependence of the dielectric permittivity, the conductivity, and the elastic parameters of various materials. Providing both a critical evaluation of characterization methods and a quantitative description of composition-dependent properties, the material surveyed is of particular interest to researchers in materials and polymer science. Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled ‘‘Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems,’’ provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker–Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole–Cole, Cole–Davidson, and Havriliak–Negami equations of anomalous dielectric relaxation from a microscopic model based on a

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kinetic equation, just as the Debye model. These kinetic equations are obtained from a generalization of the noninertial Fokker–Planck equation of conventional Brownian motion to fractional kinetics governed by the Cole– Cole, Cole–Davidson, and Havriliak–Negami relaxation mechanisms. As particular examples, approximate solutions of the fractional diffusion equation are derived for anomalous noninertial rotational diffusion in various potentials. It is shown that a knowledge of the effective relaxation times for normal rotational diffusion is sufficient to predict accurately the anomalous dielectric relaxation behavior of the system for all time scales of interest. Furthermore, the inertia-corrected Debye model of rotational Brownian motion of polar molecules is generalized to fractional dynamics (anomalous diffusion) using the fractional Klein–Kramers equation. The result can be considered as a generalization of the solution for the normal Brownian motion in a periodic cosine potential to fractional dynamics (giving rise to anomalous diffusion) and also represents a generalization of Fro¨hlich’s model of relaxation over a potential barrier. Chapter 9 by A.V. Chechkin, V.Y. Gonchar, J. Klafter, and R. Metzler, entitled ‘‘Fundamentals of Le´vy Flight Processes,’’ reviews recent developments in the fractional dynamics of Le´vy flights under the influence of an external force field and for non trivial boundary conditions—in particular, first passage time problems. The Le´vy flights are formulated in terms of a space-fractional Fokker–Planck equation, in which the usual Laplacian is replaced by the Riesz–Weyl fractional operator. The authors discuss the intriguing behavior of this type of random process in external fields; for example, for potentials of harmonic or softer types, the variance diverges and the stationary solution has the same Le´vy index as the external noise. In contrast, for steeper than harmonic potentials, the solution leaves the basin of attraction of Le´vy stable densities, and multimodal structures appear. The first passage time problem of Le´vy flights exhibits a universal character in the sense that the force-free first passage time density exhibits Sparre Andersen universality. This is discussed in detail, and it is compared to the problem of first arrival in Le´vy flights. The authors also address the question of the validity of Le´vy flights as a description of a physical system due to their diverging variance—for example, arguing that for a massive particle, dissipative nonlinearities may lead to a finite variance. Now on decreasing temperature or increasing pressure a noncrystallizing liquid will vitrify; that is, the structural relaxation time,  , becomes so long that the system cannot attain an equilibrium configuration in the time available. Such theories as exist, including the well-known free volume and configurational entropy models, explain the glass transition by invoking a single quantity governing  . Thus the dispersion of the structural relaxation is either so not addressed at all or else derived merely as afterthought and so

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is independent of  . Thus, in these models the time dependence of the relaxation process bears no fundamental relation to   and its dynamical properties. In Chapter 10 by K. Ngai, R. Casalini, S. Capaccioli, M. Paluch, and C. M. Roland, entitled ‘‘Dispersion of the Structural Relaxation and the Vitrification of Liquids,’’ the authors show from disparate experimental data that the dispersion (i.e., time dependence of the relaxation time or distribution of relaxation times) of the structural relaxation originating from many-molecule dynamics is a fundamental parameter governing   and so controls its various properties. Large bodies of experimental data are presented or cited in order to support this conclusion in a convincing fashion. It appears that without considering dispersion as a fundamental physical entity at the outset of any theory of vitrification, many general experimental features of the molecular dynamics of supercooled liquids will remain unexplained. Glass-forming systems have been studied for decades using a variety of experimental tools measuring microscopic or macroscopic physical quantities. Thus the conjecture that the glass transition has an inherent length scale has led to numerous studies on confined glassy dynamics. In this context, thin polymer films are of special interest. Chapter 11 by F. Kremer and A. Serghei, entitled ‘‘Molecular Dynamics in Thin Polymer films,’’ contributes to this discussion. The authors address from an experimental point of view many interesting topics such as ensuring both reproducible preparation and reproducible measurements of thin polymer films, the influence of the molecular architecture of polymers on their dynamics in thin layers, the effect of confinement in thin polymer films giving rise to novel dynamic modes, methods for the determination of the glass transition temperature, and so on. The Guest Editors and authors are very grateful to the Series Editior, Stuart A. Rice, for the opportunity to produce this anthology. We would like to thank Dr. Sergey V. Titov and Ms Christine Moore for their excellent help in the preparation of the manuscripts. We would also like to thank Dr. David Burns and Dr. Michael Milligan of USAF, EOARD London for facilitating Window on Science visits to the United States during the course of which this project was conceived, as well as Professor Werner Blau of Trinity College Dublin for financial support from the HEA Ireland PRTLI Nanomaterials project and the Trinity College Dublin Trust. November 2005 WILLIAM T. COFFEY YURI P. KALMYKOV Dublin and Perpignan

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References 1. Proceedings available as special issue (Diffusion and Relaxation in Disordered Complex Systems) of Journal of Molecular Liquids 114, No. 1–3 (2004), guest editor W. T. Coffey. 2. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, London, 1974. 3. B. B. Mandelbrot, Fractals: Form, Chance and Dimension, Freeman, San Francisco, 1982; The Fractal Geometry of Nature, Freeman, San Francisco, 1982. 4. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993. 5. I. Sokolov, J. Klafter, and A. Blumen, Fractional Kinetics. Physics Today, Nov. 2002, p. 48. 6. R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339, 1 (2000). 7. C. J. F. Bo¨ttcher and P. Bordewijk, Theory of Electric Polarization, Vol. 2, Elsevier, Amsterdam, 1973. 8. J.-P. Bouchaud and A. Georges, Anomalous diffusion in disordered media: Statistical mechanisms, models and physical applications. Phys. Rep. 195, 127 (1990). 9. N. G. McCrum, B. E. Read, and G. Williams, Anelastic and Dielectric Effects in Polymeric Solids Dover, New York, 1991. 10. M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Strange Kinetics, Nature 363, 31 (1993). 11. R. Richert and Blumen, eds., Disorder Effects on Relaxation Processes, Springer-Verlag, Berlin, 1994. 12. A. Bunde and S. Havlin, eds., Fractals in Disordered Systems, Springer-Verlag, Berlin, 1996. 13. A. K. Jonscher, Universal Relaxation Law, Chelsea Dielectric Press, London, 1996. 14. W. Paul and J. Baschnagel, Stochastic Processes from Physics to Finance, Springer-Verlag, Berlin, 1999. 15. R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientific, River Edge, N J, 2000. 16. E. Donth, The Glass Transition: Relaxation Dynamics in Liquids and Disordered Materials, Springer-Verlag, Berlin, 2000. 17. G. M. Zaslavsky, Chaos, fractional kinetics and anomalous transport. Phys. Rep. 371, 461 (2002). 18. F. Kremer and A. Scho¨nhals, eds., Broadband Dielectric Spectroscopy, Springer, Berlin, 2002. 19. A. A. Potapov, Fractals in Radiophysics and Electromagnetic Detections, Logos, Moscow, 2002. 20. B. J. West, M. Bologna, and Grigonlini, Physics of Fractal Operators, Springer, New York, 2003. 21. W. T. Coffey, Y. P. Kalmykov, and J. T. Waldron, The Langevin Equation, 2nd ed, World Scientific, Singapore, 2004. 22. R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent development in the description of anomalous transport by fractional dynamics. J. Phys. A: Math Gen 37, 1505 (2004) 23. J. Klafter and I. Sokolov, Anomalous diffusion spreads its wings. Physics Today, Aug. 2005 p. 29.

CONTENTS PART B CHAPTER 6 FRACTAL PHYSIOLOGY, COMPLEXITY, FRACTIONAL CALCULUS

AND THE

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By Bruce J. West CHAPTER 7

PHYSICAL PROPERTIES OF FRACTAL STRUCTURES

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By Vitaly V. Novikov CHAPTER 8 FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS

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By William T. Coffey, Yuri P. Kalmykov, and Sergey V. Titov CHAPTER 9

FUNDAMENTALS OF LE´VY FLIGHT PROCESSES

439

By Aleksei V. Chechkin, Vsevolod Y. Gonchar, Joseph Klafter, and Ralf Metzler CHAPTER 10 DISPERSION OF THE STRUCTURAL RELAXATION AND THE VITRIFICATION OF LIQUIDS

497

By Kia L. Ngai, Riccardo Casalini, Simone Capaccioli, Marian Paluch, and C.M. Roland CHAPTER 11

MOLECULAR DYNAMICS IN THIN POLYMER FILMS

595

By Friedrich Kremer and Anatoli Serghei

Author Index

633

Subject Index

671

xix

CONTENTS PART A CHAPTER 1

DIELECTIC RELAXATION PHENOMENA IN COMPLEX MATERIALS

1

By Yuri Feldman, Alexander Puzenko, and Yaroslav Ryabov CHAPTER 2 EVOLUTION OF THE DYNAMIC SUSCEPTIBILITY IN SUPER COOLED LIQUIDS AND GLASSES

127

By Thomas Blochowicz, Alexander Brodin, and Ernst Rossler CHAPTER 3 SLOW RELAXATION, ANOMALOUS DIFFUSION, AND AGING IN EQUILIBRATED OR NONEQUILIBRATED ENVIRONMENTS

257

By Noe¨lle Pottier CHAPTER 4 POWER-LAW BLINKING QUANTUM DOTS: STOCHASTIC PHYSICAL MODELS

AND

327

By Gennady Margolin, Vladimir Protasenko, Masaru Kuno, and Eli Barkai CHAPTER 5 THE CONTINUOUS-TIME RANDOM WALK VERSUS GENERALIZED MASTER EQUATION

THE

357

By Paolo Grigolini

Author Index

475

Subject Index

513

xxi

CHAPTER 6 FRACTAL PHYSIOLOGY, COMPLEXITY, AND THE FRACTIONAL CALCULUS BRUCE J. WEST Mathematical & Information Sciences Directorate, U.S. Army Research Office, Research Triangle Park, NC 27709, USA

CONTENTS I. Introduction II. Scaling in Physiological Time Series A. Allometric Aggregation Data Analysis B. Fractal Heartbeats C. Fractal Breathing D. Fractal Gait E. Fractal Neurons III. Dynamical Models of Scaling A. Scaling in Time Series 1. Simple Random Walks and Scaling 2. Fractional Random Walks and Scaling 3. Various Inverse B. Dichotomous Fluctuations with Memory 1. The Exact Solution 2. Early Time Behavior 3. Late Time Behavior C. Fractals, Multifractals, and Data Processing 1. Multifractal Special 2. Diffusion Entropy Analysis (DEA) IV. Fractional Dynamics A. Fractional Calculus 1. Derivative of a Fractal Function 2. Fractional Brownian Motion B. Fractional Langevin Equations 1. Physical/Physiological Models C. Fractional Diffusion Equations D. Langevin Equation with Le´vy Statistics

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

1

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bruce j. west

V. Summary, Conclusions, and Speculations References

I.

INTRODUCTION

The title of this chapter targets two central themes. The first theme is a modern view of physiology that explicitly takes into account the complexity of living systems, since physiology is that branch of biology that deals with the functions and activities of life and of living matter such as organs, tissues, or cells. Complexity in this context incorporates the recent advances in physiology concerned with the applications of the concepts ranging from fractal geometry, fractal statistics, and nonlinear dynamics to the formation of a new kind of understanding in the life sciences. The second theme has to do with a parallel development on understanding the dynamics of fractal processes. For a number of years the study of fractals was restricted to the determination of the fractal dimension of structure—in particular, the static structure of objects and the stationary structure of time series. However, now we explore the dynamics of fractal processes using the fractional calculus, applying this dynamical approach to both regular and stochastic processes. In our discussion we motivate the applications of the fractional calculus using physiological time series. The fractal concept was formally introduced into the physical sciences by Benoit Mandelbrot over 20 years ago and has since then captured the imagination of a generation of scientists. Mandelbrot had, of course, been working on the development of the idea for over a decade before he was finally willing to expose his brainchild to the scrutiny of the scientific community at large. His monograph [1] brought together mathematical, experimental, and physical arguments that undermined the traditional picture of the physical world. It has been accepted that celestial mechanics and physical phenomena are, by and large, described by smooth, continuous, and unique functions, since before the time of Lagrange (1736–1813). This belief is part of the conceptual infrastructure of the physical sciences. The changes in physical processes are modeled by systems of dynamical equations, and the solutions to such equations are continuous and differentiable at all but a finite number of points. Therefore the phenomena being described by these equations were thought to have these properties of continuity and differentiability as well. Thus, the solutions to the equations of motion such as the Euler–Lagrange equations, or Hamilton’s equations, are analytic functions, and such functions were thought to represent physical phenomena in general. At the turn of the last century, there were two opposing points of view in physics regarding continuity: those held by the atomists and those held by the anti-atomists. The latter camp believed in the continuity of nature and saw no reason why matter should stop being divisible at the level of the atom and should, they reasoned, continue indefinitely to smaller and smaller scales. The

fractal physiology, complexity, and the fractional calculus

3

atomists, on the other hand, with the successes of the periodic table and the kinetic theory of gases, had Boltzmann as their chief proponent. Boltzmann was such an ‘‘extreme’’ atomist that he did not even accept the continuity of time. In his St. Louis lecture in 1904 he stated [2] the following: Perhaps our equations are only very close approximations to average values that are made up of much finer elements and are not strictly differentiable.

It is the atomist’s view of the classical microscopic world and its influence on the macroscopic world that we endorse in this chapter. From the phenomenological side, Mandelbrot called the accuracy of the traditional perspective into question, by pointing to the failure of the equations of physics to explain such familiar phenomena as turbulence and phase transitions. In his books [1,3], Mandelbrot catalogued and described dozens of physical, social, and biological phenomena that cannot be properly described using the familiar tenets of dynamics from physics. The functions required to explain these complex phenomena have properties that for 100 years had been thought to be mathematically pathological. Mandelbrot argued that, rather than being pathological, these functions capture essential properties of reality and are therefore better descriptors of the physical world than the traditional analytic functions of theoretical physics. Living organisms are immeasurably more complicated than inanimate objects, so we do not have available fundamental laws and principles governing biological phenomena equivalent to those in physics. Some may object to this harsh characterization, but there are no equivalents of Newton’s Laws, Maxwell’s equations, and Boltzmann’s Principle in physiology. Part of the goal of biophysics, in fact, is to seek out and establish the existence of such biological laws and relate them to known physical laws, so that both the physical and biological aspects of living matter can be better understood. In this chapter our aim is much more modest than identifying such fundamental biophysical principles; it is merely to present a strategy for understanding a diverse set of complex phenomena in physiology and suggest that this strategy reveals an underlying symmetry that can be exploited. Schro¨dinger, in his book What is Life? [4], laid out his understanding of the connection between the world of microscopic and macroscopic based on the principles of equilibrium statistical physics. In that discussion he asked why atoms are so small relative to the dimension of the human body. The answer to this question is both immediate and profound. The high level of organization necessary for life is only possible in a macroscopic system; otherwise the order would be destroyed by microscopic (thermal) fluctuations. A living system must be sufficiently large to maintain its integrity in the presence of thermal fluctuations that disrupt its constitutive elements. Thus, macroscopic phenomena are characterized by averages over ensemble distribution functions characterizing

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bruce j. west

microscopic fluctuations. Consequently, any strategy for understanding physiology must be based on the probabilistic description of complex phenomena and, as we shall see, on our understanding of phenomena lacking characteristic scales. There are three types of fractals that appear in the life sciences: geometrical fractals, which determine the spatial properties of the tree-like structures of the mammalian lung, arterial, and venous systems and other ramified structures [5]; statistical fractals, which determine the properties of the distribution of intervals in the beating of the mammalian heart, in breathing, in walking, and in the firing of certain neurons; and finally dynamical fractals, which determine the dynamical properties of systems having a large number of characteristic time scales. In the complex systems found in physiology, the distinctions between these three kinds of fractals blur, but we focus our attention on the dynamical rather than the geometrical fractals, in part, because the latter have been reviewed in a number of places and we have little to add to that understanding. In this chapter we lay the foundation for how such concepts as complexity, fractals, diverging moments, nonlinear dynamics, and other related mathematical topics are used in understanding physiology. Of course, a number of books have been written about any one of these ideas—books for the research expert, books for the informed teacher, books for the struggling graduate student, and books for the intelligent lay person. Different authors stress different characteristics of complex phenomena, from the erratic data collected by clinical researchers to the fluctuations generated by deterministic dynamical equations used to model such systems. Some authors have painted with broad brushstrokes, indicating only the panorama that these concepts reveal to us, whereas others have sketched with painstaking detail the structure of such phenomena and have greatly enriched those that could follow the arguments. Herein we view our efforts midway between the two because fractal physiology is still a work in progress and much of what we present may prove to be irrelevant, whereas some of it might be even more significant than we can now appreciate. II.

SCALING IN PHYSIOLOGICAL TIME SERIES

We begin our discussion of physiological processes with an empirical study of the scaling behavior of time series obtained from the quantitative measurements of certain physiologic systems—for example, the cardiovascular and respiratory systems. We pursue this approach in order to convince the reader of the ubiquity of scaling in physiological time series. Once this property is established, we turn to the mathematical modeling of the mechanisms that generate such scaling. The attention is on scaling statistics because physiological measures are usually given to us in the form of time series. Whether it is the electrocardiogram (ECG) for the beating heart from which the interbeat intervals are extracted to determine heart rate variability (HRV) [6–8], the electrogastrogram (EGG) for gastric activity of

fractal physiology, complexity, and the fractional calculus

5

the stomach in which the contraction intervals are used to determine the gastric rate variability (GRV) [9], the stride intervals during walking that determines the stride rate variability (SRV) [10–13], or the interbreath interval which determines the breathing rate variability (BRV) [14], they all appear at first sight to be random processes with no underlying pattern. However, upon processing the data, they reveal long-term memory indicative of fractal time series, as we review in Section III. A.

Allometric Aggregation Data Analysis

Note that the term scaling denotes a power-law relation between two variables x and y, y ¼ Axa

ð1Þ

as Barenblatt [15] explained in his excellent inaugural lecture delivered before the University of Cambridge on May 3, 1993. He points out that such scaling laws are not merely special cases of more general relations; they never appear by accident; they always reveal self-similarity, a very important property of the phenomenon being studied. In biology, Eq. (1) is historically referred to as an allometric relation between the two observables. Allometric relations are not new in science. Such relations were introduced into biology in the nineteenth century. Typically, an allometric relation interrelates two properties of a given organism. For example, the total mass of a deer y is proportional to the mass of the deer’s antlers x raised to a specific power a. Thus, on doubly logarithmic graph paper, this relation would yield a straight line with a slope given by the power-law index. Huxley summarized the experimental basis for this relation in his 1931 book, Problems of Relative Growth [16], and developed the mathematics to describe and explain allometric growth laws. In biological systems, he reasoned, two parts of an organism grow at different rates, but the rates are proportional to one another. Consequently, how rapidly one part of the organism grows can be related to how rapidly the other part of the organism grows, and the ratio of the two rates is constant. In this section the notion of an allometric relation is generalized to include measures of time series. In this view, y is interpreted to be the variance and x the average value of the quantity being measured. The fact that these two central measures of a time series satisfy an allometric relation implies that the underlying time series is a fractal random process and therefore scales. It was first determined empirically that certain statistical data satisfy a power-law relation of the form given by Taylor [17] in Eq. (1), and this is where we begin our discussion of the allometric aggregation method of data analysis. Taylor was a scientist interested in biological speciation. For example, he was curious about how many species of beetle can be found over a given area of

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bruce j. west

land. He answered this question by sectioning off a large field into plots and in each plot sampling the soil for the variety of beetles that were present. This enabled him to determine the distribution in the number of new species of beetle spatially distributed across the field. From the distribution he could then
and the variance in the number of determine the average number of species X species Var X. After this first calculation, he partitioned his field into smaller plots and redid his sampling, again determining the mean and standard deviation in the number of species at this increased resolution. This process was repeated a number of times, yielding a set of values of means and variances. In the ecological literature a graph of the logarithm of the variance versus the logarithm of the average value is called a power curve, which is linear in the logarithms of the two variables, and b is the slope of the curve. The algebraic form of the relation between the variance and mean is
b Var X ¼ aX

ð2Þ

where the two parameters a and b determine how the variance and mean are related to one another. Taylor was able to exploit the curves obtained from data in a number of ways using the slope and intercept parameters [17]. If the slope of the curve and the intercept are both equal to one, a ¼ b ¼ 1, then the variance and mean are equal to one another. This equality is only true for a Poisson distribution, which, when it occurred, allowed him to interpret the number of species as being randomly distributed over the field, with the number of species in any one plot being completely independent of the number of species in any other plot. If, however, the slope of the curve was less than unity, the number of species appearing in the plots was interpreted to be quite regular. The spatial regularity of the number of species, in this case, was compared with the trees in an orchard and given the name evenness. Finally, if the slope of the variance versus mean curve was greater than one, the number of new species was interpreted as being clustered in space, like disjoint herds of sheep grazing in a meadow. Of particular interest to us here was the mechanism that Taylor and Taylor [18] postulated to account for the experimentally observed allometric relation: We would argue that all spatial dispositions can legitimately be regarded as resulting from the balance between two fundamental antithetical sets of behaviour always present between individuals. These are, repulsion behaviour, which results from the selection pressure for individuals to maximise their resources and hence to separate, and attraction behaviour, which results from the selection pressure to make the maximum use of available resources and hence to congregate wherever these resources are currently most abundant.

fractal physiology, complexity, and the fractional calculus

7

Consequently, they postulated that it is the conflict between the attraction and repulsion, migration and congregation, which produces the interdependence (scaling) of the spatial variance and the average population density. We can now interpret Taylor’s observations more completely because the kind of clustering he observed in the spatial distribution of species number, when the slope of the power curve is greater than one, is consistent with an asymptotic inverse power-law distribution of the underlying data set. Furthermore, the clustering or clumping of events is due to the fractal nature of the underlying dynamics. Willis, some 40 years before Taylor, established the inverse power-law form of the number of species belonging to a given genus [19]. Willis used an argument associating the number of species with the size of the area they inhabit. It was not until the decade of the 1990s that it became clear to more than a handful of experts that the relationship between an underlying fractal process and its space filling character obeys a scaling law [1,3]. It is this scaling law that is reflected in the allometric relation between the variance and mean. It is possible to test the allometric relation of Taylor using computergenerated data. But before we do so, we note that Taylor and Woiwod [20] were able to extend the discussion from the stability of the population density in space, independent of time, to the stability of the population density in time, independent of space. Consequently, just as spatial stability, as measured by the variance, is a power function of the mean population density over a given area at all times, so too the temporal stability, as measured by the variance, is a power function of the mean population density over time at all locations. With this generalization in hand we can apply Taylor’s ideas to time series. The correlation of discrete time series data is here determined by grouping the data into aggregates of two or more of the original data points and calculating the mean and variance at each level of aggregation. Consider the jth data element of an aggregation of n-adjacent data points: ðnÞ

Yj

¼

n
1 X

Ynj
k

ð3Þ

k¼1

In terms of these new data the average is defined by Y
ðnÞ 

1 X ðnÞ Y ¼ nY
ð1Þ ½N=n j¼1 j ½N=n

ð4Þ

For example, when n ¼ 3 each value of the new variable, defined by Eq. (3), consists of the sum of three non-overlapping original data points, and the number

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bruce j. west

of new data points is given by [N/3], where the brackets denote the closest integer value and N is the original number of data points. The variance, for a monofractal random time series, is similarly given by [21] Var Y ðnÞ ¼ n2H Var Y
ð1Þ

ð5Þ

where the superscript (1) on the average variable indicates that it was determined using all the original data without aggregation, and the superscript (n) on the average variable indicates that it was determined using the aggregation of nadjacent data points and H is referred to as the Hurst exponent. Thus, comparing Eq. (5) with Eq. (4), we obtain the allometric relation given by Eq. (2): Var Y

ðnÞ




Y
ðnÞ Y
ð1Þ

2H

Y
ð1Þ

¼ aY
ðnÞ

b

ð6Þ

with the parameters given by the theoretical values a¼

Var Y ð1Þ ðY
ð1Þ Þb

and

b ¼ 2H

ð7Þ

It is well established that the exponent in such scaling equations is related to the fractal dimension [21] D of the underlying time series by D ¼ 2
H, so that D ¼ 2
b=2

ð8Þ

A simple mono-fractal time series, therefore, satisfies the power-law relation of the allometric form given by Eq. (2). This allometric aggregation technique has been applied to a number of data sets implementing the method of linear regression using the equation log Var Y ðnÞ ¼ log a þ b log Y
ðnÞ

ð9Þ

Fitting the parameters a and b in Eq. (9) to time series data gives the best leastsquares estimates of the parameters a and b in the allometric relation, respectively. We find that the Gaussian distribution is a more useful exemplar for time series than is the Poisson distribution used by Taylor. In Fig. 1 we apply Eq. (9) to one million computer-generated data points with Gaussian statistics. The far left dot in Fig. 1 contains all the data in the calculation of the aggregated mean and variance so that n ¼ 1 in Eq. (9); the next point to the right in the figure

fractal physiology, complexity, and the fractional calculus

9

Figure 1. The logarithm of the variance is plotted versus the logarithm of the mean for the successive aggregation of 106 computer-generated random data points with Gaussian statistics. The slope of the curve is essentially one, determined by a linear regression using Eq. (9), so the fractal dimension of the time series is 1.5 using Eq. (8).

contains the nearest-neighbor data points added together to define a data set with 500,000 data points from which to calculate the aggregated mean and variance. Next we take the original data and add the three nearest-neighbor data points to define a data set with 333,333 data, and so on. The jth data element after aggregating n nearest-neighbor data points is given by Eq. (3). Consequently, this process of aggregating the data is equivalent to decreasing the resolution of the time series, and as the resolution is systematically decreased, the adopted measure, the relationship between the mean and variance, reveals an underlying property of the time series. The increase in the variance with increasing mean for increasing aggregation number shown in the figure is not an arbitrary pattern. The relationship indicates that the aggregated uncorrelated data points are interconnected. The original data points are not necessarily correlated, but the addition of data in the aggregation process induces a correlation, one that is completely predictable. The induced correlation is linear if the original data are uncorrelated, but not linear if the original data are correlated. The aggregated variance versus the aggregated mean falls along a straight line in Fig. 1 with a slope of b ¼ 1 for the uncorrelated random process with computer-generated Gaussian statistics. Therefore, in the case of Gaussian statistics, with b ¼ 1, we have D ¼ 1:5 corresponding to the fractal dimension of Brownian motion. In the same way, a completely correlated time series would

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have b ¼ 2, so that D ¼ 1. The fractal dimension for most time series fall somewhere between the two extremes; the closer the fractal dimension is to one, the more regular the process; the closer the fractal dimension is to 1.5, the more it is like an uncorrelated random process. The Gaussian process depicted in Fig. 1 is certainly a mono-fractal in the sense we have defined it here; that is, the computer-generated time series is characterized by a single fractal dimension. We point out here that the allometric aggregation method is just one of many procedures designed to take advantage of the scaling properties of the central moments of time series. We refer to such methods collectively as finite variance statistical methods (FVSM). However, it should be emphasized that not all time series that scale have finite variance. Time series having Le´vy a-stable statistics exemplify processes with diverging variance, but they are described by probability density functions that scale. We review these matters after some discussion of the scaling properties of physiological time series. B.

Fractal Heartbeats

The mechanisms producing the observed variability in the size of a heart’s interbeat intervals apparently arise from a number of sources. The sinus node (the heart’s natural pacemaker) receives signals from the autonomic (involuntary) portion of the nervous system which has two major branches: the parasympathetic, whose stimulation decreases the firing rate of the sinus node, and the sympathetic, whose stimulation increases the firing rate of the sinus node pacemaker cells. The influence of these two branches produces a continual tug-of-war on the sinus node, one decreasing and the other increasing the heart rate. It has been suggested that it is this tug-of-war that produces the fluctuations in the heart rate of healthy subjects, but alternate suggestions will be pursued subsequently. Consequently, heart rate variability (HRV) provides a window through which we can observe the heart’s ability to respond to normal disturbances that can affect its rhythm. The clinician focuses on retaining the balance in regulatory impulses from the vagus nerve and sympathetic nervous system and in this effort requires a robust measure of that balance [22]. A quantitative measure of HRV time series, such as the fractal dimension, serves this purpose. HRV time series have become very well known over the past two decades as a quantitative indicator of autonomic activity. The medical community became interested in developing such an indicator of heart rate because experiments indicated a relationship between lethal arrhythmias and such activity. The importance of HRV to medicine became widely apparent when a task force was formed by the Board of the European Society of Cardiology and the North American Society of Pacing and Electrophysiology and was charged with the responsibility of developing the standards of measurement, physiological interpretation and clinical use of HRV. The task force published their findings in

fractal physiology, complexity, and the fractional calculus

11

1996 [22]. It is one of the few times that the members of such a task force were drawn from the fields of mathematics, engineering, physiology and clinical medicine in recognition of the complexity of the phenomenon they were charged to investigate and actually worked together. When heart rate is atypical, say 120 bpm, in contrast to its usual 60 bpm, quantifying the variation in heart rate becomes very important. The degree of deviation from normality is determined by the interpretation of the size of the variation and how it is used to identify associated patterns. There are a number of ways to assess HRV, some 16 in all, each related to scaling in one way or another and most being FVSM, but we do not want to go into all of them here. Instead we identify the quantity that is the most revealing of the nature of HRV, but again we shall not go into a detailed discussion of the many ways of estimating this quantity. Instead we present a single explanatory technique that is relatively straightforward and that allows us to introduce the measure of interest. We use the allometric aggregation technique on real data to relate the variance and mean, as we discussed. Now we apply the allometric aggregation approach to the beat-to-beat intervals shown in Fig. 2, a typical HRV time series for a healthy young adult male. The data points in the figure are connected to aid in visualizing how the time intervals between heartbeats are changing. It is evident that the variation in the time intervals between heartbeats is relatively small, the mean being 1.0 s and the standard deviation being 0.06 s. This modest variance supports the frequently used Heartbeat Intervals 1.1

Seconds

1.05 1 0.95 0.9 0.85 0.8 0

100

200 300 Interval Number

400

500

Figure 2. The time series of heartbeat intervals of a healthy young adult male is shown. It is clear that the variation in the time interval between beats is relatively modest, but certainly not negligible.

12

bruce j. west Interbeat Interval

0

Log standard deviation

–0.2

–0.4

regular

–0.6

random

–0.8

–1

–1.2 0

0.2

0.4

0.6 0.8 Log average

1

1.2

Figure 3. The logarithm of the standard deviation is plotted versus the logarithm of the average value for the heartbeat interval time series for a young adult male, using sequential values of the aggregation number. The solid line is the best fit to the aggregated data points and yields a fractal dimension of D ¼ 1:24 midway between the curve for a regular process and that for an uncorrelated random process as indicated by the dashed curves.

medical term ‘‘normal sinus rhythm.’’ So the question of what is learned by applying the allometric aggregation technique to these data and constructing the standard deviation and mean as a function of aggregation number, becomes important. In Fig. 3 the logarithm of the standard deviation is plotted versus the logarithm of the mean value for the HRV time series depicted in Fig. 2. Note that we use the standard deviation in the figure and not the variance used in the discussion of Taylor’s Law. We use the standard deviation because we are primarily interested in whether the time series is fractal or not and not particularly in the actual value of the fractal dimension. At the leftmost position the data point indicates the standard deviation and mean, using all the data points. Moving from left to right, the next data point is constructed from the time series with two nearest-neighbor data points added together, and the procedure is repeated moving right until the rightmost data point has 20 nearestneighbor data points added together. The solid line is the best linear representation of the scaling and intercepts most of the data points with a positive slope of 0.76. We can see that the slope of the HRV data is midway

fractal physiology, complexity, and the fractional calculus

13

between the dashed curves depicting an uncorrelated random process (slope ¼ 1/2) and one that is deterministically regular (slope ¼ 1). We emphasize that the conclusions we draw here are not from this single figure or set of data presented; these are only representative of a much larger body of work. The conclusions are based on a large number of similar observations [23,24] made using a variety of data processing techniques, all of which yield results consistent with the scaling of the HRV time series indicated in Fig. 3. So we conclude that the heartbeat intervals do not form an uncorrelated random sequence. Instead we see that the HRV time series is a statistical fractal, indicating that the heartbeats have a long-time memory. The implications of this long-time memory concerning the underlying physiological control system will be taken up later when we discuss the mathematical models of these phenomena. Phenomena obeying a scaling relation, such as shown for the HRV time series data in Fig. 3, are said to be self-similar. The fact that the standard deviation and mean values change as a function of aggregation number implies that the magnitudes of the standard deviation and mean values depend on the size of the ruler used to measure the time interval. Recall that this is one of the defining characteristics of fractal curves; the length of the curve becomes infinite as the size of the ruler goes to zero. The dependence of the mean and standard deviation on the ruler size, for a self-similar time series, implies that the statistical process is fractal and consequently defines a fractal dimension for the HRV time series. The average scaling exponent obtained by Peng et al. [25] for a group of 10 healthy subjects having a mean age of 44 years, using 10,000 data points for each subject, was a ¼ 0.19 for the difference in heartbeat interval time series. They interpreted this value to be consistent with a theoretical value of a ¼ 0, which they conjectured would be obtained for an infinitely long time series. The latter scaling implies that the scaling exponent for the beat intervals themselves would be 1.0. However, all data sets are finite, and it was determined that the asymptotic scaling coefficients for the heartbeat interval time series of healthy young adults lie in the interval 0.7  a  1.0. The value of the scaling coefficient obtained using much shorter time series and the relatively simple processing technique of allometric aggregation is consistent with these results. We also comment on certain speculations made by Peng et al. [25] regarding their analysis of the set of HRV time series; speculations that have been supported by subsequent research. They suggested that the scaling behavior is adaptive for two reasons: (i) The long-range correlations serve as an organizing principle for highly complex, nonlinear processes that generate fluctuations on a wide range of time scales.

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bruce j. west

(ii) The lack of a characteristic scale helps prevent excessive mode-locking that would restrict the functional responsiveness of the organism. C.

Fractal Breathing

Breathing is a function of the lungs, whereby the body takes in oxygen and expels carbon dioxide. The smooth muscles in the bronchial tree are innervated by sympathetic and parasympathetic fibers, much like the heart, and produce contractions in response to stimuli such as increased carbon dioxide, decreased oxygen, and deflation of the lungs. Fresh air is transported through some 20 generations of bifurcating airways of the lung, during inspiration, down to the alveoli in the last four generations of the bronchial tree. At this tiny scale there is a rich capillary network that interfaces with the bronchial tree for the purpose of exchanging gases with the blood. In this section we are interested in the dynamics of breathing; the apparently regular breathing as you sit quietly reading this paper. Here evolution’s design of the lung may be closely tied to way in which the lung carries out its function. It is not by accident that the cascading branches of the bronchial tree become smaller and smaller, nor is it good fortune alone that ties the dynamics of our every breath to this biological structure. We argue that, like the heart, the lung is made up of fractal processes, some dynamic and others now static. However, both kinds of processes lack a characteristic scale, and a simple argument establishes that such lack of scale has evolutionary advantages [26]. An early application of fractal analysis was made by Szeto et al. [27] to fetal lamb breathing. The changing patterns of breathing in 17 fetal lambs and the clusters of faster breathing rates, interspersed with periods of relative quiescence, suggested to them that the breathing process was self-similar. The physiological property of self-similarity implies that the structure of the mathematical function describing the time series is repeated on progressively finer time scales. Clusters of faster rates were seen within the fetal breathing data, what Dawes et al. [28] called breathing episodes. When the time series were examined on even finer time scales, clusters could be found within these clusters, and the signature of this scaling behavior emerged as an inverse power-law distribution of time intervals. Consequently, the fractal scaling was found to reside in the statistical properties of the fluctuations and not in the geometrical properties of the dynamic variable. As with the heart, the variability of breathing rate using breath-to-breath time intervals is denoted by breathing rate variability (BRV), to maintain a consistent notation. Examples of HRV and BRV time series data on which scaling calculations are based are shown in Fig. 4. A typical BRV time series for a senior citizen at rest is shown at the top of the figure; the simultaneous HRV time series for the same person is depicted at the bottom of the figure. Because

fractal physiology, complexity, and the fractional calculus

15

Interbreath Intervals

Interval [sec]

7 6 5 4 3 0

20

40 60 Interval Number

80

Interbeat Intervals

Interval [sec]

0.85 0.8 0.75 0.7 0.65 0.6 0

100

200

300

400

500

600

Interval Number

Figure 4. Typical time series from one of the 18 subjects in the study conducted by West et al. [14], while at rest, is shown for the interbreath intervals (BRV) and the interbeat intervals (HRV) time series.

heart rate is higher than respiration rate, in the same measurement epoch there is a factor of five more data for the HRV time series than there is for the BRV time series. These data were collected under the supervision of Dr. Richard Moon, the Director of the Hyperbaric Laboratory at Duke Medical Center. Looking at these two time series together, one is struck by how different they appear. It is not apparent that both physiological phenomena scale in essentially the same way, but they do [14]. The allometric aggregation method applied to the various time series obtained by West et al. [14] yields the typical results depicted in Fig. 5, where the logarithms of the aggregated variance versus the aggregated means are plotted for the HRV and BRV data depicted in Fig. 4. At the extreme left of each graph in Fig. 5 (m ¼ 1), all the data points are used to calculate the variance and mean, and at the extreme right the aggregated quantities use m ¼ 10 data points. Note that we stop the aggregation at 10 points because of

16

bruce j. west Interbreath Interval

Log standard deviation

0.8

0.6

0.4

0.2

0 0.8

1.2 1.4 Log average

1.6

Interbeat Interval

–0.2

Log standard deviation

1

–0.4

–0.6

–0.8

–1

–1.2 0

0.2

0.4 0.6 Log average

0.8

Figure 5. A typical fit to the aggregated variance versus the aggregated mean for BRV and HRV time series obtained by West et al. [14]. The points are calculated from the data and the solid curve is the best least-square fit to the data. The upper curve is the fit to the BRV data (slope ¼ 0.86), and the lower curve is the best fit to the HRV data (slope ¼ 0.80). It is evident from these two graphs that the allometric relation given by Eq. (9) does indeed fit both data sets extremely well and lies well within the regular and random boundaries, indicated by the dashed curves.

fractal physiology, complexity, and the fractional calculus

17

the small number of data in the breathing sequence. The solid curve at the top of Fig. 5 is the best least-square fit to the aggregated BRV data and has a slope of 0.86, which is the scaling index. A similar graph is constructed for the HRV data in the lower curve, where we obtain a slope of 0.80 for the scaling index. The scaling index of both the HRV and BRV time series increase with increasing levels of exercise, but the data are not shown here, mainly because we can’t show everything all the time. The 18 subjects in the experiment rode a stationary bicycle with various levels of load on the wheels to mimic cycling uphill. The breathing rate, breathing volume, and heart rate were monitored for each of the individuals in the study. The consistent result was that as the level of exercise increased, the amount of variability in both HRV and BRV decreased, indicating that the associated time series were becoming more ordered. This increase in scaling index was determined to be statistically significant [14]. The scaling indices and fractal dimensions obtained from these curves are consistent with the results obtained by other researchers. Such observations regarding the self-similar nature of breathing time series have been used in a medical setting to produce a revolutionary way of utilizing mechanical ventilators. Historically, ventilators have been used to facilitate breathing after an operation and have a built-in frequency of ventilation. The single-frequency ventilator design has recently been challenged by Mutch et al. [29], who have used an inverse power-law spectrum of respiratory rate to drive a variable ventilator. They demonstrated that this way of supporting breathing produces an increase in arterial oxygenation over that produced by conventional control-mode ventilators. This comparison indicates that the fractal variability in breathing is not the result of happenstance, but is an important property of respiration. A reduction in variability of breathing reduces the overall efficiency of the respiratory system. Altemeier et al. [30] measured the fractal characteristics of ventilation and determined that not only are local ventilation and perfusion highly correlated, but they scale as well. Finally, Peng et al. [31] analyzed the BRV time series for 40 healthy adults and found that under supine, resting, and spontaneous breathing conditions, the time series scale. This result implies that human BRV time series have ‘‘long-range (fractal) correlations across multiple time scales.’’ Some particularly long records of breathing were made by Kantelhardt et al. [32], for 29 young adults participating in a sleep study where their breathing was recorded during REM sleep, non-REM sleep, and periods of being awake. They showed that the stages of healthy sleep (deep, light, and REM) have different autonomic regulation of breathing. During deep or light sleep, both of which are non-REM, the scaling index was determined to be a ¼ 1/2 and consequently the BRV time series was essentially an uncorrelated random process. On the other hand, during REM sleep and during periods of wakefulness, the scaling index was

18

bruce j. west

found to be in the interval 0.9  a  0.8, so that the BRV time series has a longtime memory. Webber [33] also investigated the nonlinear physiology of breathing, pointing out that the central rhythm and pattern generators within the brainstem/ spinal cord are subject to afferent feedback and suprapontine feed-forward inputs of varying coupling strengths. In short, there are multiple signals that influence one another in the respiratory controller, and the dynamics are nonlinear; that is, a given response is not proportional to a given stimulus. But putting this aside for the moment, Webber applied a number of techniques from nonlinear dynamics systems theory to the analysis of breathing. He recorded 4500 consecutive respiratory cycles for the spontaneous breathing patterns of anaesthetized, unrestrained rats (10 in all). He then constructed phase-space plots using continuous thoracic pressure fluctuations. His plots reveal patterns in the breathing data that are consistent with chaos and fractal time series. D. Fractal Gait Walking is one of those things we do without giving it much thought, day in and day out. We walk confidently with a smooth pattern of strides and without apparent variation in gait. This seeming lack of pattern is remarkable considering that the motion of walking is created by the loss of balance, as pointed out by Leonardo da Vinci (1452–1515) in his treatise on painting. da Vinci considered walking to be a sequence of fallings; consequently, it should come as no surprise that there is variability in this sequence of falling intervals, even if such variability is usually masked. The regular gait cycle, so apparent in everyday experience, is not as regular as we believed. Gait is no more regular than is normal sinus rhythm or breathing. The subtle variability in the stride characteristics of normal locomotion were first discovered by the nineteenth-century experimenter Vierordt [34], but his findings were not exploited for over 120 years. The random variability he observed was so small that the biomechanical community has historically considered these fluctuations to be an uncorrelated random process. In practice this means that the fluctuations in gait were thought to contain no information about the underlying motorcontrol process. The followup experiments to quantify the degree of irregularity in walking was finally done in the middle of the last decade by Hausdorff et al. [35] and involved observations of healthy individuals, as well as of subjects having certain diseases that affect gait and also the elderly. Additional experiments and analyses that both verified and extended the earlier results were subsequently done by West and Griffin [11,36]. Human gait is a complex process, since the locomotor system synthesizes inputs from the motor cortex, the basal ganglia, and the cerebellum, as well as

fractal physiology, complexity, and the fractional calculus

19

feedback from vestibular, visual, and proprioceptive sources. The remarkable feature of this complex phenomenon is that although the stride pattern is stable in healthy individuals, the duration of the gait cycle is not fixed. Like normal sinus rhythm in the beating of the heart, where the intervals between successive beats change, the time interval for a gait cycle fluctuates in an erratic way from step to step. The gait studies carried out to date concur that the fluctuations in the stride-interval time series exhibit long-time inverse power-law correlations indicating that the phenomenon of walking is a self-similar fractal activity. Walking consists in a sequence of steps, and the corresponding time series is made up of the time intervals for these steps. These steps may be partitioned into two phases: a stance phase and a swing phase. It has been estimated, using blood flow to skeletal muscles, that the stance phase muscles consume three times the energy as do the swing phase muscles, independently of speed [37]. The stance phase is initiated when a foot strikes the ground and ends when it is lifted. The swing phase is initiated when the foot is lifted and ends when it strikes the ground again. The time to complete each phase varies with the stepping speed. A stride interval is the length of time from the start of one stance phase to the start of the next stance phase. It is the variability in the time series made from these intervals that is probed in this analysis of the stride interval time series. One definition of the gait cycle or stride interval is the time between successive heel strikes of the same foot [10]. An equivalent definition of the stride interval uses successive maximum extensions of the knee of either leg [36]. The stride interval time series for a typical subject is shown in Fig. 6, where it is seen that the variation in time interval is on the order of 3–4%, indicating that the stride pattern is very stable. The stride interval time series is referred to as stride rate variability (SRV) for name consistency with the other two time series we have discussed. It is the stability of SRV that has historically led investigators to decide that not much could go wrong by assuming that the stride interval is constant and that the fluctuations are merely biological noise. The second set of data in Fig. 6 is computer-generated Gaussian noise, having the same mean and standard deviation as the experimental data. Note that it is not easy to distinguish between the two data sets. However, the experimental data fluctuate around the mean gait interval and, although small, are nonnegligible because they indicate an underlying complex structure and, as we show, these fluctuations cannot be treated as an uncorrelated random noise. Using a 15-minute SRV time series, from which the data depicted in Fig. 6 were taken, we apply the allometric aggregation procedure to determine the relation between the standard deviation and mean of the time series as shown in Fig. 7. In the latter figure, the curve for the SRV data is, as we did with the other data sets, contrasted with an uncorrelated random process (slope ¼ 0.5) and a regular deterministic process (slope ¼ 1.0). The slope of the data curve is 0.70,

20

bruce j. west

Figure 6. Real data compared with computer-generated data. At the top, the time interval between strides for the first 500 steps made by a typical walker in an experiment [36] is depicted. At the bottom a computer-generated time series having uncorrelated Gaussian statistics is shown, with the same mean and variance as in the data shown at the top.

midway between the two extremes of regularity and uncorrelated randomness. So, as in the cases of HRV and BRV time series, we again find the erratic physiological time series to represent a random fractal process. In a previous section we argued that if the power-law index, the slope of the aggregated variance versus aggregated average curve on a log–log graph, is greater than one, then the data are clustered. In the SRV context, indicated by a slope greater than the random dashed line, this clustering indicates that the intervals between strides change in clusters and not in a statistically uniform manner over time. This result suggests that the walker does not smoothly adjust his/her stride from step to step. Rather, there are a number of steps over which adjustments are made followed by a number of steps over which the changes in stride are completely random. The number of steps in the adjustment process and the number of steps between adjustment periods are not independent. The

fractal physiology, complexity, and the fractional calculus

21

Interstride Interval

Log standard deviation

–0.4

–0.6

–0.8 regular –1

random

–1.2

–1.4

–1.6

0

0.2

0.4

0.6 0.8 Log average

1

1.2

Figure 7. The SRV data for a typical walker in the experiment [36] is used to construct the aggregated variance and mean as indicated by the dots. The logarithm of the aggregated variance is plotted versus the logarithm of the aggregated mean, starting with all the data points at the lower left to the aggregation of 20 data points at the upper right. The SRV data curve lies between the extremes of uncorrelated random noise (lower dashed curve) and a regular deterministic process (upper dashed curve) with a fractal dimension of D ¼ 1.30.

results of a substantial number of stride interval experiment supports the universality of this interpretation. The SRV time series for 16 healthy adults were downloaded from PhysioNet and the allometric aggregation procedure carried out. Each of the curves looked more or less like that in Fig. 7, with the experimental curve being closer to the indicated regular or the random limits (dashed curves). On average, the 16 individuals have fractal dimensions for gait in the interval 1.2  D  1.3 and an average correlation on the order of 40% [38]. The fractal dimension obtained from the analysis of an entirely different dataset, obtained using a completely different protocol, yields consistent results [36]. The narrowness of the interval around the fractal dimension suggests that this quantity may be a good quantitative measure of an individual’s dynamical variability. We suggest the use of the fractal dimension as a quantitative measure of how well the motor control system is doing in regulating locomotion. Furthermore, excursions

22

bruce j. west

outside the narrow interval of fractal dimension values for apparently healthy individuals may be indicative of hidden pathologies. Further analysis was done on the SRV time series of 50 children, also downloaded from PhysioNet. One of the features of the time series were large excursions in the fractal dimension for children under the age of five, 1.12  D  1.36, unlike the narrower range of values for mature adults. The interval expands from 0.08 for adults to 0.24 for children, a factor of three decrease of the interval from childhood to adulthood. It is clear that the average fractal dimension over each group is the same, approximately 1.24, but the range of variation in the fractal dimension decreases significantly with age [38]. This would seem to make the fractal dimension an increasingly reliable indicator of the health of the motorcontrol system with advancing age. It should not go unnoticed that people use pretty much the same control system when they are standing still, maintaining balance, as they do when they are walking. This observation would lead one to suspect that the body’s slight movements around the center of mass of the body—that fictitious point at which all of the body’s mass is located in any simple model of locomotion—would have the same statistical behavior as that observed during walking. These tiny movements are called postural sway in the literature and have given rise to papers with such interesting titles as ‘‘Random walking during quiet standing’’ by Collins and De Lucca [39]. It has been determined that postural sway is really chaotic [40], so one might expect that there exists a relatively simple dynamical model for balance regulation that can be used in medical diagnosis. Here again the fractal dynamics can be determined from the scaling properties of postural sway time series, and it is determined that a decrease of postural stability is accompanied by an increase of fractal dimension. E.

Fractal Neurons

Up to this point the discussion has been focused primarily on time series generated by various complex physiological phenomena. Now let us examine a class of basic building blocks used to construct these phenomena, the individual neurons in the various control systems of the body. The neuron is in most respects quite similar to other cells in that it contains a nucleus and cytoplasm. However, it is distinctive in that long, threadlike tendrils emerge from the cell body, and those numerous projections branch out into still finer extensions. These are the dendrites that form a branching tree of ever more slender threads not unlike the fractal trees discussed by West and Deering [5]. One such thread does not branch and often extends for several meters even though it is still part of a single cell. This is the axon that is the nerve fiber in the typical nerve. Excitations in the dendrites always travel toward the cell body in a living system, whereas excitations in the axon always travel away from the cell body.

fractal physiology, complexity, and the fractional calculus

23

The activity of a nerve is invariably accompanied by electrical phenomena. The first systematic observation of this effect was made by Luigi Galvani in 1791. He observed that when a muscle (frog’s leg) was touched with a scalpel and a spark was drawn nearby, but not in direct physical contact with the scalpel, the muscle contracted. In 1852 the German physician/physicist Helmholtz first measured the speed of the nerve impulse by stimulating a nerve at different points and recording the time it took the muscle to which it was connected to respond. Electrical impulses are observed along the corresponding axon, whether it is an external excitation of a nerve or the transmission of a message from the brain. The properties of individual nerves seem to be ubiquitous because there is apparently no fundamental difference in structure, chemistry, or function between the neurons and their interconnections in humans and those in a squid, a snail, or a leach. However, neurons do vary in size, position, shape, pigmentation, firing patterns, and chemical substances by which they transmit information to other cells. There are two different ways in which neurons can be fractal. The first way is through their geometrical structure. The shape of nerve cells may be fractal in space just as observed for the cardiac conduction system and the architecture of the lung [5]. The fractal dimension has been used to classify the different shapes of neurons and to suggest mechanisms of growth responsible for these shapes. The second way neurons can be fractal is through the time intervals between the action potentials recorded from nerves, again as was observed for the interbeat interval distribution in cardiac time series, the interbreath interval distribution for breathing, and the interstride interval for walking discussed herein. The statistical properties of these time intervals have the same strange properties observed earlier, in that collecting more data does not improve the accuracy of the statistical distribution of the intervals measured for some neurons as characterized by the width of the distribution. The realization that the statistics of these intervals are fractal helps in understanding these surprising properties. Bassingthwaighte et al. [21] review how the fractal dimension can be used to classify neurons into different functional types. The statistical properties of the time intervals between action potentials display three different types of distributions: (1) Some neurons are well described by a Poisson process, in which the probability that there is an action potential in a time interval t is proportional to t. The durations of subsequent intervals between action potentials are statistically independent of one another. (2) Some neurons fire almost, but not exactly, at a constant rhythm. The time series for the action potentials in this case are well described by a Gaussian model, where the intervals have a small dispersion around the mean value. (3) Some neurons show self-similar fractal patterns on the scaled interval distributions. In the last case, neurons occasionally have very long intervals between action potentials. These long intervals occur with sufficient frequency

24

bruce j. west

so that they can be the most important in the determination of the average interval length. As more and more data are collected, longer and longer intervals are found. Thus, the average interval increases with the duration of the data record analyzed, in such a way that if an infinitely long record could be analyzed, then the average interval would be infinite. Such infinite moments are characteristic of a type of stable statistical fractal process called a Le´vy process [5]. Gernstein and Mandelbrot [41] were the first to quantitatively analyze the scaling properties of neuronal action potentials, and from their analysis a number of conclusions were drawn. The first conclusion concerns the fact that, as more data were analyzed, the values found for the average interval and the variance in the average increase. That is, in the limit of an infinitely long data record, both the mean and its variance could become infinite. Since the variance is, in principle, infinite, the averages measured for different segments of the same data record may be markedly different. It is commonly thought that if moments, such as the average, measured from data are constant in time, then the parameters of the process that generate the data are constant, and that if these moments vary, then the parameters that generate the data also vary. This commonly held notion is wrong, as stressed by Gernstein and Mandelbrot. Processes, especially fractal processes, where the generating mechanism remains unchanged can yield time series whose moments, such as the average, vary with time. The second conclusion concerns the fact that as additional data are analyzed, increasingly long intervals are found. Hence the inclusion of these intervals increases rather than decreases the variance in the measured distribution. That is, the statistical irregularities in the measured distribution become larger as more data are collected. As stated by Gernstein and Mandelbrot [41]: Thus, in contradiction to our intuitive feelings, increasing the length of available data for such processes does not reduce the irregularity and does not make the sample mean or sample variance converge.

Action potentials can be recorded from the primary auditory neurons that transmit information about sound and from the pulse vestibular neurons that transmit information about head position to the brain. Fractal behavior is ubiquitous in these and other such sensory systems [42]. Without including the references given by Teich et al. [42], we quote their review of the evidence for this observation: Its presence has been observed in cat striate-cortex neural spike trains, and in the spike train of a locust visual interneuron, the descending contralateral movement detector. It is present in the auditory system of a number of species; primary auditory (VIII-nerve) nerve fiber in the cat, the chinchilla, and the chicken all exhibit fractal behavior. It is present at many biological levels, from the

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25

microscopic to the macroscopic; examples include ion-channel behavior, neurotransmitter exocytosis at the synapse, spike trains in rabbit somatosensorycortex neurons, spike trains in mesencephalic reticular-formation neurons, and even the sequence of human heartbeats. In almost all cases the upper limit of the observed time over which fractal correlations exist is imposed by the duration of the recording.

It would probably be considered bad form in a discussion of neurons and action potentials if we did not comment on the electrical properties of the human brain. It has been known for well over a century that the activity of a nerve is based on electrical phenomena and that the mammalian brain generates a small but measurable electrical signal. The electroencephalograms (EEGs) of small animals were measured by Caton in 1875, and those of humans were measured by Berger in 1925. The mathematician N. Wiener thought that generalized harmonic analysis would provide the mathematical tools necessary to penetrate the mysterious relations between EEG time series and the functioning of the brain. The progress along this path has been slow, and the understanding and interpretation of EEG’s remains quite elusive. After 130 years, one can only determine intermittent correlations between the activity of the brain and that found in EEG records. There is no taxonomy of EEG patterns that delineates the correspondence between those patterns and brain activity. It probably bears repeating that the traditional methods of analyzing EEG time series rely on the paradigm that all temporal variations consist of a superposition of harmonic and periodic vibrations, in the tradition of Wiener. The attractor reconstruction technique reinterprets the time series as a multidimensional geometrical object generated by a deterministic dynamical process in phase space. If the dynamics are reducible to deterministic laws, then the phase portraits of the system converge toward a finite region of phase space containing an attractor. Figure 8 shows EEG time series for a variety of brain states, including quiet resting with eyes open and closed, three stages of sleep, and a petit mal epileptic seizure. Adjacent to each of these time series is depicted a projection of the EEG attractor onto a two-dimensional subspace using the attractor reconstruction technique [43]. The brain wave activity of an individual during various stages of sleep is depicted in Fig. 8. Here the standard division of sleeping into four stages is used. In stage one, the individual drifts in and out of sleep. In stage two, the slightest noise will arouse the sleeper, whereas in stage three a loud noise is required. The final stage, level four, is one of deep sleep. This is the normal first sequence of stages one goes through during a sleep cycle. Afterwards the cycle is reversed back through stages three and two, at which time dreams set in and the individual manifests rapid eye movement (REM). The dream state is followed by stage two, after which the initial sequence begins again. It is clear that whatever the form of the cognitive attractor, if such an object exists, it is not static but varies with the

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bruce j. west

Figure 8. Typical episodes of the electrical activity of the human brain as recorded from the electroencephalogram (EEG) time series together with the corresponding phase space portraits. The portraits are two-dimensional projections of the actual attractors. (From Babloyantz and Destexhe [43] with permission.)

level of sleep. In fact, the fractal dimension decreases as sleep deepens, from a fractal dimension of eight during REM to half that in deep sleep level four [43]. The dimension drops further, to a value of approximately two, during petit mal epileptic seizure. The seizure corresponds to highly organized discharges between the right and left hemispheres of the brain. III.

DYNAMICAL MODELS OF SCALING

The physiological time series processed in the previous section clearly show that the complex phenomena supporting life, although they appear to be random, do in fact scale in time. This scaling indicates that the fluctuations that occur on multiple time scales are tied together, and the way we understand such interdependency in the physical sciences is through underlying mechanisms

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27

that are coupled one to the other. This coupling is typically done through the equations of motion governing the dynamical description of the process. Unfortunately, we do not have the dynamic equations to describe the physiologic phenomena in which we are interested. Therefore we must take a more phenomenological tack and develop mathematical models to explain the data based on heuristic reasoning. However, to start we need to know the formal properties of the models we propose to use. The best physical model is the simplest one that can ‘‘explain’’ all the available experimental time series, with the fewest number of assumptions. Alternative models are those that make predictions and which can assist in formulating new experiments that can discriminate between different hypotheses. We start our discussion of models with a simple random walk, which in its simplest form provides a physical picture of diffusion—that is, a dynamic variable with Gaussian statistics in time. Diffusive phenomena are shown to scale linearly in time and generalized random walks including long-term memory also scale, but they do so nonlinearly in time, as in the case of anomalous diffusion. Fractional diffusion operators are used to incorporate memory into the dynamics of a diffusive process and leads to fractional Brownian motion, among other things. The continuum form of these fractional operators is discussed in Section IV. The continuous limit of a simple random walk model leads to a stochastic dynamic equation, first discussed in physics in the context of diffusion by Paul Langevin. The random force in the Langevin equation [44], for a simple dichotomous process with memory, leads to a diffusion variable that scales in time and has a Gaussian probability density. A long-time memory in such a random force is shown to produce a non-Gaussian probability density for the system response, but one that still scales. Finally we show that physiologic time series are not mono-fractal, but have a fractal dimension that changes over time. The time series are multifractal, and as such they have a spectrum of dimensions. We review the procedure for constructing the multifractal spectrum and apply the technique to the SRV time series data obtained in our walking experiment [36] as a typical example of physiologic variability. A.

Scaling in Time Series

Time series analysis is the backdrop against which most theoretical models are developed in the life sciences and their analysis employs the traditional engineering assumption of signal plus noise. The signal plus noise model postulates that the time series variable X(t) consists of a slowly varying part S(t) and a randomly fluctuating part x(t): XðtÞ ¼ SðtÞ þ xðtÞ

ð10Þ

28

bruce j. west

The slow, regular variation of the time series S is called the signal, and the rapid erratic fluctuations represented by x is called the noise. The implication of this separation of effects is that S(t) contains only information about the system of interest, whereas x(t) is a property of the environment and does not contain any information about the system. In this model the noise can therefore be removed, by means of such techniques as filtering, without influencing what can be learned about the system. In more complex phenomena the separation of effects implied by Eq. (10) may no longer be appropriate. The low-frequency, slowly varying part of the spectrum may be coupled to, and exchange energy with, the high-frequency, rapidly varying part of the spectrum; a fact that often results in fractal statistical processes. For these latter processes the traditional view of a deterministic, predictable signal given by the smooth part of the time series, on which random, unpredictable noise is superposed, distorts the dynamics of the underlying process, see, for example, Biodynamics [38] for a complete discussion. Herein we study techniques that purport to isolate and separate the deterministic part from the scaling part of the time series, without distorting the mutual influence of the low-frequency and high-frequency components of the same time series. The science of complexity, in so far as it can be said to be a science, has relinquished the signal plus noise paradigm for a different perspective. Physiological time series invariably contain fluctuations, so that when sampled N times the data set {Xj}, j ¼ 1, . . . , N, appears to be a sequence of random points. Examples of such data are the interbeat intervals of the human heart, interstride intervals of human gait, brain wave data from EEGs and interbreath intervals, to name a few. The processing of time series in each of these cases has made use of random walk concepts in both the processing of the data and in the interpretation of the results. So let us review some of what is known about random walks. 1.

Simple Random Walks and Scaling

We define the variable of interest as Xj, where j ¼ 0,1,2, . . . indexes the time step, and in the simplest model a step is taken in each increment of time, which for convenience we set to one. The operator B lowers the index by one unit such that BXj ¼ Xj
1 so that a simple random walk can be written ð1
BÞXj ¼ xj

ð11Þ

where xj is þ1 or
1 and is selected according to some random process characterized by the probability density pðxÞ. The solution to this discrete equation is given by the position of the walker after N steps, the sum over the sequence of steps XðNÞ ¼

N X j¼1

xj

ð12Þ

fractal physiology, complexity, and the fractional calculus

29

and the total number of steps N can be interpreted as the total time t over which the walk unfolds, since we have set the time increment to one. For N sufficiently large and a symmetric probability density pðxÞ with a finite width, the central limit theorem determines that the statistics of the dynamic variable X(t) are Gaussian: 2 1

x pN ðxÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e 2hXðNÞ2 i 2phXðNÞ2 i

ð13Þ

Assuming that the random steps are statistically independent hxj xk i ¼ hx2 idjk , we have for the second moment of the diffusion variable hXðtÞ2 i ¼

N X N X

hxj xk i ¼ hx2 iN ! 2Dt

ð14Þ

j¼1 k¼1

In the continuum limit the second moment increases linearly with time and in direct proportion to the diffusion coefficient, so that the probability density becomes the familiar Gaussian distribution for Einstein diffusion: 1 2 pðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi e
x =2Dt 4pDt

ð15Þ

Of particular interest to us here is the scaling property of the Gaussian distribution, Eq. (15). The joint probability distribution for two statistically independent processes X1 and X2, occurring in sequential time intervals, each of length t, is the product of the separate probability densities. Consequently the probability density for the aggregate process X ¼ X1 þ X2 , occurring in the time interval 2t, is obtained by integrating over the two-variable constrained integral: 1 ð

Pðx; 2tÞ ¼

1 ð

dx2 Pðx1 ; tÞPðx2 ; tÞdðx
x1
x2 Þ

dx1
1

ð16Þ


1

Integrating out the delta function constraint and substituting the Gaussian distribution from Eq. (15) into Eq. (16) yields 1 ð

pðx; 2tÞ ¼

2

dx1
1

e
ðx
x1 Þ =2Dt e
x2 =2Dt e
x =2Dð2tÞ pffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pDt 4pDt 4pDð2tÞ 2

2

ð17Þ

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bruce j. west

Thus, when viewed with only half the time resolution, that being 2t rather than t, the increments of the Brownian particle position are still zero-centered Gaussian random. More generally, whatever the number of the microscopic time steps between observations M, one always finds that the increments in the particle position constitute a zero-centered Gaussian process with a variance that increases linearly with M. The above property means that one can sample the process of Brownian motion at any level of resolution and still observe a zero-centered Gaussian process. The time series obtained by sampling the process at every time t, or at every time bt, where b is an integer, would be statistically indistinguishable. This is the scale invariance of Brownian motion. This scaling property is manifest by writing ^x ¼ l1=2 x and ^t ¼ lt, yielding the scaling result pð^x; ^tÞ ¼ l
1=2 pðx; tÞ

ð18Þ

so that the distribution for the random variable l1=2 XðltÞ is the same as that for X(t). This scaling relation establishes that the random irregularities are generated at each scale in a statistically identical manner; that is, if the fluctuations are known in a given time interval, they can be determined in a second larger time interval by scaling. This is the property manifest in the allometric aggregation method used in the previous section. In the simple random walk the steps are statistically independent of one another. The simplest generalization of this model is to make each step dependent on the preceding step in such a way that the second moment is hXðtÞ2 i ¼ 2Dt2H

ð19Þ

where H 6¼ 1/2 corresponds to anomalous diffusion. A value of H < 1/2 is interpreted as an antipersistent process in which case a step in one direction is preferentially followed by a reversal of direction. A value of H > 1/2 is interpreted as a persistent process in which case a step in one direction is preferentially followed by another step in the same direction. A value of H ¼ 1/2 is interpreted as ordinary diffusion in which case the steps are statistically independent of one another. This interpretation of anomalous diffusion would be compatible with the concept of environmental noise and the signal plus noise paradigm. In the science of complexity the system response X(t) is expected to depart from the totally random condition of the simple random walk model, since such fluctuations are expected to have memory and correlation. In the physics literature, anomalous diffusion has been associated with phenomena with longtime memory such that the autocorrelation function is Cðt1 ; t2 Þ ¼ hXðt1 ÞXðt2 Þi / jt1
t2 jb

ð20Þ

fractal physiology, complexity, and the fractional calculus

31

and the brackets denote an average over an ensemble of realizations of fluctuations in the random variable. Here the power-law index is given by b ¼ 2H
2. Note that the two-point correlation function depends only on the time difference, thus, the underlying process is stationary in time. The Fourier transform of the autocorrelation function yields the spectrum, which in this case has the inverse power-law form SðoÞ /

1 obþ1

ð21Þ

and could also have been determined by applying a Tauberian theorem to Eq. (20). These power-law properties of the spectrum and the autocorrelation function, as well as a number of other properties involving long-time memory, are clearly discussed for discrete time series by Beran [45]. 2.

Fractional Random Walks and Scaling

One way of introducing long-term memory into a random walk model is by means of fractional differences. The concept of fractional differences is most readily introduced through the shift operator introduced in the previous subsection. Following Hosking [46], we define a fractional difference process as ð1
BÞa Xj ¼ xj

ð22Þ

and the exponent a is not an integer. As it stands, Eq. (22) is just a formal definition without content. To make this equation usable, we must determine how to represent the operator acting on Xj, and this is done using the binomial expansion [45,46]. The inverse operator in the formal solution of Eq. (22), Xj ¼ ð1
BÞ
a xj

ð23Þ

has the binomial series expansion
a

ð1
BÞ

¼

 1
X
a k¼0

k

ð
1Þk Bk

ð24Þ

Expressing the binomial coefficient as the ratio of gamma function in the solution given in Eq. (23), we obtain after some algebra [23] Xj ¼ ¼

1 X

ðk þ aÞ Bk x j ðk þ 1ÞðaÞ k¼0

1 X k¼0

k xj
k

ð25Þ

32

bruce j. west

The solution to the fractional diffusion equation is clearly dependent on fluctuations that have occurred in the remote past; note the time lag k in the index on the fluctuations and the fact that it can be arbitrarily large. The extent of the influence of these distant fluctuations on the system response is determined by the relative size of the coefficients in the series. Using Stirling’s approximation on the gamma functions determines the size of the coefficients in Eq. (25) as the fluctuations recede into the past, that is, as k ! 1 we obtain k 

ðk þ a
1Þkþa
1 kk ða
1Þ

k

¼

ka
1 ða
1Þ!

ð26Þ

since k  a. Thus, the strength of the contributions to Eq. (25) decrease with increasing time lag as an inverse power law asymptotically in the time lag as long as a < 1=2. The spectrum of the time series in Eq. (25) is obtained using its discrete Fourier transform ðp 1 ^ o do eiko X ð27Þ Xk ¼ 2p
p

in the discrete convolution form of the solution, Eq. (25), to obtain ^ o^ ^o ¼  xo X

ð28Þ

D E D E ^ o j2 ^ o j2 ¼ j^ SðoÞ ¼ jX x o j2 j

ð29Þ

yielding the power spectrum

The strength of the fluctuations is assumed to be constant—that is, independent of the frequency. On the other hand, the Fourier transform of the strength parameter is given by o ¼

1 X

k e
iko ¼

k¼0

¼

1 ð1
e
io Þa

1 X ðk þ a
1Þ!
io k ðe Þ k!ða
1Þ! k¼0

ð30Þ

so that rearranging terms in Eq. (30) and substituting that expression into Eq. (29), we obtain 1 SðoÞ / ð31Þ ð2sino=2Þ2a

fractal physiology, complexity, and the fractional calculus

33

for the spectrum of the fractional-differenced white noise process. In the lowfrequency limit we therefore obtain the inverse power-law spectrum SðoÞ /

1 o2a

ð32Þ

Thus, since the fractional-difference dynamics are linear, the system response is Gaussian, the same as the statistics for the white noise process on the right-hand side of Eq. (22). However, whereas the spectrum of fluctuations is flat, since it is white noise, the spectrum of the system response is inverse power law. From these analytic results we conclude that Xj is analogous to fractional Brownian motion. The analogy is complete if we set a ¼ H
1=2 so that the spectrum in Eq. (32) can be expressed as SðoÞ /

1 o2H
1

as

o!0

ð33Þ

Taking the inverse Fourier transform of the exact expression in Eq. (31) yields the autocorrelation coefficient [23] rk  ¼

hXj Xjþk i ð1
aÞ 2a
1 k  ðaÞ hXj2 i ð1:5
HÞ 2H
2 k ðH
0:5Þ

ð34Þ

as the lag time increases without limit k ! 1. The probability density function (pdf) for the fractional-difference diffusion process in the continuum limit p(x,t) satisfies the scaling condition pðx; tÞ ¼

1 x F td td

ð35Þ

where d ¼ H ¼ a
1=2. The deviation from ordinary statistical mechanics, and consequently the manifestation of complexity, is indicated by two distinct quantities. The first indicator is the scaling parameter d departing from the ordinary value d ¼ 0:5, which it would have for a simple diffusion process. But for fractional Brownian motion the value of the scaling index can be quite different. A second indicator of the deviation from ordinary statistical mechanics is the function F(y) in Eq. (35) departing from the conventional Gaussian form. The scaling index is usually determined by calculating the second moment of a time series. This method of analysis is reasonable only when F(y) has the Gaussian form, or some other distribution with a finite second moment—that is, if it is a member of the FVSM class. If the scaling condition Eq. (35) is realized,

34

bruce j. west

it is convenient to measure the scaling parameter d by the method of Diffusion Entropy Analysis (DEA) [47], which, in principle, works independently of whether the second moment is finite or not. The DEA method affords many advantages, including that of being totally independent of a constant bias. However, before we review the DEA method, let us examine another way in which the diffusion variable may scale—that is, another mechanism to generate long-time memory. 3.

Various Inverse Power-Law Autocorrelation Functions

Consider the following form of the autocorrelation function, CðtÞ ¼ hXðtÞXðt þ tÞi ¼

1 ð1 þ jtja Þ

b=a

ð36Þ

where the random process X(t) is still Gaussian, see Gneiting and Schlather [48] for a complete discussion of this correlation function and its implications. Any combination of parameters 0 < a  2 and b > 0 is allowed, in which case (36) is referred to as the Cauchy class of correlation functions. Now consider the two asymptotic limits. In the short time limit t ! 0 we expand the autocorrelation function in a Taylor series and obtain the power-law form lim CðtÞ  1
jtja

t!0

ð37Þ

for 0 < a  2. The autocorrelation function in this case indicates realizations of the random function X(t) in an E-dimensional Euclidian space that has a fractal dimension given by D ¼ E þ 1
a=2

ð38Þ

with probability one [48]. In the one-dimensional case (E ¼ 1) the power spectrum corresponding to Eq. (25) is the inverse power law SðoÞ /

1 jojaþ1

as

o!1

ð39Þ

Consequently the inverse power-law spectrum obtained in this way has a slope related to the fractal dimension by a þ 1 ¼ 5
2D. At the long time extreme, the autocorrelation function can again be expanded in a Taylor series to yield the long-time memory indicated by the inverse powerlaw correlation function lim CðtÞ  jtj
b

t!1

ð40Þ

fractal physiology, complexity, and the fractional calculus

35

when 0 < b < 1. In this case we obtain, as found earlier, 2H ¼ 2
b, relating the scaling index with the Hurst exponent. Here again the Fourier transform of the autocorrelation function yields the power spectrum SðoÞ / jojb
1 ¼

1 joj2H
1

as o ! 0

ð41Þ

which is still an inverse power law for the index in the range 0 < b < 1, or equivalently the Hurst exponent in the same range. It cannot be too strongly emphasized that the fractal dimension and the Hurst exponent can vary independently of one another. From the example presented we see that the fractal dimension is a local property of time series ðt ! 0Þ, whereas the Hurst exponent is a global property of time series ðt ! 1Þ and, although not proven here, these are general properties of D and H [48]. Therefore, returning to our DEA argument, the scaling behavior d ¼ H would be the exception and not the rule for fractal stochastic processes. Note that these two scaling exponents are obtained by comparing the scaling of the second moment with that of the probability distribution (DEA). Most methods for determining the scaling index of time series, including the allometric aggregation method, rely on the central moments of the time series having finite values (FVSM). So let us consider the case where such moments do not exist and therefore are of no help in determining the scaling exponents. B.

Dichotomous Fluctuations with Memory

The time interval between steps were assumed to be a constant finite value in the simple random walk model. If, however, we explicitly take the limit where the time interval vanishes, then the discrete walk is replaced with a continuous rate. We begin our discussion of the scaling of statistical processes by considering one of the simplest stochastic rate equations and follow the development of Allegrini et al. [49]. dXðtÞ ¼ xðtÞ dt

ð42Þ

where x(t) is a two-state random process taking the values W. If fðx; x; R; tÞ is the phase-space distribution function, then the equation of evolution corresponding to the dynamical equation (42) is qfðx; x; R; tÞ ¼ qt


 q ^ fðx; x; R; tÞ
^ x þ qx

ð43Þ

36

bruce j. west

^ is an operator where we are adopting a quantum-like formalism. Thus,  characterizing the dynamics of the x-process and ^x is an operator having the eigenvalues W, namely, ^ xji ¼ Wji: ð44Þ The underlying process generating x(t) need not be specified, but one realization of it could be a Hamiltonian system with a set of variables R. These latter variables can be infinitely many so as to result in the relaxation of the correlation properties of the system. At equilibrium, the two states jþi and j
i must have the same statistical weight. Thus we assume that the bath equilibrium corresponds to the state 1 jp0 i ¼ pffiffiffi fjþi þ j
igðRÞ 2

ð45Þ

where ðRÞ denotes the equilibrium distribution of the variables responsible for the stochastic dynamics of the variable x. The state jp0 i is one of the eigenstates ^ In fact, we set of the operator . ^ jmi ¼
m jmi

ð46Þ

and jp0 i ¼ jm ¼ 0i, 0 ¼ 0. Within this quantum-like formalism the variable x(t), as mentioned earlier, corresponds to the operator ^x. This operator, applied to the equilibrium state jp0 i, yields the excited state jp1 i ¼

^ xjp0 i ðRÞ W

ð47Þ

Thus the operator ^ x does not affect the distribution of R but has the effect of making the transitions jþi þ j
i ! jþi
j
i and jþi
j
i ! jþi þ j
i, without affecting the other bath variables. The ‘‘excited’’ state jp1 i is not an ^ but it is a linear combination of the states jmi, with m 6¼ 0. The eigenstate of , ^ operator  applied to the ‘‘excited’’ state jp1 i has the effect of relaxing it through coupling the state jp1 i to infinitely many other eigenstates jmi. The autocorrelation function hxxðtÞi, within this quantum-like formalism, reads   ^ ^xjp0 i hxxðtÞi ¼ hp0 j^ xexp t

ð48Þ

^ this correlation function On the basis of the properties of the operators ^ x and , can also be expressed as X   hxxðtÞi ¼ W 2 hp1 jmihmjp1 iexp
m t ð49Þ m6¼0

fractal physiology, complexity, and the fractional calculus

37

It is convenient to define the phase space distribution function sm ðtÞ  hmjfðx; x; R; tÞi

ð50Þ

with m ¼ 0,1,2, . . . . We are interpreting the distribution f of Eq. (43) as a sort of ket vector jfi. By multiplying Eq. (43) on the left by the states jmi, we obtain 1 X qs0 ðx; tÞ qsm ðx; tÞ ¼
W am qt qx m6¼0

ð51Þ

1 X qsm ðx; tÞ qs0 ðx; tÞ ¼
W
m sm ðx; tÞ am qt qx m6¼0

ð52Þ

and, for m>0,

with am ¼ hmj^ xjp0 i. Let us make the assumption that at t ¼ 0 all the sm ’s but the one with m ¼ 0 vanish. This condition is equivalent to assuming the spatial distribution is statistically independent of the ‘‘velocity’’ distribution and results in an equation of motion without an inhomogeneous term. By solving Eq. (52) and placing the solution into Eq. (51), we obtain ð 1 X qs0 ðx; tÞ q q2 s0 ðx; tÞ ¼ W2 am jam j2 dt0 exp½
m ðt
t0 Þ qt qx qx2 m6¼0 t

ð53Þ

0

From now on we shall focus on the reduced density matrix s0 ðx; tÞ and for the sake of simplicity we omit the subscript zero. Using Eq. (48), we can rewrite Eq. (53) in the form ðt qsðx; tÞ q2 sðx; t0 Þ ¼ dt0 hxðtÞxðt0 Þi qt qx2

ð54Þ

0

where again the brackets denote an average over an ensemble of realizations of the statistical fluctuations. In Section IV the form of Eq. (54) is determined to be that for a fractional diffusion equation when the two-point correlation function is appropriately chosen. In the case where the correlation function in Eq. (54) is an exponential hxðtÞxðt0 Þi ¼ hx2 ie
gt

ð55Þ

38

bruce j. west

taking the time derivative of Eq. (54) yields 2 q2 sðx; tÞ qsðx; tÞ 2 q sðx; tÞ
hx þ g i ¼0 qt2 qt qx2

ð56Þ

This is the celebrated telegrapher’s equation, whose phenomenological pedigree dates back to Maxwell. His (Maxwell’s) argument was to include relaxation into the wave equation and did not require the invocation of microscopic dynamics. However, his use of dissipation was compatible with the action of infinitely many degrees of freedom in the medium supporting the wave motion. The equation of motion for the Liouville density from Eq. (54) is ðt ðt 2 2 0 0 qsðx; tÞ 0 0 q sðx; t Þ 0 0 q sðx; t
t Þ ¼ dt x ðt
t Þ ¼ dt  ðt Þ x qt qx2 qx2 0

ð57Þ

0

In the case when the correlation function x is integrable, using the last term of the equality of Eq. (57), we can make use of the Markov approximation. This approximation is based on replacing the second derivative in space containing the time argument (t
t0 ) with the second derivative in space containing the time argument t and extending the upper bound of the time integration from t to infinity. This can be justified by expanding the probability density in a Taylor series in time and neglecting all but the first term. In this, the Markov approximation in Eq. (57) reduces to the ordinary diffusion equation qsðx; tÞ q2 sðx; tÞ ¼D qt qx2

ð58Þ

where the diffusion coefficient D is given by D ¼ W 2 tC

ð59Þ

and the correlation time is given by 1 ð

tC ¼

dt0 ðt0 Þ

ð60Þ

0

Notice that in the Continuous-Time Random Walk (CTRW) as used in Klafter et al. [50], in the case where the waiting time distribution is exponential, cðtÞ ¼ a exp½
at, the same evolution for the probability density p(x,t) and the phase-space distribution sðx; tÞ occurs as that resulting from Eq. [57]. This can

fractal physiology, complexity, and the fractional calculus

39

be established by noticing that in the case of the dichotomous variable x used here, the waiting-time distribution is related to the correlation function by the exact relation [51] 1 ð 1 ðt0
tÞcðt0 Þdt0 ð61Þ x ðtÞ ¼ tW t

where tW denotes the mean sojourn time. In the exponential case this sojourn time becomes identical to the correlation time; that is, since x ð0Þ ¼ 1, in the exponential case a ¼ 1=tW and tC ¼ tW . 1.

The Exact Solution

Here we are interested in the asymptotic behavior of the exact solution to Eq. (57) and we follow the analysis of Bologna et al. [52]. The most direct way to determine these properties is to take the Laplace transform in time and Fourier transform in space to obtain the Fourier–Laplace transform of the Liouville density 1 ^ ~ ðk; sÞ ¼ ð62Þ s ~ s þ x ðsÞk2 where we have imposed the initial conditions sðx; tÞjt¼0 ¼ dðxÞ

and

 qsðx; tÞ ¼0 qt t¼0

ð63Þ

The inverse Fourier transform of Eq. (62) yields s sffiffiffiffiffiffiffiffiffiffiffi
jxjpffiffiffiffiffiffi ~ ðsÞ  x s e ~ ðx; sÞ ¼ s ~ x ðsÞ 2s 

ð64Þ

which we can integrate over space to obtain 1 ð

~ ðx; sÞdx ¼ s
1

1 s

ð65Þ

indicating the conservation of normalization over time. To go beyond the formal solution in Eq. (64), we must specify the autocorrelation function. We select an inverse power-law autocorrelation function, x ðtÞ ¼ W 2

Tb ðT þ tÞb

ð66Þ

40

bruce j. west

with 0 < b < 1 and T is a positive constant. The Laplace transform of the autocorrelation function given by Eq. (66) becomes ~ x ðsÞ ¼ ð1
bÞTW ½esT
EsT   b
1 ðsTÞ1
b 2

ð67Þ

where the generalized exponential function is defined by [23,53] Egx 

1 X

x n
g ðn þ 1
gÞ n¼0

ð68Þ

and we subsequently define the generalized exponential using the fractional derivative operator. 2.

Early Time Behavior

Let us first consider the behavior of the autocorrelation function at early times. In this domain, t ! 0, we have s ! 1, so that the generalized exponential becomes sT Eb
1  esT


1 ð1
bÞðsTÞb

ð69Þ

~ x ðsÞ  W 2 =s so that the Laplace which when substituted into Eq. (67) yields  transform of the early time solution for the phase-space equation of evolution is ~ ðx; sÞ  s

e
jxjs=W 2W

ð70Þ

The inverse Laplace transform of Eq. (70) yields the delta function for the phasespace distribution function
 1 jxj d t
sðx; tÞ  2W W

ð71Þ

Thus, for times shorter than T, the evolution of the Liouville density consists of two peaks traveling in opposite directions at the same speed, W. Note that this is the same early-time solution one would obtain for the solution to the telegrapher’s equation [51]. 3.

Late Time Behavior

Now let us now consider the time asymptotic behavior of the exact solution. In the late time domain t ! 1, we have s ! 0, so examining the behavior of

fractal physiology, complexity, and the fractional calculus Eq. (67) in this domain we obtain ~ x ðsÞ  ð1
bÞTW  ðsTÞ1
b

2

"

ðsTÞ1
b 1
ð1
bÞ

41

# ð72Þ

Note that as s ! 0 the leading term in this expansion diverges for b < 1, corresponding to the fact that there is no correlation time for this process. Inserting this expression for the Laplace transform of the correlation function into Eq. (64), keeping only the diverging term, yields

~ ðx; sÞ ¼ s

jxjs1
b=2 exp
ð1
bÞWT b=2 2ð1
bÞWðsTÞb=2

ð73Þ

The inverse Laplace transform of Eq. (73) yields the phase-space distribution function
n 1 X 1 ð
1Þn jxj ð74Þ sðx; tÞ  2hxit1
b=2 n¼0 n!ð1
ðn þ 1Þð1
b=2ÞÞ hxit1
b=2 where the average of the system variable is hxi ¼ WT b=2 ð1
bÞ

ð75Þ

as had been obtained previously [54]. Straightforward dimensional analysis indicates that the space variable in Eq. (74) scales as x ta where a ¼ 1
b=2

ð76Þ

as had also been found by other authors [55]. To further support this conclusion, note that in the asymptotic limit s ! 0, Eq. (62) yields 1 ^ ~ ðk; sÞ ¼ s ð77Þ s þ constant sb
1 k2 The scaling condition x ta implies k ¼ sa , which when inserted into the righthand-side term of Eq. (77) makes the left-hand side of the same equation proportional to 1/s when the scaling condition of (76) applies. We note that 1/s is the Laplace transform of a constant in accordance with the fact that scaling is a reflection of stationarity. For this reason we are inclined to believe that the density perspective yields in the asymptotic limit a unique scaling and that our solution correctly reflects this condition.

42

bruce j. west C.

Fractals, Multifractals, and Data Processing

The salient property of mathematical random fractals processes is the existence of long-time correlations, here measured by the correlation index r, which can be related to the fractal dimension by [21] r ¼ 23
2D
1

ð78Þ

Successive increments of mathematical fractal random processes are independent of the time step. Here D ¼ 1:5 corresponds to a completely uncorrelated random process r ¼ 0, such as Brownian motion, and D ¼ 1:0 corresponds to a completely correlated process r ¼ 1, such as a regular curve. Studies of various physiologic time series have shown the existence of strong long-time correlations in healthy subjects and demonstrated the breakdown of these correlations in disease; see, for example, the review by West [56]. Complexity decreases with convergence of the Hurst exponent H to the value 0.5 or equivalently of the fractal dimension to the value 1.5. Conversely, system complexity increases as a single fractal dimension expands into a spectrum of dimensions. 1.

Multifractal Spectrum

The spectrum of fractal dimensions can be calculated in a number of ways. One way is to cover the time axis with cells of size d such that the time is given by t ¼ Nd and N  1. Following Falconer [57] we can define the partition function Zðq; dÞ 

X

mðCj Þd

ð79Þ

j

where Cj is the jth box in the d-coordinate mesh that intersects with the measure m. We can construct the measure using the time series obtained from the physiologic interval data. This measure is made by aggregating the observed time intervals {tj}, j ¼ 1,2, . . . , N, where tj denotes the time interval between the end points of stride j
1 and j, Tðn; dÞ ¼

n X

tj

ð80Þ

j¼1

such that T(n, d) is interpreted as a random walk trajectory. In this way we can construct the partition function in Eq. (79) using jTð j þ n; dÞ
Tð j; dÞj mðCj Þ ¼ N
n P jTðk þ n; dÞ
Tðk; dÞj k¼1

ð81Þ

fractal physiology, complexity, and the fractional calculus

43

where the integer n lags the trajectory by n steps. The typical scaling behavior of the partition function in the limit of vanishing grid scale [57] is Zðq; dÞ  dtðqÞ

ð82Þ

where t(q) is the mass exponent [58]. The mass exponent is related to the generalized dimension D(q) by the relation tðqÞ ¼ ð1
qÞDðqÞ

ð83Þ

where D(0) is the fractal or box counting dimension, D(1) is the information dimension, and D(2) is the correlation dimension [57,58]. The q-moment therefore accentuates different aspects of the underlying dynamical process. For q > 0, the partition function emphasizes large fluctuations and strong singularities through the generalized dimensions, whereas for q < 0, the partition function stresses the small fluctuations and the weak singularities. This property of the partition function deserves a cautionary note because the negative moments can easily become unstable, introducing artifacts into the calculation. Thus the interpretation of the trajectory approach must be judged with some caution for q < 0. A mono-fractal time series is characterized by a single fractal dimension. In general, time series have a local Ho¨lder exponent h that varies over the course of the trajectory and is related to the fractal dimension by D ¼ 2
h [57]. Note that for an infinitely long time series the Ho¨lder exponent h and the Hurst exponent H are identical; however, for a time series of finite length they need not be the same. We stress that the fractal dimension and the Ho¨lder exponent are local quantities, whereas the Hurst exponent is a global quantity; consequently the relation D ¼ 2
H is only true for an infinitely long time series. The function f(h), called the multifractal or singularity spectrum, describes how the local Ho¨lder (fractal) exponents contribute to such time series. Here h and f are independent variables, as are q and t. The general formalism of Legrendre transform pairs interrelates these two sets of variables by the relation [58], f ðqÞ ¼ qh þ tðqÞ

ð84Þ

The local Ho¨lder exponent h varies with the q-dependent mass exponent through the equality dtðqÞ hðqÞ ¼
¼
t0 ðqÞ ð85Þ dq so the singularity spectrum can be written as f ðhðqÞÞ ¼
qt0 ðqÞ þ tðqÞ where the mass exponent and its derivative are determined by data.

ð86Þ

44

bruce j. west

Figure 9. (a) The mass exponent as a function of the q-moment obtained from a numerical fit to the partition function using Eq. (87) for a typical walker. (b) The singularity spectrum f(h) obtained from a numerical fit to the mass exponent and its derivative using Eq. (86) for a typical walker [36].

As mentioned above, a time series is mono-fractal when the mass exponent is linear in q, otherwise the underlying process is multifractal. We apply the partition function measure to numerically evaluate tðqÞ ¼


lnZðq; dÞ lnd

ð87Þ

and the results are depicted in Fig. 9a. Rigorously speaking, the expression for the mass exponent requires d ! 0, but we cannot do that with data, so there is

fractal physiology, complexity, and the fractional calculus

45

TABLE I The Fitting Parameters for the Mass Exponent [Eq. (88)]a Walker

a0

1 2 3 4 5 6 7 8 9 10 Average

1.03 0.99 1.05 1.05 1.00 1.01 1.02 1.09 1.02 1.01 1.03  0.03


a1 1.26 1.14 1.32 1.26 1.12 1.07 1.17 1.29 1.14 1.17 1.19  0.08

a2 0.13 0.08 0.14 0.12 0.07 0.05 0.09 0.14 0.08 0.09 0.10  0.03

a The column
a1 is the fractal dimension for the SRV time series. In each case these numbers agree with those obtained earlier using a different method [36].

always some error in the results. The significance of that error remains to be determined. In Fig. 9 the mass exponent for a typical subject in the walking experiment [11] is shown and the individual mass exponents do not look too different from the one shown. It is clear from the figure that the mass exponent is not linear in the moment index q and therefore the SRV time series is multifractal. In Table I we record the fitting coefficients for each of the 10 SRV time series using the quadratic polynomial in the moments interval
4  q  4 tðqÞ ¼ a0 þ a1 q þ a2 q2

ð88Þ

The fit to the data using Eq. (88) is indicated by the solid curve in Fig. 9a. A second method for determining the singularity spectrum, the one we use here, is to numerically determine both the mass exponent and its derivative. In this way we calculate the multifractal spectrum directly from the data using Eq. (86). It is clear from Fig. 9b that we obtain the canonical form of the spectrum; that is, f(h) is a convex function of the scaling parameter h. The peak of the spectrum is determined to be the fractal dimension, as it should. Here again we have an indication that the interstride interval time series describes a multifractal process. We stress that we are only using the qualitative properties of the spectrum for q < 0, due to the sensitivity of the numerical method to weak singularities. The singularity spectrum can now be determined using the Legendre transformation by at least two different methods. One technique is to use the fitting equation substituted into Eq. (86). We do not do this here, but we note in

46

bruce j. west

passing that if Eq. (88) is inserted into Eq. (85), the fractal dimension is determined by the q ¼ 0 moment to be hð0Þ ¼
t0 ð0Þ ¼
a1

ð89Þ

The values of the parameter a1 listed in Table I agree with the fractal dimensions obtained earlier using a scaling argument for the same data [11,36]. The multifractal behavior of time series such as SRV, HRV, and BRV can be modeled using a number of different formalisms. For example, a random walk in which a multiplicative coefficient in the random walk is itself made random becomes a multifractal process [59,60]. This approach was developed long before the identification of fractals and multifractals and may be found in Feller’s book [61] under the heading of subordination processes. The multifractal random walks have been used to model various physiological phenomena. A third method, one that involves an integral kernel with a random parameter, was used to model turbulent fluid flow [62]. Here we adopt a version of the integral kernel, but one adapted to time rather than space series. The latter procedure is developed in Section IV after the introduction and discussion of fractional derivatives and integrals. 2.

Diffusion Entropy Analysis (DEA)

So far in this section we have focused on mathematical models that generate time series with scaling properties. In Section II we introduced a simple data aggregation procedure to reveal the scaling of physiologic time series. The allometric aggregation method offered some insight into the scaling of the underlying process, but now we turn our attention to a method that can reveal both the statistical and the correlation properties of a time series. To do this, we interpret the physiologic time series as the generator of a diffusion process and replace the random elements of the right-hand side of the simple random walk in Eq. (90) with the time series data. Thus,  even though we do not know the a priori statistical properties of the data set xj , we can deduce them from the probability density function pðx; tÞ for the diffusion variable X(t). Note that X(t) is the dynamic variable that aggregates the time series data into a ‘‘random walk’’ trajectory. If the time series is stationary, the scaling property of the probability density function for the diffusive process is given by Eq. (35), where d is the scaling exponent. We now offer a way to independently determine the scaling exponent from the time series data using the Shannon entropy for a diffusive process pðx; tÞ: 1 ð

SðtÞ ¼


pðx; tÞln pðx; tÞ dx
1

ð90Þ

47

fractal physiology, complexity, and the fractional calculus

Now if the probability density function satisfies the scaling condition Eq. (35) substituting this functional form of pðx; tÞ into Eq. (90) yields 1 ð

SðtÞ ¼

1



1 x 1 x ln d F d dx F td td t t

x using the transformation y ¼ d simplifies this equation to t SðtÞ ¼ A þ dlnt

ð91Þ

and the constant A is determined by the time-independent distribution F(y) 1 ð

A¼


FðyÞlnFðyÞ dy
1

It is obvious from Eq. (91) that a graph of the entropy S(t) versus the logarithm of the time t yields a straight line with positive slope d. Consequently using time series data to generate a diffusive process we can construct a histogram of the probability density enabling us to numerically determine the scaling parameter using the entropy. This procedure is called diffusion entropy analysis [47] (DEA). The theoretical scaling index for ordinary diffusion is d ¼ H ¼ 1=2. To test this prediction using a known data set, we generate a diffusive trajectory from Eq. (90) using a computer-generated uncorrelated Gaussian time series for 104 data points on the right-hand side of the equation. We consider a time series with a maximum length of 200 data points and construct a histogram from the nearly 104 realizations of such a time series X(t) obtained using Eq. (90). The histogram constructed from the realizations of the trajectories is inserted into Eq. (90), and the resulting entropy is calculated as a function of time. Figure 10 shows that the entropy calculated this way increases linearly with the logarithm of time with a slope of 0.48, very close to the theoretical value of 0.50 one would obtain for an infinitely long time series. Recall that fractional Brownian motion, with the distribution given by Eq. (19), satisfies the scaling relation [Eq. (35)] for the probability density. Consequently, we have the equality d ¼ H 6¼ 1=2, so that the scaling exponent d, determined by DEA, is given by the Hurst exponent H. We emphasize that this equality is not true in general, and it is quite possible that d 6¼ H, indicating that there is scaling in the time series, but the statistics need not be Gaussian. We examine this case now. Consider the limit distribution first studied by Paul Le´vy for processes having diverging central moments and which consequently violate the central limit

48

bruce j. west

Figure 10. The entropy for an uncorrelated Gaussian random process generating random walk trajectories calculated using DEA is graphed versus the natural logarithm of the time. The data indicate a linear relationship between S(t) and ln t as predicted by Eq. (91).

theorem. The generalized central limit theorem yields the probability density for the symmetric stable Le´vy process in terms of the Fourier transform of the characteristic function [63] 1 ð

pL ðx; tÞ ¼
1

dk ikx
gtjkja e e 2p

ð92Þ

where the Le´vy index is in the interval 0 < a  2 and g > 0. The only explicit expression for the Le´vy stable distribution is an infinite series whose lowestorder term is given by [64] pL ðx; tÞ /

t aþ1

jxj

ð93Þ

It is evident from the inverse power-law form of the probability density given by Eq. (93) that the second moment hx2 i of the Le´vy a-stable distribution diverges since a þ 1 < 3. Equally clear is the fact that the first moment for this distribution diverges for a þ 1 < 2. The first and second moments converge for a ¼ 2, in which case the Le´vy stable distribution becomes a Gaussian distribution and the central limit theorem again applies to the time series. We use the parameters l and k to scale the random walk phase-space variable x and time t in the Le´vy stable distribution [Eq. (92)] and obtain after

fractal physiology, complexity, and the fractional calculus some algebra

 x  1 p ; 1 L lt1=a t1=a 1  x  ¼ 1=a F 1=a t t

49

pL ðlx; ktÞ ¼

ð94Þ

when the parameters are related by k ¼ la. Thus, the Le´vy a-stable distribution satisfies the scaling relation in Eq. (35), so that the underlying process is fractal, but without memory. This lack of memory is a consequence of the Le´vy a-stable distribution being a Markov process [63,64]. The scaling behavior of a time series described by Le´vy a-stable statistics is determined using the DEA by substituting Eq. (92) into the equation for the entropy. The scaling index is determined to be d ¼ 1/a, but d in this case is not related to a Hurst exponent H. Therefore we obtain the remarkable result that the scaling index can be determined from statistical processes even when the central moments of such processes diverge and the traditional scaling methods fail. Using DEA, we have established that there are statistical processes for which d ¼ H and statistical processes for which d 6¼ H, both of which scale. However, there is a third class of processes for which the scaling index is a function of the Hurst exponent, but the relation is not one of their being equal. This third class is the Le´vy random walk process (Le´vy diffusion) introduced by Shlesinger et al. [65] in their discussion of the application of Le´vy statistics to the understanding of turbulent fluid flow. Let us distinguish between a Le´vy flight and a Le´vy walk. In a Le´vy flight the jumps taken by the random flyer each take the same amount of time, so that the statistical properties of the trajectories are determined solely by the length of the steps. If the distribution of step lengths is inverse power law, with index less than three, then using the generalized central limit theorem the resulting distribution is Le´vy [Eq. (92)]. If the step-distribution index is greater than three, the distribution converges to that of Gauss and the usual central limit theorem is recovered. A Le´vy walk, on the other hand, takes into account the fact that longer steps take longer times to complete than do shorter steps. The recognition of this simple fact ties the distribution of step sizes to the distribution of time intervals, which in the case of turbulence was determined by the fluctuations in the fluid velocity [62]. In the present example the continuum form of the Le´vy walk process is described by Eq. (42), with the autocorrelation function for the random driver being given by the inverse power law Eq. (66) and W is the constant speed of the walker. The asymptotic form of the second moment for this process is ðt

ðt

2

hXðtÞ i / dt1 dt2 x ðjt1
t2 jÞ ¼ t2
b 0

0

ð95Þ

50

bruce j. west

and in terms of the index for the autocorrelation function the Hurst exponent is 2H ¼ 2
b

ð96Þ

Using the relation between the waiting-time distribution function Eq. (61) and the autocorrelation function Eq. (66), we obtain the inverse power-law waitingtime distribution cðtÞ ¼

aT a ðT þ tÞaþ1

ð97Þ

where the autocorrelation index and the waiting-time index are related by b¼a
1

ð98Þ

Consequently, by means of the delta function dðjxj
WtÞ which ties space and time together, the resulting distribution for the diffusion process is Le´vy a-stable and the scaling parameter in Eq. (98) is the Le´vy index. Therefore, using Eq. (96) and the fact that d ¼ 1=a for a Le´vy a-stable process, we obtain the relation between exponents: d¼

1 3
2H

ð99Þ

Scafetta and Grigolini [47] established that the DEA scaling does, in fact, yield the scaling relation given by (99) for Le´vy diffusion. We give two examples of fractal time series. The first is fractal Gaussian intermittent noise characterized by a long-time correlated waiting-time sequence, and the second is a Le´vy-walk intermittent noise. These examples were developed in an environmental context to explain the observed distribution of earthquakes in California [66]. The Gaussian intermittent noise has an autocorrelation function given by Eq. (20) and power-law index given by 2H
2. The statistics of this sequence is determined by a finite variance waiting time distribution function c(t) whose form may be, for example, that of a Gaussian, exponential, or Poisson. The diffusion generated by a fractal Gaussian intermittent noise is a particular type of fractional Brownian motion and satisfies the asymptotic scaling relation between indices d ¼ H. Figure 11A is based on a realization of this process using an exponential waiting-time distribution function. The parallel lines through the computer-generated data show the two scaling exponents, one determined by DEA, indicated by the entropy S(t) minus the constant term, and the other determined directly from the second moment, denoted by D(t). If the long-time correlations are destroyed by means of shuffling—that is, by

fractal physiology, complexity, and the fractional calculus

51

Figure 11. DEA and SDA for (A) a fractal Gaussian intermittent noise with c(t) ¼ exp[
t/g] with g ¼ 25 and H ¼ d ¼ 0.75; the fractal Gaussian relation of equal exponents is satisfied. (B) A Le´vy-walk intermittent noise with cðtÞ / t
m and m ¼ 2.5; note the bifurcation between H ¼ 0.75 and d ¼ 0.67 caused by the Le´vy-walk diffusion relation [66].

randomly interchanging the positions of the elements of the sequence—the new intermittent random time series is characterized by the value d ¼ H ¼ 0:50. This latter result is not shown. The Le´vy-walk intermittent noise is characterized by an uncorrelated waiting-time sequence and a Le´vy or an inverse power-law waiting-time distribution function such as given by Eq. (97) with 1 < a < 2. This interval for the scaling index insures that although the second moment diverges, the first moment is finite. The presence of a Le´vy-walk process in a given time series can be detected by means of the asymptotic relation Eq. (99), which we refer to as

52

bruce j. west

the Le´vy-walk diffusion relation [66]. Figure 11B is based on a realization of this process using an inverse power-law waiting-time distribution function. This figure shows the scaling properties of a computer-generated random Le´vy-walk intermittent noise with a ¼ 1.5 that has H ¼ 0.75 and d ¼ 0.67 in agreement with the Le´vy-walk diffusion relation. We stress that the Le´vy-walk diffusion relation is fulfilled if the waiting times are uncorrelated, in which case any shuffling of the elements in data sequence would not alter the scaling exponents H and d. In fact, the superdiffusion scaling exponent 0.5 < d < H < 1 of a Le´vy-walk intermittent noise are related to the fatness of the waiting-time inverse power-law tail, as measured by the exponent a. Contrary to a fractal Gaussian intermittent noise, this Le´vy scaling does not imply a temporal correlation, or a historical memory, among events because the occurrence of future events is independent of the frequency of past events. It should be stressed that even though the second moment for a Le´vy walk is finite, the scaling index obtained from the second moment does not give the correct scaling properties of the time series. This is a word of caution regarding the application of FVSMs to determine the scaling properties of a time series. Even when the second moment has the form hXðtÞ2 i / t2m

ð100Þ

this does not mean that the index so determined has any interesting implications regarding the underlying dynamics of the time series. It is only after the statistics of the time series are determined, by using DEA or some other technique, that can one begin to interpret the anomalous diffusion equation, Eq. (100). IV.

FRACTIONAL DYNAMICS

In the late nineteenth century, most mathematicians felt that a continuous function must have a derivative ‘‘almost everywhere,’’ which means that the derivative of a function is singular only on a set of points whose total length (measure) vanishes. However, some mathematicians wondered if functions existed that were continuous, but did not have a derivative at any point (continuous everywhere but differentiable nowhere). The motivation for considering such pathological functions was initiated within mathematics and not in the physical or biological sciences, the insights of Boltzmann and Perrin notwithstanding. In 1872, Karl Weierstrass (1815–1897) gave a lecture to the Berlin Academy in which he presented functions that had the aforementioned continuity and nondifferentiability properties; consequently, these functions had the symmetry of self-similarity. Twenty-six years later, Ludwig Boltzmann, who elucidated the microscopic basis of entropy, said that physicists could have

fractal physiology, complexity, and the fractional calculus

53

invented such functions in order to treat collisions among molecules in gases and fluids. Boltzmann had a great deal of experience thinking about such things as discontinuous changes of particle velocities that occur in kinetic theory and to wonder about their proper mathematical representation. He had spent many years trying to develop a microscopic theory of gases and he was successful in developing such a theory, only to have his colleagues reject his contributions. Although kinetic theory led to acceptable results (and provided a suitable microscopic definition of entropy), it was based on time-reversible dynamic equations; that is, entropy distinguishes the past from the future, whereas the equations of classical mechanics do not [2]. This basic inconsistency between analytic dynamics and thermodynamics remains unresolved today, although there are indications that the resolution of this old chestnut lies in microscopic chaos. It was assumed in the kinetic theory of gases that molecules are materially unchanged as a result of interactions with other molecules, and collisions are instantaneous events as would occur if the molecules were impenetrable and perfectly elastic. As a result, it seemed quite natural that the trajectories of molecules would sometimes undergo discontinuous changes. Robert Brown, in 1827, observed the random motion of a speck of pollen immersed in a water droplet. Discontinuous changes in the speed and direction of the motion of the pollen mote were observed, but the mechanism causing these changes was not understood. Albert Einstein published a paper in 1905 that, although concerned with diffusion in physical systems, ultimately explained the source of Brownian motion as being due to the net imbalance of the random collisions of the lighter particles of the medium with the surface of the pollen mote. Jean Baptiste Perrin, of the University of Paris, experimentally verified Einstein’s predictions and received the Nobel Prize for his work in 1926. Perrin [67], giving a physicist’s view of mathematics in 1913, stated that curves without derivatives are more common than those special, but interesting ones, like the circle, that have derivatives. In fact he was quite adamant in his arguments emphasizing the importance of nonanalytic functions for describing complex physical phenomena, such as Brownian motion. Thus, there are valid physical reasons for looking for these types of functions, but the scientific reasons became evident to the general scientific community only long after the mathematical discoveries made by Weierstrass. On the theoretical physics side, the Kolmogorov–Arnold–Moser (KAM) theory for conservative dynamical systems describes how the continuous trajectories of a particle break up into a chaotic sea of randomly disconnected points. Furthermore, the strange attractors of dissipative dynamical systems have a fractal dimension in phase space. Both these developments in classical dynamics—KAM theory and strange attractors—emphasize the importance of nonanalytic functions in the description of the evolution of deterministic nonlinear dynamical systems. We do not discuss the details of such dynamical systems herein, but refer the reader to a number of excellent books on the

54

bruce j. west

subject ranging from the mathematical rigorous, but readable [68], to provocative picture books [69], to extensive applications [23]. In this section we describe some of the essential features of fractal functions starting from the simple dynamical processes described by functions that are fractal (such as the Weierstrass function) and that are continuous everywhere but are nowhere differentiable. This idea of nondifferentiability leads to the introduction of the elementary definitions of fractional integrals and fractional derivatives starting from the limits of appropriately defined sums. We find that the relation between fractal functions and the fractional calculus is a deep one. For example, the fractional derivative of a regular function yields a fractal function of dimension determined by the order of the fractional derivative. Thus, the changes in time of phenomena that are best described by fractal functions are probably best described by fractional equations of motion, as well. In any event, this latter perspective is the one we developed elsewhere [52] and discuss herein. Others have also made inquiries along these lines [70]: It is interesting to investigate whether fractional calculus, which generates the operation of derivation and integration to fractional order, can provide a possible calculus to deal with fractals. In fact there has been a surge of activity in recent times which supports this point of view. This possible connection between fractals and fractional calculus gives rise to various interesting questions. . . .

The separation of time scales in physical phenomena allows us to smooth over the microscopic fluctuations and construct a differentiable representation of the dynamics on large space scales and long time scales. However, such smoothing is not always possible, examples of physical phenomena that resist this approach include turbulent fluid flow [71], the stress relaxation of viscoelastic materials such as plastics and rubber [72,73], and finally phase transitions [74,75]. Metaphorically, these complex phenomena, whose evolution cannot be described by ordinary differential equations of motion, leap and jump in unexpected ways to obtain food; they unpredictably twist and turn to avoid capture, and they suddenly change strategy to anticipate environmental changes. To understand these and other analogous processes in physiology, we find that we must adopt a new type of modeling, one that is not in terms of ordinary or partial differential equations of motion. It is clear that the fundamental elements of complex physical phenomena, such as phase transitions, the deformation of plastics, and the stress relaxation of polymers, satisfy Newton’s laws. In these phenomena the evolution of individual particles are described by ordinary differential equations that control the dynamics of individual particle trajectories. It is equally clear that the connection between the fundamental laws of motion controlling the individual particle dynamics and the observed large-scale dynamics cannot be made in any straightforward way.

55

fractal physiology, complexity, and the fractional calculus

In previous sections we have investigated the scaling properties of processes described by certain stochastic differential equations. The scaling in the system response was a consequence of the inverse power-law correlation in the fluctuations driving the system. The fractal statistics of the dynamic model in Section III are suggestive of the scaling observed in the physiologic time series presented in Section II. Of course this is not the only way the system variable, as characterized by measured time series, can manifest scaling. Another is through the internal dynamics of the system, and that is what we explore in this section. We construct a fractional Langevin equation in which the fractional derivatives give rise to the long-time memory in the system dynamics. It is determined that the solutions to such equations describe multifractal statistics, and we subsequently apply this model to a number of physiological phenomena, including cerebral blood flow and migraines. A.

Fractional Calculus

It is useful to have in mind the formalism of the fractional calculus before embarking on the interpretation of models using this formalism to explain the complexity of physiological phenomena. What we call the fractional calculus dates back to a question L’Hoˆpital asked Leibniz in 1695, where in a letter he asked the meaning of the expression d n y=dxn if n ¼ 1=2, that is: . . .what if n is fractional?

Leibniz replied in part [76]: pffiffiffiffiffiffiffiffiffiffiffi Thus it follows that d 1=2 x will be equal to 2 dx : x . . . . John Bernoulli seems to have told you of my having mentioned to him a marvelous analogy which makes it possible to say in a way the successive differentials are in geometric progression. One can ask what would be a differential having as its exponent a fraction. You see that the result can be expressed by an infinite series. Although this seems removed from Geometry, which does not yet know of such fractional exponents, it appears that one day these paradoxes will yield useful consequences, since there is hardly a paradox without utility.

After 310 years of sporadic development, the fractional calculus is now becoming so sufficiently well developed and well known that books and articles are being devoted to its consequences in the physical sciences [53,77,78]. The simplest way to introduce fractional derivatives is to consider the ordinary derivative of a monomial, say the nth derivative of tm for m > n: Dnt ½tm  ¼ mðm
1Þ . . . ðm
n þ 1Þtm
n m! tm
n ¼ ðm
nÞ!

ð101Þ

56

bruce j. west

where the operator Dt is the ordinary derivative. We can generalize the form of Eq. (101) by recognizing that the ratio of factorials can be expressed as the ratio of gamma functions: Dnt ½tm  ¼

ðm þ 1Þ m
n t ðm þ 1
nÞ

ð102Þ

We can extend these considerations to fractional derivatives by means of analogy. We define a real indexed derivative of a monomial t: Dat ½tb  ¼

ðb þ 1Þ b
a t ðb þ 1
aÞ

ð103Þ

where b þ 1 6¼ 0;
1;
2; . . . ;
n; this is, the monomial index is not integervalued. With this analogy we can solve the ½-derivative problem posed to Leibniz. Consider the definition (103) for a ¼ b ¼ ½ yielding ð
1=2 þ 1Þ t
1=2
1=2 ð
1=2 þ 1
1=2Þ ð1=2Þ
1 t ¼0 ¼ ð0Þ

Dt ½t
1=2  ¼ 1=2

since gð0Þ ¼ 1. Thus, a particular function is effectively a constant with regard to a certain functional derivative. Consider a second example, this time with monomial index equal to zero b ¼ 0 so that we have the ½-derivative of a constant: ð0 þ 1Þ t
1=2 ð0 þ 1
1=2Þ 1 ¼ pffiffiffiffiffi pt

1=2

Dt ½1 ¼

where we see that the constant is not a constant with regard to fractional derivatives. Finally, there is the ½-derivative of t, b ¼ 1: 1=2 Dt ½t

ð1 þ 1Þ 1
1=2 t ¼ ¼ ð1 þ 1=2Þ

the result obtained by Leibniz.

rffiffiffi t p

fractal physiology, complexity, and the fractional calculus

57

Another way to introduce fractional operators is by generalizing Cauchy’s formula for a n-fold integration over a fixed time interval (a,t): tn
1 ðt ðt ðt1 ð 1 ð
nÞ n
1 ðt
xÞ f ðxÞdx ¼

f ðxn Þ dxn dx1  a Dt ½ f ðtÞ ð104Þ ðn
1Þ! a a

a

a

ð
nÞ ½  a Dt

denotes the n-fold integration operation. The where the operator fractional integral analogue to this equation is defined as ð
aÞ

a Dt

½ f ðtÞ ¼

ðt 1 ðt
xÞa
1 f ðxÞ dx; ðaÞ

ta

ð105Þ

a

where the factorial has been replaced by the gamma function, the latter having an analytic continuation into the complex domain for noninteger and nonpositive values of its argument. The corresponding fractional derivative is given by ðaÞ a Dt

¼

d n ða
nÞ aD dtn t

ð106Þ

where ½a þ 1  n  ½a and the bracket denotes the integer value n closest to a. Consequently for a < 1=2 we have n ¼ 0. Equation (105) is the Riemann– Liouville (RL) formula for the fractional operator; it is the integral operator when a < 0 and it is the differential operator interpreted as (106) when a > 0. 1.

Derivative of a Fractal Function

Richardson, in his 1926 investigation of turbulence, observed that the velocity field of the atmospheric wind is so erratic that it probably cannot be described by an analytic function [79]. He suggested a Weierstrass function as a candidate to represent the velocity field, since the function is continuous everywhere, but nowhere differentiable, properties he observed in the wind-field data. Here we investigate a generalization of the Weierstrass function in order to simplify some of the discussion: 1 X 1 ½1
cosðbn tÞ ð107Þ WðtÞ ¼ n a n¼
1 under the conditions that b > a > 1. The generalized Weierstrass function (GWF) satisfies the scaling relation WðbtÞ ¼ aWðtÞ

ð108Þ

58

bruce j. west

resulting from a simple shift of the index n in the summation. Equation (108) has the form of a renormalization group scaling relation [5], which can be solved by assuming a solution of the form WðtÞ ¼

1 X

ð109Þ

A n t Hn

n¼
1

Inserting Eq. (109) into Eq. (108) yields the equation for the scaling index: Hn ¼

ln a 2pn
i ln b ln b

ð110Þ

where the scaling exponent is seen to be complex. This exponent has been related to a complex fractal dimension in the architecture of the human lung [80], in many other physiological systems [5,23], and in earthquakes, turbulence, and financial crashes [81]. The GWF is a superposition of harmonic terms having increasing frequencies as powers of b with decreasing amplitudes as powers of 1/a. This function has a fractal dimension D if we choose a ¼ b2
D , so that in terms of the fractal dimension we write the GWF as WðtÞ ¼

1 X

1 ½1
cosðbn tÞ ð2
DÞn b n¼
1

ð111Þ

The RL -fractional integral of the GWF is given by

W

ð
aÞ

ðtÞ 

ð
aÞ ½WðtÞ
1 Dt

1 ¼ ðaÞ

ðt
1

WðxÞ ðt
xÞ1
a

dx

ð112Þ

for 0 < a < 1, which after some not so straightforward analysis [23,82] yields W ð
aÞ ðtÞ ¼

1 X

1

n¼
1

bð2
DþaÞn

½1
cosðbn tÞ

ð113Þ

Similarly the RL-fractional derivative of the GWF is given by

W

ðaÞ

ðtÞ 

ðaÞ
1 Dt ½WðtÞ

1 d ¼ ð1
aÞ dt

ðt
1

WðxÞ dx ðt
xÞa

ð114Þ

fractal physiology, complexity, and the fractional calculus

59

for 0 < a < 1, which integrates to [23,82] W ðaÞ ðtÞ ¼

1 X

1 ½1
cosðbn tÞ ð2
D
aÞn b n¼
1

ð115Þ

Consequently, we see that the fractional integral shifts the fractal dimension D ! D
a and the fractional derivative shifts the fractal dimension D ! D þ a. These results can be interpreted by noticing that the fractional dimension gives information about the degree of irregularity of the function under analysis. Carrying out a fractional integral of the GWF implies decreasing its fractional dimension and therefore smooths the process, whereas carrying out the fractional derivative means increasing the fractional dimension and therefore making the process and its increments more irregular. What is most intriguing is the fact that a fractional operator acting on a fractal function yields another fractal function; the derivative does not diverge, as does an ordinary derivative of a fractal function, like that of Weierstrass. This suggests that the fractional calculus might be the appropriate method for characterizing the dynamics of complex phenomena, particularly those that are described by fractal functions. 2.

Fractional Brownian Motion

In the previous subsection it would have been possible to extend out discussion to random processes by including random phases in the definition of the GWF: ( ) 1 X 1 n  WðtÞ ¼ Re ð116Þ 1
eib t eifn an n¼
1 where the phase is a random quantity uniformly distributed on the interval (0,2p). Equation (116) would be one way to introduce a random function that has the desired scaling properties and we could discuss the dynamics of this process in time using the fractional operators. An alternative approach is to start with continuous functions and that is what we do now. Consider a function defined by the Fourier transform 1 ð 1 xðoÞ do ð117Þ xðtÞ ¼ e
iot ^ 2p
1

Operating on this function with the RL-fractional operator, with a lower limit of negative infinity, defines a new function 1 ð 1 ð
aÞ Fa ðtÞ 
1 Dt ½xðtÞ ¼ ð118Þ e
iot ð
ioÞ
a ^xðoÞ do 2p
1

60

bruce j. west

and the weighting in the integrand is obtained by operating on the exponential in ð
aÞ the Fourier transform
1 Dt ½eiot  ¼ eiot ðioÞ
a . If we now interpret the function Eq. (117) as a stochastic quantity, then we can evaluate the correlation of the function Eq. (118) at two time points separated by an interval t hFa ðtÞFa ðt

þ tÞi ¼

1 ð

1 ð2pÞ2

1 ð

do1
1


a ^ ^ do2 e
iðo1
o2 Þt eio2 t o
a 1 o2 hxðo1 Þx ðo2 Þi


1

ð119Þ where the brackets denote an average over an ensemble of realizations of random fluctuations. If the random fluctuations correspond to a Wiener process, the average in the integrand reduces to h^ xðo1 Þ^ x ðo2 Þi ¼ Cdðo1
o2 Þ

ð120Þ

and C is a constant. Substituting Eq. (120) into Eq. (119) and integrating over one of the frequencies yields for the autocorrelation function hFa ðtÞFa ðt

þ tÞi ¼

C ð2pÞ2

1 ð

doeiot o
2a / t2a
1

ð121Þ


1

If we make the association of the order of the fractional operator with the Hurst exponent a ¼ H
1=2

ð122Þ

we observe that the fractional index lies in the interval
1=2  a  1=2, because the Hurst exponent is confined to the interval 0 < H  1. Therefore the solution to the fractal stochastic equation of motion ðaÞ
1 Dt ½Fa ðtÞ

¼ xðtÞ

ð123Þ

given by Eq. (115) has the same scaling properties as the dichotomous process with the inverse power-law correlations studied in Section III. Note that Eq. (118) is a colored noise representation of the dynamics expressed by Eq. (123). Even though the central moments of the solution to Eq. (123) scale in the present case in the same way as the moments did for the process in Section III, they are very different processes. The simplest way to see the difference is to note that the integral relation in Eq. (118) is linear so that the solution Fa ðtÞ and the random fluctuations x(t) have the same statistics,

fractal physiology, complexity, and the fractional calculus

61

which, by assumption, are Gaussian. Consequently the solution Fa ðtÞ is a realization of fractional Brownian motion with the fractional index restricted to the indicated region. This is certainly different from the exact solution given by the series expansion for the phase-space distribution function in Eq. (74). B.

Fractional Langevin Equations

Of course, the fractional calculus does not in itself constitute a physical/ biological theory; however, one requires such a theory in order to interpret the fractional derivatives and integrals in terms of physical/biological phenomena. We therefore follow a pedagogical approach and examine the simple relaxation process described by the rate equation d ðtÞ þ lðtÞ ¼ 0 dt

ð124Þ

where t > 0 and the relaxation rate l determines how quickly the process returns to its equilibrium state. The solution to Eq. (124) is given by ðtÞ ¼ ð0Þe
lt, which is unique in terms of the initial condition ð0Þ. An alternative way of writing Eq. (124) is in terms of the anti-derivative operator

1 d ðtÞ
ð0Þ ¼ l ðtÞ dt which suggests, for its generalization, replacing the anti-derivative with the RLfractional integral operator ð
aÞ

ðtÞ
ð0Þ ¼ la 0 Dt

½ðtÞ

ð125Þ

where the lower limit of the fractional integral is zero, corresponding to the initial value problem. Operating on the left in Eq. (125) with the fractional derivative we obtain the generalization to the relaxation equation given by [83] ðaÞ 0 Dt ½ðtÞ

þ la ðtÞ ¼

t
a ð0Þ ð1
aÞ

ð126Þ

and the initial value becomes an inhomogeneous term in this fractional relaxation equation of motion. Here the relaxation time is raised to the power a > 0 in order to maintain the correct dimensionality. Equations of the form (126) are mathematically well-defined, and strategies for solving such equations have been developed by a number of investigators, particularly in the book by Miller and Ross [84] that is devoted almost

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bruce j. west

exclusively to solving such equations when the index is rational. Here we make no such restriction and consider the Laplace transform of Eq. (126) to obtain ð0Þ sa ~ ðsÞ ¼ s la þ sa

ð127Þ

whose inverse Laplace transform is the solution to the fractional differential equation. Inverting Laplace transforms such as Eq. (127) is nontrivial and an excellent technique that overcomes many of the technical difficulties, implemented by Nonnenmacher and Metzler [83], involve the use of Fox functions [53]. The solution to the fractional relaxation equation is given by the series expansion for the standard Mittag–Leffler function ðtÞ ¼ ð0ÞEa ð
ðltÞa Þ ¼ ð0Þ

1 X

ð
1Þk ðltÞka ð1 þ kaÞ k¼0

ð128Þ

which in the limit a ! 1 yields the exponential function lim Ea ð
ðltÞa Þ ¼ e
lt

a!1

as it should, since under this condition (126) reduces to the ordinary relaxation rate equation Eq. (124). The Mittag–Leffler function has interesting properties in both the short-time and the long-time limits. In the short-time limit it yields the Kohlrausch– Williams–Watts Law from stress relaxation in rheology given by lim Ea ð
ðltÞa Þ ¼ e
ðltÞ

a

t!0

ð129Þ

also known as the stretched exponential. In the long-time limit it yields the inverse power law, known as the Nutting Law, lim Ea ð
ðltÞa Þ ¼ ðltÞ
a

t!1

ð130Þ

Figure 12 displays the Mittag–Leffler function as well as the two asymptotes, the dashed curve being the stretched exponential and the dotted curve the inverse power law. What is apparent from this discussion is the long-time memory associated with the fractional relaxation process, being an inverse power law rather than the exponential of ordinary relaxation. It is apparent that the Mittag– Leffler function smoothly joins these two empirically determined asymptotic distributions.

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63

MITTAG–LEFFLER FUNCTION

0 –0.25

log–function

–0.5 –0.75 –1 –1.25 –1.5 –1.75 –6

–4

–2 log–variable

0

2

Figure 12. The solid curve is the Mittag–Leffler function, the solution to the fractional relaxation equation. The dashed curve is the stretched exponential (Kohlrausch–Williams–Watts Law), and the dotted curve is the inverse power law (Nutting Law).

We can now generalize the fractional differential equation to include a random force xðtÞ and in this way obtain a fractional Langevin equation ðaÞ 0 Dt ½ðtÞ

þ la ðtÞ ¼

t
a ð0Þ þ xðtÞ ð1
aÞ

ð131Þ

The solution to this equation is obtained using Laplace transforms as done previously: ~ xðsÞ ð0Þsa
1 ~ ðsÞ ¼ a þ a a l þ sa l þs

ð132Þ

Note the difference in the s-dependence of the two coefficients of the right-hand side of Eq. (132). The inverse Laplace transform of the first term yields the Mittag–Leffler function as found in the homogeneous case above. The inverse Laplace transform of the second term is the convolution of the random force and a stationary kernel. The kernel is given by the series Ea;b ðzÞ 

1 X

zk ; ðak þ bÞ k¼0

a > 0; b > 0

ð133Þ

which is the generalized Mittag–Leffler function. The function defined by Eq. (133) reduces to the usual Mittag–Leffler function when b ¼ 1, so that both

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bruce j. west

the homogeneous and inhomogeneous terms in the solution to the fractional Langevin equation can be expressed in terms of these series. Note that taking the average value of Eq. (131) and observing that the average of the random force is zero, we obtain ðaÞ 0 Dt ½hðtÞi

þ la hðtÞi ¼

t
a ð0Þ ð1
aÞ

ð134Þ

which is clearly of the form of the fractional stress relaxation equation. The average response of the system is determined by the Mittag–Leffler function; that is, the average has a long-time memory (inverse power law). The explicit inverse of Eq. (132) yields the solution [53] ðt

ðtÞ ¼ ð0ÞEa ð
ðltÞ Þ þ ðt
t0 Þa
1 Ea;a ð
ðltÞa Þxðt0 Þ dt0 a

ð135Þ

0

In the case a ¼ 1, the Mittag–Leffler function becomes the exponential, so that the solution to the fractional Langevin equation reduces to that for an Ornstein– Uhlenbeck process ðtÞ ¼ ð0Þe


lt

ðt

0

þ e
lðt
t Þ xðt0 Þ dt0

ð136Þ

0

as it should. The analysis of the autocorrelation function of Eq. (135) can be quite daunting and so we do not pursue it further here, but refer the reader to the literature [53,85]. A somewhat simpler problem is the fractional Langevin equation without dissipation. Consider the second moment of the solution to the Langevin equation when l ¼ 0 giving rise to h½ðt1 Þ
ð0Þ½ðt1 Þ
ð0Þi ¼

ðt1

1 ðaÞ

ðt2 dt1 dt2

2 0

0

hxðt1 Þxðt2 Þi ðt1
t1 Þ1
a ðt2
t2 Þ1
a ð137Þ

Here again we take the random force to have Gaussian statistics and to be delta correlated in time: hxðt1 Þxðt2 Þi ¼ Cdðt1
t2 Þ

ð138Þ

fractal physiology, complexity, and the fractional calculus

65

Inserting Eq. (138) into the expression for the autocorrelation function (137) and noting that the integral is symmetric in the times, the delta function restricts the integration to the lesser of the two times, so introducing the notation for the lesser time t < and greater time t > we obtain [53] h½ðt > Þ
ð0Þ½ðt < Þ
ð0Þi ¼

a 2Cta
1 > t




ð139Þ in terms of the hypergeometric function. Note that although the statistics of the solution are Gaussian, they are also nonstationary, since the autocorrelation function depends on the lesser and the greater times separately and not on their difference. Of course, we can also use the general expression Eq. (139) to write the second moment of the solution at time t ¼ t < ¼ t > , h½ðtÞ
ð0Þ2 i ¼ ¼

2Ct2a
1

Fð1; 1
a; 1 þ a : 1Þ ðaÞ2 2Ct2a
1

ð2a
1ÞðaÞ2

ð140Þ

where the second equality results from writing the hypergeometric function as the ratio of gamma functions. The time dependence of the second moment [Eq. (140)] agrees with that obtained for anomalous diffusion in Section III, if we make the identification 2H ¼ 2a
1, where, since the fractional index is less than one, we have 1=2  H > 0. Consequently, the process described by the dissipation-free fractional Langevin equation is antipersistent. This antipersistent behavior of the time series was observed by Peng et al. [25] for the differences in time intervals between heart beats. They interpreted this result, as did a number of subsequent investigators, in terms of random walks with H < 1=2. However, we can see from Eq. (140) that the fractional Langevin equation without dissipation is an equally good, or one might say an equivalent, description of the underlying dynamics. The scaling behavior alone cannot distinguish between these two models; what is needed is the complete statistical distribution and not just the time-dependence (scaling behavior) of a moment. 1.

Physical/Physiological Models

A theoretical Langevin equation is generally constructed from a Hamiltonian model for a simple dynamical system coupled to the environment. The equations of motion for the coupled system are manipulated so as to eliminate the degrees

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of freedom of the environment from the dynamical description of the system. Only the initial state of the environment (heat bath) remains in the Langevin description, where the random nature of the driving force is inserted through the choice of distribution of the initial states of the bath. The simplest Langevin equation for a dynamical system open to the environment has the form dXðtÞ þ lXðtÞ ¼ xðtÞ dt

ð141Þ

where x(t) is a random force, l is a dissipation parameter and there exists a fluctuation–dissipation relation connecting the two [86]. Of course we cannot completely interpret Eq. (141) until we specify the statistical properties of the fluctuations, and for this we need to know the environment of the system. The random driver is typically assumed to be a Wiener process—that is, to have Gaussian statistics and no memory. When the system dynamics depends on what occurred earlier—that is, the environment has memory—Eq. (141) is no longer adequate and the Langevin equation must be modified. The generalized Langevin equation takes this memory into account through an integral term of the form ðt dXðtÞ þ Kðt
t0 ÞXðt0 Þ dt0 ¼ xðtÞ dt

ð142Þ

0

where the memory kernel replaces the dissipation parameter and the fluctuation– dissipation relation becomes generalized: KðtÞ ¼ hxðt þ tÞxðtÞi

ð143Þ

Both these Langevin equations are monofractal if the fluctuations are monofractal, which is to say, the time series given by the trajectory X(t) is a fractal random process if the random force is a fractal random process. However, neither of these models is adequate for describing multifractal statistical processes as they stand. A number of investigators have recently developed multifractal random walk models to account for the multifractal character of various physiological phenomena, and here we introduce a variant of those discussions based on the fractional calculus. The most recent generalization of the Langevin equation incorporates memory into the system’s dynamics and has the simple form of Eq. (131) with the dissipation parameter set to zero: ðmÞ 0 Dt ½XðtÞ




t
a X0 ¼ xðtÞ ð1
mÞ

ð144Þ

fractal physiology, complexity, and the fractional calculus

67

Equation (144) could also be obtained from the construction of a fractional Langevin equation by Lutz [87] for a free particle coupled to a fractal heat bath, when the inertial term is negligible. The formal solution to this fractional Langevin equation is ðt 1 xðt0 Þdt0 XðtÞ
X0 ¼ ðmÞ ðt
t0 Þ1
m 0

which can be expressed in terms of the integral kernel: ðt

XðtÞ
X0 ¼ Km ðt
t0 Þxðt0 Þ dt0

ð145Þ

0

As mentioned earlier, the form of this relation for multiplicative stochastic processes and its association with multifractals has been noted in the phenomenon of turbulent fluid flow [61], through a space, rather than time, integration kernel. The random force term on the right-hand side of Eq. (145) is selected to be a zero-centered, Gaussian random variable and therefore to scale as [21] xðltÞ ¼ l
H xðtÞ

ð146Þ

where the Hurst exponent is in the range 0 < H  1. In a similar way the kernel in Eq. (145) is easily shown to scale as Km ðltÞ ¼ lm Km ðtÞ

ð147Þ

so that the solution to the fractional Langevin equation scales as XðltÞ
X0 ¼ l
Hþm ½XðtÞ
X0 

ð148Þ

In order to make the solution to the fractional Langevin equation a multifractal, we assume that the parameter a is a random variable. To construct the traditional measures of multifractal stochastic processes, we calculate the qth moment of the solution (148) by averaging over both the random force and the random parameter to obtain hjXðltÞ
X0 jq i ¼ lðq
1ÞH hlqm ihjXðtÞ
X0 jq i ¼ hjXðtÞ
X0 jq ilrðqÞ

ð149Þ

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bruce j. west

The scaling relation in Eq. (149) determines the qth order structure function exponent r(q). Note that when r(q) is linear in q the underlying process is monofractal, whereas when it is nonlinear in q the process is multifractal, because we can relate the structure function to the mass exponent [88]: rðqÞ ¼ 2
tðqÞ

ð150Þ

Consequently we have that r(0) ¼ H so that tð0Þ ¼ 2
H, as it should because of the well-known relation between the fractal dimension and the global Hurst exponent D0 ¼ 2
H. To determine the structure function exponent, we make an assumption about the statistics of the parameter a. We can always write the m-average as hlqm i ¼ heqZðlnlÞ i

ð151Þ

where Z(ln l) is the random variable. In this way the expression on the right-hand side of Eq. (151) is the Laplace transform of the probability density. We assume the random variable is an a-stable Le´vy process in which case the statistics of the multiplicative fluctuations are given by the distribution 1 Pðx; sÞ ¼ 2p

1 ð

a

eikz e
bsjkj dk

ð152Þ


1

with 0 < a  2. Inserting Eq. (152) into Eq. (151) and integrating over z yields the delta function dðk þ iqÞ, which, integrating over k, results in a

heqZðlnlÞ i ¼ e
bjqj

lnl

a

¼ l
bjqj

ð153Þ

so that comparing this result with Eq. (149) we obtain for the structure function exponent rðqÞ ¼ ðq
1ÞH
bjqja

ð154Þ

Therefore the solution to the fractional Langevin equation corresponds to a monofractal process only in the case a ¼ 1 and q > 0; otherwise the process is multifractal. We restrict the remaining discussion to positive moments. Thus, we observe that when the memory kernel in the fractional Langevin equation is random, the solution consists of the product of two random quantities giving rise to a multifractal process. This is Feller’s subordination process. We apply this approach to the SRV time series data discussed in Section II and observe, for the statistics of the multiplicative exponent given by Le´vy statistics, the singularity spectrum as a function of the positive moments

fractal physiology, complexity, and the fractional calculus

69

Singularity Spectrum

1

0.95 0.9 0.85 0.8 0

0.5

1

1.5

2

2.5

3

q-moment Figure 13. The singularity spectrum for q > 0 obtained through the numerical fit to the human gait data. The curve is the average over the 10 data sets obtained in the experiment [11].

shown by the points in Fig. 13. The solid curve in this figure is obtained from the analytic form of the singularity spectrum f ðqÞ ¼ 2
H
ða
1Þbqa

ð155Þ

which is determined by substituting Eq. (154) into the equation for the singularity spectrum [Eq. (84)], through the relationship between exponents [Eq. (150)]. It is clear from Fig. 13 that the data are well fit by the solution to the fractional Langevin equation with the parameter values a ¼ 1.45 and b ¼ 0.1, obtained through a mean-square fit of Eq. (155) to the SRV time series data. The nonlinear form of the mass exponent in Fig. 9a, the convex form of the singularity spectrum f(h) in Fig. 9b, and the fit to f(q) in Fig. 13, are all evidence that the interstride interval time series are multifractal. This analysis is further supported by the fact that the maxima of the singularity spectra coincide with the fractal dimensions determined using the scaling properties of the time series using the allometric aggregation technique. Of course, different physiologic processes generate different fractal time series, because the long-time memory of the underlying dynamical processes can be quite different. Physiological signals, such as cerebral blood flow (CBF), are typically generated by complex self-regulatory systems that handle inputs with a broad range of characteristics. Ivanov et al. [89] established that healthy human heartbeat intervals, rather than being fractal, exhibit multifractal properties and uncovered the loss of multifractality for a life-threatening condition of congestive heart failure. West et al. [90] similarly determined that

70

bruce j. west

CBF in healthy humans is also multifractal, and this multifractality is severely narrowed for people who suffer from migraines. Migraine headaches have been the bane of humanity for centuries, afflicting such notables as Caesar, Pascal, Kant, Beethoven, Chopin, and Napoleon. However, its etiology and pathomechanism have to date not been satisfactorily explained. It was demonstrated [90] that the characteristics of CBF time series significantly differs between that of normal healthy individuals and migraineurs. Transcranial Doppler ultrasonography (TCD) enables high-resolution measurement of middle cerebral artery blood flow velocity. Like the HRV, SRV, and BRV time series data, the time series of cerebral blood flow velocity consists of a sequence of waveforms. These waveforms are influenced by a complex feedback system involving a number of variables, such as arterial pressure, cerebral vascular resistance, plasma viscosity, arterial oxygen, and carbon dioxide content, as well as other factors. Even though the TCD technique does not allow us to directly determine CBF values, it helps to clarify the nature and role of vascular abnormalities associated with migraine. In particular we present the multifractal properties of human middle cerebral artery flow velocity, an example of which is presented below in Fig. 14 The dynamical aspects of cerebral blood flow regulation were recognized by Zhang et al. [91]. Rossitti and Stephensen [92] used the relative dispersion (the ratio of the standard deviation to mean), of the middle cerebral artery flow velocity time series to reveal its fractal nature; this is a technique closely related to the allometric aggregation introduced in Section II. West et al. [93] extended this line or research by taking into account the more general properties of fractal time series, showing that the beat-to-beat variability in the flow velocity has a long-time memory and is persistent with the average scaling exponent 0.85  0.04, a value consistent with that found earlier for HRV time series. They also observed that cerebral blood flow was multifractal in nature.

Figure 14.

Middle cerebral artery flow velocity time series for a typical healthy subject [90].

fractal physiology, complexity, and the fractional calculus

71

Figure 15. The average multifractal spectrum for middle cerebral blood flow time series is depicted by f(h). (a) The spectrum is the average of 10 time series measurements from five healthy subjects (filled circles). The solid curve is the best least-squares fit of the parameters to the predicted spectrum using Eq. (157). (b) The spectrum is the average of 14 time series measurements of eight migraineurs (filled circles). The solid curve is the best least-squares fit to the predicted spectrum using Eq. (157). (Taken from [90].)

In Fig. 15 we compare the multifractal spectrum for middle cerebral artery blood flow velocity time series for a healthy group of five subjects and a group of eight migraineurs [90]. A significant change in the multifractal properties of the blood flow time series is apparent. Namely, the interval for the multifractal distribution on the local scaling exponent is greatly constricted. This is reflected in the small value of the width of the multifractal spectrum for the migraineurs (0.013), which is almost three times smaller than the width for the control group (0.038); for both migraineurs with and without aura the distribution is centered at 0.81, the same as that of the control group, so the average scaling behavior would appear to be the same. However, the contraction of the spectrum suggests that the underlying process has lost its flexibility. The biological advantage of

72

bruce j. west

multifractal processes is that they are highly adaptive, so that in this case the brain of a healthy individual adapts to the multifractality of the interbeat interval time series. Here again we see that disease, in this case migraine, may be associated with the loss of complexity and consequently the loss of adaptability, thereby suppressing the normal multifractality of cerebral blood flow time series. Thus, the reduction in the width of the multifractal spectrum is the result of excessive dampening of the cerebral flow fluctuations and is the manifestation of the significant loss of adaptability and overall hyperexcitability of the underlying regulation system. West et al. [90] emphasize that hyperexcitability of the CBF control system seems to be physiologically consistent with the reduced activation level of cortical neurons observed in some transcranial magnetic simulation and evoked potential studies. Regulation of CBF is a complex dynamical process and remains relatively constant over a wide range of perfusion pressure via a variety of feedback control mechanisms, such as metabolic, myogenic, and neurally mediated changes in cerebrovascular impedance respond to changes in perfusion pressure. The contribution to the overall CBF regulation by different areas of the brain is modeled by the statistics of the fractional derivative parameter, which determines the multifractal nature of the time series. The source of the multifractality is over and above that produced by the cardiovascular system. The multifractal nature of CBF time series is here modeled using a fractional Langevin model. We again implement the scaling properties of the random force and the memory kernel to obtain Eq. (148) as the scaling of the solution to the fractional Langevin equation. Here when we calculate the qth moment of the solution we assume Gaussian, rather than the more general Le´vy, statistics. Consequently we obtain the quadratic function for the singularity spectrum f ðqÞ ¼ 2
H
bq2

ð156Þ

which can be obtained from Eq. (155) by setting a ¼ 2. Another way to express Eq. (156) is b f ðhÞ ¼ f ðHÞ
ðh
HÞ2 4

ð157Þ

where we have used the fact that the fractal dimension is given by 2
H, which is the value of the function at h ¼ H. It seems that the changes in the cerebral autoregulation associated with migraine can strongly modify the multifractality of middle cerebral artery blood flow. The constriction of the multifractal to monofractal behavior of the blood flow depends on the statistics of the fractional derivative index. As the distribution of this parameter narrows down to a delta function, the nonlocal

fractal physiology, complexity, and the fractional calculus

73

influence of the mechanoreceptor constriction disappears. On the other hand, the cerebral autoregulation does not modify the monofractal properties characterized by the single global Hurst exponent, presumably that produced by the cardiovascular system. C.

Fractional Diffusion Equations

The change in time of a stationary stochastic process using the conditional transition probability density Pðx; tjx0 ; t0 Þ for the dynamical variable X(t) to lie in the range (x; x þ dx) conditional on X(t0 ) ¼ x0 is given by the chain condition ð Pðx; tjx0 ; t0 Þ ¼ Pðx; tjx0 ; t0 ÞPðx0 ; t0 jx0 ; t0 Þ dx0

ð158Þ



where  is the domain of the variate. This equation is often used as the starting equation for the analysis of Brownian motion. Here Pðx; tjx0 ; t0 Þ is the probability that the process undergoes a transition from the initial value x0 to a final value x at time t through a sequence of intermediate values. Equation (158) was introduced by Bachelier in 1900 in his Ph.D. thesis on speculation in the French stock market. The nonphysical application of this equation was probably the reason why his work went unnoticed for nearly 50 years, even though the mathematical content was equivalent to that found in the Einstein papers on diffusion published five and more years later. The chain condition is the general description of the evolution of the probability density for an infinitely divisible stable process and the solution to which has the most general form of a Markov probability density. When the range of the variate is unbounded  ¼ ð
1; 1Þ and the process under consideration has translational invariance, so the probability density is independent of the origin of the coordinate system Pðx; tjx0 ; t0 Þ ¼ Pðx
x0 ; t
t0 Þ, the chain condition becomes ð Pðx
x0 ; t
t0 Þ ¼ Pðx
x0 ; t
t0 ÞPðx0
x0 ; t0
t0 Þ dx0

ð159Þ



The stationary chain condition (159) is more simply expressed in terms of characteristic functions, the Fourier transform of the probability density, as the product fðk; t
t0 Þ ¼ fðk; t
t0 Þfðk; t0
t0 Þ

ð160Þ

using the convolution property of Fourier transforms. Montroll and West [64] noticed that, since the probability density resulting from the characteristic function satisfies the product form, its solution yields an infinitely divisible

74

bruce j. west

distribution. The most general form of the characteristic function for infinitely divisible distributions was first obtained by Paul Le´vy in 1937. Here we merely sketch how to obtain the general solution to Eq. (160). Take the logarithm of the equation to obtain logfðk; t
t0 Þ ¼ logfðk; t
t0 Þ þ logfðk; t0
t0 Þ

ð161Þ

from which it is clear that the characteristic function factors into a function of k, say g(k), and a function of time. In order for the intermediate time to vanish from the solution, the function of time must be linear. Thus, the form of the solution to Eq. (161) is fðk; tÞ ¼ egðkÞt

ð162Þ

Since the probability density is normalizable at all times, the real part of g(k) must be negative definite. In order for the characteristic function to retain the product form at all spatial scales, it must be infinitely divisible. If we scale the Fourier variable k by a constant factor b, then in order for the probability density to be infinitely divisible, g(k) must be homogeneous: ð163Þ

gðbkÞ ¼ ba gðkÞ The homogeneity requirement implies that gðkÞ ¼
bðaÞjkja

ð164Þ

where b(a) is a complex function dependent on the parameter a, with a positive definite real part. Thus, we have for the characteristic function fðk; tÞ ¼ e
bðaÞjkj

a

t

ð165Þ

The symmetric solution to the chain condition is obtained by setting the constant in the exponential to be independent of a, b(a) ¼ b. The most general solution is obtained using

k ð166Þ bðaÞ ¼ b 1 þ iCoðk; aÞ jkj where C is a real parameter, o(k,a) is a real function, and the imaginary part of the coefficient determines the skewness of the distribution. The functional form of the characteristic function in Eq. (165) gives


k fðk; tÞ ¼ exp
btjkja 1 þ iCoðk; aÞ jkj

ð167Þ

fractal physiology, complexity, and the fractional calculus

75

so that the inverse Fourier transform sets the conditions 0 < a  2, so that it is positive definite, b > 0 so that it is normalizable, and
1  C  1 indicating the degree of skewness. Finally the function o(k,a) is defined by ( tanðap=2Þ if a 6¼ 1 ð168Þ oðk; aÞ ¼ 2 ln jkj if a¼ 1 p whose derivation can be found in Gnedenko and Kolmogorov [63]. The equation of evolution for the probability density is obtained by taking the time derivative of the characteristic function in Eq. (165): qfðk; tÞ ¼
bðaÞjkja fðk; tÞ qt

ð169Þ

The inverse Fourier transform of Eq. (169) yields qPðx; tÞ ¼
qt

1 ð

bðaÞe
ikx jkja fðk; tÞ


1

dk 2p

ð170Þ

which Gorenflo and Mainardi [94] identify with Le´vy–Feller diffusion through the Feller pseudodifferential operator Day, the Feller fractional derivative of order a: qPðx; tÞ ¼ Day ½Pðx; tÞ ð171Þ qt In this notation the inverse Fourier transform of the characteristic function in Eq. (167) is Green’s function for Eq. (171), but in a notation where h i fðk; t; a; yÞ ¼ exp
tjkja eiypsignðkÞ=2

ð172Þ

and the Feller pseudodifferential operator acting with respect to the spatial variable x has the Fourier representation ^ ay ¼
jkja eiypsignðkÞ=2 D

ð173Þ

In the symmetric case where C ¼ 0 Eq. in (167), using the convolution property of the product of Fourier amplitudes in Eq. (170), we obtain [95] qPðx; tÞ b ¼ ða þ 1Þsinðap=2Þ qt p

1 ð


1

Pðx0 ; tÞ dx0 jx
x0 jaþ1

ð174Þ

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bruce j. west

Note that the integral term in Eq. (174) is the Reisz fractional derivative, first applied in this context by Seshadri and West [96] and whose solution is the symmetric Le´vy distribution. It is worth stressing that in the last few years the approaches based on fractional derivatives, of which Eq. (174) is an early example [95], have received an ever-increasing interest, as shown by the excellent review articles by Metzler and Klafter [97] and Sokolov et al. [78]. The symmetric Le´vy distribution that solves Eq. (174) is 1 Pðx; tÞ ¼ 2p

1 ð

a

eikx e
btjkj dk

ð175Þ


1

which satisfies the scaling relation  1  1 Pðx; tÞ ¼ ga P ga x; gt

ð176Þ

as does the more general form of the Le´vy a-stable distribution. From Eq. (176) it is clear that the Le´vy or Le´vy–Feller diffusion diffusion process has the scaling, x td , with d ¼ 1=a

ð177Þ

as we demonstrated in Section III. This scaling is consistent with the process generated by the fluctuations of the variable x, as verified by numerical simulation [98]. D.

Langevin Equation with Le´vy Statistics

We now examine the response of a linear dissipative system to Le´vy fluctuations using the ordinary Langevin equation, dVðtÞ þ lVðtÞ ¼ xðtÞ dt

ð178Þ

where V(t) is the dynamical variable—say, the velocity of a particle—l is the dissipation parameter, and the fluctuations are represented by a stationary differential Markov process whose statistics are assumed to be given by a symmetric Le´vy distribution 1 pðx; tÞ ¼ 2p

1 ð


1

a

eikx e
btjkj

ð179Þ

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As we know, if the random force had Gaussian statistics and was delta-correlated in time, we would have an Ornstein–Uhlenbeck process. The variance of the system response would increase linearly in time for early times and be constant at late times. However, when the random force is Le´vy-stable the second moment of the system response is infinite. The linear dynamical equation can be formally integrated to yield ðt 0 VðVð0Þ; tÞ ¼ V0 e
lt þ e
lðt
t Þ xðt0 Þ dt0

ð180Þ

0

where V0 ¼ Vð0Þ is the initial value of the velocity variable. West and Seshadri [95] used the phase-space equations to determine the conditional probability density for this process. However, it is somewhat easier to use the characteristic function [53], given by 1 ð

fðk; tjVð0ÞÞ ¼

eiku Pðu; tju0 Þ du

ð181Þ


1

to determine the complete dynamical properties of the system response. Another way to express the characteristic function is in terms of the solution to the Langevin equation fðk; tjVð0ÞÞ ¼ he
ikVðtÞ i ¼ exp½iku0 e

*


lt

" ðt #+
lðt
t0 Þ 0 0  exp ik e xðt Þ dt

ð182Þ

0

where V(0) ¼ u0. Doob [99] has shown that a differential Le´vy process described by Eq. (179), for an arbitrary analytic function q(t), satisfies the equation *

" ðt " #+ # ðt   0 0 0 2 0 a 0 exp i qðt Þxðt Þdt  ¼ exp
s jqðt Þj dt  0

ð183Þ

0

Thus, the characteristic function in (182) can be evaluated to yield

  fðk; tjVð0ÞÞ ¼ exp
iku0 e
lt exp
s2al ðtÞjkja

ð184Þ

where the time-dependent ‘‘variance’’ is s2al ðtÞ 

 s2  1
e
alt al

ð185Þ

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which agrees with the result obtained by Doob [99] and also by West and Seshadri [95]. The conditional probability density is then given by 1 ð

Pðu; tju0 Þ ¼
1

 dk exp ikðu
u0 e
lt Þ
s2al ðtÞjkja 2p

ð186Þ

where the Fourier transform is taken with respect to the centered variable u
u0 e
lt . Hence, the solution to the linear Langevin equation driven by a random force with Le´vy statistics has Le´vy-stable statistics in the variable u
u0 e
lt with Le´vy index a and parameter given by Eq. (185). In the long time limit the characteristic function in Eq. (184) reduces to the asymptotic form

s2 fðk; 1jVð0ÞÞ ¼ exp
jkja al

ð187Þ

a characteristic function that is independent of both time and the initial state of the system. At long times the probability distribution given by the inverse Fourier transform of Eq. (187) attains the steady-state form 1 Pss ðuÞ ¼ 2p

1 ð

a

dke
iku e
bjkj

ð188Þ


1

where the Le´vy parameter is given by b ¼ s2 =al. Thus, the dissipation in the linear Langevin equation leads to a steady state in the presence of Le´vy fluctuations. The variance of the system response is, however, infinite for all t > 0 and, in particular, for the steady-state distribution given by Eq. (188). A scale-invariant biological process that has been shown to possess such Le´vy statistics is the human heartbeat time series, see Peng et al. [25]. The data consist of digitized electrocardiograms of beat-to-beat heart rate fluctuations over approximately 24 hours or 105 beats recorded with an ambulatory monitor. The time series is constructed by recording the interval between adjacent beats as data, for example, let f(n) be the interval between the n and n þ 1 beat. A great deal of variability is observed in the interbeat interval as we discussed in Section II. Peng et al. [25] graph the histogram for the differences in the beatto-beat intervals I(n) ¼ f(n þ 1)
f(n) and find that this is a stationary Le´vy process such as given by Eq. (188) and shown in Fig. 16. They find that the statistics of healthy and diseased (dilated cardiomyopathy) conditions are the same, that being Le´vy-stable with an index a ¼ 1.7; however, the spectra for the two cases are quite different.

fractal physiology, complexity, and the fractional calculus 1.0

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(a)

P(I) / P(0)

0.8 0.6 0.4 0.2 0 –3

P(I) / P(0)

10 0

–2

–1

0 I / S.D.

1

2

3

(b)

10 –1

10 –2

10 –3 –3.0 –2.0 –1.0

0.0 1.0 I / S.D.

2.0

3.0

Figure 16. The histogram in the differences between interbeat intervals I for healthy (circles) and diseased (triangles) subjects P(I) is the probability of finding an interbeat increment in the range [I
I=2; I þ I=2]. To facilitate comparison, we divide the variable I by the standard deviation of the increment data and rescale the probability with P(0). In Le´vy-stable distributions, a is related to the power-law exponent describing the distribution for large values of the variable, while the width of the distribution is characterized by b. Both histograms are well fitted by a Le´vy-stable distribution with a ¼ 1:7 (solid line). The dashed line is a Gaussian distribution and is shown for comparison purposes only. Similar fits were obtained for 8 of the 10 normal subjects with heart disease. The slow decay of Le´vy-stable distributions for large increment values may be of physiological importance and relate to the dynamics of the system: (a) linear–linear scale; (b) log–linear scale. (From Peng et al. [25] with permission.)

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The power spectrum S(f), the square of the Fourier transform of I(n), yields Sðf Þ f b , where b ¼ 1
2H and the mean-square level of the interbeat fluctuations increases as n2H. Here again H ¼ 0:5 corresponds to Brownian motion, so that b ¼ 0 indicates the absence of correlations in the time series I(n) (‘‘white noise’’). They observed that for a diseased data set, b is approximately zero in the low-frequency regime, confirming that the I(n) are not correlated over long times. On the other hand, they observed that for the healthy data set, b is approximately equal to 1, indicating a long-time correlation in the interbeat interval differences. The anticorrelated property of I(n) are consistent with a nonlinear feedback system that ‘‘kicks’’ the heart rate away from extremes. This tendency operates on a wide range of time scales, not on a beat-to-beat basis. The conclusion is that the different scaling pattern must be a consequence of the ordering of the differences, rather than their statistics, which is to say in the correlations produced by the underlying dynamics. The power-law spectrum has been observed in a number of dynamical systems having chaotic solutions, see, for example, Reichl [100]. Goldberger and West suggested that the observed spectrum may be a consequence of such dynamics and that one may interpret the modifications in the inverse power-law spectrum as being indicative of pathology and therefore of diagnostic and prognostic value. Loss of heart rate variability has already been described in numerous settings, including multiple sclerosis [101], fetal distress [102], bed-rest deconditioning [103], aging [104], and in certain patients at risk for sudden cardiac death [105]. Presumably, the more severe pathologies will be associated with the greatest loss of spectral power, which we have referred to as loss of spectral reserve [106]. V.

SUMMARY, CONCLUSIONS, AND SPECULATIONS

If one were to form a hierarchy of understanding of complex phenomena, it would surely start with physics as the most basic, expand into chemistry as large aggregates of atoms and molecules form, become biology as life is breathed into these chemical aggregates, and then form physiology as the phenomenology of human life is explored. Mathematical rigor is demanded at the base of this hierarchy, but mathematical models become less familiar and more suspect as we climb the ladder of complexity from physics to physiology. The standard in physics is the high-order accuracy to which scientists can now theoretically predict the measured value of the fine structure constant or the gravitational constant. The comfortable mathematical models that physicists rely on are missing in the studies of biomedical phenomena, and even where such models exist they do not have the degree of agreement with data that physicists have come to expect. Thus, it might seem to some that the application of mathematical concepts such as fractals, scaling, inverse power-law distributions, and the fractional calculus to physiology, such as done in this chapter, is premature.

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However, unless one is willing to neglect the serious attempts that have been and are being made to apply these ideas to the understanding of physiology and medicine, and dismiss the entire activity as misguided, overviews, such as the one given herein, of how these ideas are being implemented are useful. We have seen in Section II that physiologic time series, such as the interbeat intervals of the human heart, the interstride intervals of human gait, and the interbreath intervals in human breathing, although apparently random, do in fact have long-time memory. This combination of randomness and order has been used as the defining characteristic of complexity in this chapter. In a medical context, this complexity is encountered when attempting to understand physiological phenomena from a hotistic perspective, rather than looking at specific mechanisms. We have used the allometric aggregation technique to establish that such dynamic phenomena are complex, at least in the sense that they generate time series that are statistical fractals. The scaling behavior of such time series determine the overall properties such complex systems must have, much like the older analysis of errors and noise in physical systems. The historical view of complexity involved having a large number of variables, each variable making its individual contribution to the operation of the system and each variable responding in direct proportion to the changes in the other variables. The small differences in the contributions produced the fluctuations in the observed outcome. The linear additive statistics of measurement error or biological noise is not applicable to complex medical phenomena discussed here. The elements in complex physiologic systems are too tightly coupled, so instead of a linear additive process, nonlinear multiplicative statistics more accurately represent the fluctuations. In this chapter we examined how intersystem interactions in a generic physiologic system may give rise to the observed scaling. The individual mechanisms giving rise to the observed statistical properties in physiologic systems are very different, so we did not attempt to present a common source to explain the observed scaling in walking, breathing, and the beating heart. On the other hand, the physiologic time series for each of these phenomena scale in the same way, so that at a certain level of abstraction the separate mechanisms cease to be important and only the relations matter and not those things being related. It is the relation between blood flow and heart function, between locomotion and postural balance, and between breathing and respiration, which are important. The thesis of complexity theory, insofar as such a theory can be said to exist, is that such relations have a common form for complex phenomena. This assumption is not so dramatic as it might first appear. Consider that traditionally such relations have been assumed to be linear, in which case their control was assumed to be in direct proportion the disturbance. Linear control theory has been the backbone of homeostasis, but fails miserably in describing, for example, the full range of HRV from the running child to the sedate senior, with all the pathologies that await them along the way.

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The issue we finally address is how to control complexity. Such control is one of the goals of medicine—in particular, understanding and controlling physiologic networks in order to ensure their proper operation. We distinguish between homeostatic control and allometric control mechanisms. Homeostatic control is familiar and has as its basis a negative feedback character which is both local and instantaneous. Allometric control, on the other hand, is a relatively new concept that can take into account (a) long-term memory, (b) correlations that are inverse power law in time, and (c) long-range interactions in complex phenomena as manifest by inverse power-law distributions in the system variable. Allometric control introduces the fractal character into otherwise featureless random time series to enhance the robustness of physiologic networks by introducing either fractional Brownian motion or fractional Le´vy diffusion into the control of the network. It is not merely a new kind of control that is suggested by the scaling of physiologic time series. Scaling also suggests that the historical notion of disease, which has loss of regularity at its core, is inadequate for the treatment of dynamical diseases. Instead of loss of regularity, we identify the loss of variability with disease, so that a disease not only changes an average measure, such as heart rate, which it does in late stages, but is manifest in changes in heart rate variability at very early stages. Loss of variability implies a loss of physiologic control, and this loss of control is reflected in the change of fractal dimension—that is, in the scaling index of the corresponding time series [56]. The proper operation of physiologic processes is manifest through the scaling of appropriate time series. A measured function denoted by X(t) is said to be homogeneous when the time axis being scaled by a constant g yields the original function modified by an overall scale X(gt) ¼ X(t)=gH. This scaling behavior generalizes to time series when the measured function is stochastic, and the scaling relation is interpreted in terms of the probability density function rather than the dynamic function itself. This latter scaling is evident in Eq. (35). In one of the stochastic models discussed Section III, the scaling behavior of the process of interest is a consequence of the two-point stochastic process driving the system having an inverse power-law autocorrelation function. The scaling of the autocorrelation function is only approximate, in that lim x ðgtÞ ¼ x ðtÞ=gb

t!1

so the scaling arises asymptotically in the noise. The correlation in the noise however, gives rise to scaling in the exact solution to the phase-space equation of evolution given by Eq. (74): XðgtÞ ¼ XðtÞ=g1
b=2

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The overall scaling exponent is therefore given by H ¼ 1
b=2 where H is the Hurst exponent for the measured time series. In Section II we argue for the ubiquity of such behavior in physiologic phenomena. Section III steps back from physiology to review various mathematical models that can generate the fractal time series uncovered in Section II for various physiologic time series using allometric aggregation. We began with a brief discussion of simple random walks and showed how the resulting stochastic process for diffusion has a second moment that scales linearly in time. The arguments were extended to fractional random walks to explain anomalous diffusion defined by the second moment [Eq. (19)] scaling nonlinearly in time: hXðtÞ2 i / t2H This leads us to one of the standard, but often inappropriate, explanations of anomalous diffusion using fractional Brownian motion with the probability density

x2 exp
2Dt2H pðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4pDt2H It is clear that this distribution function satisfies the scaling relation Eq. (35) with the scaling index given by d ¼ H. We note that the variance calculated using fractional Brownian motion distribution is given by the equation for anomalous diffusion. However, we can see that this explanation of anomalous diffusion is not unique and there are multiple statistics that lead to this type of scaling. Consequently, we referred to all such models collectively as fractal stochastic processes and subsequently discussed alternative measures that can distinguish among them. An alternative to the random walk model in describing diffusion is the Langevin equation, where the microscopic dynamics are linked to a macroscopic rate equation through a stochastic driving force. A simple dichotomous driver with an inverse power-law memory having an index b was shown to yield an asymptotic system response that has the scaling given by Eq. (35) with scaling index d ¼ 1
b=2. However, these mathematically generated fractal random processes are not in themselves sufficient to properly describe the physiological processes considered herein. The physiologic time series are shown to be multifractal rather than mono-fractal. An example of the multifractal character

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of walking is shown in Fig. 9 and that of others (such as heartbeats) are referenced. A way to distinguish among different ways of generating anomalous diffusion processes is by using diffusion entropy analysis (DEA), where the scaling of the probability density Eq. (35) yields a scaling of the Shannon entropy given by Eq. (91). Consequently, we determined that there are time series for which the scaling index satisfies d ¼ H, such as fractional Brownian motion, and other time series for which d 6¼ H, such as the exact solution to the dichotomous Langevin equation. Also we observed that there is a third class of processes where d ¼ (3
2H)
1, which is valid for Le´vy random walks [66] as distinct from Le´vy flights. We discussed in Section IV how the fractional calculus could embody a number of the properties so prevalent in physiologic phenomena, not the least of which being that the evolution of a fractal processes can be described by a fractional differential equation. We showed that the evolution of fractal stochastic process can be described by a fractional Langevin equation in which a fractional differential equation is driven by a stochastic force. In particular, we demonstrated that the fractional calculus could provide a description of the dynamics of an anomalous diffusion process in which the long-time memory is not part of the stochastic driver, as it was in the earlier models, but is actually part of the system’s nonlocal dynamics through the fractional derivatives. The multifractal character of certain physiological time series, such as gait and cerebral blood flow, is described by fractional Langevin equations with random indices. The multifractal spectrum is shown to be related to the statistical properties of these random indices. Finally, the fractional calculus was used to construct fractional diffusion equations. One such equation, in particular, models the evolution of the Le´vy astable probability density describing Le´vy diffusion, another mechanism for generating anomalous diffusion. It was shown that this probability density satisfies the scaling relation [Eq. (35)] with the Le´vy index a such that d ¼ 1=a. The dynamics of a Le´vy diffusion process, using a Langevin equation, were also considered. The probability density for a simple dissipative process being driven by Le´vy noise is also Le´vy but with a change in parameters. This is a possible alternate model of the fluctuations in the interbeat intervals for the human heart shown to be Le´vy stable over a decade ago [25]. The well-being of the body’s system-of-systems is measured by the fractal scaling properties of the various dynamic subsystems, and such scaling determines how well the overall complexity is maintained. Once the perspective that disease is the loss of complexity has been adopted, the strategies presently used in combating disease must be critically examined. Life support equipment is one such strategy, but the tradition of such life support is to supply blood at the average rate, of the beating heart, to ventilate the lungs at their average rate,

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and so on. So how does the new perspective regarding disease influence the traditional approach to healing the body? Alan Mutch, of the University of Manitoba, argues that both blood flow and ventilation are delivered in a fractal manner in both space and time in a healthy body. However, during critical illness, conventional life support devices deliver respiratory gases by mechanical ventilation or blood by cardiopulmonary bypass pump in a monotonously periodic fashion. This periodic driving overrides the natural a´periodic operation of the body. Mutch [107] speculates that these devices result in the loss of normal fractal transmission and consequently: . . . life support systems do more damage the longer they are required and are more problematic the sicker the patient . . . . We hypothesize that loss of fractal transmission moves the system through a critical point . . . to transform a cohesive whole to one where organ systems are no longer as well connected.

Disease as the loss of complexity is consistent with the view that complex phenomena have a multiplicity of failure modes. These failure modes result in phenomena changing character, invariably becoming simpler with an accompanying inability to carryout their function. A cascade of failures is not so much a consequence of the initiating event as it is the result of the state of the network when the event is initiated. It is, in part, the irreversibility of failure cascades that makes them so formidable. In medicine such failure cascades may be manifest as multiple organ dysfunction syndrome (MODS) that rapidly accumulates following a minor insult; MODS is the leading cause of death in intensive care units. As Buchman [108] points out: Despite timely and appropriate reversal of the enticing insult . . . many patients develop the syndrome. Mortality is proportional to the number and depth of system dysfunction and the mortality of MODS after (for example) repair of ruptured abdominal aortic aneurysm is little changed despite three decades of medical progress.

One of the consequences of the traditional view of disease is what Buchman [108] calls ‘‘fix-the-number’’ imperative: If the bicarbonate level is low, give bicarbonate; if the urine output is low, administer a diuretic; if the bleeding patient has a sinking blood pressure, make the blood pressure normal. Unfortunately, such interventions are commonly ineffective and even harmful. For example, sepsis—which is a common predecessor of MODS—is often accompanied by hypocalcaemia. In controlled experimental conditions, administering calcium to normalize the laboratory value increases mortality.

Consequently, one’s first choice of options, based on an assumed simple linear causal relationship between input and output as in homeostatsis, is probably wrong.

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A number of scientists [109] have demonstrated that the stability of hierarchal biological systems is a consequence of the interactions among the elements of the system. Furthermore, there is an increase in stability resulting from the nesting of systems within systems—organelles into cells, cells into tissue, tissues into organs, and so on, up from the microscopic to the macroscopic. Each system level confers additional stability on the overall fractal structure. The fractal nature of the system suggests a basic variability in the way systems are coupled together. For example, the interaction between cardiac and respiratory cycles is not constant, but adapts to the physiologic challenges being experienced by the body. A number of scientists have arrived at remarkably similar conclusions regarding the nature of disease that is quite different from the traditional one. Take, for example, the following observation made by Buchman [108]: . . . Herein, we have suggested that breakdown of network interactions may actually cause disease, and when this breakdown is widespread the clinical manifestation is the multiple organ dysfunction syndrome. If the hypothesis is correct, then network dysfunction might be expected at multiple levels of granularity, from organ systems to intracellular signal molecules. Restoration of network integrity may be a reasonable therapeutic goal, and a more permissive approach to clinical support (including algorithms that simulate biological variability) might facilitate restoration of network complexity that now appears essential to health.

Or those made by Mutch [107]: The layer upon layer of fractal redundancy in scale-free biological systems suggests that attack at one level does not place the organism at undue risk. But attack at vital transmission nodes can cause catastrophic failure of the system. The development of multiple organ dysfunction syndrome (MODS) in critically ill humans may be such a failure. Once devolved, death almost inevitably ensues. The similarity to concerted attack on vital Internet router nodes is evident. Patients managed by conventional non-fractal life support may sustain further unintentional attack on a devolving scale-free system due to loss of normal fractal transmission. Returning fractal transmission to life support devices may improve patient care and potentially offer benefit to the sickest of patients.

We conclude with a number of observations: 1. The empirical evidence overwhelmingly supports the interpretation of the time series analysis that complex physiologic phenomena are described by fractal stochastic processes. Furthermore, the fractal nature of these time series is not constant in time but changes with the vagaries of the interaction of the system with its environment, and therefore these phenomena are multifractal.

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2. The scaling index or fractal dimension marks the system’s response and can be used as an indicator of the system’s state of health. Since the fractal dimension is also a measure of the level of complexity, the change in dimension with disease suggests a new definition of disease as a loss of complexity, rather than the loss of regularity [56]. This observation was first made by Goldberger and West, see, for example, Ref. [110]. 3. The fractal dynamics of complex physiologic systems can be modeled using the fractional rather than the ordinary calculus because the changes in the fractal functions necessary to describe physiologic complexity remain finite in the former formalism but diverge in the latter [53].

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36. B. J. West and L. Griffin, Allometric control of human gait. Fractals 6, 101–108 (1998); B. J. West and L. Griffin, Allometric control, inverse power laws and human gait. Chaos, Solitons & Fractals 10, 1519–1527 (1999). 37. R. L. Marsh, D. J. Ellerby, J. A. Carr, H. T. Henry and C. I. Buchanan, Partitioning the energetics of walking and running: Swinging the limbs is expensive. Science 303, January (2004). 38. B. J. West and L. Griffin, Biodynamics: Why the Wirewalker Doesn’t Fall, John Wiley & Sons, New York (2003). 39. J. J. Collins and C. J. De Lucca, Random walking during quiet standing. Phys. Rev. Lett. 73, 764–767 (1994). 40. J. W. Blaszczyk and W. Klonowski, Postural stability and fractal dynamics. Acta Neurobiol. Exp. 61, 105–112 (2001). 41. G. L. Gernstein and B. Mandelbrot, Random walk models for the spike activity of a single neuron, Biophys. J. 4, 41–68 (1968). 42. M. C. Teich, C. Heneghan, S. B. Lowen, T. Ozaki, and E. Kaplan, Fractal character of the neural spike train in the visual system of the cat, Opt. Soc. Am. 14, 529–546 (1997). 43. A. Babloyantz and A. Destexhe, Low-dimensional chaos in an instance of epilepsy. Proc. Natl. Acad. Sci. USA 83, 3515–3517 (1987). 44. P. Langevin, C. R. Acad. Sci. Paris, 530 (1908). 45. J. Beran, Statistics for Long-Memory Processes, Chapman & Hall, New York, 1994. 46. J. T. M. Hosking, Fractional differencing. Biometrika 68, 165–176 (1982). 47. N. Scafetta and P. Grigolini, Scaling detection in time series: Diffusion entropy analysis. Phys. Rev. E 66, 036130 (2002). 48. T. Gneiting and M. Schlather, Stochastic models that separate fractal dimension and the hurst effect. SIAM Rev. 46, 269–282 (2004). 49. P. Allegrini, P. Grigolini, and B. J. West, Dynamical approach to Le´vy processes. Phys. Rev. E 54, 4760 (1996). 50. J. Klafter, M. F. Shlesinger, and G. Zumofen, Phys. Today 49(2), 33 (1996); Lect. Notes in Phys. 519, 15 (1998). 51. T. Geisel, in Le´vy Flights and Related Topics in Physics, Proceedings, Nice, France; Editors, M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, eds., Lecture Notes in Physics 450, 153 (1995). 52. M. Bologna, P. Grigolini, and B. J. West, Strange kinetics: Conflict between density and trajectory description. Chem. Phys. 284, 115–128 (2002). 53. B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer, New York, 2003. 54. E. Barkai, R. Metzler, and J. Klafter, Phys. Rev. E 61, 132 (2000). 55. R. Metzler and T. F. Nonnenmacher, Phys. Rev. E 57, 6409 (1998). 56. B. J. West, Where Medicine Went Wrong, Rediscovering the Path to Complexity, to be published (2005). 57. K. Falconer, Fractal Geometry, John Wiley & Sons, New York, 1990. 58. J. Feder, Fractals, Plenum Press, New York, 1988. 59. J. F. Muzy, E. Bacry, and A. Arnedo, Phys. Rev. E 47, 875 (1993). 60. B. J. West, M. Latka, M. Glaubic-Latka, and D. Latka, Multifractality of cerebral blood flow. Physica A 318, 453–460 (2003).

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61. W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, John Wiley and Sons, New York, 1966. 62. D. Schertzer, S. Lovejoy, F. Schmitt, Y. Chigirinskays, and D. Marsan, Multifractal cascade dynamics and turbulent intermittency. Fractals 5, 427 (1997). 63. B. V. Gnedenko and A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, MA, 1954. 64. E. Montroll and B. J. West, An enriched collection of stochastic processes, in Fluctuation Phenomena, E. W. Montroll and J. L. Lebowitz, eds., North Holland Personal Library, Amsterdam, 1987; 1st ed. 1979. 65. M. Shlesinger, J. Klafter, and B. J. West, Le´vy dynamics of enhanced diffusion: Applications to turbulence. Phys. Rev. Lett. 58, 1100–1103 (1987). 66. N. Scafetta and B.J. West, Multiscale Comparative Analysis of Time Series and a Discussion on ‘Earthquake Conversations’ in California. Phys. Rev. Lett. 92, 138501 (2004). 67. J. Perrin, Mouvement brownien et moleculaire culaire. Ann. chim. Phys. VIII 18, 5–114; translated by F. Soddy as Brownian Movement and Molecular Reality, Taylor and Francis, London. 68. E. Ott, Chaos in Dynamical Systems, Cambridge University Press, New York, 1993. 69. R. J. Abraham and C. D. Shaw, Dynamics-The Geometry of Behavior, Part 1 (1982), Part 2 (1983), Part 3 (1985), and Part 4 (1988), Aerial Press, Santa Cruz, CA. 70. K. M. Kolwankar, Studies of Fractal Structures and Processes using Methods of the Fractional Calculus, unpublished thesis, University of Pune, 1997. 71. D. Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, Cambridge, 1989. 72. M. F. Shlesinger, Fractal time and 1/f noise in complex systems. Ann. N.Y. Acad. Sci. 504, 214 (1987). 73. Y. N. Rabotnov, Elements of Hereditary Solid Mechanics, MIR , Moscow, 1980. 74. H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, Oxford, 1979. 75. P. Meakin, Fractals, Scaling and Growth Far from Equilibrium, Cambridge Nonlinear Science Series 5, Cambridge University Press, Cambridge, 1998. 76. This letter was translated by B. Mandelbrot and is contained in the Historical Sketches of his second book [3]. 77. R. Hilfer, ed., Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. 78. I. Sokolov, J. Klafter, and A. Blumen, Fractional Kinetics. Phys. Today Nov. (2002). 79. L. F. Richardson, Atmospheric diffusion shown on a distance-neighbor graph. Proc. R. Soc. Lond. A 110, 709–737 (1926). 80. M. F. Shlesinger and B. J. West, Complex fractal dimension of the bronchial tree. Phys. Rev. Lett. 67, 2106–2109 (1991). 81. D. Sornette, Discrete scale invariance and complex dimensions. Phys. Rep. 297, 239–270 (1994). 82. A. Rocco and B. J. West, Fractional calculus and the evolution of fractal phenomena. Physica A 265, 535 (1999). 83. T. F. Nonnenmacher and R. Metzler, On the Riemann-Liouville fractional calculus and some recent applications. Fractals 3, 557 (1995).

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84. K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, 1993. 85. V. Kobelev and E. Romanov, Fractional Langevin equation to describe anomalous diffusion. Prog. Theor. Phys. Suppl. 139, 470–476 (2000). 86. See, for example, K. Lindenberg and B. J. West, The Nonequilibrium Statistical Mechanics of Open and Closed Systems, VCH, New York, 1990. 87. E. Lutz, Fractional Langevin equation. Phys. Rev. E 64, 051106 (2001). 88. B. Rajagopalon and D. G. Tarboton, Fractals 1, 6060 (1993). 89. P. C. Ivanov, L. A. N. Amaral, A. L. Goldberger, S. Havlin, M. G. Rosenblum, Z. R. Struzik, H. E. Stanley, Multifractality in human heartbeat dynamics. Nature 399, 461 (1999). 90. B.J. West, M. Latka, M.Galaubic-Latka, and D. Latka, Multifactility of cerebral blood flow. Physica A 318, 453–460 (2003). 91. R. Zhang, J. H. Zuckerman, C. Giller, and B. D. Levine, Am. J. Physiol. 274, H233 (1999). 92. S. Rossitti and H. Stephensen, Acta Physiol. Scand. 151, 191 (1994). 93. B. J. West, R. Zhang, A. W. Sanders, J. H. Zuckerman, and B. D. Levine, Fractal fluctuations in transcranial Doppler signals. Phys. Rev. E 59, 3492 (1999). 94. R. Gorenflo and F. Mainardi, Feller fractional diffusion and Le´vy stable motion. Preprint 1999. 95. B. J. West and V. Seshadri, Linear systems with Le´vy fluctuations. Physica A 113, 2030216 (1982). 96. V. Seshadri and B. J. West, Fractal dimensionality of Le´vy processes. Proc. Natl. Acad. Sci. USA 79, 4051 (1982). 97. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). 98. M. Annunziato and P. Grigolini, The Markov approximation revisited: Inconsistency of the standard quantum Brownian motion model. Phys. Lett. A 269, 31 (2000). 99. J. L. Doob, Stochastic Processes, John Wiley & Sons, New York (1953). 100. L. E. Reichl, The Transition to Chaos, Springer, New York, 1992. 101. B. Neubauer and H. J. G. Gundersen, Analysis of heart rate variations in patients with multiple sclerosis. A simple measure of autonomic disturbances using an ordinary ECG. J. Neurol. Neuosurg. Psychiatry 41, 417–419 (1978). 102. V. Kariniemi and P. Amma¨la¨, Short-term variability of fetal heart rate during pregnancies with normal and insufficient placental function. Am. J. Obster. Gynecol. 139, 33–37 (1981). 103. A. L. Goldberger, D. Goldwater, and V. Bhargava, Atrophine unmasks bed-rest deconditioning effect in healthy men: A spectral analysis of cardiac interbeat intervals. J. Appl. Physiol. 61, 1843–1848 (1986). 104. J. L. Waddington, M.J. MacCulloch and J. E. Sambrooks, Resting heartrate variability in man declines with age. Experientia 35, 1197–1198 (1979). 105. A. L. Goldberger, L. Findley, M. J. Blackburn, and A. J. Mandell, Nonliear dynamics of heart failure: implications of long-wavelength cardiopulmonary osciallations. Am. Heart J. 107, 612–615 (1984); G. A. Myers, G. J. Martin, and N. M. Magrid et al. Power spectral analysis of heart rate variability in sudden cardiac death: Comparison to other methods. IEEE Trans. Biomed. Eng. 33, 1149–1156 (1986). 106. A. L. Goldberger, V. Bhargava, B. J. West, and A. J. Mandell, On a mechanism of cardiac electrical stability: The fractal hypothesis. Biophys. J. 48, 525–528 (1985); A. L. Goldberger and B. J. West, Applications of nonlinear dynamics to clinical cardiology. Ann. NY Acad. Sci. 504, 195–213 (1987).

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107. A. Mutch, Health, ‘‘small-worlds’’, fractals and complex networks: An emerging field. Med. Sci. Monit. 9, MT55–MT59 (2003). 108. T. G. Buchman, Physiologic failure: Multiple organ dysfunction syndrome. Preprint. 109. G. A. Chauvet, Hierarchical functional organization of formal biological systems: A dynamical approach. I. The increase of complexity by self-association increases the domain of stability of a biological system. Philos. Trans. R. Soc. Lond. B Biol. Sci. 339(1290), 425–44 (1993). 110. A. L. Goldberger, D. R. Rigney and B. J. West, Chaos and Fractals in Human Physiology, Scientific American, Feb., 42–49, (1990).

CHAPTER 7 PHYSICAL PROPERTIES OF FRACTAL STRUCTURES VITALY V. NOVIKOV Odessa National Polytechnical University, 65044 Odessa, Ukraine

CONTENTS I. II.

III.

IV.

Introduction Elements of Fractal Theory A. Continuous, Nowhere Differentiable Functions and Deterministic Fractals B. Fractal Sets C. Fractional Hausdorff–Besicovich Dimensions D. Multifractals E. Fractal Set Constructed on a Square Lattice F. Cayley Tree. Ultrametric space Chaotic Structures A. Percolation Systems 1. Percolation Cluster 2. Critical Indices 3. Renormalization-Group Transformations 4. Physical Properties B. Fractal Structure Model 1. Properties of Finite Lattices 2. Appendix. The probability functions Physical Properties A. Conductivity 1. Maxwell Model 2. The Effective Medium Theory 3. Variational Approach 4. Iterative Averaging Method for Conductivity B. Frequency Dependence of Dielectric Properties 1. Iterative Averaging Method for Dielectric Properties C. Galvanomagnetic Properties 1. Iterative Averaging Method for Hall’s Coefficient 2. Results and Discussion 3. Appendix. Galvanomagnetic Properties of the Cube Inside a Cube Cell

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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94 D.

E. F.

G.

Elastic Properties 1. Iterative Averaging Method for Elastic Properties 2. Results of Calculation Negative Poisson’s Ratio 1. Results of Calculations of Poisson’s Ratio Frequency Dependence of Viscoelastic Properties 1. Iterative Averaging Method for Viscoelastic Properties 2. Results of Calculations for Viscoelastic Media 3. Negative Shear Modulus 4. Appendix. Fractal Model of Shear Stress Relaxation Relaxation and Diffusion Processes 1. Non-Debye Relaxation 2. Anomalous Diffusion 3. Distribution Function of a Brownian Particle with Memory 4. Inertial Effects of a Brownian Particle 5. Appendix. Derivative of Fractal Functions

References

I.

INTRODUCTION

Recently much attention has been paid to materials with a random structure. They possess an internal hierarchical organization depending on the scale; moreover, the physical properties of such materials depend on mesoscopic parameters. A mesoscopic inhomogeneous material can be considered as a subensemble of a wider class of inhomogeneous media. Materials with a fractal structure also belong to this class. Various structures can be thought of as fractal—for example, aggregates of particles in colloids, as well as the structures of certain binary solutions, polymers, and composites. Fractal structures are formed, in particular, in diffusion-controlled aggregation (polymerization). It is worth noting that media with fractal structure can exhibit properties different from those of uniform matter, like crystals, ordinary composites or homogeneous fluids. In this chapter, numerical calculations for the elastic properties of a random medium are carried out using an iterative method of averaging developed by the author and his co-workers. This method is based on the results of fractal geometry and renormalization—group transformation methods. Our averaging method has been shown to be efficient in the development of the physical properties of composites. The chapter consists of three main sections. In Section II the elements of fractal theory are given. In Section III the basis of percolation theory is described; moreover, a model of fractal structures conceived by us is described. Fractal growth models, constructed using small square or rectangular generating cells as representative structural elements, are considered. Fractal dimensions of structures generated on various unit cells ð2
1; 2
2; 2
3; 2
4; 3
1; 3
2; 3
3; 3
4; 4
1; 4
2; 4
3; 4
4Þ are calculated. Probability

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functions and critical indices for the percolation threshold and percolation cluster density are also derived. It is shown that use of anisotropic (rectangular) initial cells in place of ‘‘isotropic’’ (square) cells increases the range of possible values of critical indices characterizing the modeled systems. In the third Section IV the results of our calculations for physical properties of inhomogeneous media with a fractal structure are presented. Hierarchical structure models are applied to study the conductivity, elastic properties and Poisson’s ratio of a two-component inhomogeneous medium with a chaotic, fractal structure. Elastic properties of non-uniform, two-component systems are studied using a model of percolation on a simple cubic lattice. It is shown that as the ratio of the bulk modulus K of the components tends to zero, namely, K2 =K1 ! 0 (where 1, 2 denote the harder and softer phase, respectively), then Poisson’s ratio for the system tends to 0.2 at the percolation threshold of the harder phase in any individual Poisson ratio of the components. A qualitatively new, collective mechanism leading to negative Poisson’s ratio is suggested. Moreover, a Poisson ratio calculation for a composite with Hashin–Strikman structure has been carried out. The hierarchical structure model is generalized and applied to study the viscoelastic properties of a two-component inhomogeneous medium with chaotic, fractal structure. It is shown that just as the results obtained recently using the Hashin–Strikman model, the present model predicts the possibility of obtaining composites with an effective shear and dumping coefficient much higher than those characterizing the individual component phases. The viscoelastic properties of the fractal medium, however, differ qualitatively from the properties of the Hashin–Strikman medium. In the following section the power of the fractional derivative technique is demonstrated using as example the derivation of all three known patterns of anomalous, nonexponential dielectric relaxation of an inhomogeneous medium in the time domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of the microstructure of disordered media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. Different relaxation functions are derived assuming that the real (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the relaxation should be of classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The analysis of diffusion for a Brownian particle, where the assumption that the

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Maxwellian distribution of velocities is instantaneously attained is abandoned, i.e. inertial effects are included, has also been carried out. The material described in this chapter can be used in the analysis of the dependence of the conductivity and elastic parameters of various polymeric materials. Providing both a critical evaluation of characterization methods and a quantitative description of composition-dependent properties the material given in this chapter should have broad appeal in both the academic and industrial sectors, being of particular interest to researchers in materials and polymer science. II.

ELEMENTS OF FRACTAL THEORY

Progress in the physics of disordered media—that is, in the physics of media with a random distribution of microheterogeneity—is mainly made via the solution of problems involving the connection between the microscopic structure and the macroscopic behavior. This problem properly belongs to the realm of the kinetic theory of matter and is analogous to the problem of locking in the theory of fluids, hydrodynamic turbulence, the theory of phase transitions, and so on. Using the methods of classical statistical physics one may more or less rigorously solve problems where the system on a microscopic level is either in a state of complete chaos (perfect gas) or total order (solid perfectly crystalline bodies). In contrast, disordered media and processes in which there is neither crystalline order nor complete chaos on the microscopic level have not yet had an adequate description. This problem is connected with the condition that the macroscopic variables must considerably exceed the correlation scales of microscopic variables, a condition which is not met by disordered media. Consequently in order to describe such systems, fractal models and phased averaging on different scale levels (meso-levels) should be effective. The success of fractal models applied to the physics of disordered media may be explained first of all by the fact that fractal forms are characteristic of a huge number of processes and structures because many diverse models of the formation and growth of disordered objects of disparate nature may ultimately be reduced to a transition model—namely a connected set and an unconnected set—and to a limited diffusive aggregation [1–6]. In the first case a fractal percolation cluster is formed; in the second case a fractal aggregate is formed. A typical situation is realized in such cases – fractal structure properties appear on a range scale which is limited by the dimensions of the particles forming the aggregate at the beginning, and at the end, by the dimensions of the initial fractal clusters. Typical particle dimensions are 1–10 nm; aggregate dimensions are 10– 1000 nm. The fractal dimension, df , depends on the conditions of aggregate formation and, as a rule, lies within a range of df ¼ 2  2:9. Another class of materials with fractal structure are amorphous polymers. Here fractal properties manifest themselves on scales exceeding the dimensions

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of monomer molecules and are restricted at the top end to a scale of several tens of angstroms. A linear monomer molecule subject to random link shifts has already a fractal curve shape. Referring to ramified polymer molecules, these form fractal nets similar to those forming fractal aggregates in gels from macroscopic particles. The main characteristic of fractal structures is the dependence of their properties, C, on some linear scale, L: C  La

ð1Þ

where a is a constant. The scale dependence of the properties is the result of the self-similarity property of a fractal structure. A percolation cluster is a typical fractal. It is formed in a ‘‘geometric phase transition’’ region when unconnected sets become connected. In actual media, this dependence is usually limited to the so-called intermediate asymptotic region, which is defined as l0  L  x

ð2Þ

where l0 is the lattice constant (microscopic constant) and x is the correlation length. In the range of scales L  x, a microheterogeneous medium is homogeneous (the self-similarity property disappears) and can be characterized by the effective properties. Fractal structures have been examined, in particular, in diffusion-controlled aggregation process (polymerization) [7–9], in colloids (aggregates of particles) [10–13], and in percolation clusters [1–3]. The regularities revealed in the theory of fractals and percolation have turned out to be generally true for heterogeneous stochastic media and, in particular, for composite materials. Fractals are geometric objects, which have a fractal dimension and where the constituent small parts are similar to the whole object. Fractals became well known following the publications of Mandelbrot in 1977 [14,15]. Note that fractals (self-similar sets with fractal dimension) were first studied and described by mathematicians long before the publications of Mandelbrot, when such fundamental definitions as function, line, surface, and shape were analyzed. In mathematics, fractals appear as a result of the opposition and unity of two fields of mathematics: One of these fields studies numbers (discrete objects), while the other studies shapes (continuous objects). Use of the concept of the fractal set allows one to examine the dependence of physical properties on the behavior of hierarchical structures. Such structures appear in stochastic inhomogeneous medium.

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Algorithms for the formation of fractal sets and the determination of fractal dimension have a significant place in fractal theory. Simple models of fractal sets are considered below to illustrate formation algorithms and the calculation of the fractal dimension. A.

Continuous, Nowhere Differentiable Functions and Differentiable Fractals

In the second half of the nineteenth century, the theories of real numbers and sets were created (Weierstrass, Dedekind, Cantor [17]). These allow one to give a general and rigorous mathematical definition of a function. This definition can be formulated as follows. A function y ¼ f ðxÞ given on a set X is a rule according to which each element x from a set X may be mapped onto an element from set Y. It is known, however, that defining mathematical notions more precisely is often rather risky because the visual connection between the object under investigation and its mathematical model can disappear in tandem with the increase of accuracy, clearness, and rigor of mathematical proofs. For example, the following definition of a function f ðxÞ can be used as a Dirichlet function:
0; if x is an irrational number f ðxÞ ¼ ð3Þ 1; if x is a rational number which is a function discontinuous at all points. This function cannot be realized, and mathematicians were unable to study such a function until the second half of the nineteenth century. Along with functions discontinuous at all points, continuous functions having no derivatives at any point were discovered—that is, functions which are almost everywhere continuous but nowhere differentiable. We consider a few nowhere differentiable functions. Some of them have such amazing properties that they have been given the names of the great mathematicians who invented them: Bolzano, Cantor, Peano, Weierstrass, Koch, Van der Waerden, Sierpinski, and others. Bolzano Function About 1830, Bolzano showed that continuous, nowhere differentiable functions exist [16, 17]. The manuscript of Bolzano’s was discovered only in 1920. So that the example of a nowhere differentiable function found by Weierstrass in 1871 was deemed the first example of such a function. Nowadays many examples of nowhere differentiable functions exist. Let us consider a nowhere differentiable Bolzano function paying tribute to Bolzano as the first scientist who formulated a nowhere differentiable function (Fig. 1).

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Figure 1. Constructing a graph of the Bolzano function: (a) Graph of function B0 (x) and B1 (x), (b) Graph of function B1 (x). and B2 (x), (c) Graph of function B2 (x) and B3 (x).

According to Bolzano, we shall call the B-operation an operation on two points, Aðx; yÞ and Bðx þ a; y þ bÞ, namely,    a b a  ; A23 x þ ; y A11 ðx; yÞ; A22 x þ ; y  4 2 2   3a b A24 x þ ; y þ ; A25 ð x þ a; y þ bÞ 4 2

ð4Þ

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Consider the function with graph the straight line segment connecting points A011 ð0; 0Þ; A022 ð0; 0Þ as B0 ðxÞ. Applying the B-operation to these points, we obtain the additional points       1 1 1 3 1 0 0 0 0 A21 ð0; 0Þ; A22 ;  ; A23 ; 0 ; A24 ; ; A025 ð1; 1Þ ð5Þ 4 2 2 4 2 Then we define a function B1 ðxÞ whose graph is the broken line A021 ; A022 ; A023 ; A024 ; A025 : Using the B-operation at each pair of points, we again obtain the additional points       1 1 2 3 A31 ð0; 0Þ; A32 2 ;  ; A33 2 ; 0 ; A34 2 ; 0 4 4 4 4       ð6Þ 4 1 5 3 6 1 A35 2 ;  ; A36 2 ;  ; A37 2 ;  4 2 4 2 4 2 Then a function B2 ðxÞ with graph the broken line A31 ; A32 , . . . , A37 is now defined. Continuing this process, we find a function Bn ðxÞ with graph the broken line with peaks at the points having abscissae (see Fig. 1a): 0;

1 2 4n  1 ; n ;...; n 4 4 4n

ð7Þ

It is now possible to define the graph of Bolzano function BðxÞ at the values of x: k ðk ¼ 0; 1; 2; . . . ; 4n ; n ¼ 0; 1; . . .Þ ð8Þ x¼ n 4 taking account of B

    k l ¼ B p n 4 4p

ð9Þ

Thus, the graph of Bolzano function BðxÞ traverses the peaks of all broken lines Bn ðxÞðn ¼ 0; 1; 2; . . .Þ (Table I). Any value of x, if it is different from values (1.4), can be exhibited as the limit of a sequence of numbers. The limit is: lim Bðam Þ ¼ BðaÞ

m!1

ð10Þ

Thus, the Bolzano function BðxÞ is defined on the whole segment [0, 1]. It can be rigorously proved that BðxÞ is a nowhere differentiable function [16, 17].

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TABLE I Values of the Argument of Bn ðxÞ n 0

X 0, 1

1

0; 14 ; 24 ; 34 ; 1

2 .. . n

0; 412 ; 422 ; 432 ; . . . ; 15 42 ; 1 .......................................................... ............................. n 0; 41n ; 42n ; . . . ; 4 41 n ;1

The Bolzano construction for obtaining a nowhere differentiable function can be simplified and made more graphic [18]. B.

Fractal Sets

Van der Waerden Function and Koch Snowflake If we divide segment [0,1] into four equal parts and construct an isosceles triangle without a base over the two central parts, we obtain a line which is the graph of a function y ¼ f1 ðxÞ (Fig. 2b). Next we construct the graph of the function y ¼ f2 ðxÞ. To do this, we divide segment [0,1] first into four parts and then each part again into four parts. According to the previous algorithm, we construct an isosceles triangle on each 1/8 part of the segment [0,1] and obtain the graph of a function y ¼ f2 ðxÞ (Fig. 2c). Adding up these functions, y ¼ f1 ðxÞ þ f2 ðxÞ, we obtain the graph shown in Fig. 2d, Next, we divide each part into four parts again and construct 16 isosceles triangles. The graph so obtained, y ¼ f5 ðxÞ, is added to the previous construction. Continuing this process, we obtain a nowhere differentiable function (Van der Waerden function) [18].

Figure 2.

Generating the graph of a Van der Waerden function.

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Another example of a nowhere differentiable function is the Weierstrass function, defined by [19] n X bk cos p ak x ð11Þ f ðxÞ ¼ lim n!1

k¼1

where 0 < b < 1, a is an odd whole number satisfying: 3 ab > 1 þ p 2 The following function is a little simpler:
f ðxÞ ¼

x cosðp=xÞ;

if x 6¼ 0

0;

if x ¼ 0

ð12Þ

ð13Þ

It has infinitely many maxima and minima in the segment [0,1]. Along with the discovery of nowhere differentiable functions, geometric shapes were created and it was difficult to say whether they were lines, surfaces, or volumes. Koch’s ‘‘Snowflake’’ From an analogous procedure to that which has been used for the formation of the graph of Van der Waerden function from an isosceles triangle, it is possible to obtain Koch’s ‘‘snowflake’’ (Fig. 3). For this purpose, sides of an isosceles triangle are divided into three equal parts. Then the middle parts are rejected, and above them ‘‘small houses’’ are built up. In accordance with Koch’s ‘‘snowflake’’ 1n construction, the length of a link at the nth iteration step is equal to l ¼ n 3 , and the total length of the broken line  n is equal to Ln ¼ 43 . Therefore, lim ln ¼ 0;

n!1

Figure 3.

lim Ln ¼ 1

n!1

Koch’s ‘‘snowflake.’’

ð14Þ

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…

(a)

…

(b)

Figure 4.

Constructing a Sierpinski carpet: (a) quadrangle; (b) triangle.

that is, a line of unlimited length results from the iteration procedure. This line has no tangent at any point. Sierpinski Carpet Consider a square with side 1. We divide it into nine equal squares and remove the central part. Then we repeat this procedure with each of the eight remaining squares (Fig. 4). Repeating the procedure n times (n ! 1), we obtain a shape that is called the Sierpinski carpet (Fig. 4). Cantor Set Consider a one-unit segment. We divide it into three equal parts and remove the central one. Each of the two remaining parts is also divided into three parts and the middle parts are removed (Fig. 5). Continuing this procedure n times, find that the length of the link at the  we n nth iteration stage is equal to ln ¼ 13 , and the total length of the segments remaining at the nth iteration stage is Ln ¼

 n 2 3

ð15Þ

Thus, when n ! 1, we obtain Cantor ‘‘dust’’: lim ln ¼ 0;

n!1

lim Ln ¼ 0

n!1

ð16Þ

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vitaly v. novikov

Figure 5. Cantor set.

The function shown in Fig. 6 is connected with the Cantor set (Cantor ‘‘dust’’). This function has been called the ‘‘devil’s stairs.’’ Constructing the ‘‘Devil’s Stairs’’ This function is obtained as follows. Again, we divide segment [0,1] into three equal parts and assume that the value of the function y ¼ f ðxÞ equals 1=2 at all points of the middle part. Then the left and right thirds of the segment are again divided into three equal parts. Now the function y ¼ f ðxÞ is assumed to equal 1=4

Figure 6.

The ‘‘devil’s stairs.’’

physical properties of fractal structures

105

over the middle of the left part and 3=4 over the middle of the right part. Therefore, 1=4 is subtracted from 1=2 for the left part, and 1=4 is added to 1=2 for the right part; that is, increment in the function is 1=2. Each of the remaining segments are also divided into three equal parts. The function y ¼ f ðxÞ over the middle parts of the segments is assumed to equal 1=8; 3=8; 5=8; 7=8, respectively; that is, the increment in the function is equal to ð1=2Þ3 . Continuing this process, as n ! 1 we obtain a function that is defined at all points of segment [0,1], except at the points belonging to the Cantor set (Cantor ‘‘dust,’’ Fig. 6). Here the jump in the function is lim ð1=2Þn ¼ 0

n!1

ð17Þ

This means that the function is continuous as n ! 1. Thus, the graph of the function y ¼ f ðxÞ looks like a staircase with an infinite number of steps whose total length is 1 because the length of Cantor ‘‘dust’’ (where there are no steps) is zero. This function goes up by 1, although it only increases over the set of zero length and does not make any jumps. Jordan first gave a definition of a line as the trajectory of a point moving continuously without any jumps. The Italian scientist Peano showed that it is possible to construct a curve, in Jordan’s sense, traversing all points of a square. Peano Function Peano constructed the representation of the set of points of a straight line segment on the set of points of a square (Fig. 7). In this, adjacent points of the square corresponded to adjacent points on the segment. That is, Peano constructed a line that encompassed all points of the square. It must be stated

Figure 7.

Constructing a Peano function.

vitaly v. novikov

106

9

8

7

6

4

5

1

2

3

Figure 8. Constructing a Peano function.

that Peano’s curve passed through certain points of the square several times. It was proved that no continuous curve passing over all points of the square only once exists. Let us consider the square K in the plane: 0  x  1;

0y1

ð18Þ

We divide it into nine equal squares and enumerate them as shown in Figs. 8 and 9. The squares having immediately adjacent numbers have a common side.

9

8

7

52 51

53

4

6 54 55

56

58 57

59

1

2

3

Figure 9. Constructing a Peano function.

physical properties of fractal structures

107

The squares so constructed will be called first rank squares. Each first-rank square is divided into nine second-rank squares. The second-rank squares are numbered in such a way that square number one is adjacent to square number nine (see Fig. 8). This process can be continued infinitely. The length of the nthrank square side will be ð1=3Þn . If point A belongs to the main square, K, it also belongs to at least one first-rank square, at least one second-rank square, and so on (Fig. 7). If we take two different points, A and B, of the main square K, then they will belong to two different nrank squares beginning with some large n. Let us take some point Aðx; yÞ belonging to the main square K. Let s1 ; s2 , . . . , sn , . . . be the numbers of 1st, 2nd, . . . , nth-rank squares respectively to which point A belongs. Thus, at least one sequence of numbers s1 , s2 , . . . , sn , . . . corresponds to point A and, on the contrary, one point of square K corresponds to each such sequence. Then, two different sequences will correspond to two different points of square K, however even one point of the square may correspond to two different sequences. For example, the sequences 2; 3; 5; 6; 7; 1; 1; . . . ; 1 2; 3; 5; 5; 9; 9; 9; . . . ; 9 define the same point of square K. Consider segment [0,1]: 0  t  1 We can show that each point of segment [0,1] can be assigned a corresponding point of square K. We divide the segment [0,1] into nine equal parts, number them from left to right with 1; 2; . . . ; 9, and call them first-rank segments. Each first-rank segment is divided into nine equal second-rank segments that are also numbered from left to right 1,2,3,4,5,6,7,8,9. This procedure is carried out n times (n ! 1). Thus, at least one sequence of numbers s1 ; s2 ; . . . ; sn ; . . . (si ¼ 1; 2; . . . ; 9Þ will correspond to each A point of segment [0,1]; vice versa, a point of segment [0,1] corresponds to each point of the sequence. Now take some point B belonging to segment [0,1]. If it is not the end of an n-rank segment, then only one sequence a1 ; a2 ; . . . ; an ; . . . corresponds to it. If this point is the end of an n-rank segment, then two sequences will correspond to it, namely: 1. a1 ; a2 ; . . . ; an , 1; 1; . . . ; 1; . . . . 2. a1 ; a2 ; . . . ; an1 , 9; 9; . . . ; 9; . . . . Thus, each point B 2 ½0; 1 can be assigned the corresponding point A 2 K.

108

vitaly v. novikov

Let a point B belonging to the segment have abscissa t, and point A belonging to the square have coordinates x and y. Then we have single-valued functions x ¼ jðtÞ and

y ¼ cðtÞ

ð19Þ

We prove that these functions are continuous. Thus, if we take two values t1 and t2 , such that jt1  t2 j
1, then xn ! 1; n ! 1.  If x0 ¼ 1; then the xn lie on the unit circle. Therefore, the plane is divided into two domains of influence. The boundary of these domains is the circle. Everything becomes more interesting if complex representations are used instead of the former iteration functions. znþ1 ¼ f ðzn Þ;

zn ¼ xn þ iyn

ð30Þ

For such an iteration function scheme, a fractal attractor exists. The best known example for obtaining a fractal set is the square representation in the complex plane znþ1 ¼ z2n þ c

ð31Þ

f ðzn Þ ¼ z2n þ c

ð32Þ

that is,

where c ¼ a þ ib is a complex number. Therefore, xnþ1 ¼ x2n  y2n þ a;

ynþ1 ¼ 2xn yn þ b

ð33Þ

The fixed point of representation (30) is defined as the root of the equation f ðzÞ ¼ z

ð34Þ

Developing f ðzÞ in a Taylor series about the fixed point z0 , we obtain f ðzÞ ¼ f ðz0 Þ þ f 0 ðz0 Þðz  z0 Þ þ   

ð35Þ

f ðz0 Þ ¼ z0

ð36Þ

znþ1 ¼ f ðzn Þ

ð37Þ

Then, noting that

and

in the linear approximation we obtain znþ1  z0 ¼ ðz  z0 Þ f 0 ðz0 Þ

ð38Þ

vitaly v. novikov

112 Thus:

(a) If jf 0 ðz0 Þj < 1, then the distance to the fixed point z0 decreases as a result of iteration: jznþ1  z0 j < 1

ð39Þ

that is, z0 is a stable fixed point. (b) If jf 0 ðz0 Þj > 1, then jznþ1  z0 j > 1

ð40Þ

The distance increases and z0 is an unstable fixed point. Periodic points and cycles may consist of several points. So, for example, a cycle of period two consists of two points, z1 and z2 : f ðz1 Þ ¼ z2

and

f ðz2 Þ ¼ z1

ð41Þ

It is obvious that points z1 and z2 are fixed points of the reflection: znþ1 ¼ f ðf ðzn ÞÞ

ð42Þ

f ðz1 Þ ¼ z2

ð43Þ

f ðf ðz2 ÞÞ ¼ z2

ð44Þ

f ðf ðz1 ÞÞ ¼ z1

ð45Þ

Also,

Concerning f ðz1 Þ ¼ z2 we obtain

Analogously,

If z0 is a fixed point of period n, then it is a fixed point of the function: f ðnÞ ðzÞ f ð. . . ðf ðf ðzÞÞÞ . . .Þ ¼ z0

ð46Þ

The point to which the iteration process converges is called an attractor in the complex plane znþ1 ¼ f ðzn Þ as n ! 1; that is, a stable fixed point can act as an attractor.

ð47Þ

physical properties of fractal structures

Figure 11.

113

Julia sets.

If, for example, one chooses c ¼ 0:12375 þ i0:56508 in Eq. (30), then the sequence fzn þ 1g admits of three possibilities, however the inner attractor is not zero, and the boundary is no longer smooth (Fig. 11a). The boundary is a broken self-similar curve, a Julia set [14,15,21]. Given different values of c one can obtain different Julia sets (Fig. 11). The important feature of all geometric shapes examined in this chapter is their self-similarity, that is, scaling invariance. The dimensions of such geometric objects can be defined using the Hausdorff–Besicovitch measure. Verhulst Dynamics Consider the model of growth of a bond set in
f . Let p0 be the initial relative number of the complete bonds and pn their number after n iteration steps. The growth coefficient, K, is K¼

pnþ1 1 pn

ð48Þ

If this is equal to a constant r, then the evolution law controlling the growth dynamics is pnþ1 ¼ f ðpn Þ ¼ ð1 þ rÞpn

ð49Þ

After n iteration steps, the number of complete bonds is pn ¼ ð1 þ rÞn p0 With a maximum value ðpn Þmax ¼ 1.

ð50Þ

114

vitaly v. novikov

Verhulst assumed that K is proportional to 1  pn . He took K ¼ rð1  pn Þ, where r is the growth parameter. Hence, the evolution law is pnþ1 ¼ f ðpn Þ ¼ ð1 þ rÞpn  rp2n

ð51Þ

If p0 ¼ 0 or 1, then pn does not change. If 0 < p0 1, then at r > 0 number p1 increases at the next iteration step p1 ffi p0 þ rp0 ¼ ð1 þ rÞp0

ð52Þ

So, p0 , p1 , . . . , pn :; . . . increase until 1. We examine the stability of the balanced condition for p0 6¼ 1; p0 < 1: For small deviations dn ¼ pn  p0

ð53Þ

according to Eq. (51), on linearization we find dnþ1 ð1  rÞdn

ð54Þ

Hence jdnþ1 j < jdn j, if 0 < r < 2. When r > 2, jdnþ1 j > jdn j, that is, the deviations increase, and the point p ¼ 1 is now unstable. For r ¼ 2:5, stable periodic oscillations occur. When r ¼ 2:570 the process becomes chaotic. The ratio dn ¼

rn  rn1 rnþ1  rn

ð55Þ

converges to 4:669 . . . dn jn!1 ! 4; 669 . . .

ð56Þ

This regularity refers to the length interval of the parameter values at which periodic motion with some definite period is stable. These intervals are reduced at each reduplication of the period, the multiplier characterizing the reduction approaching the universal value: d ¼ 4; 669201660910 . . .

ð57Þ

which was first published by Grossman and Tome in 1977 and is called the Feigenbaum number [20].

physical properties of fractal structures

115

In general, when we have the equation xnþ1 ¼ kxn ; ðk > 0Þ

ð58Þ

the solution xn ¼ xn ehn ;

h ¼ ln k

ð59Þ

exhibits instability with respect to perturbation of the initial condition. Whence ðaÞ k > 1; h > 0 and lim dn ¼ 1 n!1

ðbÞ k < 1; h < 0 C.

and

lim dn ¼ 0

n!1

ð60Þ ð61Þ

Fractional Hausdorff–Besicovich Dimensions

We now consider the definition of the dimensions of such common geometric objects as a straight line segment, a square, a cube, and so on. We divide the segment of length of L into Nn equal parts. Then each part of the division of length ln can be considered as a copy of the whole segment 1=Nn times reduced. It is obvious that N and ln are connected via the correlation (Table III) ð62Þ Nn  ln ¼ L TABLE III Geometric Object

Unit

Straight line

ln

ln 2

Dimension 1 ln 3 n ¼ 3; Nn ¼ 3; ln ¼ ; d ¼  ¼1 3 lnð1=3Þ

1 ln 9 n ¼ 3; Nn ¼ 9; ln ¼ ; d ¼  ¼2 3 lnð1=3Þ

Square

ln 3

1 ln 27 n ¼ 3; Nn ¼ 27; ln ¼ ; d ¼  ¼3 3 lnð1=3Þ

Cube

d

lnf

Koch’s snowflake

n ¼ 3; Nn ¼ 64; ln ¼

 3 1 ln 64 ; df ¼  ffi 1:26 3 lnð1=3Þ3

vitaly v. novikov

116

Dividing the square with area S into Nn equal squares with areas l2n , the correlation becomes Nn  l2n ¼ S

ð63Þ

Nn  lnd ¼ M

ð64Þ

In general, for a set


where M is the measure of the set
(of a geometric object). Hence, the dimension d of the geometric object with a finite measure M (M ¼ constant, in particular M ¼ 1) is defined via Nðln Þand ln as d ¼  lim log M=logð1=ln Þ þ lim log Nðln Þ=logð1=ln Þ

ð65Þ

d ¼  lim log Nðln Þ=logð1=ln Þ

ð66Þ

ln !0 ðN!1Þ

ln !0

or ln !0

Let the fractal set
f be in a Euclidean space with a dimension of d. It is possible to generalize the above result to a covering set
f comprised of elementary geometric objects with unequal dimensions ln . Thus, the dimension procedure applied to sets of arbitrary metric space
f generally consists of the following: First Nðln Þ is defined—the minimum number of cubes with linear dimension ln < e ðe > 0Þ needed to cover the set
f . Then the dimension of the set is defined as the limit df ¼ lim log Nðln Þ=logð1=ln Þ e!0

ð67Þ

If the limit (67) exists, then df is called the Hausdorff, or Hausdorff– d Besicovitch, dimension. When covering the Cantor set with segments l nf , the covered area (Cantor set measure) is equal to Nðln Þ  ldnf ¼ 1

ð68Þ

2n  ð3n Þdf ¼ ð2  3df Þn ¼ 1

ð69Þ

or

It follows from Eq. (69) that df ¼ log3 2 ¼ 0; 63093.

physical properties of fractal structures

117

Thus, df is a fractional number and defines the Cantor set dimension. Thus, d the function of the set measure, Mf ¼ Nðln Þl nf represents the main characteristics of the fractal. We first examine the result obtained more thoroughly. Let us find the summed length of the remaining segments at the nth division of the Cantor set. We have   1 2 22 2n þ þ þ    þ nþ1 þ    Ln ¼ 1  3 32 33 3 " #  2  n 1 2 2 2 ¼1 1þ þ þ þ þ 3 3 3 3

ð70Þ

The expression in square brackets is a geometric series with partial sum  n  2 Sn ¼ 3 1  3

ð71Þ

Thus, the summed length of the remaining segments, Ln , at the nth stage is  n 2 Ln ¼ 3

ð72Þ

lim Ln ¼ 0

ð73Þ

Whence n!1

that is, due to the division of the unitary segment according to the above mentioned procedure, we obtain the remainder with a length of zero. The set obtained in this way is called a Cantor set, or Cantor ‘‘dust.’’ It follows from the construction that Cantor ‘‘dust’’ is infinitely divisible and absolutely broken. It is obvious intuitively that after such a division some remainder whose length is not equal to zero must exist. Let us show that with a corresponding ‘‘ruler’’ used to measure the length of the remaining segment, the summed length of Cantor ‘‘dust’’ is not equal to zero. If we choose a ‘‘ruler’’ (a unit of d measurement) to measure the summed length of Cantor ‘‘dust’’ as l nf ¼ ð3n Þdf , then the summed length Ln (Cantor set measure) is Ln ¼

Nn lndf

n df

¼ 2 ð3 Þ ¼ n



2 3d f

n ð74Þ

vitaly v. novikov

118 and

 lim Ln ¼ lim ðNn lndf Þ ¼ lim

n!1

n!1

n!1

2 3 df

n ð75Þ

Therefore, if we take df ¼ ln 2= ln 3, then we obtain the finite dimension of the Cantor dust summed length. Thus, the measure (measurement) of set
f depends on the dimensions of the objects covering it. Measuring a set gives a ‘‘smart’’ result if the ‘‘ruler’’ used to measure the set corresponds to the geometry (dimensions) of the set. The idea that every fractal set needs its own ruler in order to be measured will be used when analyzing the physical properties of heterogeneous media. Generally, for the measure Mf of the set of points in space
f with dimension d 8 9 0 . . . if . . . d > df < = Mf ¼ lim Nn ðln Þldn ¼ constant . . . if . . . d ¼ df ð76Þ n!1 : ; 1; . . . if . . . d < df The main feature of the definition in Eq. (76) is that max ln ! 0. Whence in general the Hausdorff–Besicovitch dimension is a local characteristic. For deterministic self-similar sets, the local Hausdorff–Besicovitch dimension coincides with the dimension of the set itself. For statistically homogeneous sets, however, the local Hausdorff–Besicovitch dimension may not coincide with the dimension of the whole set. The similarity transformation of metric space
is the representation, g, of a space
on itself whereby all distances between the points change with the same ratio, k > 0. Now, the number k is called the similarity transformation coefficient. The non empty limited set E
is called a self-similar set if it may be represented as the union of a limited number of two by two nonoverlapping subsets Ei ; i ¼ 1; n ðn > 1Þ, such that E is similar to E0 with coefficient k. An arbitrary segment, the Sierpinski carpet and sponge are examples of self-similar sets. For fractal sets, the Hausdorff–Besicovitch dimension coincides with the self-similar dimension. Consider a set of points on the limited straight line, L. Any other set of points of limited segment L0 is self-similar (scale multiplier) KðNÞ ¼ 1=N

ð77Þ

where N is a whole number. The set of points of a rectangular area of plane L will contain any other set L0 consisting of the points of the rectangular area as a selfsimilar subset; that is, the self-similarity coefficient is equal to KðNÞ ¼ 1=N 2

ð78Þ

physical properties of fractal structures

119

In general, the scale similarity coefficient is KðNÞ ¼ N dn

ð79Þ

where dn is the similarity dimension. Thus, the similarity dimension is defined as dn ¼ ln N= ln KðNÞ

ð80Þ

For fractal systems, the Hausdorff–Besicovitch dimension is equal to the similarity dimension, that is, df ¼ dn . We consider the triangular Sierpinski carpet as an example (Fig. 4). The iteration process means that the triangle is replaced by N ¼ 3 triangles diminished with similarity coefficient K ¼ 1=2: Thus, the fractal dimension and the triangular Sierpinski carpet similarity dimension are given by df ¼ dn ¼

D.

ln 2 ln 3

ð81Þ

Multifractals

The notion of generalized (multifractal) dimension is introduced by examining heterogeneous fractal sets [22,23]. We consider the definition of multifractal dimension. Let some fractal set
f .be given. We divide this set into nonoverlapping subsets An , such that the diameter ln of the set An is less than e > 0ðln < eÞ. The information (configurational) Shannon entropy of such a division is Hs ðln Þ ¼ 

n X

pi ln pi

ð82Þ

i¼1

where pi is the probability of An containing n points. The entropy in Eq. (82) is the measure of the information needed in order to estimate the place of the point in the ith cell. If a set
f is continuous, then the probability density rðri Þ of the set
f can be defined as pi ðrÞ ¼ rðri ÞVi

ð83Þ

vitaly v. novikov

120

where r 2 Ai , Vi  lnd  e d is the volume of Ai . Then Hs ðln Þ ¼ 

X

pi ln pi ¼

i

¼

X i

X

pi ln rðri ÞVi ¼ 

X

i

pi ln ri 

X

pi ðln ri þ ln Vi Þ

i

pi ln ldn ¼ hln ri  d ln ln

ð84Þ

i

and Hs ðln Þ ¼ hln ri  d ln ln

ð85Þ

Thus, the configurational dimension of the set
f is Hs ðln Þ ln !0 lnðln Þ

ds ¼  lim

ð86Þ

or n P

pi ln pi ds ¼ lim i¼1 ln !0 lnðln Þ

ð87Þ

It follows from Eq. (87) that the information (configuration) Shannon entropy, Hs ðln Þ, of a fractal set depends on the scale ln : s Hs ðln Þ  ld n

ð88Þ

According to (88), the information needed to define the position of the point increases when the cell dimension, ln , approaches zero. The result obtained can be generalized if the notion of Rennie entropy is introduced, Hq [22,23] Hq ¼ ð1  qÞ1 ln

Nðl nÞ X

pqi

ð89Þ

i¼1

Hence, the Rennie dimension is dq ¼  lim

ln !0

Hq ðln Þ lnðln Þ

ð90Þ

physical properties of fractal structures

121

or ln dq ¼ ð1  qÞ1 lim

Nðl Pn Þ

pqi

i¼1

ð91Þ

lnðln Þ

ln !0

Thus, the Rennie dimension, dq , is a function of the variable q taking on values in the interval 1 < q < 1. We demonstrate that if q ! 1, then the Rennie entropy Hq is equal to the Shannon entropy Hs. In fact, if q ! 1, then Nðl nÞ X

pqi ¼

i¼1

Nðl nÞ X

ð92Þ

pi

i¼1

expanding the exponent and noting the normalization namely, Nðl nÞ X

pqi

i¼1

Nðl Pn Þ

pi we obtain

i¼1

,

i¼1

Nðl nÞ X

Nðl nÞ X

i¼1

i¼1

½pi þ ðq  1Þpi ln pi  ¼ 1 þ ðq  1Þ

Nðl Pn Þ

pi ln pi

ð93Þ

Hence: ln

Nðl nÞ X

" pqi

¼ ln 1 þ ðq  1Þ

i

Nðl nÞ X

# pi ln pi ¼ ðq  1Þ

Nðl nÞ X

pi ln pi

ð94Þ

i

i¼1

that is, ðq  1Þ1 ln

Nðl nÞ X

pqi ¼

i

Nðl nÞ X

pi ln pi

ð95Þ

i

Concerning Eq. (95), it follows that if q ! 1, then the Rennie entropy Hq [Eq. (89)] coincides with the Shannon entropy Hs [Eq. (82)], and the Rennie dimension [Eq. (91)] is the information dimension [Eq. (87)], that is, ds ¼ lim dq q!1

ð96Þ

The Rennie dimension [Eq. (91)] can be rewritten as dq ¼ 

1 ln Zðln ; qÞ lim l !1 1q n lnð1=ln Þ

ð97Þ

vitaly v. novikov

122 where

Zðln ; qÞ ¼

Nðl nÞ X

pqi

ð98Þ

i

Zðln ; qÞ is the generalized statistical sum. If q ¼ 0, then Zðln ; qÞ ¼ 1. Now, as shown above [Eq. (87)], the Rennie dimension is equal to the information dimension. If q ¼ 0, then Zðln ; qÞ ¼ Nðln Þ, where Nðln Þ is the number of elementary geometric objects (cells) with a linear dimension ln covering the set
. According to Eq. (66) f Nðln Þ  ld n

ð99Þ

Thus, if q ¼ 0, then the Rennie dimension dq is the fractal dimension df of the set: df ¼ lim dq

ð100Þ

q!1

The correlation dimension dv can be defined as dv ¼ lim dq

ð101Þ

q!2

Moreover, the correlation sum Cðln Þ, or the number of pairs of points the distance between which does not exceed ln , is [22,23] Cðln Þ ¼ lim

N!1 ln !2

N N X 1 X Zðl  r Þ ¼ lim p2i n ij N!1 N 2 i; j¼1 i¼1

ð102Þ

ln !2

where ZðxÞ is the Heaviside unit step function; rij is the distance between points ri and rj . Thus, Cðln Þ defines the probability for two points selected at random in order to be divided by a distance less than ln : According to Eq. (98), we obtain lim ln Zðln ; qÞ ¼  ln Cðln Þ

q!2

ð103Þ

Hence, we obtain the critical index for correlation length: dv ¼ lim dq ¼ lim q!2 ln !0

ln !0

ln Cðln Þ ln l1 n

ð104Þ

physical properties of fractal structures

123

The correlation sum (integral) for a fractal set Cðln Þ depends on the scale ln as Cðln Þ  ldnn ð105Þ In conclusion, we note that pi is the probability for the point to lie in the region li < e; that is, pi defines the weight (contribution) of different regions in the set measure, Md ð
Þ. Thus, the choice of large q  1 contributes to putting elementary geometric objects with relatively great weights pi into the set measure, Md ð
Þ. If q 1, then the contribution of cells with small weights pi is increased in Md ð
Þ Lognormal Distribution Let us consider a volume V (diameter L) with dimension d. Let N particles of dimension l0 be distributed in V. We divide the whole volume V into cells with sides ln > l0 and volume ldn . We will examine only cells, occupied by at least one particle. Let the number of occupied cells k change within the limits ½ð1; 2; . . . ; Nðln Þ. This is the total number of occupied cells which depends on the cell dimension. If the medium is homogeneous, the number of particles per unit cell volume is equal. Hence f Nðln Þ  ld n

ð106Þ

where df is the fractal dimension of the set formed by the particles. The number of particles in the cells is a random variable, thus the relative filling of a cell is defined by a probability pk. The smaller the cell dimension ln , the smaller is the probability pk for a particle to enter the cell. For self-similar sets, the dependence pk on the cell dimensions has a power law character ð107Þ pk ðln Þ  lan i where pk is the probability for a particle to be in cell number k, and ai is generally different for different k cells. For a regular (homogeneous) fractal, all exponents ai are equal to df . pi ¼

1  ldnf Nðln Þ

where Nðl1n Þ is the probability of an elementary event. Moreover, we note that here X p i ¼ c 1 e a 1 þ c 2 e a2 þ    þ c n e an ¼ 1 i

ð108Þ

ð109Þ

vitaly v. novikov

124

We consider the probability distribution of ai . Let nðaÞda be the probability for ai to lie in the interval from a to a þ da; that is, nðaÞda is the relative number of k cells having the same measure pk with ak . In a multifractal, various values of a exist with a probability characterized not only by the single value a ¼ df but also by various other values, with the power law exponent f ðaÞ constituting a spectrum of fractal dimensions of the homogeneous subsets
0 of the basic set
: nðaÞ  ef ðaÞ

ð110Þ

So, the basic set consists of the subset of different homogeneous sets, each having its own fractal dimension f ðaÞ. In accordance with Eq. (107), we have ai  ln pi = ln ln

ð111Þ

nðaÞ  exp½ f ðaÞ ln e

ð112Þ

f ðaÞ ¼ df  Zða  a0 Þ2

ð113Þ

According to (111), we have

Now [36,37]

00

where the curvature Z ¼ f ðaÞ=2 ¼ const= ln ln ln is defined by the value of the second derivative of f ðaÞ at a0 . Then h i ð114Þ nðaÞ  exp constða  a0 Þ2 Noting that ai ¼  ln pi = ln ln , we obtain "  2 # ln pi nðaÞ  exp const þ a0 ln ln

ð115Þ

This is a lognormal distribution, it is the probability density function of the random variable pi characterizing the relative filling of the cells. We consider the asymptotic behavior of pi . pi ¼ lim ni ðln Þ=N  mi =Mi

ð116Þ

where ni ðln Þ is the number of points in the cell with number i from the general set of N points belonging to the volume V, mi is the mass of ni ðln Þ points of the ith kind, and Mi is the mass of all points of the ith kind.

physical properties of fractal structures

125

Taking account of Eq. (115), we obtain "

  # const ln M0 2 pðmi Þ  exp  2 ln mi  ln Mi ln ln

ð117Þ

where M0  lna0 . It can be shown that the probability density function of the random variable ri characterizing the dimension of the ith component is "

  # const  di 1 ln M0 2 pðri Þ  exp  ln ri  di ln Mi ln2 ln

ð118Þ

where di is the fractal dimension of the ith element (component) of a heterogeneous medium. The lognormal distribution is not unlimitedly divisible. Hence, the resulting distribution for all components corresponding to the manifold of generating functions, or in other words, to the sum of lognormal distributions of the masses or dimensions of the ith components, will not be the lognormal distribution of the sum of these distributions. The lognormal distribution of the ith component of a multifractal of dimension ri obtained shows that if 1–3 components are isolated in a real multifractal, then 1–3 lognormal modes corresponding to these characteristic scales can be isolated in an experiment connected with the diffusion of radiation in a heterogeneous medium. E.

Fractal Set Constructed on a Square Lattice

By using the above-described algorithms to generate fractal sets, fractal sets constructed on square lattices have been obtained [24]. The principal or main set of bonds,
n , is obtained using an iteration process, whereby at the initial step (k ¼ 0) a finite lattice is investigated in the space d ¼ 2 or d ¼ 3 with a probability p0 for the bond between the neighboring knots of the lattice to be complete and to have specific color. Bonds of like color are attributed the same properties. At the next stage of the process ðk ¼ 1; 2; . . . ; nÞ, each bond in the lattice is replaced by the lattice obtained at the previous stage (Fig. 12), The iteration process ends when the properties of the lattice no longer depend on the iteration number k. Thus, a lattice with linear dimensions l greater than the correlation length x can be obtained; that is, a lattice on which the effective macroscopic properties may be defined. The set of bonds
n obtained with the aid of the iteration procedure depends on the initial lattice dimension l0 , the probability p0 ; and is a self-similar fractal one.

vitaly v. novikov

126

Finite rectangular models flx ; ly ; lz g in the space of dimension d ¼ 2; 3 have also been studied, where lx ; ly ; lz are the dimensions of the lattice sides in the units of bond length (we have a constant lattice); lz for a planar lattice is zero. Formation of a connecting set (CS) is manifested by the presence in the set of complete (black) bonds connecting two opposite sides of the lattice in the direction of side lx . The fractal dimension dfk of the main set of bonds (i.e., of the frame) (Fig. 12) obtained from the iteration process for p0 ¼ 1 (all bonds colored black) can be defined from the dependence of the set mass (i.e., the number of the constituent ðnÞ bonds of the frame), Mk at iteration stage number n on the linear dimension of the lattice ln : dk

ðnÞ

Mk  lnf

ð119Þ

For example, for a planar square lattice with lx ¼ ly ¼ l0 , lz ¼ 0 ðnÞ

Mk ¼ ð2  l20 Þn

ð120Þ

Noting that ln ¼ ln0 , we obtain  ðnÞ Mk

¼ ln



2þ lnlnl2

0

ð121Þ

Hence dfk ¼ 2 þ

ln 2 ln l0

ð122Þ

The dimension dfk of the geometric set at p0 ¼ 1 is greater than the topological dimension of space d ¼ 2. For arbitrary dimension d of Euclidean space in which the fractal set
f is embedded and arbitrary form of the initial element, the mass (the number of bonds) can be defined by singling out the factorial geometric coefficient F: ðnÞ

Mk ¼ ðF  ld0 Þn ¼ F n  ldn

ð123Þ

Then Eq. (123) will read dfk ¼ d þ

ln F ln l0

ð124Þ

physical properties of fractal structures

127

Figure 12. Scheme for an iteration process on a square lattice with l0 ¼ 2: (a) p0 ¼ 1; (b) p0 ¼ 0:75 at the fourth iteration step.

It follows from Eq. (124) that the dimension of the bond set coincides with the Euclidean dimensions only in the limit of infinitely large dimensions of the initial lattice, l0 : lim dfk ¼ d

l0 !1

which is obviously correct for any model.

ð125Þ

vitaly v. novikov

128

ðnÞ

If p0 6¼ 1 ð0 < p0 < 1Þ, then the mass MCS of the fractal, which is a connecting set within the scales (l0 < ln < x), depends on ln in accordance with the law: ðnÞ

MCS  ldnf

ð126Þ

The density of the connecting set is ðnÞ

ðnÞ

rCS ¼

MCS

ðnÞ

Mk

df dfk

 ln

ð127Þ

Now introduce Yp ¼ df  dfk ¼ a 

ln F ln l0

ð128Þ

where a ¼ d  df . Because Yp < 0, we see that the connecting set density decreases to zero with increase of the iteration number—that is, with increase of ln : lim rCS ðln Þ ¼ 0

ln !1

ð129Þ

If p0 > pc , on scales higher than the correlation length x (ln > x), the connecting set becomes homogeneous with constant density, and Eq. (126) reads lim rCS ðln Þ ¼ r0 > 0

ln !1

ð130Þ

Thus, by giving the dimensions of the initial lattices, l0 , and the probability p0 for the bond to have a definite color, various disordered fractal sets can be obtained. F.

Cayley Tree. Ultrametric Space

The fractal dimension of the set obtained by the iteration process is df ¼

ln j ln K 1

ð131Þ

where j is the number of blocks taking part in the construction of the elementary shape of a fractal (for the Koch curve, j ¼ 4; for Cantor dust, j ¼ 2), and K is the similarity exponent showing by how much the size of the block decreases at each

physical properties of fractal structures

129

stage of the construction. If certain blocks are removed at each stage, then K  1¼)df < 1. If some block is added (Koch shape), then K  1¼)df > 1. ðK > j1 Þ The procedure of fractal set construction can be shown using the Cayley tree so that each fractal set has its own Cayley tree [25,26]. We show a Cayley tree with branch characteristic j ¼ 4. The Cayley tree is a pictorial representation of a space that is called ultrametric. Each point of the ultrametric space can be put into correspondence with an element of the fractal set; that is, the fractal set and ultrametric space are topologically equivalent sets. We remark that the main feature of an ultrametric space, as well as that of a fractal set, is its hierarchical property. The following constitutes the definition of the distance between two points in an ultrametric space. The points in an ultrametric space on a given hierarchical level are the ends of the Cayley tree branches (Fig. 13). The number of points on the nth level of the Cayley tree is equal to Nn ¼ jn . Each point on the nth level can be numbered: a1 ; a2 ; . . . ; a n ;

0  ak  j  1; 1  k  n

n=0

n=1

n=2

n=3

Figure 13.

Cayley tree.

ð132Þ

130

vitaly v. novikov

Thus, each point on the nth level of an ultrametric space corresponds to an n-digit number in the j-digit system of calculation (Fig. 13): fal gjn ¼ a0 ; a1 ; . . . ; al ; . . . ; an1 ; al ¼ 0; 1; . . . ; j  1

ð133Þ

These points constitute a space with ultrametric topology. The distance between two points in the ultrametric space is defined by the number of steps from these points to the common limit. For example, the distance between points 00 and 03 equals 1, and the distance between points 02 and 12 equals 2 (Fig. 13). Thus, the distance between two points in the ultrametric space with coordinates given by n-digit numbers in the j-digit system of calculation only depends on which digit these numbers first differ and does not depend on the specific values of this difference. The points of the discrete ultrametric space (Cayley tree junctions) on the nth level, namely, Nn , are divided into clusters (groups). Each cluster contains j points the distance between which is l ¼ 1 and has its progenitor on the ðn  1Þth level. The number of such clusters is Nn =j ¼ jn1 . The unity of clusters corresponds to an arbitrary distance l between points in the ultrametric space. All points are united in j subclusters with distance l  1 and having jl points. Thus, the group of clusters formed on the hierarchical level n  1 corresponds to an arbitrary distance l. If the limit transition is made when n ! 1, then the number of the points attaining level n approaches infinity ðNn ! 1Þ; that is, the intervals between points xn ¼ 1=Nn become infinitely small, and the ultrametric space itself becomes continuous. On the Cayley tree, the transition to a continuous ultrametric space indicates a condensation of the hierarchical levels. The distance between two points in an ultrametric space in the conventional Euclidean sense can be defined as ða  b Þ jn1 þ    þ ða  b Þ jnl þ    þ 1 l l 1 rða; bÞ ¼ jra  rb j ¼ þ ðan1  bn1 Þ j þ ðan  bn Þ



ð134Þ

Two arbitrary points in an ultrametric space belong to any cluster characterized by the distance l  n; hence the first terms of the sum in Eq. (134) equal zero because ai ¼ bi , i ¼ 1; 2; . . . ; n  l. For a continuous ultrametric space (n ! 1), knowing that j > 1, we obtain that the dominant member in the series is the member with multiplier jl, because the other members have multiplier j k, where k ¼ l  1; l  2; . . . ; 0. The values of coefficients (al ¼ bl ) in Eq. (134) are limited by the number j. Representation (134) corresponds to Cayley tree division jn into n groups, each of them consisting of clusters. Each of the clusters of the group is

physical properties of fractal structures

131

characterized by the same value l of the maximum distance between the knots of the cluster. For example, the first member in Eq. (134) describes: the contribution of those clusters with knots divided by the distance l ¼ 1. Based on the above statements, the distance rða; bÞ ffi jðal  bl Þj j l

ð135Þ

ln rða; bÞ ffi l ln j

ð136Þ

or

This approximate equation means that the ultrametric space has a logarithmic metric. Thus, when constructing a fractal set, each element corresponds to a point of the ultrametric space with geometric image represented by the Cayley tree.

III.

CHAOTIC STRUCTURES A. 1.

Percolation Systems Percolation Cluster

On analyzing the filtration of air through a porous medium, Broadbent and Hammersley described a novel process differing considerably from the wellknown phenomenon of diffusion [27]. Broadbent and Hammersley named such processes ‘‘percolation processes.’’ The phenomena described by the theory of percolation belong to so-called critical phenomena. These are characterized by a critical point at which the physical properties of a system dramatically change. Investigations of percolation systems are often carried out numerically on lattices, which are aggregates (sets) of junctions (sites) and bonds. Here the roles of bonds and the roles of sites are quite distinct. In the former, the transition of the set of bonds out of the unconnected domain into the connected one on increase of bond concentration p is examined, in the latter, such a transition is examined on the set of lattice sites. (We will further consider the role of bonds.) The concentration p ¼ pc at which the transition from the unconnected set of isolated clusters to the connected set, (the infinite cluster), occurs is called the threshold of percolation (Table IV). If p ¼ pc þ 0, then an infinite cluster exists. If p ¼ pc  0, then all clusters are isolated and finite. Applying fractal geometry to description of disordered media allows one to use the properties of scaling invariance—that is, to introduce macroscopic

vitaly v. novikov

132 (a)

Figure 14.

(b)

The percolation clusters: (a) Isolated cluster. (b) Infinite cluster (schematic).

values depending on the scale of averaging. In its turn, this allows one to construct the theory of such media using the renormalization group transformation method [1,28–33], which was developed in the theory of temperature phase transitions [34]. We describe briefly the main properties of a percolation cluster [35–37]. 2.

Critical Indices

As stated above, the most characteristic feature of percolation is bonding. The dimension of bonding domains (a bonding cluster)—that is, the regions in which it is possible via black bonds to go from one point of the region to another— rapidly increases with the growth of black bond concentration p. When p ¼ pc , an infinite (percolation) cluster spreading over the entire lattice first appears. The correlation length defines the connectivity of clusters. It defines the scale range within which percolation clusters behave self-similarly and, consequently, are characterized by a fractal dimension [38,39]. The correlation length x for a percolation lattice can be defined as ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi+ *v u N u1 X x¼ t ðri  r0 Þ2 N i¼1

ð137Þ

TABLE IV Percolation Thresholds for Lattices d

Lattice

Bond

Site

2 2 2 3 3 3 3

Square Triangular Honeycomb Simple cubic Body-centered cubic Face-centered cubic Diamond

0.5 [42} 0.33 [42,43] 0.66 [42] 0.24 [42] 0:18025  0:00015 [48] 0:119  0:002 [44] 0:388  0:05 [44]

0:590  0:010 [44] 0:500  0:005 [45] 0:70  0:01 [44] 0:320  0:004 [45] 0:243  0:010 [44] 0:208  0:0035 [45] 0:4299  0:0008 [46]

physical properties of fractal structures

133

Here N 1X ðri  r0 Þ2 N i¼1

ð138Þ

is the averaged squared radius of the cluster measured from its center of gravity; ! r0 is the conditional center of gravity of the cluster ! r0 ¼

N ! X ri i¼1

N

ð139Þ

where N is the number of knots in the cluster, and ri is the coordinate of the ith bond in the cluster. The summation is over all N bonds of the cluster. Thus, according to Eq. (137) the correlation length xðpÞ is the average dimension of those clusters that contribute most to the second moment of the distribution of cluster dimensions near the percolation threshold, pc . As p ! 0 ( p ¼ p  pc ) the correlation length xðpÞ ! 1 [1], namely, xðpÞ  jp  pc jn

ð140Þ

where n is the correlation length critical index and depends only on the dimension d of the space. Two linear dimensions, the minimum length l0 (the lattice constant) and the correlation length x, play a key role in the behavior of a percolation system. For real systems, an intermediate asymptotic region exists such that l0 < l < x where l0 is the lattice constant. In the domain x > l0 , that is in the domain of intermediate asymptotics, on the interval l all characteristics of the cluster are similar to its characteristics at the most critical point. On all scales l > x the system is homogeneous, and the self-similarity disappears. To quantitatively characterize the percolation cluster as a whole, the notion of infinite cluster density is introduced, P1 ðpÞ It is the ratio of the number of the bonds belonging to the infinite cluster, nk to all black bonds on the percolation lattice N: P1 ðpÞ ¼

nk N P ni i¼1

ð141Þ

vitaly v. novikov

134

l

ξ

ξ

pc Figure 15.

p

Composition dependence of the correlation length near the percolation threshold

(schematic).

Thus, P1 ðpÞ is the probability of an arbitrary bond belonging to the infinite cluster. The increase of infinite cluster density P1 ðpÞ near the percolation threshold pc ðp=p 1Þ is characterized by a critical index b P1 ðpÞ  pb ;

p ¼ p  pc > 0

ð142Þ

where b as well as v depends only on the dimension d of the space. The Typical dependence P1 ðpÞ is as shown in Fig. 16.

1

P∞

0 Figure 16.

pc

p

1

Dependence of infinite cluster density on concentration.

physical properties of fractal structures

135

TABLE V Percolation Critical Exponents Exponent

d¼2

d¼3

n

4/3 [40, 41]

P1  ðpc  pÞb

b

SðpÞ  jpc  pjg

g ¼ nd  2b

5=3 ½40; 41 0:15  0:03 [47] 43/18 [40, 41] 2:43  0:04 [52]

0.82  0.05 ½47 0:905  0:023 [48] 0:39  0:07 [47] 0:454  0:008 [46] 1:70  0:11 [47] 1:91  0:01 [50]

Function xðpÞ  jpc  pj

n

The correlation length, xðpÞ, is defined by Eq. (140). Hence j p  pc j ¼ jpj  x1=n ;

p ! 0

ð143Þ

Here, the percolation cluster (infinite cluster) density P1 ðpÞ can be represented as P1 ðpÞ ¼ xb=n

ð144Þ

The critical indices of correlation length v and percolation cluster densities b constitute the main percolation parameters. In two dimensions (d ¼ 2), these indices may be found analytically [40,41], and are, n ¼ 4=3 and b ¼ 5=36. For d ¼ 3, only numerical estimations are available: n ¼ 0:90 and b ¼ 0:40 (see Table V) The critical percolation indices depend only on the spatial dimensions and do not depend on the type or other parameters of a lattice. Another value characterizing a percolation system is the average cluster dimension SðpÞ. The critical behavior near the percolation threshold is defined by the critical index g [1]: SðpÞ  jpc  pjg ; p < pc

ð145Þ

The density of clusters of finite dimensions, ns ðpÞ is connected to the percolation cluster density P1 ðpÞ by the equation 1 X

sns ðpÞ þ P1 ðpÞ ¼ p

ð146Þ

s¼1

The critical indices n; b; g for the percolation cluster are given in Table V.

136

vitaly v. novikov

Figure 17. Percolation cluster density as a scaled function of its dimension, l, for three values of pðp p  pc Þ [24]: 1  p ¼ 5  104 ; 2  p ¼ 1:5  103 ; 3  p ¼ 2:5  103 .

The fractal set mass, Mf (of the percolation cluster), depending on the scale x can be defined as Mf  xdf

ð147Þ

where df is the fractal dimension. Therefore, the percolation cluster density P1 ðpÞ is P1 ðpÞ ¼

Mf  xdf d ; M

p ffi pc

ð148Þ

where M  xd is the mass of the domain containing in the percolation cluster. From Eq. (144) and (148) we have df  d ¼ b=n

ð149Þ

The percolation cluster includes those bonds forming lines, loops, dead ends and other configurations. Thus to characterize percolation cluster structure in detail, all bonds belonging to the percolation cluster are colored ‘‘red,’’ ‘‘blue’’ and so on. For example, ‘‘blue’’ bonds are a set of bonds in which current will flow if the percolation cluster is placed between electrodes subject to a potential difference (Fig. 14). The set of ‘‘blue’’ bonds generates an infinite cluster.

physical properties of fractal structures

137

TABLE VI Fractal Dimension of the Percolation Cluster and Its Components Fractal Dimension

d¼2

d¼3

Percolation cluster df ¼ d  b=n

91/48 [39]

‘‘Blue’’ bond set

1:62  0:02 [53] 1:60  0:05 [54] 0:75  0:01 [55] 1.75 [58] 1.18 [56]

2:484  0:012 [46] 2:529  0:016 [51] 1:74  :04 [54] 1:77  0:007 [53] — 2.54 [51] 1.35 [54] 1.26 [57]

‘‘Red’’ bond set Perimeter Minimum path

The set of ‘‘red’’ bonds consists of those bonds whereby removing one bond disturbs the bonding of the percolation cluster (removing a ‘‘red’’ bond leads to an open circuit, Fig. 14), while removing a single ‘‘blue’’ bond does not lead to disturbance of infinite cluster bonding. The outer boundary of the percolation cluster is the set of bonds belonging to the boundary of the percolation cluster. The fractal dimensions of these sets are shown in Table VI. 3.

Renormalization-Group Transformations

The macroscopic behavior of physical systems is determined by the microscopic behavior of these systems. Usually the microscopic fluctuations are averaged, and on larger scales the averaged values satisfy the classical equations. A number of extreme (critical) situations exist when fluctuations extend to macroscopic scales and exist on all intermediate scales. Temperature phase transitions and percolation processes concern such situations. In these systems a critical point exists which separates two different phase states of the system. The difficulties in the theoretical study of such systems are caused by the large numbers of interacting degrees of freedom, since it is necessary to have many variables in order to characterize such systems near by a critical point. For the purpose of overcoming difficulties in the study of critical phenomena the following method was used: sequential averaging was carried out on all scales, starting with fluctuations at an atomic level and then moving step-bystep to larger scales. Such stage-by-stage modification of scales allows one to reduce the number of degrees of freedom. This method is known as the renormalization group method. The result of the renormalization group method as applied to percolation problems consists of the following: a physical state characterized, for instance, by the parameters fY0 g evolving via a set of equations Y forms a continuous sequence of new effective equations Y ðlÞ, characterized by the new parameters

138

vitaly v. novikov

fYðlÞg [28–33]. The new parameters fYðlÞg are functions of the initial parameters fp0 g and a scale factor l0, that is, Yk ðlÞ ¼ Y ðl0 ; fp0 gÞ

ð150Þ

In the limits of large l a trajectory of renormalization group transformation terminates in a fixed point. The dimensionless correlation length x can be used as a measure of the remoteness of a percolation system from a critical point x ¼ x=l0

ð151Þ

Under renormalization group transformation the correlation length x decreases so that x

 0  1  0  pi ¼ xp pi l0

ð152Þ

where p0 is the probability of making a connection (probability that connection will be black) after renormalization group transformation. This transformation has a fixed point related to a percolation threshold pi ¼ pc where pi is determined from f p g ¼ Y ð f p gÞ

ð153Þ

The procedure of renormalization group transformation (153) for finite lattices is approximate as it does not consider the surface effects on a lattice properly because the connecting configurations gained after renormalization group transformation, differ from connecting configurations in a real lattice. Under renormalization group transformation some connecting configurations are lost and new connecting trajectories appear. As the initial cell l0 grows in size, boundary effects decrease [24]. To illustrate the basics of renormalization group transformation we shall consider a triangular lattice, which is featured prominently in the literature. The initial cell is connecting if all three knots are made or two of them are made, and one is vacant. In this case the probability, for the cell to be connecting, is [1] Y ð pÞ ¼ p3 þ 3p2 ð1  pÞ

ð154Þ

0

p ¼ Y ð pÞ It follows from the last equation, that three fixed points: 0; 1=2; 1 exist. Two points ð0; 1Þ are unstable; however, point p ¼ 1=2ðp ¼ Y ðp Þ:Þ is stable and equal to the percolation threshold pc ¼ 1=2.

physical properties of fractal structures

139

Expansion in a Taylor series about p ¼ 1=2, neglecting ð p  p Þ2 and higher yields dp0 p ¼p þ dp 0



ð p  p Þ þ   

ð155Þ

p¼p

or p 0  p  ¼ lð p  p  Þ þ   

ð156Þ

where dp0 l¼ dp

  ¼ 6p  6p2 p¼p ¼ 3=2

ð157Þ

p¼p

When p ¼ pc the correlation length tends to infinity. On small scales L < x we may write x0 ¼ x and for a renormalized lattice we have x=l0 ¼ cjp0  pc jn

ð158Þ

where l0 is a lattice distance, c and n are constants being an amplitude and a critical exponent, respectively. Thus, the following equality is valid: l0 jp  pc jn ¼ jp0  pc jn

ð159Þ

This relation is the basic equation governing a renormalization group in real space. According to Eq. (159) we shall obtain the critical exponent n for correlation length: n¼

ln l 0 p0 p ln ppcc

ð160Þ

Consequently, we have 1 ln l ¼ n ln l0

ð161Þ

pffiffiffi n ¼ ln 3= lnð3=2Þ ’ 1:355

ð162Þ

that is,

140

vitaly v. novikov

If as initial cell we choose a square cell with 8 bonds [1], then Y ð pÞ ¼ p5 þ 5p4 ð1  pÞ þ 8p3 ð1  pÞ2 þ2p2 ð1  pÞ3 ; p0 ¼ Y ð p Þ

ð163Þ

It follows from Eq. (163), that p ¼ 1=2 is a nontrivial fixed point and that the critical index for correlation length is n ¼ 1:43 . . .. In such calculations of n and other critical indexes it is impossible to determine the accuracy of the calculation and growth of cells of size l0 > 4 poses essential calculation difficulties [33]. The percolation probability of a lattice Yðp; l0 Þ with initial concentration of black connections p is calculated as the ratio of number of the number of connecting configurations to the number of all possible scatters. Typical Yðp; lÞ for a cell of size l0 ¼ 2 is presented in Fig. 18. Intersections of the bisector of coordinates axes with curves Yðp; lÞ define the percolation thresholds pc for models of size l0 (see Fig. 18). The function Yðp; lÞ converges to a step function when l0 ! 1 and the derivative dY ¼ f ðpÞ dp now converges to a Dirac delta function (f ðpÞ ! dðpÞ; if l0 ! 1).

Figure 18. The percolation lattice probability Yðp; l0 Þ for a cell of sizes l0 ¼ 2:

ð164Þ

physical properties of fractal structures

141

ξ

Figure 19.

Framework of an infinite cluster according to the Scala–Shklovsky model.

By using the function Yðp; lÞ it is possible to obtain a probability density function f ðp; lÞ for the percolative configurations Z Y ð p; lÞ ¼ f ð p; lÞdp ð165Þ Thus the relation dY f ðp ; lÞ ¼ dp 

¼ lp

ð166Þ

p¼p

is satisfied. Let us consider two models that have been used in percolation theory to describe the properties of a percolation cluster. The first is the Scala–Shklovsky model [64]. 4.

Physical Properties

In the Scala–Shklovsky model [64] it was assumed that the structure of an infinite cluster is a net with the characteristic geometric distance between knots being the percolation length x. The sites of the net are connected by single-core macrobonds with length L  ðp  pc Þ


ð167Þ

which can be greater than distance x between the sites of the net (Fig. 19). It was shown [65,66] that the critical index
is 1 and does not depend on the dimension d of the space. The disadvantage of this model is that, according to Eq. (167), the distance x between the knots of the infinite cluster net increases faster than length L of the macrobond. In the Coniglio–Sarychev–Vinogradov model, this disadvantage was removed [65].

vitaly v. novikov

142

Figure 20.

Blob model of an infinite cluster framework.

In the Coniglio–Sarychev–Vinogradov model, it is assumed that the infinite cluster consists of blobs connected to macrobonds (Fig. 20). The blob of dimension b consists of several blobs of dimension b=2 connected by singlecore macrobonds, and so on. Therefore, the system is self-similar on any scale. It has been shown [64–67] that the average summed length L of nondoubled bonds in a fragment of the infinite cluster is defined as L  b1=v . Whence, if b ¼ x, then L  x1=n  ðp  pc Þ1 ; that is, the critical index
¼ 1. The relative length of a macrobond L=b  b1=ðn1Þ  ðp  pc Þn1

ð168Þ

increases with an increase of scale b, so that this model does not have the contradictions of the Scala–Shklovsky model. We now examine the main ideas of how we define the conductivity and elasticity of fractal structures. Conductivity. Let sðlÞ be the conductivity of a fragment of the fractal structure of dimension l, where l0 < l < x. Because of the self-similarity of the structure, the ratio of conductivities on different scales l and l0 is defined only by the ratio of the scales:   sðlÞ l ¼ f ð169Þ 0 sðl Þ l0 For three different scales, l; l0 and l00 , the following equations are appropriate:    0   sðlÞ l sðl0 Þ l sðlÞ l ¼ f ¼ f ¼ f ; ; ð170Þ sðl0 Þ l0 sðl00 Þ sðl00 Þ l00 l00

physical properties of fractal structures

143

Hence any function f ðxÞ of the scale ratio must satisfy the equation f ðx  yÞ ¼ f ðxÞ  f ðyÞ

ð171Þ

It follows from Eq. (171) that f ðxÞ ¼ xg

ð172Þ

where g is an arbitrary number. Therefore, sðllÞ ¼ lg sðlÞ

ð173Þ

where l belongs to interval l > x. Now we divide the percolation cluster into d-dimensional cubes (d > 1). The conductivity of one cube with side l is designated by sðlÞ. The number of cubes per unit length in the column is equal to l1 , and the number of parallel columns is l1d . So, the conductivity of the percolation cluster is s ¼ sðlÞl2d

ð174Þ

The conductivity of a single cube, sðlÞ can be defined as sðlÞ ¼ s1 l1 , (s1 is the conductivity of a ‘‘black’’ bond). Thus, we obtain s ¼ s1 l1d . Since l  x, we obtain s  s1 ðp  pc Þnðd1Þ ;

ðp  pc Þ > 0

ð175Þ

Introducing a critical index t for the conductivity s  ðp  pc Þt ; ðp  pc Þ > 0

ð176Þ

t ¼ nðd  1Þ

ð177Þ

we obtain

Taking account of the values of n (n ¼ 1; 33 at d ¼ 2 and n ¼ 0; 8  0; 9 at d ¼ 3), we obtain t ¼ 1:3 if d ¼ 2

and

t ¼ 1:6  1:8

if d ¼ 3

The approximate estimations of the critical index of conductivity, t, so obtained, agree with the numerical evaluations (see Table VII). The critical index s for conductivity in the concentration region ðp  pc Þ < 0, ð p ! pc  0Þ is defined by s  ðp  pc Þs ;

ðp  pc Þ < 0

ð178Þ

vitaly v. novikov

144

TABLE VII Conductivity Critical Indexes Function

d¼2

Index

t

s  ðp  pc Þ ðp  pc Þ > 0

t

s  ðpc  pÞs ðp  pc Þ < 0

s

1:10  0:05 [58] 4/3 [60] 1:32  0:05 [59] 1:15  0:25 [71] 1:0  0:1 [62]

d¼3 1:6  0:1 [67] 2:00 [62] 1:95  0:1 [63] 2:46 [28] 0:7  0:05 [62] 0:9  0:01 [68] 0:7  0:05 [72]

Linear Elasticity. Here the task of defining the elastic properties of a percolation system will be formulated completely if the relevant Hamiltonian is defined on that set of sites and bonds with geometric parameters (numbers of sites and bonds, distances to the most remote elements, sinuosity, etc.) given statistically. The Hamiltonian describing the elastic properties of a percolation system must satisfy the following criteria:  Elastic bonding must exist: At p > pc , the lattice must have finite elastic macromodulus becoming zero at p ! pc þ 0.  The tensor properties of the elasticity of long chains must be generated properly.  Invariance must be preserved in relation to rotations in free states of the Hamiltonian. The Born model [74], for example, satisfies the first condition; however, it does not satisfy the second one because in it the longitudinal and transverse elastic constants of the linear chain of bonds (the lattice analog of a rod) decrease  N 1 ; however, the rods must behave more pliably in relation to transverse shifts (the elastic constant decreases  L3 ). Therefore, the Born scalar model leads to enhanced rigidity in the vicinity of pc .

j bi

i

Figure 21.

Φijk k Elastic chain of vectors.

physical properties of fractal structures

145

U. Cantor and I. Webman [75] have suggested a Hamiltonian describing the elasticity of continuous chains correctly, namely, H¼

N N GX Q X aij ajk dijk þ 2 ðui  uj Þ2 4 i; j; k 4a i; j

ð179Þ

where dijk is the change of the angle between the bonds fðijÞ; ðjkÞg; a is the lattice constant; aij is a random variable lying in f0; 1g with probabilities ð1  pÞ and p, respectively; Q and G are local elastic constants; ðui  uj Þ is the difference of the shift of sites i and j in the direction parallel to bond ði; jÞ: The rigidity of the percolation net ‘‘is supported’’ by the infinite cluster system which consists of comparatively straight and single bonds connecting compact multibond domains. Thus, the elastic behavior of a percolation system can be investigated based on the behavior of chains with N vectors (bonds) fbi g (Fig. 21). The corresponding Hamiltonian is H¼

N N GX Q X 2 d2i þ 2 dbi 2 i¼1 2a i¼1

ð180Þ

In order to define the elastic constant of such a vector chain, it is necessary to define the relative change of angle di in the orientation of the bonds when ! force F is applied to the end of the chain. This relative change, di , can be found by minimizing the equation for the system energy, namely, ! 0 W ¼ H  F ðRN  RN Þ

ð181Þ

0

where ðRN  RN Þ is the shift of the chain end from the balanced state. ! 0 The work done by the force F  ðRN  RN Þ can be represented as the PN summation i¼1 , and the exact expression for the change of the angles between two bonds after minimization of W reads !X N N X X F N ! ! 0 di bi þ bi dbi ; F  ðRN  RN Þ ¼ ð F
zÞ a i¼1 i¼1 i¼1 ! ! N F
zX F
z a! ðRN  Ri1 Þ; d ¼ F bi di ¼ bi ¼ G i¼1 G b

ð182Þ

where z is a single vector perpendicular to the plane of the chain, and Ri is the balanced position of the end of vector bi.

vitaly v. novikov

146

Substituting di and bi into (180), we have H ¼ F 2 NS2? =2G  F 2 aL=2Q

ð183Þ

where S2? is the square inertia of the projection of angle R onto the direction F
z: S2? ¼

N N 1 X 1 X ! ! fð F
zÞðRi1  RN Þg; L ¼ 2 ð F  bÞ2 2 NF i¼1 aF i¼1

ð184Þ

Thus, the force constant (rigidity) of the chain is defined as K ¼ G=ðNS2? Þ

ð185Þ

Thus, the elastic constant of a long chain depends not only on the length ð1=NÞ but also on its geometry, S2? . The analysis given for d ¼ 2 can also be applied to higher dimensions. As before, the deformation of the chain may be represented by a sequence of transformations, where the ith transformation includes the stretching of the bond bi and the rotation of bonds bi . . . bn about Ri¼1 : The result obtained can be used to estimate the critical index t of the macroelastic constant at p ! pc þ 0. K  ðp  pc Þt

ð186Þ

Dividing the percolation system into cubes with linear dimension l, the infinite cluster system macroscopic elasticity can be defined as K ¼ KðlÞl2d

ð187Þ

KðlÞ is the elasticity of single cube with linear dimension l. As elasticity on any scale is assumed to behave in a self-similar mode, KðlÞ can be defined as KðlÞ  K1 ðp  pc Þt ¼ K1 lt=n

ð188Þ

(K1 is the elasticity of a bond). Moreover, according to Eq. (185), 1 ¼ l2 l1=n KðlÞ  S2 b L

ð189Þ

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147

TABLE VIII Elasticity Critical Indexes d

2

t=n S=n

2.97 0.92

3 4.3 0.74

Comparing the two latter expressions for KðlÞ, we can conclude that t=n ¼ 2 þ 1=n, that is, t ¼ 2n þ 1. Using the obtained expression for n, the infinite cluster system macroscopic elasticity can be defined as KðlÞ  K1 lð1þndÞ=n ¼ K1 ðp  pc Þndþ1

ð190Þ

Thus, the volume elastic modulus of the percolation system near the percolation threshold ðp ! pc þ 0Þ may be defined as K  K1 ðp  pc Þt ;

t ¼ nd þ 1;

p ! pc þ 0

ð191Þ

and the critical index is t ¼ nd þ 1

ð192Þ

Noting the values of n (n ¼ 1; 33 at d ¼ 2 and n ¼ 0; 8  0; 9 at d ¼ 3), we obtain t ¼ 3; 6 if d ¼ 2 and t ¼ 3:4  3:7 if d ¼ 3. The approximate estimations of the critical index of volume elasticity t obtained agree well with the numerical evaluations (see Table VIII). Comparing t and t, we see that tt ¼1þn

ð193Þ

Hence, the critical index t for elasticity is different from the critical index t for conductivity at the value of 1 þ n. Critical indices S and t, obtained numerically, are shown in Table VIII (according to Ref. 76). The critical index S for the volume elastic modulus in the concentration domain ðp  pc Þ < 0 ðp ! pc  0Þ is defined by K  ðp  pc ÞS ; B.

ðp  pc Þ < 0ðp ! pc  0Þ

ð194Þ

Fractal Structure Model

Stauffer and Aharony [1] have studied chaotic fractal ensembles on square lattices where all bonds were identically colored at the initial stage and later

148

vitaly v. novikov

Figure 22. Schematic for constructing a self-similar lattice via iterative growth of a square generation cell.

randomly change their color. These chaotic fractal ensembles
f depend on the initial lattice size l0 and on the probability of p0 . At each stage of the growth process, each thin bond of the generating cell is replaced by a structure obtained at the previous stage. The sides marked by thin lines are assumed to be always connecting. The growth of the fractal ensemble was initiated on the two-dimensional (d ¼ 2) finite-size lattice l0
l0 ; next each bond of this lattice at the kth

physical properties of fractal structures

149

stage was replaced by a lattice generated at the preceding, (k  1)st stage. The growth was considered completed after the properties of the fractal ensemble became independent of the linear scale ln ¼ ln0 . The fractal dimension, df0 ðl0 Þ, of the principal ensemble
0 ðl0 Þ (with all bonds of the same color) generated on square lattices (lx ¼ ly ¼ l0 ) was determined, as usual, from the relationship between its mass (i.e., number of bonds) and ln , that is, [24], d0 ðl0 Þ

Mn ðl0 Þ ¼ lnf

ð195Þ

Noting that for square lattices  n Mn ðl0 Þ ¼ 2l20

ð196Þ

ln 2 ln l0

ð197Þ

one obtains df0 ðl0 Þ ¼ 2 þ

It follows from Eq. (197) that fractals of dimensions 2 < df0 ðl0 Þ < 3 can be obtained by altering the size l0 of the generating (initial) square cell. For example, df0 ðl0 Þ ¼ 3 for l0 ¼ 2, while df0 ðl0 Þ ! 2 for l0 ! 1. Consider now another ensemble of structures,
0 ðl0 ; p0 Þ, with bonds that may be colored black and white with p0 as the probability of a black bond. Hence, one may distinguish between two basic states, in which the black bonds form either a bonded ensemble (connecting set) spanning the entire lattice space between two opposite faces, or via non-bonded ensemble (non connecting set) with no trajectory available to connect two opposite lattice faces. The mass of the connecting set of black bonds in the vicinity of the critical point of transition nonconnecting set!connecting set depends on the linear scale ln , Mf ðl0 ; p0 Þ  lndf ðl0 ; p0 Þ

ð198Þ

where the connecting set density may be defined as PCS ðl0 ; p0 Þ ¼

df ðl0 ; p0 Þdf0 ðl0 Þ M f ð l 0 ; p0 Þ ¼ ln Mn ðl0 Þ

ð199Þ

150

vitaly v. novikov

The correlation length x of the connecting set is confined to the range of intermediate asymptotics, which may be defined as l0 pc

ð202Þ

where p0 ! pc þ 0. Using Eqs. (201) and (202), we have the dependence of P1 CS ðl0 ; p0 Þ on the linear size of the system 

P1 CS ðl0 ; p0 Þ  ln

bðl0 Þ nðl0 Þ

;

when p0 ! pc þ 0

ð203Þ

Here, the critical indices for the connecting set correlation length and density are related via the fractal dimension df ðl0 Þ and bðl0 Þ; nðl0 Þ as bðl0 Þ ; nðl0 Þ df ðl0 Þ ¼ lim df ðl0 ; p0 Þ; p0 !pc þ0   bðl0 Þ ¼ lim log10 P1 CS ðl0 ; p0 Þ =log10 ½ p0  pc ;

df ðl0 Þ ¼ df0 ðl0 Þ 

ð204Þ

p0 !pc þ0

vðl0 Þ ¼  lim log10 ½xðl0 ; p0 Þ=log10 jp0  pc j p0 !pc

An important characteristic of the chaotic fractal ensemble
0 ðl0 ; p0 Þ is the probability that a given configuration belongs to the connecting set (i.e., the percolation probability). At the 0th (i.e., initial) stage, this probability depends on the initial density of black bonds p0 and on the size l0 of the generating cell, and it may be defined as the ratio of the number of bonding configurations to the total number of possible configurations. At the first growth stage the length of the lattice rib is l1 ¼ l20 , and the density of black bonds is p1 ¼ Yðl0 ; p0 Þ. In the

physical properties of fractal structures

151

next stages ln ¼ l0 ln1 and the probability of a bond to belong to the connecting set will depend on p0 as follows: p2 ¼ Yðl1 ; p1 Þ; ... pn ¼ Yðln1 ; pn1 Þ

ð205Þ

The unstable critical point p ¼ Yðl0 ; p Þ (pc ¼ p ) may be determined from the equality
1; p0 > pc lim pn ¼ ð206Þ n!1 0; p0 < pc In any practical (finite precision) calculations the growth trajectory of the chaotic fractal ensemble
f ðl0 ; p0 Þ ends at the nth growth step (level) reaching a point indistinguishable from one of the fixed points 0 or 1 of the bounding probability function Yðl0 ; p0 Þ. ðnÞ The probability PCS ðl0 ; p0 Þ that a bond belongs to the connecting set at the nth growth step is ðnÞ

PCS ðl0 ; p0 Þ ¼

n Y

Yðl0 ; pi Þ

ð207Þ

i¼0

Equation (207) shows that a complete (conducting) bond belongs to the infinite cluster only when it belongs to a cluster connecting the two opposite sides of the lattice at each iteration step i ¼ 1; . . . ; n. It follows from Eqs. (206) and (207) that
p0 > pc PCS ðl0 ; p0 Þ; ðnÞ lim PCS ðl0 ; p0 Þ ¼ ð208Þ n!1 0; p 0 < pc Thus, a knowledge of the function Yðl0 ; p0 Þ is crucial for determining the properties of the fractal model. For small initial lattices this function can be calculated exactly. The results for square generating cells of l0 ¼ 2; 3; 4 as well for the more general case of rectangular generating cells are given in the Appendix. Consider now fractal ensembles grown on rectangular subsets of the square lattice, lx
ly ðlx 6¼ ly Þ, further referred to as rectangular generating cells. Hence, the characteristic length of the system, l0 , can be chosen in various ways. For lx > 1 ( lx ¼ 1 is trivial) the simplest and most natural choice is l0 ¼ lx . As can

vitaly v. novikov

152

A

B

A

B

Figure 23. Schematic for construction of a self-similar lattice by iterative growth of a rectangular generation cell: the meaning of the thin and thick lines is as in Fig. 22.

be seen in Fig. 23, for such a choice each iteration step increases the length of the system by the factor l0. Thus, it is natural to choose ln ¼ ln0 as the characteristic length of the system at the nth step. At the nth growth step, the mass of the ‘fractal’ ensemble so obtained will be   n ðnÞ  ð209Þ M0 lx ; ly ¼ 2lx ly þ lx  ly

Figure 24.

The relatively dual model.

physical properties of fractal structures

153

TABLE IX The Percolation Threshold pc , Fractal Dimension of the Ensemble at p ¼ 1, df0 ðlx ; ly Þ, Mean Fractal Dimension for p ¼ pc, df , and Critical Indices b bðlx ; ly Þ, a1 , and n nðlx ; ly Þ for Various Initial Rectangular Cellsa. lx
ly

pc

df0 ðlx ; ly Þ

df

lp

n

2
1 2
2 2
3 2
4 3
1 3
2 3
3 3
4 4
1 4
2 4
3 4
4

0.500 0.304 0.223 0.178 0.696 0.500 0.410 0.358 0.777 0.590 0.500 0.451

2.322 3.000 3.459 3.807 1.893 2.335 2.631 2.854 1.730 2.085 2.322 2.500

1.322 1.282 1.293 1.314 1.563 1.704 1.819 1.919 1.548 1.704 1.822 1.926

1.625 1.823 1.912 1.962 1.823 2.217 2.441 2.590 1.911 2.441 2.766 3.097

1.428 1.154 1.069 1.028 1.829 1.380 1.231 1.154 2.139 1.553 1.363 1.226

a

a1 1.000 1.718 2.166 2.493 0.330 0.631 0.811 0.935 0.182 0.381 0.500 0.574

b 1.428 1.983 2.315 2.564 0.603 0.870 0.999 1.079 0.389 0.591 0.681 0.704

ln l The lp is calculated from lp ¼ dY dp jp¼pc and n is calculated from n ¼ ln lp .

  Thus, using Eq. (195) one obtains the fractal dimension df0 lx ; ly of the system as d f0



lx ; ly



  ln 2lx ly þ lx  ly ¼ ln lx

ð210Þ

  One can easily check that 1 < df0 lx ; ly < 1, where the limits are reached for ly =lx ! 0 (yields df0 lx ; ly ¼ 1) and lx =ly ! 0 (yields df0 lx ; ly ! 1). Probability functions Yðlx ; ly ; pÞ for fractal ensembles grown on several lattices (of the generating cells lx
ly where 2  lx  4; 1  ly  4) are presented in the Appendix, while calculated values  ofthe percolation threshold 0 pc , fractal dimension of the ensemble at p ¼ 1, d lx ly , mean fractal dimension f  at p ¼ pc df , and critical indices bðlx ; ly Þ and nðlx ; ly Þ are listed in Table IX. The index a1 in this table is calculated from   1 PCS lx ; ly ; p  la n

ð211Þ

that is, a1 ¼ bðlx ; ly Þ=nðl  x ;ly Þ. The  results  presented in Table IX were obtained in the limit p0 ! pc and df ¼ df0 lx ; ly  bðlx ; ly Þ=nðlx ; ly Þ: Ending these calculations, one should note that, according to the construction procedure outlined, the lattices considered are inhomogeneous, that is, the

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154

coordination number, Z, for these lattices depends on the lattice coordinates. By introducing the average coordination number hZ i ¼ 2ðtotal number of bondsÞ=ðtotal number of sitesÞ

ð212Þ

and noting that the lattices considered are self-similar, the average coordination number can be expressed via the length of the sides of the generating cell as hZ i ¼ 2

2lx ly þ lx  ly  1 lx ly þ lx  ly  1

ð213Þ

It follows from Eq. (213) that for rectangular generating cells the average coordination number can vary in the range 3  hZ i  6. This range is a factor 9=2 larger than that for square generating cells (lx ¼ ly ), for which 4  hZ i  14=3

ð214Þ

Table X present results obtained for the overage coordination number hZ i and for the average number of complete bonds around a lattice site at the percolation threshold (i.e., the product of hZ i and p at the critical point) for the lattices considered. Thus, for the inhomogeneous lattices considered this product can be much different from the value 2 which has been obtained for the infinite uniform square lattice [3].

TABLE X The Average Coordination Number, hZ i, and the Average Number of Complete Bonds Around a Lattice Vertex, pc hZ i, Calculated for Various Initial Rectangular Cells at the Percolation Threshold, pc lx
ly

pc

2
1 2
2 2
3 2
4 3
1 3
2 3
3 3
4 4
1 4
2 4
3 4
4

1/2 0.3039 0.2227 0.1776 0.6961 1/2 0.4100 0.3580 0.7772 0.5900 1/2 0.4509

hZ i 4 14/3 5 26/5 7/2 4 17/4 22/5 10/3 34/9 4 62/15

pc hZ i 2 1.418 1.114 0.924 2.436 2 1.743 1.575 2.591 2.229 2 1.864

physical properties of fractal structures 1.

155

Properties of Finite Lattices

It follows from the results obtained above that the regularities in the statistical properties of a percolation cluster can be studied even at length intervals of the order of the lattice constant. Hence, we shall analyze in greater detail the finite scales method [78] which was implemented on the smallest models. Thus, we separate the set of all possible initial cells in two-dimensional space into classes according to the following criterion: A model belongs to class CðnÞ if the difference lx  ly for the model equals n, where n 2 Z. We choose a finite-dimensional representation for the density of an infinite cluster as   b=n P1  a1 þ a2 gp ð l Þ 1 ¼l

ð215Þ

where the function gp ðlÞ is a correction to the scaling and l characterizes the size of the square model flx ; ly ; lz ¼ 0g in the sense that the number of bonds in the class increase as a power-law function (215) with exponent d. A computer calculation performed on small lattices ðlx < 10Þ in two dimensional space suggests that G ln P1 1 ðp ðlÞÞ = ln l is a linear function of 1= ln l (Fig. 25):   b ln a1 þ a2 gp ðlÞ b B ¼ þ f ðnÞ þ þ ; ln l n n ln l

ð216Þ

G 3

0.3

2 0.2

1 0.1

0

Figure 25.

0.3

0.6

0.9

l/ln l

G ¼ ln½P1 1 ðlÞ= ln l versus 1= ln l for the model classes 1, Cð2Þ 2, Cð1Þ; and 3, Cð0Þ.

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156

where f ðnÞ is the difference of the coordinates of the point of intersection of the experimental straight line for a given class CðnÞ, and the G axis and the ratio b=n are singled out on the right-hand side for the purpose of generalization. Then the assumption of linearity determines the form of gp ðlÞ in Eq. (215), namely, a1 þ a2 gp ðlÞ ¼ lf ðnÞ expðBÞ

ð217Þ

During self-similar growth of the model, the similarity dimension of the statistically homogeneous fraction that now arises can be estimated right at the percolation threshold from DðlÞ ¼ d þ

ln½P1 ðp ðlÞÞ ln½F ðp ðlÞÞ þ ln l ln l

ð218Þ

which, generally speaking, takes on different values for different models. Using Eqs. (215) and (218), we obtain DðlÞ  d þ

b B ln½F ðp ðlÞÞ ¼ f ðnÞ  þ n ln l ln l

ð219Þ

The fractal dimension df of an infinite cluster (limiting value of DðlÞ in the limit l ! 1) satisfies Eq. (219), which determines the asymptotic behavior of the function p ðlÞ, namely, lim

l!1

ln½p ðlÞ ¼ f ð nÞ ln l

ð220Þ

It is evident from Eq. (220) that the decrease, recorded during numerical modeling, of the nonzero quantity dðlÞ ¼ jp ðlÞ  pc j (power-law decrease with exponent  1n [1]) satisfies Eq. (217) only if f ðnÞ ¼ 0 [hence a2 ¼ 0 in Eq. (216)]. Then the curves 1 and 3 in Fig. 24 are convex and concave, respectively, with respect to the horizontal axis, since for the classes Cð0Þ and Cð2Þ the linearity assumption leads to f ð2Þ; f ð0Þ ¼ 6 0. Thus the assumption (216) can generally  be expressed as follows: There exists a value of F such that the curves G ln1 ðlÞ for classes C ðnÞ and Cð2  nÞ, where n 2 Z, are symmetric with respect to one another relative to the x axis. The x axis itself will now constitute the plot for the class Cð1Þ, which is distinguished from the other classes by the property   pc lx ; ly ; 0 jCð1Þ ¼ 0:5

ð221Þ

To prove Eq. (221), we erect from the geometric centers of the interstitial squares of the model flx ; ly g, which we call the initial model, perpendiculars to

physical properties of fractal structures

157

all its bonds. We continue the perpendiculars to the periphery of the model and connect them as in Fig. 24, forming in this manner a dual square model, whose bonds are in one-two-one correspondence with the bonds of the initial model (the intersection

 of

bonds in Fig.  24 indicates this correspondence). The new model lx ; ly d ly þ 1; lx  1 has the property that       Yd p; lx ; ly ¼ Y p; ly þ 1; lx  1 ¼ 1  Y 1  p; lx ; ly

ð222Þ

that is, the percolation probabilities of mutually dual models are symmetric relative to the point ð0:5; 0:5Þ. A general argument in the proof of the property (221) is as follows. We form with the dual model a configuration starting from a connected (unconnected) configuration of the initial model according to the following principle: An unbroken bond in the initial model transforms into a broken bond corresponding to it and vice versa. Then, the resulting ‘‘symmetric’’ configuration will be disconnected (connected). The formal proof is constructed for the corresponding site models which are subsets of the socalled covering lattice [80]. Since self-dual models are of class Cð1Þ, the property (221) is proved (the exact equality pc ¼ 0:5 is proved simultaneously for bonds on a square lattice). It is obvious that the inverse assertion will also be true: Any model for which pc ¼ 0:5 will belong to class Cð1Þ. The result that P1 ¼ const  lb=n for class Cð1Þ is confirmed because of the result b=n ¼ 0:1041  0:0013 [the average value over the data obtained using the four models with lx ¼ 3; 4; 5, and 9 [24] from Cð1Þ agrees with the exact value 5/48 [81]. Therefore small experimental models, together with models in which the number of sites is  106 or larger, are suitable for calculating the finite-dimensional scaling ratios of the critical exponents (it can be inferred that b=n is only one such ratio). 2.

Appendix. The Probability Functions.

The probability functions Yðp; lx ; ly Þ derived for various nucleating cells are presented below. Unit cell 2
1: Yðp; 1; 2Þ ¼ 2p2 ð1  pÞ3 þ 8p3 ð1  pÞ2 þ 5p4 ð1  pÞ þ p5 ; Unit cell 2
2: Yðp; 2; 2Þ ¼ 3p2 ð1  pÞ6 þ 22p3 ð1  pÞ5 þ 56p4 ð1  pÞ4 þ 54p5 ð1  pÞ3 þ 28p6 ð1  pÞ2 þ 8p7 ð1  pÞ þ p8 ;

vitaly v. novikov

158 Unit cell 2
3:

Yðp; 2; 3Þ ¼ 4p2 ð1  pÞ9 þ 42p3 ð1  pÞ8 þ 178p4 ð1  pÞ7 þ 382p5 ð1  pÞ6 þ 442p6 ð1  pÞ5 þ 328p7 ð1  pÞ4 þ 165p8 ð1  pÞ3 þ 55p9 ð1  pÞ2 þ 11p10 ð1  pÞ þ p11 ; Unit cell 2
4: Yðp; 2; 4Þ ¼ 5p2 ð1  pÞ12 þ 68p3 ð1  pÞ11 þ 398p4 ð1  pÞ10 þ 1298p5 ð1  pÞ9 þ 2575p6 ð1  pÞ8 þ 3288p7 ð1  pÞ7 þ 2977p8 ð1  pÞ6 þ 2000p9 ð1  pÞ5 þ 1001p10 ð1  pÞ4 þ 364p11 ð1  pÞ3 þ 91p12 ð1  pÞ2 þ 14p13 ð1  pÞ þ p14 ; Unit cell 2
5: Yðp; 2; 5Þ ¼ 6p2 ð1  pÞ15 þ 100p3 ð1  pÞ14 þ 743p4 ð1  pÞ13 þ 3225p5 ð1  pÞ12 þ 9036p6 ð1  pÞ11 þ 17220p7 ð1  pÞ10 þ 23402p8 ð1  pÞ9 þ 24084p9 ð1  pÞ8 þ 19416p10 ð1  pÞ7 þ 12374p11 ð1  pÞ6 þ 6188p12 ð1  pÞ5 þ 2380p13 ð1  pÞ4 þ 680p14 ð1  pÞ3 þ 136p15 ð1  pÞ2 þ 17p16 ð1  pÞ þ p17 ; Unit cell 3
1: Yðp; 3; 1Þ ¼ 2p3 ð1  pÞ5 þ 14p4 ð1  pÞ4 þ 34p5 ð1  pÞ3 þ 25p6 ð1  pÞ2 þ 8p7 ð1  pÞ þ p8 ; Unit cell 3
2: Yðp; 3; 2Þ ¼ 3p3 ð1  pÞ10 þ3 8p4 ð1  pÞ9 þ 209p5 ð1  pÞ8 þ 627p6 ð1  pÞ7 þ 1089p7 ð1  pÞ6 þ 1078p8 ð1  pÞ5 þ 677p9 ð1  pÞ4 þ 283p10 ð1  pÞ3 þ 78p11 ð1  pÞ2 þ 13p12 ð1  pÞ þ p13 ; Unit cell 3
3: Yðp; 3; 3Þ ¼ 4p3 ð1  pÞ15 þ 72p4 ð1  pÞ14 þ 594p5 ð1  pÞ13 þ 2936p6 ð1  pÞ12 þ 9582p7 ð1  pÞ11 þ 21470p8 ð1  pÞ10 þ 33494p9 ð1  pÞ9 þ 36774p10 ð1  pÞ8 þ 29642p11 ð1  pÞ7 þ 18119p12 ð1  pÞ6 þ 8514p13 ð1  pÞ5 þ 3057p14 ð1  pÞ4 þ 816p15 ð1  pÞ3 þ 153p16 ð1  pÞ2 þ 18p17 ð1  pÞ þ p18 ;

physical properties of fractal structures

159

Unit cell 3
4: Yðp;3;4Þ ¼ 5p3 ð1  pÞ20 þ 116p4 ð1  pÞ19 þ 1264p5 ð1  pÞ18 þ 8544p6 ð1  pÞ17 þ 39915p7 ð1  pÞ16 þ 135919p8 ð1  pÞ15 þ 346869p9 ð1  pÞ14 þ 672995p10 ð1  pÞ13 þ 1001865p11 ð1  pÞ12 þ 1158842p12 ð1  pÞ11 þ 1064880p13 ð1  pÞ10 þ 793300p14 ð1  pÞ9 þ 485136p15 ð1  pÞ8 þ 244390p16 ð1  pÞ7 þ 100877p17 ð1  pÞ6 þ 33646p18 ð1  pÞ5 þ 8855p19 ð1  pÞ4 þ 1771p20 ð1  pÞ3 þ 253p21 ð1  pÞ2 þ 23p22 ð1  pÞ þ p23 ; Unit cell 4
1: Yðp; 4; 1Þ ¼ 2p4 ð1  pÞ7 þ 20p5 ð1  pÞ6 þ 80p6 ð1  pÞ5 þ 152p7 ð1  pÞ4 þ 123p8 ð1  pÞ3 þ 51p9 ð1  pÞ2 þ 11p10 ð1  pÞ þ p11 ; Unit cell 4
2: Yðp; 4; 2Þ ¼ 3p4 ð1  pÞ14 þ 54p5 ð1  pÞ13 þ 445p6 ð1  pÞ12 þ 2182p7 ð1  pÞ11 þ 6984p8 ð1  pÞ10 þ 15126p9 ð1  pÞ9 þ 22288p10 ð1  pÞ8 þ 22242p11 ð1  pÞ7 þ 15628p12 ð1  pÞ6 þ 7974p13 ð1  pÞ5 þ 2988p14 ð1  pÞ4 þ 812p15 ð1  pÞ3 þ 153p16 ð1  pÞ2 þ 18p17 ð1  pÞ þ p18 ; Unit cell 4
3: Yðp; 4; 3Þ ¼ 4p4 ð1  pÞ21 þ 102p5 ð1  pÞ20 þ 1230p6 ð1  pÞ19 þ 9272p7 ð1  pÞ18 þ 48718p8 ð1  pÞ17 þ 188512p9 ð1  pÞ16 þ 553496p10 ð1  pÞ15 þ 1252416p11 ð1  pÞ14 þ 2198498p12 ð1  pÞ13 þ 3001802p13 ð1  pÞ12 þ 3204984p14 ð1  pÞ11 þ 2715264p15 ð1  pÞ10 þ 1854463p16 ð1  pÞ9 þ 1032857p17 ð1  pÞ8 þ 471428p18 ð1  pÞ7 þ 175870p19 ð1  pÞ6 þ 53028p20 ð1  pÞ5 þ 12646p21 ð1  pÞ4 þ 2300p22 ð1  pÞ3 þ 300p23 ð1  pÞ2 þ 25p24 ð1  pÞ þ p25 ;

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Unit cell 4
4: Yðp; 4; 4Þ ¼ 5p4 ð1  pÞ2 8 þ 164p5 ð1  pÞ2 7 þ 2582p6 ð1  pÞ2 6 þ 25910p7 ð1  pÞ25 þ 185667p8 ð1  pÞ24 þ 1009026p9 ð1  pÞ23 þ 4311522p10 ð1  pÞ22 þ 14818844p11 ð1  pÞ21 þ 41566143p12 ð1  pÞ20 þ 95995718p13 ð1  pÞ19 þ 183464428p14 ð1  pÞ18 þ 291036648p15 ð1  pÞ17 þ 384352578ð1  pÞ16 þ 424714914p17 ð1  pÞ15 þ 395869210p18 ð1  pÞ14 þ 314074078p19 ð1  pÞ13 þ 213777310p20 ð1  pÞ12 þ 125503512p21 ð1  pÞ11 þ 63685924p22 ð1  pÞ10 þ 27896896p23 ð1  pÞ9 þ 10497184p24 ð1  pÞ8 þ3363764p25 ð1  pÞ7 þ906060p26 ð1  pÞ6 þ 201372p27 ð1  pÞ5 þ 35960p28 ð1  pÞ4 þ 4960p29 ð1  pÞ3 þ 496p30 ð1  pÞ2 þ 32p31 ð1  pÞ þ p32

IV.

PHYSICAL PROPERTIES A.

Conductivity

Theoretical investigations of the dielectric properties of inhomogeneous media stem from works published as far back as the 1870s–1930s [82–84]. Based on these investigations, the concept of an effective medium was developed [85]. It is in essence the replacement of an inhomogeneous medium consisting of two composites with conductivities s1 and s2 by a continuous medium with an effective conductivity s. Note that such an approximation is only applicable when the wavelength of the electromagnetic wave interacting with the medium is much greater than the sizes of the inhomogeneities and the spacings between them. The effective-medium method has been sufficiently widely used for the description of the physical properties of inhomogeneous media [85}; however, it does not permit one to predict the behavior of the system at the metal-insulator transition near the percolation threshold [1–4]. An exact and complete description of the effective properties of a composite may be obtained if the detailed distribution of physical fields in all components of the composite is known—for example, based on the concept of a quasi-homogeneous medium [49]. The problem of defining the distribution of physical fields in components of a composite with chaotic structure is rather

physical properties of fractal structures

161

difficult. Hence certain assumptions are made when defining the effective properties of the composite. Now we shall briefly describe the main results of defining the effective conductivity of the composite. The effective conductivity s and effective resistivity r are defined according to the formulae hji ¼ shEi; hEi ¼ rhji

ð223Þ

where sr ¼ 1 and hji; hEi are the flow current density and the corresponding electric field, averaged over the volume V: ððð hji ¼ V 1 jðrÞ dV; hEi ¼ V 1

V ððð

EðrÞ dV

ð224Þ

V

we have locally in a composite jðrÞ ¼ sðrÞEðrÞ; EðrÞ ¼ rðrÞjðrÞ

ð225Þ

where jðrÞ, EðrÞ, sðrÞ and rðrÞ are functions of space. The constitutive relations equations, Eqs. (223–225), may also be applied to other properties–for example, dielectric permittivity and magnetic permeability. Using Eq. (223–225), the effective conductivity s of a two-component composite can be obtained via s ¼ s1 pA1 þ s2 ð1  pÞA2 ; pA1 þ ð1  pÞA2 ¼ 1

ð226Þ

where Ai ði ¼ 1; 2Þ is defined by hEi ðrÞi ¼ Ai hEðrÞi; ððð 1 EðrÞ=dV hEi ðrÞi ¼ Vi

ð227Þ

Vi

where Vi is the volume occupied by the ith component (i ¼ 1; 2), and p is the volume concentration of component 1.

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A similar procedure can be used to define r, namely, r ¼ r1 pB1 þ r2 ð1  pÞB2 ; pB1 þ ð1  pÞB2 ¼ 1

ð228Þ

where Bi is defined as

hji i ¼ Vi1

ððð

hji i ¼ Bi hji; jðrÞ dV ;

i ¼ 1; 2

ð229Þ

Vi

Unfortunately, Eq. (226) and (228) cannot be used for direct calculations of s and r because the number of unknowns—three (s, A1 , A2 in the first case and r, B1 , B2 in the second case)—is more than the number of equations available (two in each case). Thus, it is necessary to have some additional information about the structure of the composite. We consider the simplest structure—that is, parallel layers. When the flow hji is directed along the layer, we have hE1 i ¼ hE2 i ¼ hEi

ð230Þ

Therefore A1 ¼ A2 ¼ 1. From Eq. (226) we obtain sjj ¼ s1 p þ s2 ð1  pÞ

ð231Þ

where sjj is the conductivity parallel to the layers. If the flow hji is directed perpendicular to the layers, that is, hj1 i ¼ hj2 i ¼ hji

ð232Þ

B1 ¼ B2 ¼ 1

ð233Þ

we have

Moreover, from Eq. (228) we obtain r? ¼ r1 p þ r2 ð1  pÞ

ð234Þ

and finally, the conductivity perpendicular to the layers s? s? ¼ ðr? Þ1

ð235Þ

physical properties of fractal structures is



p ð1  pÞ þ s? ¼ s1 s2

163

1 ð236Þ

The effective conductivity of a composite s with random structure lies between the limits s?  s  sjj [87]. Subtracting Eq. (231) from Eq. (236) we obtain sjj  s? ¼

ðs1  s2 Þ2 pð1  pÞ s2 p þ s1 ð1  pÞ

ð237Þ

Generalizing the above result, we have s ¼ hsi  K

ðs1  s2 Þ2 pð1  pÞ s2 p þ s1 ð1  pÞ

ð238Þ

where 0  K  1 is the structurally dependent coefficient, and hsi ¼ sjj . Thus, a knowledge of the structural characteristics of a composite must constitute the first step in the theoretical analysis of its effective conductivity. 1.

Maxwell Model

One of the first models proposed in order to calculate the generalized conductivity of a composite was the Maxwell model, namely a spherical insertion (component 1) in a continuous matrix (component 2). Thus the following equation was obtained [82]: s ¼ s2 þ 2.

ps2 ðs1  s2 Þ ðs2 þ ð1  pÞðs1  s2 Þ=3Þ

ð239Þ

The Effective Medium Theory

The effective medium model has been described in [84,86]. It constitutes an isolated spherical insertion (component 1) in a continuous medium with effective (to be determined) properties. Thus, the following formula was obtained:  1 1 2 2 s0 ¼ s1 f ð x; pÞ; f ð x; pÞ ¼ a þ a þ x ; 2

   1 3 1 1 p  ð1  x Þ þ x ð240Þ a¼ 2 2 2 2 where x ¼ s2 =s1 .

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164 3.

Variational Approach

This method is appropriate in order to estimate the upper and lower limits of the effective conductivity of a composite. The initial estimations were [87] hs1 i1  s  hsi

ð241Þ

Later they were improved [88]: hsi 

pð1  pÞðs1  s2 Þ2 pð1  pÞðs1  s2 Þ2  s  hsi  ps2 þ ð1  pÞs1 þ s1 ps2 þ ð1  pÞs1 þ s2

ð242Þ

The formulae obtained, Eqs. (239)–(242), conform quite well to the experimental data of effective conductivity s if the conductivities of the components (s2 ; s1 ) of the composite are not more than two orders of magnitude differs (102 < s2 =s1 < 1) or at low concentrations of one of the components (e.g., p 1). If the ratio of the properties of the system is s2 =s1 < 102 , then the result yielded by Eqs. (239)–(242) and the experimental data [68,69] are essentially different. If the ratio of the properties of the system is s2 =s1 ! 0, then the results of the theory of percolation can be used to predict the effective conductivity of a composite [1–4] (Section III, A). If the properties of the system are within the range 0 < s2 =s1 < 102 , so far no suitable theory exists which can predict the effective conductivity of a heterogeneous medium. As shown below, an attempt is made to solve this problem using the ideas of the renormalization group transformation method and the theory of fractals, which is also called the geometry of chaos. 4.

Iterative Averaging Method for Conductivity

We consider inhomogeneous media with chaotic structure (Fig 26a). On Fig. 26b an illustration of how such inhomogeneous media may be split into hierarchical levels is presented. A lattice with a random distribution of parameters was chosen as an appropriate model of the chaotic structure of an inhomogeneous material Spatial microinhomogeneities (i.e., system components) were modelled by the lattice junctions, and the interjunction bonds simulated their contacts with neighbors (Fig. 27). Thus, in view of the dominant contribution of contact conditions between the components of the macroscopic properties of an medium, the general problem was again reduced to a problem of bonds. The main ensemble of bonds
was derived by an iteration process in which the initial step (k ¼ 0) involved

physical properties of fractal structures (a)

165

(b) 1st step

2nd step

nth step

Figure 26. (a) A disordered cluster of particles. (b) Illustration of splitting of inhomogeneous media into hierarchical levels.

(a)

(b)

(c)

Figure 27.

A lattice junctions, with interjunction bonds simulating their contacts with neighbors.

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166

(a) 1 0.8

1

3 2

Y(p)

0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

p (b) 2.5

3

f(p)

2

2 1

1.5

1 1

3

0.5 0 0

0.2

0.4

0.6

0.8

1

p Figure 28.

(a) Probability function Yk ðpÞ; (b) Derivative f ðpÞ ¼ dYðpÞ=dp.

treatment of a finite lattice in a space of dimension d ¼ 2 or d ¼ 3 and a probability p0 for a bond between neighboring lattice junctions to be unbroken (or ‘‘colored’’ with a definite color, so that bonds of the same color were assumed to have identical properties). At the next step (k ¼ 1; 2; . . . ; n), each bond of the lattice was replaced by a lattice generated at the previous step (Fig. 12) The eventual independence of the lattice properties from the iteration number n was recognized as the termination of the iteration process. Lattices with linear dimensions l (assumed to exceed by far the correlation length) generated in this way were used to calculate the effective physical properties. It is clear that the ensemble of bonds
n ðl0 , p0 Þ

physical properties of fractal structures

167

derived by the iteration process is a function of both the size of the initial lattice l and the probability p. Three probability functions YðpÞ were taken to illustrate the influence of the function YðpÞ on the calculation of the effective conductivity of a composite s. These functions YðpÞ have different values of p —that is, different thresholds of percolation. This function [62] was obtained in Y1 ðpÞ ¼ p2 ½4 þ 8p  14p2  40p3 þ 16p4 þ 288p5  655p6 þ 672p7  376p8 þ 112p9  14p10 

ð243Þ

According to Y1 ðpÞ ¼ p, the percolation threshold p ¼ pc1 for Y1 ðpÞ is equal to pc1 0:20846 . . ., i.e. the non-connecting sets change into connecting sets at pc1 0:20846 The value of the derivative Y1 ðpÞ at the point p ¼ pc1 is equal to f1 ðpÞjp¼pc ¼ l1 ; l1 ffi 1:9582

ð244Þ

We construct the probability functions YðpÞ using rectangular lattices [79] (see Appendix, Section III.): The function Y2 ðpÞ ¼ 5p2 ð1  pÞ12 þ 68p3 ð1  pÞ11 þ 398p4 ð1  pÞ10 þ 1298p5 ð1  pÞ9 þ 2575p6 ð1  pÞ8 þ 3288p7 ð1  pÞ7 þ 2977p8 ð1  pÞ6 þ 2000p9 ð1  pÞ5 þ 1001p10 ð1  pÞ4 þ 364p11 ð1  pÞ3 þ 91p12 ð1  pÞ2 þ 14p13 ð1  pÞ þ p14

ð245Þ

has an unstable fixed point, that is the percolation threshold is pc2 0:178

ð246Þ

l2 ffi 1:9578

ð247Þ

the derivative at pc2 is

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The function (see Appendix, Section III.) Y3 ðpÞ ¼ 6p2 ð1  pÞ15 þ 100p3 ð1  pÞ14 þ 743p4 ð1  pÞ13 þ 3225p5 ð1  pÞ12 þ 9036p6 ð1  pÞ11 þ 17220p7 ð1  pÞ10 þ 23402p8 ð1  pÞ9 þ 24084p9 ð1  pÞ8 þ 19416p10 ð1  pÞ7 6

5

þ 12374p ð1  pÞ þ 6188p ð1  pÞ þ 2380p ð1  pÞ 11

12

13

ð248Þ

4

þ 680p14 ð1  pÞ3 þ 136p15 ð1  pÞ2 þ 17p16 ð1  pÞ þ p17 has an unstable fixed point; that is, the percolation threshold is pc3 ¼ 0:14850

ð249Þ

l3 ffi 1:9839

ð250Þ

the derivative Y3 ðpÞ at pc3 is

The dependence of the functions Yk ðpÞ and their derivatives fk ðpÞðk ¼ 1; 2; 3) on the concentration p is shown in Fig 28. Now consider a two-phase system with the distribution function ð0Þ

ð0Þ

P0 ðCÞ ¼ ð1  p0 ÞdðC  C2 Þ þ p0 dðC  C1 Þ

ð251Þ

where dðxÞ is the Dirac function, p0 is the probability of a given local area to ð0Þ possess the property C1 (black color), and 1  p0 is the probability to possess ð0Þ the property C2 (white color). After k steps of renormalization group transformation, the density function becomes ðkÞ

ðkÞ

Pk ðCÞ ¼ ð1  pk ÞdðC  C2 Þ þ pk dðC  C1 Þ

ð252Þ

Here pk ¼ Yðpk1 ; lk1 Þ is the density of the connecting set of links; it is the ratio of the number of connecting sets to the total number of scatters (colors) on a square lattice. In general, the effective properties can be defined according to the following scheme: Initially the properties of different configurations are defined; they are then averaged; these properties are then attributed to the next stage. Defining the properties of possible configurations of a set of links leads to rather cumbersome calculations. Hence we use an approximate method, meaning that we do not calculate the properties of the configurations obtained by the scatters of links on

physical properties of fractal structures

169

Figure 29. Simulation of (a, c) a connecting set and (b, d) a nonconnecting set (schematic).

a lattice. We select two kinds of link configuration sets, namely, connecting sets and nonconnecting sets. A cube inside a cube cell is used as a model of connecting sets and nonconnecting sets (Fig 29); that is, the calculations of the structure of connecting sets and nonconnecting sets are formed by the cube inside a cube

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170

cell at each step of the iteration process: The connecting sets constitute a continuous medium from the well-conducting phase (outer cube) with an insertion of a cube from the poorly conducting phase (inner cube); the nonconnecting sets constitute a continuous medium from the poorly conducting phase (outer cube) with an insertion of a cube from the wellconducting phase (inner cube). The conductivity of the cube inside a cube cell was defined in Ref. 49. The result obtained in Ref. 49 for the cell in which the cube with conductivity s2 is in the cube with conductivity s1 can be written as s¼

s1 ðc1 þ c2 Þ 2

ð253Þ

where c1 ¼

s1 þ ðs2  s1 Þð1  pÞ2=3 h i; s1 þ ðs2  s1 Þð1  pÞ2=3 1  ð1  pÞ1=3 1=3

c2 ¼

s2 þ ðs1  s2 Þð1  pÞ

h

1  ð 1  pÞ

2=3

i

ð254Þ

s2 þ ðs1  s2 Þð1  pÞ1=3

Here p is the volume concentration of a phase of conductivity s1. sðckÞ ¼

 sðck1Þ  ðk1Þ ðk1Þ 1 þ 2 2

ð255Þ

where ðk1Þ 1

ðk1Þ

2

  sðck1Þ þ sðnk1Þ  sðck1Þ ð1  pk1 Þ2=3   h i; ¼ ðk1Þ ðk1Þ ðk1Þ sc þ sn  sc ð1  pk1 Þ2=3 1  ð1  pk1 Þ1=3 h i   sðck1Þ þ sðck1Þ  sðnk1Þ ð1  pk1 Þ1=3 1  ð1  pk1 Þ2=3   ¼ ðk1Þ ðk1Þ ðk1Þ sn þ sc  sn ð1  pk1 Þ1=3

ð256Þ

where sðc0Þ ¼ s1 ; sðn0Þ ¼ s2 , and p0 ¼ p. The magnitudes of pk were determined using the formulas (243)–(248). To determine the conductivity of the nonconnecting sets snðkÞ it is necessary   c; 1! 2, and to replace the indices in Eqs. (255) and (256) as follows: n !  ð1  pÞ! p: The lower indices n and c mean that the given value corresponds to: n, nonconnecting set; c, connecting set; the upper index k shows the iteration

physical properties of fractal structures

171

0

Log10(s/s1)

–1

–2

–3

–4

–5 0

0.2

0.4

0.6

0.8

p

Figure 30. Comparison of the calculation of the effective conductivity of a composite using the iterative method (continuous) and using effective medium method – effective medium theory (dotted line).

step number. The iteration procedure leads finally to the effective conductivity of the composite, s: ðkÞ lim sðkÞ c ¼ lim sn ¼ s

k!1

k!1

ð257Þ

The Calculation Results. The calculations were made for a two-component medium. Calculations were executed for a two-component 3D composite with random structure. First we shall consider a comparison of the outcome for the effective conductivity calculated by means of the iterative method with the calculation using formulas (240) obtained on the basis of the effective medium theory model. Figure 30 shows a comparison between the results for effective conductivity obtained by means of the iterative method (continuous) and a calculation using the formula (240) obtained by the effective medium theory model (dashed). The figure compares the results of the calculation of the effective conductivity using the iteration method (the continuous line) to the calculation by formula (240) (the dotted line) obtained from the effective medium model. The comparison

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172 0

Log10(s/s1)

–1

–2

–3

–4

–5 0

0.2

0.4

0.6

0.8

p Figure 31. Comparison of the calculation of the effective conductivity of a composite based on the iterative method (continuous), on the effective medium method (dotted line) and numerical modeling (dots).

(Fig. 30) shows good agreement between the two methods if s2 =s1 > 102 . If s2 =s1  102 , the calculations differ considerably in the concentration range 0:1 < p < 0:5. The difference increase as s2 =s1 ! 0: Figure 31 compares the calculation (for s2 =s1 ¼ 105 ) according to the iteration method (the continuous line), the self-coordinated field method [Eq. (240)] (the dotted line), and numerical modeling (the dots) [68]. In the numerical modeling, the percolation threshold is pc ¼ 0:17  0:01.The function (245), for which the threshold is pc2 ffi 0:178, is used for the calculation. The comparison shows good agreement between the iteration method of calculation (the continuous line) and the numerical experiment (the dots). In Fig. 32, comparison of the results of calculation using the iteration method (the continuous line) with the experimental data is shown (alkali–tungsten bronzes at 300 K, s2 =s1 ¼ 103 [68,69]). The comparison shows good agreement between them. The function Y1 ðpÞ is used (pc1 0:20846). Thus, the iteration method of calculation developed here agrees well with the calculation using self-coordinated field method if s2 =s1 ¼ 102 and to the numerical experiment if s2 =s1 ¼ 104 ðs2 =s1 ! 0Þ. This gives us hope that the iteration method of calculation can be used to estimate the generalized conductivity of composites with chaotic structure for any values of the conductivity of phases s1 and s2 and in the entire range of concentrations 0  p  1.

physical properties of fractal structures

173

0

Log10(s/s1)

–0.5 –1 –1.5 –2 –2.5 –3 0

0.2

0.4

0.6

0.8

1

p

Figure 32. Comparison of the calculation of the effective conductivity of a composite by the iterative method (continuous) and experimental data (dots).

In Fig. 33, the comparison of the calculation of the effective conductivity s with various functions YðpÞ is given. The calculation shows that the values of s closely depend on the type of function YðpÞ (that is on the values of percolation thresholds pc ) at the concentration values 0:1  p  0:5. 0

Log10(s/s1)

–1

–2

–3

3

2

–4

1

–5 0

0.2

0.4

0.6

0.8

1

p

Figure 33. Comparison of the calculation of the effective conductivity of a composite based on the iterative method with various probability functions YðpÞ : 1  Y1 ðpÞ; 2  Y2 ðpÞ; 3  Y3 ðpÞ.

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Chaotic fractal sets on rectangular lattices have been used to the define the effective conductivity of the composite material. The effective conductivity of the composite material is defined using the fractal random structure model of a composite and the iteration method of averaging. Comparison of the calculation with experimental data is also given. B.

Frequency Dependence of Dielectric Properties

If we assume that locally the strength of the electric field Eðr; tÞ varies periodically with frequency o as Eðr; tÞ ¼ E0 ðr; oÞ expðiotÞ

ð258Þ

then the following constitutive relation may be written [90] jðr; oÞ ¼ s ðr; oÞEðr; oÞ

ð259Þ

where jðr; oÞ is the current density and s ðr; oÞ is the complex conductivity: s ðr; oÞ ¼ sðr; oÞ þ ioeðr; oÞ

ð260Þ

For an inhomogeneous medium with a chaotic structure, the permittivity eðr; oÞ and the conductivity sðr; oÞ are random (stochastic) functions of the coordinates r. Note that from Eq. (260) we can determine the scaling expressions for the conductivity of a lattice of resistors of finite dimensions l, first obtained in [91,92]:   s ¼ s1 xt=n Gþ s1 =s2 xðtþsÞ=n ; x=l ; p > 0;   p < 0; ð261Þ s ¼ s2 xs=n G s2 =s1 xðtþsÞ=n ; x=l ; where Gþ ðx; yÞ and G ðx; yÞ are functions of two variables describing the frequency and scale dependence of the conductivity above and below the percolation threshold. In recent years, great attention has been paid to the analysis of the dependence of the properties of metal-insulator composites on frequency [91–109], which is related to the difficulties in describing the anomalous behavior of dielectric properties in the low-frequency limit. The nature of the anomalous behavior of the frequency dependence of the dielectric properties can be clarified if we consider a model medium consisting of small spherical metallic particles described by the Drude dielectric function e1 ðoÞ ¼ 1 

op1 oðo þ i=t1 Þ

ð262Þ

physical properties of fractal structures

175

embedded in a matrix with a relative permittivity equal to unity (e2 ¼ 1). In Eq. (362), op1 is the plasma frequency and t1 is the relaxation time of the metallic phase. If such a medium is subjected to an external uniform electric field E0 [Eq. (258)], then the electric field E1 inside a spherical particle is by quasi electrostatics E1 ¼ 3E0 =ðe1 ðoÞ þ 2Þ

ð263Þ

and so the electric pffiffiffi field E1 inside the sphere tends to infinity at frequencies close to o ’ op1 = 3: At such frequencies, the applied field resonates with the corresponding natural mode of the small metallic particle; as a result, a strong absorption appears at this frequency; that is, the imaginary part of the effective relative permittivity of the medium strongly increases in the vicinity of the frequency pffiffiffi op = 3. For inhomogeneous media—for example, a metal–insulator composite with a chaotic structure—the behavior becomes even more complex. In the majority of numerical calculations of the anomalous frequency behavior of such composites (in particular, near the percolation threshold pc ) under the action of an alternating current, lattice (discrete) models have been used, which were studied in terms of the transfer-matrix method [91,92] combined with the Frank–Lobb algorithm [93]. Numerical calculations and the theoretical analysis of the properties of composites performed in Refs. 91–109 have allowed significant progress in the understanding of this phenomenon; however, the dielectric properties of composites with fractal structures virtually have not been considered in the literature. 1.

Iterative Averaging Method for Dielectric Properties

Each kth bond in the set of bonds
n ðl0 ; p0 Þ (obtained using the iteration procedure) possesses an impedance Zk ðoÞ which consists of an ohmic resistance Rk , an inductance Lk , and a capacitance Ck in parallel so that Zk1 ðoÞ ¼ ðRk þ ioLk Þ1 þioCk

ð264Þ

In what follows, each bond will be characterized by the complex conductivity sk .noting, that the equality sk ¼ Zk1 ðoÞ must be satisfied.

ð265Þ

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Consider a two-phase system with a distribution function     ð0Þ ð0Þ þ p0 d s  s1 P0 ðs Þ ¼ ð1  p0 Þd s  s2

ð266Þ

where dð xÞ is the Dirac delta function, p0 is the probability that a given local ð0Þ region possesses the property s1 ¼ s1, and ð1  p0 Þ is the probability that this ð0Þ region possesses the property s2 ¼ s2. After k iteration steps, the density function becomes     ðkÞ ðkÞ þ pk d s  s1 ð267Þ Pk ðs Þ ¼ ð1  pk Þd s  s2 In what follows, we will again distinguish two types of sets of bond configurations: connecting sets and nonconnecting sets. To determine the dielectric properties of the connecting sets and nonconnecting sets, we used a cell of the cube-in-cube type (Fig. 29c, d); that is, at each step of the iteration process of the calculation of the properties, the structures of the connecting sets and nonconnecting sets were simulated by a cube-in-cube cell as follows: A connecting set comprises a continuous body of a well-conducting phase including a cube of a poorly conducting phase (Fig. 29c); a nonconnecting set comprises, a continuous body of a poorly conducting phase including a cube of a well-conducting phase (Fig. 29d). According to Eqs. (259) and (260), the effective characteristics of a medium in the quasi-stationary approximation differ from the static case only in the replacement of the conductivity s (dc conductivity) by the complex conductivity s Noting Eqs. (255) and (256), the complex conductivity of connecting sets at the kth step of the calculations was determined using the formulas sc ðkÞ ¼

 sc ðk1Þ  ðk1Þ ðk1Þ 1 þ 2 2

ð268Þ

where

ðk1Þ 1

ðk1Þ

2

  sc ðk1Þ þ snðk1Þ  sc ðk1Þ ð1  pk1 Þ2=3   h i; ¼ ðk1Þ ðk1Þ ðk1Þ sc þ sn  sc ð1  pk1 Þ2=3 1  ð1  pk1 Þ1=3 h i   sc ðk1Þ þ sc ðk1Þ  snðk1Þ ð1  pk1 Þ1=3 1  ð1  pk1 Þ2=3   ¼ ðk1Þ ðk1Þ ðk1Þ sn þ sc  sn ð1  pk1 Þ1=3 ð269Þ

physical properties of fractal structures

177

where sc ð0Þ ¼ s1 ; snð0Þ ¼ s2 , and p0 ¼ p. The magnitudes of pk were determined using formula (243). In Eqs. (268) and (269), the subscripts n and c denote that a given quantity refers to the nonconnecting set and connecting set, respectively, and the index k indicates the order number of the iteration step. To determine the complex conductivity of an nonconnecting set snðkÞ it is necessary to replace the indices in Eqs. (268) and (269) as follows:    c; 1! 2, and ð1  pÞ! p: n ! Calculation Results. The calculations were performed for a two-phase (twocomponent) medium and the probability function, YðpÞ [Eq. (243)], was used in the calculations. Calculations of the dielectric properties of inhomogeneous media at various frequencies and concentrations of the phases using Eqs. (243), (268), and (269) then show that the iteration process converges; that is, lim sc ðkÞ ¼ lim snðkÞ ¼ s

k!1

k!1

ð270Þ

However, the complex local conductivity for the metallic phase with Drude dielectric function (262) was determined as   1  s1 ðoÞ ¼ s1 þ io e1  2 ð271Þ x þ g2 where x ¼ o=op ; g ¼ 1=op t1

ð272Þ

The complex local conductivity of the insulating phase was determined as s2 ðoÞ ¼ s2 þ ioe2

ð273Þ

It was assumed in the calculations that 1 ; op t1 ¼ 30; 30 s2 =s1 ¼ 102 ; t1 ¼ 1; 0:001  o=op  1:5 e1 ¼ 1; e2 ¼ 10; g ¼

ð274Þ

Figures 34 and 35 show the dependence of the effective dielectric constant e ¼ Imðs Þ=o and the effective conductivity s ¼ Reðs Þ on the concentration of the metallic phase p and the relative frequency o=op. The zeros of the effective dielectric constant e determine the plasma frequencies of the system— that is, the metal–insulator transition.

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178

It follows from the calculations (Figs. 34 and 35) that at low frequencies, a divergence arises in the effective dielectric constant and in the effective conductivity (a sharp increase in losses). This is explained by the fact that finite clusters of the metallic phase now arise in the system which are separated by thin insulating interlayers. Such structures form a hierarchical self-similar chaotic capacitance net that generates a system of resonance frequencies. (a)

0 –1000 e /e2

0.01

–2000

0.008

–3000 0.006 0 0.2

0.004

w /w p

0.4 p

0.6

0.002

0.8 1

(b)

e /e2

0 –50 –100 –150 –200

0.1 0.08 0.06 w /w p 0.04

0 0.2 0.4 p

0.6

0.02 0.8 1

Figure 34. Variation of the dielectric constant e=e2 of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

physical properties of fractal structures

179

(c)

e /e2

2 1 0 –1 –2

0.5 0.4 0.3

0 0.2 0.4

w /w p

0.2 p

0.6 0.8 1

0.1

(d)

e /e2

1 0.8 0.6 0.4 0.2 0

1.4 1.2 0

1 0.2 0.8

0.4 p

w /w p

0.6 0.6

0.8 1

Figure 34.

(Continued ).

In addition, the frequency dependence of the effective properties is affected by the configurations of the finite clusters [110]. This may be illustrated by considering a pair of inclusions that have the form of a circle with the associated set of discrete frequencies: o21m ¼ o2p tanhðmx0 Þ; o22m ¼ o2p cothðmx0 Þ

ð275Þ

vitaly v. novikov

180 where

m ¼ 1; 2; . . . ;

r þ ðr2  4RÞ1=2 x0 ¼ ln 2R

ð276Þ

r is the spacing between the centers of the circles, and R is the radius of the circles. Thus, if such regions are formed in a composite, they create resonant circuits.

(a)

log10s

1 0 –1 –2

0.01 0.008 0.006

0 0.2

w /w p

0.004

0.4 p 0.6

0.002

0.8 1 (b)

2 1 log10s 0 –1 –2

0.1 0.08 0.06

0 0.2

0.04

0.4 p 0.6

w /w p

0.02

0.8 1

Figure 35.

Variation of the logarithm of the conductivity of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

physical properties of fractal structures

181

(c)

2 1 log10s 0 –1 –2

0.5 0.4 0.3

0

w /w p

0.2 0.2

0.4 p

0.6 0.8 1

0.1

(d)

2 1 log10s 0 –1 –2

1. 4 1. 2 1

0 0. 2

w /w p

0. 8

0. 4 p

0. 6

0. 6

0. 8 1

Figure 35.

(Continued ).

It also follows from Eq. (275) that, at r ! 2R, we have x0 ! 0 and the frequencies (275) form a quasi-continuous spectrum [110]. It was shown in Refs. 91 and 92 that ring-shaped structures (ring clusters) generate double peaks in the frequency dependence of the conductivity. Such ring structures in the system at hand are located chaotically and hierarchically, in a self-similar way, and also lead to peaks in the conductivity. Figure 36 shows the dependence of the modulus of the ratio of the capacitance conductivity to the active conductivity h ¼ jeoj=s on p and o=op . Calculations show that the displacement current in the low-frequency region ðo=op < 1Þ behaves nonmonotonically. In the high-frequency range

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182

30 0.1

20 |∈ω |/s

10

0.08

0 0

0.06 0.2

0.04

0.4 p

0.6

w /w p

0.02

0.8 1

10 7.5 |∈ω |/s 5 2.5 0 0

0.5 0.4 0.3 0.2 0.4

w /w p

0.2 p

0.6 0.8 1

0.1

Figure 36. Variation of the ratio of the capacitance conductivity to the active conductivity h ¼ jeo=sj of a metal–insulator composite as a function of the concentration of the metallic phase p and the frequency o=op.

ðo=op > 1Þ and at concentrations of the metallic phase below the percolation threshold ðp < pc Þ, the displacement current exceeds the current through the active conductors ðh  1Þ and the surface dielectric properties become smooth. For p > pc, the current through the active conductors exceeds the displacement current ðh 1Þ.

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183

Now, we discuss one of the possible applications of the above model concepts to the dielectric properties of fractal systems. The optical properties of colloidal systems have not yet been explained in terms of the classical theory (e.g., in terms of the Mie theory [111–113]). In this theory, the change in the color of a solution was assumed to be due to the appearance of metallic (silver) particles of various sizes in the solution; the change in the color was attributed to the dependence of the resonance (plasma) frequency on the particle radius. However, experimental investigations show that the frequency-dependent behavior of colloidal solutions does not correlate with the statistical particle-size distribution function; that is, the role of the particle size seems to be insignificant [114]. The appearance of a longwavelength wing in the spectrum of the colloidal solution can be explained by the aggregation of particles into fractal structures. Now, a small silver particle has a frequency of plasma vibrations with a wavelength l ¼ 2pc=op ¼ 140 nm. To explain the presence of a peak at 650 nm, the classical (Lorentz) theory [111–113] requires the presence in the colloid solution of silver with a volume concentration of p ’ 0:86 (Fig. 37a), whereas the experiment yields p values that are much smaller [115], which agrees with our calculations (Fig. 37b). Thus, the shift of the peak in colloidal solutions toward the region of smaller concentrations of metal can be explained by the formation of fractal structures in these solutions. We mention some other systems that have fractal structures. For example, using sputtering regimes that correspond to the diffusional aggregation model [82], thin films consisting of metallic fractal clusters can be obtained. Fractal structures are also characteristic of percolation clusters near the percolation threshold, as well as certain binary solutions and polymer solutions. The dielectric properties of all these systems can be predicted using the above fractal model. Conclusion. Calculations of the dependence of the conductivity and the relative permittivity of chaotic hierarchical self-similar structures of composites were performed using a fractal model in the entire range of concentrations of inhomogeneities at various frequencies of an external field. The metal-insulator transition was shown to occur not only near the percolation threshold. It was also shown that the transition depends on the concentration of the metallic phase and the frequency of the external field.

C.

Galvanomagnetic Properties

There has been a number of attempts [116,117] to analyze galvanomagnetic b was properties of inhomogeneous media. The effective conductivity tensor s

184

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Figure 37. Variation of the dielectric constant e=e2 of a metal–insulator colloid solution as a function of the metallic phase (silver) p: (a) calculation based on the Lorentz model; (b) calculation based on the fractal model.

introduced to evaluate the effective galvanomagnetic properties: b hEi hji ¼ s s11 s12 0 b ¼ s12 s22 0 s 0 0 s33

ð277Þ ð278Þ

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185

The angular brackets again mean an average over a volume V: hj i ¼

ð 1 jðrÞd3 r; V

hEi ¼

ð 1 EðrÞd3 r V

ð279Þ

where jðrÞ and EðrÞ are random functions of the coordinates. Suppose that Ohm’s law is locally satisfied so that b ðrÞEðrÞ j ðrÞ ¼ s s11 ðrÞ b ðrÞ ¼ s12 ðrÞ s 0

s12 ðrÞ s22 ðrÞ 0

ð280Þ 0 s33 ðrÞ 0

ð281Þ

b ðrÞ is expressed as follows: The conductivity tensor s b ij ðrÞ ¼ s b sij ðrÞ þ s b aij ðrÞ; s

ð282Þ

b sij ðrÞ is the symmetric part of the tensor (3.59) ðb b sji ðrÞÞ and where s ssij ðrÞ ¼ s b aij ðrÞ is the antisymmetric part of the tensor (3.59). ðb s saij ðrÞ ¼ b saji ðrÞÞ. In field notation Ohm’s law in such a medium is: j þ j
bðrÞ ¼ s 0 ðrÞE

ð283Þ

where bðrÞ ¼ bðr Þn - Hall parameter which is directed along the magnetic field H ¼ Hn. The relations between the Hall coefficient R, the Hall parameter b, the mobility m and the density n of carriers with charge e is R¼

b m 1 ¼ ¼ sH s en

ð284Þ

b ðrÞ according to Eq. (280) and (283) will then The conductivity tensor s read 1 bðrÞ 0 s0 ðrÞ b ðr Þ ¼ ð285Þ s b ð r Þ 1 0 2 1 þ b ðrÞ 0 0 1 þ b2 ðrÞ This tensor satisfactorily describes the conductivity of noncompensated metals and semiconductors.

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186

The first successful attempt to get an exact solution for the effective Hall properties of a composite was made in [118,119]. Various authors acting independently of each other have given a computational method for the effective Hall properties for two-dimensional (2D) two-phase systems with statistically equivalent and isotropic allocations of the first and second phase ðp1 ¼ p2 ¼ 0:5Þ. It was supposed, that each of the phases is characterized by two parameters: the ohmic conductivity s0 ðrÞ and the Hall factor bðrÞ. However each of properties s0 ðrÞ and bðrÞ from the conductivity tensor (285) admit of only two values: s0 ¼ s1 and b ¼ b1 in the first phase, s0 ¼ s2 and b ¼ b2 in the second phase. The essence of ideas described in [118,119] consists in linear transformations from the old fields ðj; EÞ to new fields ðj0 ; E0 Þ such that the macroscopic properties of the new system are equivalent to those of the original system. These transformations can be applied only to a two-dimensional system, since they do not then change the laws governing a direct current: j ¼ bn
E; 0

0 00

0

E ¼ dn
j 00

j ¼aj þbn
E ;

0

0

00

0

E ¼cE þdn
j

00

ð286Þ

The transformations (286) allow one [118,119] to calculate the effective galvanomagnetic properties of a 2D inhomogeneous medium when conductivity fluctuates only, and the Hall factors of the components are equal; that is, s1 6¼ s2 ; b1 ¼ b2 . If we apply complementarity—that is, in the first phase 0 0 ðs1 ; b1 Þ we have s ¼ s2 ; b ¼ b2 , and in the second phase ðs2 ; b2 Þ 0 0 we have s ¼ s1 ; b ¼ b1 —then we shall obtain the following results for the effective Hall properties:


 

   12 1 1 2 þ b1 h s i 1 s ¼ hsi hsi ; s s
 

   12 1 1 2 b ¼ b1 hsi þ b1 hsi 1 s s

ð287Þ

Here the symbol hi again means volume average. The transformations (286) also allow one [118,119] to obtain results for yet another case, namely only the Hall parameter fluctuates; however, the conductivities of the components are equal, that is, s1 ¼ s2 ;

b1 6¼ b2

ð288Þ

In Ref. 120 the solution for the more general case ðs1 6¼ s2 ; b1 6¼ b2 Þ has been obtained. It was given by inserting the additional coefficients into the

physical properties of fractal structures

187

transformation (286): 0

0

j ¼ aj þ bn
E ;

0

0

E ¼ cE þ dn
j

ð289Þ

This transformation then allows one [120] to obtain the following result for the effective Hall properties: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s1 s2 s¼  ; 1 þ ðs1 b2  s2 b1 Þ2 = s21 þ s22 b¼s

ð290Þ

b1 þ b2 s1 þ s2

In the absence of a magnetic field ðb1 ¼ b2 ¼ 0Þ, both Eq. (287) and Eq. (290) lead to the classical result s¼

pffiffiffiffiffiffiffiffiffiffiffi s1 s2

ð291Þ

Numerical computations for a 2D composite were made in [123]. We remark, that a novel perturbation analysis was proposed in [124]. Moreover, a composite with a columnar microstructure was studied in [125]. In Ref. 67 the behavior of Hall coefficient near the percolation threshold pc in a composite containing a dielectric and metal phase was examined for b1 ¼ b2 ;

R1 s2 ¼

1 R2 s1

ð292Þ

Two concentration value regions were considered: before and after the percolation threshold. On the right of the percolation threshold ð p  pc 1Þ the behavior of the Hall coefficient had the following power dependence: Rð pÞ ¼ R1 ð p  pc Þs1 ;

p > pc

ð293Þ

where s1 is a critical exponent of the Hall coefficient. s1 ¼ 0 for a two-dimensional medium ðd ¼ 2Þ and s1 ¼ 0:9 for three-dimensions ðd ¼ 3Þ. On the left of the percolation threshold ð1  pc  p  Þ the behavior of Hall coefficient indicated another type of power dependence: Rð pÞ ¼ R2 ðpc  pÞt1 ;

pc > p

ð294Þ

where t1 is a critical exponent for the Hall coefficient, which equals t1 ’ 1:1; d ¼ 2; t1 ’ 1:62; d ¼ 3 [67].

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188

Reference 70 provides the first quantitative test of the random resistor network model. In Ref. 121 the authors employed the random resistor network model to determine the behavior of the low-field Hall effect in a 3D ‘‘metal–nonmetal’’ composite near the percolation threshold. For the following power laws of effective values of ohmic conductivity s, Hall coefficient R, and Hall conductivity s12, Bergman et al. 121 have obtained the critical exponents: s=s1  ð p  pc Þt ; t ¼ 1:64  0:04;

R=R1  ð p  p c Þ t 1 ; t1 ¼ 0:29  0:05;

. ð1Þ s12 s12  ð p  pc Þt2 ; t2 ¼ 3:0  0:1

p > pc ; ð295Þ

where t2 is the critical exponent for the Hall conductivity s12. Furthermore, the following scaling assumptions have been made [122]: ð1Þ

s12  s12

ð2Þ s12



ð1Þ s12

s2

1; s1

s  s1 ¼ jp  pc jt Fs ðZ Þ; s2  s1

¼ jp  pc jt2 Fs12 ðZ Þ; jp  pc j 1;

Z¼

ð296Þ

s2 =s1 j p  pc jtþs

where Fs12 and Fs are scaling functions with scaling argument Z. A similar approach was applied in Ref. 123 to describe the behavior of three-constituent ‘‘metal–insulator–superconductor’’ composite. Webman and Jortner [68] used the following formulas obtained by combining the formulas of effective medium theory with the formulas of percolation theory to calculate the effective Hall properties of a composite: s0 =s1 ¼ ð1  p=pc Þ1 ; R=R1 ¼ ð y=xÞð1  p=pc Þ2 þ ð3  p=pc Þ2 ð1  xyÞp; m=m1 ¼ yð1  p=pc Þ2 þ xð3  p=pc Þ2 ð1  p=pc Þ1 ð1  xyÞp

ð297Þ ð298Þ

Here the concentration pc is the percolation probability pc ¼ 1=3; m is Hall mobility; x ¼ s2 =s1 is the conductivities ratio; y ¼ m2 =m1 is the mobilities ratio of components. These formulas describe the behavior of effective properties satisfactorily only in the range of concentration 0:4 < p < 1. Pellegrini and Barthelemy [128] studied effective medium theory approximations for linear composite media by means of a path integral formalism. They obtained the following values of the conductivity critical exponents: s ¼ 0; t ¼ 2 in any spatial dimension d  2. Perturbation theory for a 3D composite has been described in Refs. 116, 127, and 129.

physical properties of fractal structures

189

From the above brief review it follows, that an adequate theoretical model of galvanomagnetic properties of composites with random structure still does not exist for the three-dimensional case. We are not aware of any papers where the influence of fractal and random structure of a 3D composite on the effective galvanomagnetic properties has been analyzed. In the following section, the detailed analysis of effective Hall properties of a 3D two-component composite will be carried out based on the fractal structure model and the iterative averaging method. 1.

Iterative Averaging Method for Hall’s Coefficient

Two types of connection set are possible in a lattice, namely, a connecting set that is able to connect via black bonds two opposite sides of a lattice and a nonconnecting set that is unable to connect opposite sides. The probability Y ð pÞ of connecting set formation was calculated as a ratio of connecting set number to the number of all possible configurations. The Probability function Y ð pÞ for a three-dimensional rectangular lattice d ¼ 3; l ¼ 2 was calculated by a method similar to that of Ref. 62 and Y ð pÞ of Eq. (243). Each kth bond from the fractal random set
f ðl; pÞ possesses Hall properties ðsk ; bk Þ namely ohmic conductivity and a Hall parameter. Let us consider a two-phase 3D system with the distribution function similar to [62]     ð0Þ ð0Þ pðsÞ ¼ ð1  pÞd s  s2 þ p d s  s1 ;     ð0Þ ð0Þ pðbÞ ¼ ð1  pÞd b  b2 þ p d b  b1

ð299Þ

where dð xÞ is the Dirac delta function; p — probability that the actual local ð0Þ ð0Þ region possesses the following Hall properties: s1 ¼ s1 ; b1 ¼ b1 ; ð1  pÞ — ð0Þ probability that the actual local region possesses other Hall properties: s2 ¼ ð0Þ s2 ; b2 ¼ b2 . After k iterative steps the distribution function will be   pðsÞ ¼ p d s  sðkÞ ;   pðbÞ ¼ p d b  bðkÞ

ð300Þ

and in the limit of large k we have the desired effective values: lim sðckÞ ¼ lim sðnkÞ ¼ s;

k!1

k!1

lim bðckÞ ¼ lim bðnkÞ ¼ b

k!1

k!1

ð301Þ

vitaly v. novikov

190 2.

Results and Discussion

For the purpose of the calculation of the effective Hall properties of 3D composite the model of a cube inside a cube (see Appendix) was applied. At each step of the iterative process evaluation of the Hall properties of the connecting and nonconnecting set structure was carried out based on a rudimentary cell of a cube inside a cube. A continuous array from the well-conducting (black) phase with a cube from poorly conducting (white) phase inserted forms the connecting set (Fig. 29c), and a continuous array from the poorly conducting (white) phase with a cube from the well-conducting (black) phase inserted forms the nonconnecting set (Fig. 29d). Hence, at the kth iterative step, if lk < x (x is the correlation length) the composite will have a self-similar random structure consisting of unit cells comprised of a cube inside a cube. The analysis of effective conductivity for a 3D composite with a random structure becomes more complicated if H 6¼ 0 because for H ¼ 0 the effective conductivity depends on only two parameters: volume concentration p and conductivity ratio x of the components ð x ¼ s2 =s1 Þ. If H 6¼ 0, however, the effective conductivity depends on four parameters: ð x; pÞ mentioned above, H and the mobility ratio y ¼ m1 =m2 . In order to describe the critical behavior of the galvanomagnetic properties quantitatively, we now define the logarithmic derivatives ws ðpÞ, wR ðpÞ of the conductivity and of Hall‘s coefficient as ws ð p; x; y; HÞ ¼

log10 ½sð p þ p; x; y; HÞ  log10 ½sð p; x; y; HÞ ; log10 ½p þ p  pc   log10 ½p  pc 

log10 ½Rð p þ p; x; y; HÞ  log10 ½Rð p; x; y; HÞ wR ð p; x; y; HÞ ¼ log10 ½ p þ p  pc   log10 ½ p  pc 

ð302Þ

We note again that the galvanomagnetic properties near the percolation threshold are described by the following expressions: s  ð pc  pÞs ;

if ð pc  pÞ > 0;

t

s  ð p  pc Þ ;

if ð p  pc Þ > 0;

t1

R  ð pc  pÞ ; R  ð p  pc Þ

s1

if ð pc  pÞ > 0; ;

if ð p  pc Þ > 0

The critical exponents t; s and t1 ; s1 may be obtained if we find the values of the functions ws ð p; x; y; HÞ and wR ð p; x; y; HÞ in the percolation limit ð p ! pc ;

physical properties of fractal structures

191

x ! 0; y ! 1; xy ! 1Þ; ws ð p; x; y; HÞ; tðH Þ ¼p!p lim c þ0

t1 ðH Þ ¼p!p lim wR ð p; x; y; HÞ; c þ0

x!0 y!1

x!0 y!1

ws ð p; x; y; HÞ; sðH Þ ¼p!p lim c þ0

s1 ðH Þ ¼p!p lim wR ð p; x; y; HÞ c þ0

x!0 y!1

x!0 y!1

In Figs. 38 and 39 the results of calculations for effective conductivity, Hall coefficient and their logarithmic derivatives near the percolation threshold are

(a) –2

3 2.5 2 1.5 1 0.5 0 –0.5

–4 –6 –8 0.15 0.2 0.25 0.3 p

0.15 0.2 0.25 0.3 p

0

4

–5

3

χs (p)

log10 s /s1

(c)

–10 –15

1

–20

0 0.15 0.2 0.25 0.3 p

(e) 0 χs (p)

–5 –10 –15 –20 0.15 0.2 0.25 0.3 p

(d)

2

0.15 0.2 0.25 0.3 p

log10 s /s1

(b)

χs (p)

log10 s /s1

0

5 4 3 2 1 0

(f)

0.15 0.2 0.25 0.3 p

Figure 38. Dependence of the relative effective conductivity (a, c, e) and logarithmic derivative of the effective conductivity (b, d, f) on concentration near the percolation threshold: (a, b) H ! 0; (c, d) H ¼ 105 ; (e, f) H ! 1.

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192

Figure 39. Dependence of the relative Hall coefficient (a, c, e) and the logarithmic derivative of the Hall coefficient (b, d, f) on concentration near the percolation threshold: (a, b) the H ! 0; (c, d) H ¼ 105 ; (e, f) H ! 1.

presented for y ¼ 1010 ; x ¼ 1010, and various values of the field H. Using these results, we have approximately determined the critical exponents: H ! 0;

2:0  t  2:7;

0:85  s  0:9;

1:8  t1  2:2;

5

H ¼ 10 ;

2:6  t  3:0;

1:4  s  1:3;

0  t1  1:5;

H ! 1;

3:0  t  4:0;

1:4  s  1:1;

t 1 ’ s1 ’ 0

1:2  s1  2:2; 0:8  s1  2:0;

The critical exponents t and s have been thoroughly investigated in literature when H ¼ 0. The comparison between various authors 1:6 < t < 2:4;

physical properties of fractal structures

193

Figure 40.

The dependence of the Hall coefficient (a) and the relative effective conductivity (b) on the concentration and on the magnetic field when the Hall factors of components differs essentially, and their conductivities are equal ðx ¼ 1, y ¼ 1010Þ ).

0:7 < s < 0:9 (see Section III) and our results 2:0  t  2:7; 0:85  s  0:9 reveals that our results are slightly exceeded; hence, the results of our calculations in the percolation limit ð p ! pc ; x ! 0; y ! 1; xy ! 1Þ can be considered as the analysis of the qualitative dependence of the effective Hall properties. The accuracy of the effective Hall properties calculation depends on the probability function Y ð pÞ selected, Eqs. (243) and (245), and the actual structure of the inhomogeneous medium.

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194

Note that when 1 > x > 106 ; 1 < y < 106 ð xy ¼ 1Þ, the iteration method agrees closely with various authors (Figs. 32 and 42). Figure 40 shows the results of calculation of the effective relative Hall coefficient Fig. 40a and the relative effective conductivity Fig. 40b when the values of conductivity of components are equal and the mobility in the second component is very small ðm2 ¼ 1010 m1 Þ for various values of the magnetic field H. As the magnetic field increases, the sudden jump in dependence of the Hall coefficient at p ’ 0 departs from the percolation threshold p ¼ pc (Fig. 40a). Furthermore, in the dependence of the effective conductivity on magnetic field a minimum exists near the percolation threshold the depth of which tends to zero as H ! 1 (Fig. 40b). This is caused by the appearance of rotating currents induced by the difference in the Hall coefficients of the components. Certain terms in formula (322) correspond to these rotating currents. Figure 41 shows results of the calculation of the effective relative Hall coefficient for the same parameter values as Fig. 40, but with another value for the parameter y ¼ 0:1. Hence, the Hall coefficient varies smoothly with concentration when H ’ 0; moreover, a jump exists near the percolation threshold when H ! 1 (Fig. 41). The dependence of the effective conductivity is roughly the same as that shown in Fig. 40. Fluctuations in the dependence of the effective Hall coefficient (Fig. 41a) exist when the value of Hall coefficient in second component is about R2 ’ 105 R1 if H ’ 0 at p < pc . In the range p > pc the effective Hall coefficient decreases steadily to the value R1 . For H ¼ 104 the effective Hall coefficient decreases steadily from R2 to R1 In the range 0 < p < pc . For p > pc the effective Hall coefficient is equal to R1 and does not depend on the concentration. For the parameters given the effective conductivity is practically independent of the magnetic field H. Discrepancies in the range p > pc are unimportant. Figure 42 shows the comparison between the calculation of the effective Hall coefficient and the results of two experiments [69,130]. Fractal Properties. Figure 43 shows the dependence of the effective Hall properties of a composite on the scale (the number of iterations which according to Eq. (196) is equal to n ¼ lnlnLln  1) near the percolation threshold pc . Note that to our knowledge such a dependence is presented for the first time. According to our calculations (Fig. 43), the dependence of the effective Hall properties of a composite on the scale can be divided into two ranges. On scales n < 5 (Fig. 43) the composite exhibits the properties of a fractal object with the characteristic power law dependence of such properties (resistivity r and Hall coefficient R) on the scale: R  ðLn Þa2 ; r  ðLn Þa1 ;

a2 ’ 1:92; a1 ’ 3:0;

n < 5; n < 5;

H ! 0; H!0

ð303Þ

physical properties of fractal structures

195

0 –0.2 log10 R/R1

–0.4 –0.6 –0.8 –1 –6

1 0.8 0.6 0.4

–4 (a)

log10 H

–2

p

0.2 0 2 0

0 –2 log10 s/s1

–6 0 (b)

2

–4 0 –2 0. 2

–log10 H

–4

0. 4 p 0. 6 0. 8 1

–6

Figure 41. Dependence of the relative effective Hall coefficient on the concentration at x ¼ 1; y ¼ 0:1 for various values of the magnetic field H.

On scales n > 5 (Fig. 43) the Hall properties do not depend on the scale; that is, Euclidean geometry prevails. Here the composite can be described as a quasihomogeneous (‘‘gray’’) medium, whose properties correspond to effective values of properties. When n ’ 5, a transformation between a fractal and a quasihomogeneous mode of behavior of Hall properties exists. In other words, the scale nx ¼ 5 determines the correlation length x.

vitaly v. novikov

196 500 400

R

300 200 100 0 0

0.2

(a)

0.4

0.6

0.8

1

p

50 40

R

30 20 10 0 0.2 (b)

0.4

0.6

0.8

1

p

Figure 42. Comparison between calculations and experiments for the dependencies of the effective Hall coefficient of the composites Bi–Cd (a) and for Nax WO3 (b) on the concentration of a phase of one of components.

Note that the accuracy of calculation by the iterative method for the Hall properties of a 3D composite with chaotic structure depends on the accuracy of the calculation of the probability function Y ð pÞ (probability that the point belongs to a connecting set) which describes the random structure of a composite). A rough approximation is the computation of the Hall properties of a connecting and a nonconnecting set based on a cube inside a cube cell by formulas from the Appendix. The results of our calculations can be considered as an analysis of the qualitative dependence of the effective Hall properties

physical properties of fractal structures

197

17 16

log10 R/R1

15 14 13 12 11 (a)

1

2

3

5

nx

6

7

n

–4 –6 –8 –10 –12 –14 –16 (b)

2

4

nx

6

8

n

Figure 43. Dependence of the relative effective Hall coefficient (a) and the relative effective conductivity (b) on the scale.

(Figs. 38–43). In the future we shall calculate the probability function Y ð pÞ and the Hall properties of a connecting and nonconnecting set more accurately. Conclusion. The iterative averaging method allows us to study the Hall properties of the composite over a large range of different parameters: concentrations, conductivities, Hall coefficients of components and a magnetic field.

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A number of interesting results have been obtained (due to our fractal model of structure and the iterative averaging method) for the Hall properties of the composite; for example, use of a logarithmic derivative allows one to obtain critical exponents for the effective Hall coefficient (Fig. 39) for various values of the magnetic field H. When s1 ¼ s2 (Fig. 40) the effective conductivity is a constant if H ¼ 0 and tends to zero if H ! 1 near the percolation threshold. On the left of the percolation threshold ð p < pc Þ the rise in the Hall coefficient is more rapid as the magnetic field increases (Fig. 40). On the right of the percolation threshold ð p > pc Þ the Hall coefficient is practically independent of the concentration p. The iterative averaging method allows one to obtain the specific dependence of the effective Hall properties on the scale (number of iteration steps). This dependence yields information about the geometry prevailing at a given scale. The transformation to the regime of Euclidean geometry (quasihomogeneous medium) occurs on a characteristic scale x where the logarithm of a property ceases to depend on the scale. Note that the scale x, as well as the dependence of the effective Hall properties, is a multiparametric dependency. Our general aims in future publications are a more accurate calculation of the probability function Y ð pÞ and a less crude approximation for the elementary cell.

3.

Appendix. Galvanomagnetic Properties of the Cube Inside a Cube Cell.

Suppose, that the magnetic field H is directed vertically (along the axis Ox3 ) and the current h j1 i is directed horizontally along Ox1 , as shown in Fig. 44. Take the size of the outer cube as 1, and the size of the inner cube as d. The volume concentration and the size of the inner cube are now related by pffiffiffi the formula: d ¼ 3 p. Ohm’s law then becomes bE j¼s

ð304Þ

where 0

s11 s ¼ @ s12 0

s12 s22 0

1 0 0 A; s33

0

1 E1 E ¼ @ E2 A; E3

0

1 j1 j ¼ @ j2 A j3

We can express this equation as E¼b rj

ð305Þ

physical properties of fractal structures

199

Ox3

H

Ox1 Ox2 <j1>

Figure 44. The cube inside a cube model.

where

0

r11 r ¼ @ r12 0

1 0 0 A r33

r12 r22 0

ð306Þ

Now we carry out a conventional partition of the cube inside a cube cell into layered structures so that it is possible to compose the cell from them. Next we calculate the Hall properties of the cube inside a cube cell approximately by means of step-by-step averaging of the Hall properties of layered structures. Hence, the approximate evaluation of the Hall properties of the cube inside a cube cell is reduced to the evaluation of the properties of a layered medium (see Fig. 44). Galvanomagnetic Properties of Layered Structures. We shall consider three different cases of orientation for the magnetic field H, current h j1 i and direction of layers (see Fig. 44). orientation a. The layers are parallel to the current h j1 i and perpendicular to the field H. According to Fig. 45a the currents and fields obey the conditions D E D E9   ð1Þ ð2Þ j1 ¼ p j1 þ ð1  pÞ j1 > > > > > >   D ð1Þ E D ð2Þ E > > = E1 ¼ E1 ¼ E1 D E D E ð307Þ   ð1Þ ð2Þ > j2 ¼ p j2 þ ð1  pÞ j2 > > > > > >   D ð1Þ E D ð2Þ E > ; E ¼ E ¼ E 2

2

2

vitaly v. novikov

200 (a)

(b)

(c)

j1

Figure 45.

Layered structure of a cube inside a cube cell.

where the angular brackets hi mean as usual the space average of a function f and the upper index indicates the material of the layer (first or second component): ð ð D E 1 1 3 ðiÞ f ðrÞd r; f f ðrÞd3 r ð308Þ ¼ hfi ¼ V Vi V

Vi

According to the conditions (307), we shall express the longitudinal current hj1 i and the Hall current hj2 i in the first component as follows:   ð1Þ h j1 i ¼ w1 j1 ; ð309Þ   ð1Þ h j2 i ¼ w2 j1 where 

   ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ pr11 þ ð1  pÞr11 r11 þ pr12 þ ð1  pÞr12 r12 w1 ¼ ;     ð2Þ ð1Þ 2 ð2Þ ð1Þ 2 pr11 þ ð1  pÞr11 þ pr12 þ ð1  pÞr12     ð2Þ ð1Þ ð2Þ ð2Þ ð1Þ ð2Þ pr11 þ ð1  pÞr11 r12 þ pr12 þ ð1  pÞr12 r11 w1 ¼     ð2Þ ð1Þ 2 ð2Þ ð1Þ 2 pr11 þ ð1  pÞr11 þ pr12 þ ð1  pÞr12

ð310Þ

It follows from the conditions (307) that the expression for the Hall field hE2 i in case (a) is D E D E   ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ hE2 i ¼ r12 j1 þ r11 j2 ¼ r12 w1 þ r11 w2 h j1 i

ð311Þ

physical properties of fractal structures

201

ðaÞ

The expression in parentheses is the off-diagonal element r12 of the tensor b r for the case (a): ðaÞ

ð1Þ

ð1Þ

r12 ¼ r12 w1 þ r11 w2

ð312Þ

It follows from the conditions (307) and from the law (309) that the longitudinal field hE1 i in case (a) is D E D E   ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ ð1Þ hE1 i ¼ r11 j1  r12 j2 ¼ r11 w1  r12 w2 h j1 i

ð313Þ

ðaÞ

The expression in parentheses is the diagonal element r11 of the tensor b r for case (a): ðaÞ

ð1Þ

ð1Þ

r11 ¼ r11 w1  r12 w2

ð314Þ

orientation b. Now the layers are parallel to the current h j1 i and parallel to the field H. According to Fig. 45b the currents and fields obey the conditions D E D E ðaÞ ð2Þ hj 1 i ¼ j 1 ¼ j 1 D E D E ðaÞ ð2Þ hE1 i ¼ p E1 þ ð1  pÞ E1 D E D E ð315Þ ðaÞ ð2Þ h j 2 i ¼ p j 2 þ ð 1  pÞ j 2 D E D E ðaÞ ð2Þ hE2 i ¼ E2 ¼ E2 From now on the alphabetic upper indexes specify that the fields and currents belong to the corresponding region. As we can see in Fig. 44, the region (a) in case (b) is just a layer with the averaged properties (304) and (315). According to the conditions (315), we can express the Hall current hj2 i in layer (a) as follows: D E ðaÞ j2 ¼ w3 hj1 i ð316Þ where   ð2Þ ðaÞ ð1  pÞ r12  r12   w3 ¼ ðaÞ ð2Þ p r11 þ r11

ð317Þ

vitaly v. novikov

202

It follows from conditions (315) and from law (316) that the Hall field hE2 i in case (b) is D E D E   ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ ðaÞ hE2 i ¼ r12 j1 þ r11 j2 ¼ r12 þ r11 w3 hj1 i

ð318Þ ðaÞ

The expression in parentheses in Eq. (318) is the offdiagonal element r12 of the tensor b r for case (b): ðbÞ

ðaÞ

ðaÞ

r12 ¼ r12 þ r11 w3

ð319Þ

It follows from the conditions Eq. (304) that the expression for the longitudinal field hE1 i in case (b) is D E D E ð aÞ ð2Þ hE1 i ¼ p E1 þ ð1  pÞ E1  D E  D E ð aÞ ðaÞ ðaÞ ð2Þ ð2Þ ð2Þ ¼ p r11 hj1 i  r12 j2 þ ð1  pÞ r11 hj1 i  r12 j2

ð320Þ

According to the conditions (320) we can express the Hall currents as D

ðaÞ

E

j2 D

ð2Þ j2

E

ðbÞ

ðaÞ

¼

r12  r12

¼

r12  r12

ðaÞ

r11 ðbÞ

hj1 i; ð321Þ

ð2Þ

ð2Þ

r11

hj1 i

It follows from Eqs. (315) that hE1 i ¼ ð:::Þhj1 i and that the diagonal element ð bÞ r for case (b) is r11 of tensor b " ðbÞ r11

¼p

ðaÞ r11



ð aÞ r12

ðbÞ

ðaÞ

r12  r12 ðaÞ

r11

#

" þ ð 1  pÞ

ð2Þ r11



ð2Þ r12

ðbÞ

ð2Þ

r12  r12 ð2Þ

r11

# ð322Þ

Since r12 ¼ RH, from Eq.(322) it follows that the ohmic resistivity ð bÞ r11  H 2 . Hence the existence of a field dependence is manifested by a difference between the Hall coefficients ðR1  R2 Þ of the respective components. If this difference is not equal to zero ðR1 6¼ R2 Þ, in a given composite rotational Hall currents exist. These currents yield the contribution to the ohmic resistivity (and to the ohmic conductivity, respectively), which depends on the magnetic field H. Hence, if there are even small inhomogeneities in the substance, then saturation ðr11 ! r1 ; H ! 1Þ does not appear for ohmic resistance with increase of the magnetic field.

physical properties of fractal structures

203

orientation c. Here the layers are parallel to the current hj1 i and perpendicular to the field H. According to Fig. 45c, the currents and fields satisfy the conditions D E D E ðbÞ ð2Þ hj1 i ¼ p j1 þ ð1  pÞ j1 D E D E ðbÞ ð2Þ hE1 i ¼ E1 ¼ E1 D E D E ðbÞ ð2Þ hj 2 i ¼ j 2 ¼ j 2 D E D E ðbÞ ð2Þ hE2 i ¼ p E2 þ ð1  pÞ E2

ð323Þ

According to the conditions (323), we can express the longitudinal current hj1 iin the layer ‘‘b’’ as D E ðbÞ j1 ¼ w4 hj1 i

ð324Þ

where ð2Þ

w4 ¼

r11

ðbÞ

ð2Þ

ð1  pÞr11 þ pr11

ð325Þ

It follows from the conditions (323) and from the expression (324) that the expression for the Hall field hE2 i in case (c) is D E ðbÞ ðbÞ ð2Þ hE2 i ¼ pr12 j1 þ ð1  pÞr12 ; D E   ð2Þ ðbÞ ð2Þ j1 ¼ w4 pr12 þ w4 ð1  pÞr12 hj1 i

ð326Þ

The expression in large parentheses in Eqs. (326) is the off-diagonal ðaÞ element r12 of the tensor b r for case (c)—that is, for a complete cube inside a cube cell: h i ðcÞ ðbÞ ð2Þ r12 ¼ w4 pr12 þ ð1  pÞr12 ð327Þ It again follows from the conditions (323) that the expression for the longitudinal field hE1 i in case (c) is D E ðbÞ ðbÞ ðbÞ ðbÞ hE1 i ¼ r11 hj1 i  r12 j2 ¼ r11 w4 hj1 i

ð328Þ

vitaly v. novikov

204 that is, ð cÞ

ðbÞ

ð2Þ

r11

ðbÞ

r11 ¼ r11 w4 ¼ r11

ðbÞ

ð2Þ

ð1  pÞr11 þ pr11

ð329Þ

Note that the case shown in Fig. 44 corresponds to a nonconnecting set structure, and the expressions (327) and (329) describe the average properties of a nonconnecting set structure. To obtain similar expressions for a connecting set structure, it is necessary in all the calculations of the Appendix to consistently make the change of variables: ðiÞ

ð jÞ

r11 ! r11 ;

ðiÞ

ð jÞ

r12 ! r12 ;

p ! ð1  pÞ;

i; j ¼ 1; 2

ð330Þ

Thus we shall obtain expressions similar to (327) and (329); however, they pertain to the connecting set structure. These expressions are also used for the calculation of the effective Hall properties of 3D composites. D. Elastic Properties The elastic properties of inhomogeneous media with chaotic structure can be deduced using an iterative procedure similar to that used to define effective conductivity made in the previous section. The effective elastic properties (the bulk modulus K and the shear modulus m) of the connecting set and nonconnecting set may be calculated by using standard formulas from the physics of composite materials (e.g., Hashin– Strikman formulae [133, 134]) accounting for the tensor nature of elastic properties (Fig. 46). In order to calculate the elastic properties of fractal structures according to the iterative procedure the Hashin–Strikman double-sided estimation of the elastic properties can be used [131], namely, pðK1  K2 Þ ð1  pÞðK2  K1 Þ  K  K1 þ ; 1 þ ð1  pÞa2 ðK1  K2 Þ 1 þ pa2 ðK2  K1 Þ pðm1  m2 Þ ð1  pÞðm2  m1 Þ m2 þ  m  m1 þ 1 þ ð1  pÞb2 ðm1  m2 Þ 1 þ pb2 ðm2  m1 Þ

K2 þ

ð331Þ

where ai ¼

3 ; 3Ki þ 4mi

bi ¼

6ðKi þ 2mi Þ 5Ki ð3Ki þ 4mi Þ

ð332Þ

physical properties of fractal structures (a)

(b)

A

A

Figure 46.

205

B

C

B

C

D

D

Blob models of (a) the connecting set and (b) nonconnecting set.

The Hashin–Strikman formulae were obtained by using a variational method to determine the upper Kc ; mc and the lower Kn, mn bounds of the effective elastic properties for an inhomogeneous medium [131]. The upper bound Kc ; mc corresponds to a composite structure in which spherical inclusions with elastic constants K2 , m2 are placed in a matrix of elastic constants K1 , m1; in the following, it is assumed that K1 > K2 , m1 > m2 . The lower bound Kn , mn is obtained when the components are permuted—that is, when the matrix is described by K2, m2 and the spherical inclusions are described by K1, m1 . From the ‘‘Hashin–Strikman spheres’’ (where inside a sphere of one material a sphere of the other material is placed centrally) one can form a composite as follows [131–134]: Spheres of various sizes, down to infinitesimally small, are taken and the space V is densely packed by them so that vacancies do not occur. Only one condition is required: In each Hashin–Strikman sphere the volume concentrations of both the components must be the same; that is, all the Hashin– Strikman spheres must exhibit the same elastic properties. Such a composite will be further referred to as the ‘‘Hashin–Strikman composite.’’ The elastic properties of the Hashin–Strikman composite are described by the formulae that are obtained from the exactly solvable model of a single spherical inclusion of one phase in an infinite matrix of the second phase and depend only on the volume concentrations and elastic properties of the constituent phases. The properties of the Hashin–Strikman composite do not depend on the scale chosen. 1.

Iterative Averaging Method for Elastic Properties

The configuration elastic properties corresponding to the connecting set and nonconnecting set were calculated by applying the Hashin–Strikman formulae.

vitaly v. novikov

206

The relation connecting the modulii Kc and mc sets at step i þ 1 according to Eqs. (331) and (332) are ðiÞ

mðiþ1Þ c

¼

mðiÞ c

ðiÞ

ð1  pi ÞðKn  Kc Þ

Kcðiþ1Þ ¼ KcðiÞ þ

ðiÞ

ðiÞ

ðiÞ

1 þ pi ac ðKn  Kc Þ ðiÞ

þ

; ð333Þ

ðiÞ

ð1  pi Þðmn  mc Þ ðiÞ

ðiÞ

ðiÞ

1 þ pi bc ðmn  mc Þ

where aðiÞ c ¼

3

; ðiÞ ðiÞ 3Kc þ 4mc

bðiÞ c ¼

ðiÞ

ðiÞ

6ðKc þ 2mc Þ ðiÞ

ðiÞ

ðiÞ

5Kc ð3Kc þ 4mc Þ

ð334Þ

Kc0 ¼ K1 , m0c ¼ m1 denote the complex bulk modulus and the complex shear modulus of the first phase of the inhomogeneous medium, and Kn0 ¼ K2 and m0n ¼ m2 - denote the complex bulk modulus and the complex shear modulus of the second phase, respectively (K1 , K2 ; m1 , m2 are elastic properties for nonhomogeneous media phases). For nonbonded configurations the viscoelastic ðiþ1Þ ðiþ1Þ and mn are described by the formulae which result from the modulii Kn following replacements c !n and pi !ð1  pi Þ. According to the iterative procedure we have ðkÞ

ðkÞ

ðkÞ mc

ðkÞ mn ;

Kc  K  Kn ; m

2.

lim ðkÞ lim ðkÞ k!1 Kc ¼ k!1 Kn ¼ lim ðkÞ lim ðkÞ k!1 mc ¼ k!1 mn ¼ m

K;

ð335Þ

Results of Calculation

The calculations were made for a two-component, inhomogeneous medium: K1 ; m1 -----first component;

K2 ; m2 -----second component

Apparently from the plots of log10 K (Fig. 47a) and log10 m (Fig. 47b) of the fractal ensemble versus the iteration step, number n, all these elastic properties behave like fractals before an eventual levelling off. The latter is obviously associated with the upper limit of fractal-like asymptotics, above which the elastic properties of a system are no longer m dependent on the scale—that is, on the iteration number (the loss of the self-similarity property occurs at iteration step nx ¼ logx=logl0 which defines the correlation length x at the given concentration, p).

physical properties of fractal structures 0

207

log10K

(a)

3 2 1

0

2.5

5

7.5

10

nx

n

15

0

log10 µ

(b)

3 2 1 0

2.5

5

7.5

10 nx

15

n

Figure 47. Semilogarithmic dependence of the shear elasticity modulus (a) and the bulk elasticity modulus (b) on the iteration number n for p ¼ 0:2088 (1), 0.2092 (2), and 0.2098 (3).

In order to describe the critical behavior of the inhomogeneous medium, we now define the logarithmic derivative wðpÞ of the elasticity modulus (Fig. 48) as wðpÞ ¼

log10 ½ðKðp þ pÞÞ=KðpÞ log10 ½ðp þ p  pc Þ=ðp  pc Þ

ð336Þ

Using the function wðpÞ we have determined the indexes t, S: for the elastic region we have t ¼ lim wðpÞ p!pc þ0

ð337Þ

for the highly elastic region we have S ¼ lim wðpÞ p!pc 0

ð338Þ

vitaly v. novikov

208

Figure 48.

The result of calculation of the function wðpÞ.

The value 3:200  0:002 obtained for the critical index t describing the singular behavior of the bulk modulus K in the vicinity of the critical point pc þ 0 is about 15% smaller than that obtained for d ¼ 3 by Sahimi and Arbabi [135]: t ¼ 3:75  0:11. We remark that for d ¼ 2, Zabolitzky et al. [136] obtained t ¼ 3:96  0:04; the results obtained in Refs. 135 and 136 are in good agreement with the relation proposed by Sahimi [137]: t ¼ t þ 2v, where t is the critical conductivity exponent of percolation networks. The values of the critical index S for the superelastic regime (p < pc ) were determined by using a very precise calculation (of a few hundred digits of accuracy) for K1 =m1 ¼ K2 =m2 ¼ 5 with the requirement that K2 =K1 ! 0 and m2 =m1 ! 0; in all cases we obtained S ¼ 0:62962  0:00002, which is in excellent agreement with the result obtained for d ¼ 3 by Sahimi and Arbabi [135,138] (S ¼ 0:65); for d ¼ 2 the most reliable estimate of 1:24  0:03 [135,138]. In the vicinity of the percolation threshold, the ratio of the bulk modulus to the shear modulus K=m tends to a constant. We found that K=m ¼ 1:33 (Fig. 49). This result is in agreement with [139], where a theoretical proof of the equality K=m ¼ 4=d in the limit p ! pc is given. The calculations of elastic properties according to the iteration procedure allow one to conclude that these calculations essentially agree with (a) the results of the percolation theory at K2 =K1 ! 0 (m2 =m1 ! 0Þ and (b) the results according to effective medium field formulae at K2 =K1 > 102 (Fig. 50).

physical properties of fractal structures

209

5 4.5 4

5

K/ µ

3.5 3 4 2.5 3 2

2

1.5

1 0

0.2

0.4

0.6

0.8

1

p

Figure 49. Dependence on p of the ratio K=m for K1 =m1 ¼ K2 =m2 ¼ 5. The ratio K2 =K1 : 1,1010 ;

2,108 ; 3,106 ; 4,104 ; 5,101 .

As an example of comparison of theory and experiment the shear modulus dependence on volume concentration of phases for the system polybutadiene– polystyrene [140] is presented (Fig. 51). E.

Negative Poisson’s Ratio

For a long time it was believed that no isotropic material exists in nature with Poisson’s ratio less than zero [141]. Hence, various models exhibiting negative Poisson’s ratio [142–144] were merely considered as mathematical curiosities rather than bearing any relation to reality and, therefore, were regarded as having no practical importance. After foams with negative Poisson’s ratio were manufactured by Lakes [144], and expanded polytetrauoroethylene with limiting Poisson’s ratio by Evans and co-workers [145,146], increasing interest in studies of systems exhibiting such an unusual property has existed [147–155]. Various potential applications of such systems [156,157] encourage one to study new mechanisms which may lead to negative Poisson’s ratio [158,159]. In this work, we describe the results of theoretical studies of Poisson’s ratio in disordered structures composed of two phases of disparate elastic properties applying a renormalization group approach to a model of percolation on a hierarchical cubic lattice. Although this approach has been described in detail elsewhere [160], we present it briefly here, for completeness. At the percolation

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210

(a) 0

log01 K/K1

–1

–2

–3

–4

–5

0

0.2

0.4

0.6

0.8

1

0.6

0.8

1

p (b) 0

log01 µ/µ1

–1

–2

–3

–4

–5 0

0.2

0.4 p

Figure 50. Comparison of the calculation elastic properties of the according to the iteration procedure (continuous) and according to formulae effective medium field (dotted line).

threshold, the Poisson’s ratio we obtain is in agreement with the computer simulation results and the conjecture of Arbabi and Sahimi [161]. When an isotropic body is affected from outside, it can be characterized by two parameters [141]. The ability of a body to resist volume changes is defined

physical properties of fractal structures

211

1

Log10 m/m1

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

p Figure 51. Dependence of shear elastic modulus on the volume concentration of block copolymers of styrene and butadiene (filled circles) [140]. The solid line represents the calculation results.

by the uniform compression modulus K; its ability to resist changes of shape by the shear modulus m. The elastic properties of a body in axial tension can be characterized using Poisson’s ratio np , which is defined as the negative ratio of the transverse strain to the longitudinal strain. Poisson’s ratio np depends on K and m as follows [141]: For a three-dimensional isotropic body (d ¼ 3) np ¼

3K  2m 3  2x ¼ 6K þ 4m 6 þ 4x

ð339Þ

For a two-dimensional isotropic body (d ¼ 2) np ¼

Km 1x ¼ Kþm 1þx

ð340Þ

where x ¼ m=K. For integer d, Poisson’s ratio of a d-dimensional isotropic medium depends on the space dimension d as [162] np ¼

Kd  2m d  2x ¼ dðd  1ÞK þ 2m dðd  1Þ þ 2x

ð341Þ

vitaly v. novikov

212

Whence we see that the Poisson’s ratio can vary within the limits 1 < np


d 2

ð343Þ

For typical materials with d ¼ 2; 3, the condition is not fulfilled; that is, for typical materials np > 0. According to (314), a material with a negative Poisson’s coefficient can be obtained either if it is very rigid (i.e. if its shear modulus fulfills the above condition) or by forming a structure with dimension less than 2m=K (i.e., d < 2m=K) or by combining the first and the second methods. Change of dimension of the system is impossible for continuum structures; however, it is possible in fractal structures [163]. To illustrate this idea, we will first recall the results obtained for a planar lattice (d ¼ 2) with coordination number Z ¼ 3, bulk modulus K and shear modulus m, which are [164–169] pffiffiffi h 2 3hg ð344Þ K ¼ pffiffiffi ; m¼ h þ 6g 2 3 where h and g are parameters of the lattice, which is determined by the properties of the bonds. This results in Poisson’s ratio being given by vp ¼

Km h=g  6 ¼ ; K þ m h=g þ 18

1   np  1 3

ð345Þ

By the properties of bonds h and (h=g < 6), a Poisson’s ratio of  13 < np < 1 can be obtained. If such configurations constitute the principal contribution to the microscopic properties of the system, then this can lead to negative Poisson’s ratio. Numerical studies [167] of planar (d ¼ 2) elastic random percolation networks have shown that if their linear dimension L < 0:2x (x is the correlation length), then Poisson’s ratio for the system is negative, and if L > 0:2x, Poisson’s ratio is positive In this case, if L=x ! 1, the limiting value of Poisson’s ratio is np ¼ 0:08  0:04 and is a universal constant; that is, it does not depend on the relative values of the local elastic characteristics: If

physical properties of fractal structures

213

L=x ! 1, then np ¼  13; if L=x ¼ 5, then np ¼ 0. Kantor and Webman [165] have proposed the following relation to describe the influence of the structure on Poisson’s ratio: m Z ¼ K 8

ð346Þ

where Z is the coordination number of the percolation lattice. If we assume that Poisson’s ratio for a d-dimensional isotropic inhomogeneous medium is given by [143] np ¼

Kd  2m db ¼ ; dðd  1ÞK þ 2m dðd  1Þ þ b

b ¼ 2x

ð347Þ

we have that if b > d, then np < 0(np ¼ 1 in the limit b=d ! 1); if b ¼ d, then np ¼ 0; if b < d, then the maximum value of np is equal to (1=d  1) (in the limit b=d ! 1). Now, if we substitute relation (346) into Eq. (347), we can write np ¼

b  Z=4 ; dðd  1Þ þ Z=4

b ¼ 2x

ð348Þ

Hence it follows that the Poisson coefficient at the percolation threshold is positive (np > 0) if Z < 4d, equal to zero (np ¼ 0) if Z ¼ 4d, and negative (np < 0) if Z > 4d. It is also necessary to mention that certain configurations of bonds (local regions) can possess unusual properties—in particular, a negative Poisson coefficient np < 0. Thus, for example, the chain of bonds shown in Fig. 52, when stretched out, not only lengthens but also thickens. If such a configuration constitutes the principal contribution to the macroscopic properties of the system, then the Poisson coefficient may be negative. Thus we can conclude that by making an appropriate choice for the structure of a random medium representing the inhomogeneous medium (the coordination number Z), we can obtain a material with a negative Poisson coefficient far from the percolation threshold. 1.

Results of Calculations of Poisson’s Ratio

The calculations were performed for a two-component, inhomogeneous medium.

vitaly v. novikov

214 (a)

(b)

Figure 52.

Lattice modulus with negative Poisson coefficient: (a) lattice with coordination number N ¼ 3; (b) a chain of bonds.

We have for Poisson’s ratio vp for a three-dimensional isotropic system [see Eqs. (333) and (339)] after the kth iteration: ðkÞ ðkÞ ðkÞ ðkÞ Connecting set: nðkÞ c ¼ ð3Kc  2mc Þ=ð6Kc þ 4mc Þ; ðkÞ ðkÞ ðkÞ ðkÞ Nonconnecting set: nðkÞ n ¼ ð3Kn  2mn Þ=ð6Kn þ 4mn Þ ðkÞ lim nðkÞ c ¼ lim nn ¼ np

k!1

k!1

The results of calculations of the effective Poisson’s ratio np dependence on the bulk concentration of a rigid phase p at various values of a ¼ log10 ðK2 =K1 Þ are shown in Fig. 53. The calculations were made for Poisson’s ratios of the phases ranging from 0:1 to 0:4. It can be seen that at percolation threshold Poisson’s ratio of the isotropic fractal composite is np ¼ 0:2, when K2 =K1 ! 0 it is also independent of the Poisson’s ratios of the individual components of the composite. The Poisson’s ratio obtained by us near the percolation threshold is in agreement with computer simulation results and the conjecture of Arbabi and Sahimi [161]. It has been shown that an approximate theoretical treatment of percolation on a cubic lattice exactly reproduces the Poisson’s ratio obtained in computer simulation at the percolation threshold. This result may encourage one to use this approximation to describe various elastic properties of composites. It is worth noting that some critical indices have been calculated recently with a high degree of accuracy in the context of the present model. The considerations presented in this work suggest that at the percolation threshold the ratio of the shear modulus to the bulk modulus is a universal quantity, which does not depend on the elastic properties of the percolation phase. It is well known, however, that this ratio depends on the coordination number of the lattice on which the percolation takes place. Taking this into account, we conjecture that when the coordination number, Z, of the underlying lattice is more than four times larger than the dimension of the lattice, Z > 4d, Poisson’s ratio near the percolation threshold should be negative, irrespective of the value of Poisson’s ratio of the percolation phase.

physical properties of fractal structures 0.4

215

9 8 7

0.3 vp

6 5 0.2

4

3

2 0.1 1 0

–0.1 0

0.2

0.4 p

0.6

0.8

Figure 53. Dependence of the effective Poisson ratio on the volume concentration of the rigid component p for different values K2 =K1 for the Poisson ratio of the components np1 ¼ np2 ¼ 0:4 (curves above np ¼ 0:2) and np1 ¼ np2 ¼ 0:1 (curves below np ¼ 0:2).

Assuming that Eq. (333) is fulfilled also at the percolation threshold for random lattices with Z replaced by the average coordination number hZi i (where Zi means the coordination number of the ith lattice site), one can obtain negative Poisson’s ratios from percolation systems of particles—for example, polymer molecules, which can be due to a sufficiently large number of neighbors, < Z; > > 4d. Analytic studies and extensive computer simulations of various models in 2D and 3D are in progress in order to verify the above theoretical predictions. F.

Frequency Dependence of Viscoelastic Properties

Before we discuss the viscoelastic properties of nonuniform fractal structures we shall give some basic definitions from the theory of elasticity. If a body is subjected to an external force, strains appear in it, and the body itself becomes stressed. If this stress always exists during the action of the force and instantly disappears when this action ceases, then the body is ideally elastic. In this case, the relation between the stress tensor s and the strain tensor e is described by Hooke’s law [141]: s ¼ C  e; e ¼ S  s

ð349Þ

vitaly v. novikov

216

where C is the elastic modulus tensor and S is the compliance modulus tensor. For an isotropic medium with elastic modulus tensor C and compliance modulus tensor S we have C ¼ 3KV þ 2mD; 1 1 S¼ Vþ D 3K 2m

ð350Þ ð351Þ

where K is the volume elastic modulus, m is the shear modulus, V is the volume, and D represents deviator components of the individual tensor of the fourth rank I: I ¼ V þ D; 1 Vijkl ¼ ðdij dkl Þ; 2

1 2 Dijrl ¼ ðdik djl þ dil djk  dij dkl Þ; 2 3
1; i¼k dik ¼ 0; i 6¼ k

ð352Þ

When the strain is irreversible (i.e., when a body exhibits percolation), the stress decreases rapidly and recovers again because of the displacement of structural elements. If the shape and state of the structural elements do not undergo any variations in this case, then the body is ideally viscous, and its behavior is described by the Newtonian equation [131,132] s¼Z

de dt

ð353Þ

where Z is the viscosity of the liquid. Most real bodies are viscoelastic and obey laws (349) and (353) only under certain conditions. Hence, the concept of the stress decay time or the relaxation time t is introduced to characterize the stress-strain state of real bodies. For absolutely elastic bodies, t ! 0, whereas, for ideally viscous bodies, t ! 1. Real viscous, anomalous viscous, and viscoelastic media are described in the interval 0 < t < 1. A general relation containing the law of elasticity and the law of viscosity as extreme cases can be introduced as sðtÞ ¼ K

da eðtÞ dta

ð354Þ

Hooke’s law follows from (354) for a ¼ 0, K ¼ C and Newtons law of viscosity follows for a ¼ 1; K ¼ Z:

physical properties of fractal structures

217

If the external action depends on the time, i.e. the stresses sðtÞ and strains eðtÞ depend on time, Hooke‘s law can also be introduced via the frequency domain relation [131,132] sðoÞ ¼ C  ðoÞeðoÞ

ð355Þ

where sðoÞ and eðoÞ are the Fourier transforms of sðtÞ and eðtÞ respectively: 1 ð

sðoÞ ¼

1 ð iot

e sðtÞ dt; 1 1 ð

1 sðtÞ ¼ 2p

eðoÞ ¼ 1

e

iot

sðoÞ do;

1

ð356Þ

eiot eðtÞ dt

1 eðtÞ ¼ 2p

1 ð

eiot eðoÞ do

ð357Þ

1

The complex elastic modulus in the frequency domain is C  ðoÞ ¼ C0 ðoÞ þ iC 00 ðoÞ

ð358Þ

where the real and imaginary parts of the modulus C  ðoÞ are given by [132] 0

1 ð

C ðoÞ ¼ C1 þ o

gðtÞ sin ot dt;

00

1 ð

C ðoÞ ¼ o

0

gðtÞ cos ot dt

ð359Þ

0

Here the elastic modulus cðtÞ ¼ c1 þ gðtÞ; c1 is the asymptotic value of cðtÞ as t ! 1; gðtÞ is the response function (gðtÞ in general takes into account the history of the process) Hooke’s law for a compliance tensor S can be represented in the form eðoÞ ¼ S ðoÞsðoÞ

ð360Þ

S ðoÞ ¼ S0 ðoÞ þ iS00 ðoÞ

ð361Þ

where

00

S0 ðoÞ is the inphase (accumulation) compliance, and S ðoÞ is the quadrature (loss) compliance. One can show that the relative scattering loss of the elastic energy is related 00 only to the imaginary component S ðoÞof the elastic modulus [131,132].

218

vitaly v. novikov

We will consider below isotropic media, for which, just as Eq. (361), the concept of a complex bulk elastic modulus K  ðoÞ can be introduced [131]. The complex shear modulus m and the complex viscosity Z can be written [131,132] m ðoÞ ¼ m0 ðoÞ þ im00 ðoÞ; Z ðoÞ ¼ m =io;

ð362Þ ð363Þ

Z ðoÞ ¼ Z0 ðoÞ þ iZ00 ðoÞ

ð364Þ

The relation between m0 ðoÞ; m00 ðoÞ and Z0 ðoÞ; Z00 ðoÞ is m0 ðoÞ ¼ oZ00 ðoÞ; m00 ðoÞ ¼ oZ0 ðoÞ

ð365Þ

For a medium representing a Newtonian liquid, we have m ðoÞ ¼ ioZ0 ðoÞ Viscoelastic media have been described by a variety of models involving combinations of a spring and a piston in a viscous liquid. In this (onedimensional) case, Hooke’s and Newton’s laws are [131] FH ¼ kx dx FN ¼ Z dt

ð366Þ ð367Þ

A series combination of these elements corresponds to the Maxwell model, while their parallel combination corresponds to the Kelvin–Voigt model (Fig. 54). The transition from the models to a continuous medium is performed by replacing the force F and displacements x by stresses s and strains e. The Maxwell model conforms to the series connection of these elements, and the Voigt model conforms to the parallel connection. The main disadvantage of the Maxwell model is that the static shear modulus m0 vanishes in this model, while the drawback of the Kelvin–Voigt model is that it cannot describe the stress relaxation. The Zener model [131] lacks these disadvantages. This model combines the Maxwell and Kelvin–Voigt models and describes strains closely approximating the actual physical process. The elasticity equation for the Zener model taking account of anomalous relaxation effects can be written as [131] s þ te

  da s da e ¼ m e þ t s dta dta

ð368Þ

physical properties of fractal structures (a)

Figure 54.

219

(b)

Models of viscoelastic properties: (a) Maxwell model;(b) Kelvin–Voigt model.

where m0 ¼ mðoÞjo¼0 ;

ð369Þ

m1 ¼ lim mðoÞ;

ð370Þ

o!1

te =ts ¼ m0 =m1

ð371Þ

and o is the angular frequency of the impressed stimulus. d dtf aðtÞ is the Riemann– Liouville fractional differentiation operator a

ðt d a f ðtÞ 1 d ¼ ðt  tÞa f ðtÞ dt dta ð1  aÞ dt

ð372Þ

c

where ðxÞ is the gamma function. By Fourier transformation of (368), we have [131] s þ ðiotÞa s ¼ 2m0 ðs þ ðiotÞa eÞ

ð373Þ

where s; e are the Fourier transforms of s; e. Now the Fourier transform of a fractional derivative is

1  ð da f ðtÞ a F expðiotÞ f ðtÞ dt ¼ ðioÞ f ðtÞ; f ðtÞ ¼ dta 1

ð374Þ

vitaly v. novikov

220 (a)

(b)

1

1

(m∞– m')/ (m∞– m0)

0.4

0.8

2

m"/(m∞– µ0)

0.3

0.6 0.4 0.2 4

0 –3

–2

–1

0 1 log10wt

1 2 3 2

3

2

3

0.2

3

0.1

4

0 –3

–2

–1

0 1 log10wt

2

3

(c) 1

1

0.8 f0

0.6 0.4

2

0.2 0

4 –3

–2

–1

3

0 1 log10t

Figure 55.

Dependence of the viscoelastic properties on log t (a) the real part of the relative shear modulus for a ¼ 0:2 (curves 1); 0:4 (2); 0:7 (3); 0:9 (4).; (b) the imaginary part of the relative shear modulus for a ¼ 0:2 (curves 4); 0:4 (3); 0:7 (2); 0:9 (1).; (c) normalized relaxation time distribution function for a ¼ 0:2 (curves 4); 0:4 (3); 0:7 (2); 0:9 (1).

It follows from this that the complex shear modulus for the standard linear body is m  m0 m ðoÞ ¼ m1  1 ð375Þ 1 þ ðiote Þa Hence m1  m0 ðoÞ 1 þ ðotÞa cosðpa=2Þ ¼ ; m1  m0 1 þ ðotÞa ½2 cosðpa=2Þ þ ðotÞa  m00 ðoÞ ðotÞa sinðpa=2Þ ¼ m1  m0 1 þ ðotÞa ½2 cosðpa=2Þ þ ðotÞa 

ð376Þ ð377Þ

If the Fourier transform of mðoÞ is known, then the Fourier transform of the corresponding distribution f ðtÞ of relaxation times is [131]   1 1 ð378Þ f ¼  Im mðo expðipÞÞ o p

physical properties of fractal structures

221

By using Eqs. (375) and (378), we can then determine the normalized density of the distribution f0 ðtÞ of relaxation times [Eq. (377)]: f0 ðtÞ ¼

sinðpaÞ 2pfcosh½a lnðt=te Þ þ cosðapÞg

ð379Þ

where f0 ðtÞ ¼ m f ðtÞ : 1 m0 The dependence of the dispersion g2 of the relaxation time of the chaotic dynamics on the parameter a has the form 1 ð

g ¼

ln2 ðt=te Þ f0 ðtÞd ln t ¼

2

1

p2 1  a2 3 a2

ð380Þ

It has been shown in Ref. 170 that fractional derivatives can be obtained by assuming that a set of relaxation times has a fractal nature. The parameter a in Eqs. (376) and (377) is the fractal dimension of the fractal set of relaxation times and characterizes the localization (spread) of the relaxation spectrum [170]. 1.

Iterative Averaging Method for Viscoelastic Properties

According to Eq. (358), the static parameters (333), (334) can be converted into viscoelastic parameters by replacing the elastic modulii K and m by the corresponding complex modulii: K  ¼ K 0 þ iK 00 ; m ¼ m0 þ i m00

ð381Þ

By using this correspondence principle for a connected set, the complex volume elastic modulus Kc and the complex shear modulus mc at the (k þ 1)th step can be written

Kcðkþ1Þ ¼ KcðkÞ þ mcðkþ1Þ ¼ mcðkÞ þ

ðkÞ

ð1  pk ÞðKn 1þ

ðkÞ

 Kc

ðkÞ ðkÞ pk ac ðKn ðkÞ

ð1  pk Þðmn ðkÞ

ð382Þ

ðkÞ

 mc Þ

ðkÞ

1 þ pk bc ðmn



Þ

ðkÞ Kc Þ

ð383Þ

ðkÞ

 mc Þ

where

aðkÞ c ¼

3

; ðkÞ ðkÞ 3Kc þ 4mc

bðkÞ c ¼

ðkÞ

6ðKc ðkÞ

5Kc

ðkÞ

þ 2mc Þ ðkÞ

ð3Kc

ðkÞ

þ 4mc Þ

ð384Þ

vitaly v. novikov

222 ð0Þ

ð0Þ

where Kc ¼ K1 ; mc ¼ m1 are the complex volume elastic modulus and the complex shear modulus for the first phase of the inhomogeneous medium, ð0Þ ð0Þ respectively; Kn ¼ K2 ; mn ¼ m2 are the complex volume elastic modulus and the complex shear modulus for the second phase, respectively. ðkþ1Þ ðkþ1Þ ; mn for a disconnected set are determined The elastic properties Kn from expressions that may be obtained from Eqs. (382) and (383) following the replacements c !n and pk !ð1  pk Þ. 2.

Results of Calculations for Viscoelastic Media

The calculations were made for a two-phase (two-component) inhomogeneous medium assuming that volume strains are elastic, while shear strains are viscoelastic. The ratio of local volume modulii K10 =K20 was set equal to 104 . For convenience of calculations, the local shear modulii (phase shear modulii) were written in the form 0

ð385Þ

0

¼ m2 ð1 þ iyÞ

ð386Þ

y ¼ tanðj2 Þ ¼ m002 =m02

ð387Þ

m1 ¼ m2 xð1 þ iayÞ m2 where

x ¼ m01 =m02 ¼ Z001 =Z002 a ¼ tanðj1 Þ= tanðj2 Þ

ð388Þ ð389Þ

The complex viscosity is 0

Zj ðoÞ ¼ Zj ðoÞ  iZ00j ðoÞ

ð390Þ

where m0j ðoÞ ¼ oZ00j ðoÞ;

m00j ðoÞ ¼ oZ0j ðoÞ;

j ¼ 1; 2

ð391Þ

It follows from Fig. 56 (a ¼ 0:1; m001 =m01 1; m002 =m02 1) that the concentration dependence of the relaxed viscosity (o ! 0) is described by a monotonic curve and is independent of the ratio m002 =m02 . For a ¼ 0:01 and m002 =m02 ¼ 0:01 (Fig. 56b), both a local maximum and a local minimum appear near the percolation threshold, which strongly depend on the ratio m002 =m02 for a < 0:01 (Fig. 56b–d). The form of the dependence hardly changes before the percolation threshold (p < pc ) (Fig. 56b), whereas, after the percolation threshold (p > pc ),

physical properties of fractal structures 3

1

(a) 2

2.5

(b)

0.8 1

2

1

2

3 h'/h'

h'/h'2

4

1.5

0.6

2

0.4

3

1

4 0.2

0.5

0

0 0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

p

0.8

0.6

0.8

1

0.6

0.8

1

p

(c)

(d) 1

1

2 3

0.5

0.6

4 0

0.4

h'/h'2

2 h'/h'2

223

3

–0.5 –1

4

0.2

–1.5 –2

0 0

0.2

0.4

0.6 p

0.8

1

0

0.2

0.4 p

Figure 56. Dependence of the logarithm of the relative effective viscosity of an inhomogeneous

fractal medium Z0 =Z02 ¼ Im½m ðoÞ=Im½m2 ðoÞ (Z02 is the viscosity of the second phase) on the concentration p of the first phase calculated for various values of a: (a) a ¼ 0:1; (b) a ¼ 0:01; (c) a ¼ 0:001; (d) a ¼ 0:0001; (e) a ¼ 0:00001; (f) a ¼ 0:000001. The calculations were made for the ratio of the real parts of the shear modulus x ¼ 10000 and y ¼ 0:01; 10; 100; 1000.

the concave curve (Fig. 56d) becomes convex ( m001 =m01 1, m002 =m02  1). For a  0:0001 the minimum disappears and only the maximum in the vicinity of the percolation threshold remains, which also disappears for m002 =m02 ! 1 (Fig. 56d). These results show that the dependence log10 jZ0 =Z02 j on the concentration p of phases of the fractal structure becomes convex with a single maximum when ax  1—that is, when m001  m002 for x  1 (m01  m02 ). Figure 57 shows the calculated dependence of the logarithm of the relative effective viscosity Z00 =Z002 ¼ Re½m ðoÞ=Re½m2 ðoÞ on the concentration p of the first phase demonstrating that, for a  0:001 (Fig. 57b), the relative effective

vitaly v. novikov

224 4

4 (b) 5

(a)

3

3 2 2 h" /h"

h" /h"

2

4

2 1

1

2 3 1

2 1

0

0 0

Figure 57.

0.2

0.4

0.6 p

0.8

1

0

0.2

0.4

0.6 p

0.8

1

Dependence of the logarithm of the effective relative viscosity Z00 =Z002 ¼ Re½m on the concentration p of the first phase for the same values of x; a; and y as in

ðoÞ=Re½m2 ðoÞ Fig. 56.

viscosity Z00 =Z002 is virtually independent of a. For p < pc, the form of the dependence alters when a ! 0; however, for p > pc, it does not alter. As an example of the viscoelastic properties of an actual medium, we consider the viscoelastic properties of a charged polymeric material. Percolation properties will be exhibited by the charged polymeric composite, if the stiffness of the agglomerates of particles is greater by some orders of magnitude than the stiffness of the unperturbed polymetric compound [49]. This can occur, for example, if the polymetric compound in the vicinity of a boundary (e.g., the surface of a particle) attains the superstrong state [171–173]. The properties of agglomerates of particles (particles plus a boundary region composed of a polymetric compound) can be estimated when the region occupied by the composite is completely full [171–173]. Basing on this statement, we shall calculate the shear moduli of the charged polymeric composite. According to Ref. 172 the effective shear modulus of such a composite (real part of the modulus) at limiting doping may be estimated as m0 ¼ 2  106 [Pa] at o ¼ 0, which exceeds the shear modulus of the unperturbed polymetric compound m0p . Note that  39957:2o1:8 ¼ ½Pa 1 þ 0:40o0:8 þ 0:4183o1:6   65053o þ 0:1804o1:8 ½Pa m00p ðoÞ ¼ 1 þ 0:40o0:8 þ 0:4183o1:6 m0p ðoÞ



ð392Þ ð393Þ

In calculating of the effective shear modulus of the composite, it is assumed that the shear modulus of the particles comprising the doping compound along with its boundary region is m0f ¼ 2  106 ½Pa; m00f ¼ 0 [174] so that m f is

physical properties of fractal structures 6

225

6 5

log10 µ ¢

4

4

2

3

0

2

–2

1

–4 –4

–2

0 log10 w

2

4

Figure 58. Calculation of the frequency dependence of the effective shear (storage) modulus 0

m ðoÞ for various doping concentrations p: 1, 0.01; 2, 0.09; 3, 0.13; 4, 0.15; 5, 0.2; 6, 0.25.

independent of frequency o. The shear modulus of the polymetric compound may be determined from Eqs. (392) and (393). The calculation of the real part of the effective shear modulus m0 of a composite with fractal structure is illustrated. According to this calculation (Fig. 58) the percolation transition appears after o < 104 and at doping concentration p 0:12, i.e. for p > 0:12 in a composite with a continuous and strong skeleton composed of particles of a doping compound connected by a boundary stratum of a polymetric compound. 3.

Negative Shear Modulus

Recent studies of inhomogeneous materials containing inserts with negative stiffness indicated that such composites exhibit very interesting properties; for example, they can have much higher stiffness and higher damping coefficients than the individual phases constituting them [175–178]. The analysis [175–178] of the influence of inclusions with negative shear modulus on the effective shear modulus of a composite was using the Hashin–Strikman formulae [133,134], which were obtained by assuming that the system properties do not depend on scale; that is, the system can be considered as a homogeneous medium. The idea of underlying such materials is based on the behavior of a construction which is shown on Fig. 59. This construction (a ruler deformed into the shape of the letter 00 S00 ) is in a prestressed weighted state [178], the tangential forces will not resist exterior action, and will cause bias in the direction of

vitaly v. novikov

226

Figure 59.

Buckled plastic ruler to demonstrate negative stiffness.

application of the forces. Thus, the coefficient of proportionality (shear modulus) between the tangential forces and the bias will be negative. It is possible to realize such constructions if one places them in a matrix (in columns) with stable properties, for example, in a polymetric compound. Thus, it is possible to create a composite material having insertions with a negative shear modulus and a matrix–polymetric compound with a positive shear modulus. We have also studied the elastic properties of a nonuniform medium with chaotic structure in which one phase has a negative shear modulus. The analysis may be made using the fractal hierarchical structure model. The calculations were performed for a two-component, inhomogeneous medium. For simplicity, it is assumed that both phases are isotropic and that the first phase is purely elastic whereas the second phase is elastic from the point of view of volume deformations and viscoelastic from the point of view of shear deformations. The concentration of the purely elastic phase is denoted by p. It is convenient to write the shear modulus of the second phase, m2 , in the form m2 ¼ m01 xð1 þ iyÞ

ð394Þ

y ¼ tanðj2 Þ ¼ m002 =m02 m02 =m01 ¼ x

ð395Þ

where

physical properties of fractal structures

227

and m1 ¼ m01

ð396Þ

where m01 is the (real) shear modulus of the first phase. As mentioned earlier, fractal structures can exhibit properties different from uniform structures. To illustrate this we compare the effective shear modulus and the damping coefficient of a medium with a composite material corresponding to the Hashin–Strikman formulae (referred to as the ‘‘Hashin–Strikman composite’’). In Fig. 60, the ratio of the real part of the effective shear modulus to the shear modulus of the elastic phase is shown as a function of the concentration, p, of

Figure 60. Comparison of the ratio of the effective shear modulus to the shear modulus of the elastic phase at y ¼ 0:001 as a function of the concentration of the elastic phase: (a, b) in the fractal composite and (c, d) in the Hashin–Strikman composite.

vitaly v. novikov

228

the elastic phase and the ratio x for y ¼ 103. It is assumed that the viscoelastic phase (see Figs. 60–62) has a negative shear modulus (i.e., negative real part of the complex shear modulus) and is characterized by y ¼ tan j2 ¼ 0:001 and the Poisson’s ratios of both phases (calculated from the real parts of the elastic modulii) are equal to 0:184. (The latter assumption means that the ratio of the 0 0 real parts of the shear modulii to the bulk modulus is equal to mi =Ki ¼ 0:8, where i ¼ 1; 2 denotes the phases; as a consequence, the ratio of the bulk modulus of the second phase to the bulk modulus of the first phase is 0 0 K2 =K1 ¼ x). Apparently for the inhomogeneous fractal medium the shear modulus shows in some ranges of concentration a resonance-like behavior similar to that discussed in [175–177] whereas in the Hashin–Strikman composite there exists only one such resonance in the vicinity of the concentration p ¼ 1. To complete the illustration of the dependence of the real part of the effective 0 shear modulus m of the fractal structure composite on the ratio of the real parts of the shear modulii of the phases m02 =m01 ¼ x, the results of calculations of the 0 ratio of the modulii m =m01 for concentrations of the first phase p ¼ 0:25 and p ¼ 0:45 are plotted against x in Fig. 61.

3 2 0.25 1 0.45 0 –1 –2 –3

0.2

0.4

0.6

0.8

1

x Figure 61. Ratio of the effective shear modulus to the shear modulus of the elastic phase for y ¼ 0:001 as a function of x in the fractal composite.

physical properties of fractal structures

229

Figure 62. Ratio of the imaginary to the real part of the effective shear modulus: (a, b) in the fractal composite and (c, d) in the Hashin–Strikman composite; the other details are the same as in Fig. 60.

According to the calculations (Figs. 60 and 61) the peaks in the dependence 0 of the real part of the effective shear modulus m of the composite of fractal structure are grouped near x 0:2 and 0:8 < x < 1. In Fig. 62 the ratio of the imaginary and real parts of the effective shear modulus (i.e., the loss tangent tan j) is shown as a function of the concentration of the elastic phase and x for y ¼ 103. The parameters of the phases are the same as in Fig. 61, again depending on x, and tan j of the fractal composite the ratio attain very large values in some ranges of concentration whereas the Hashin–Strikman composite exhibits only one large value, in the vicinity of the concentration p ¼ 1, for x in the range considered. The above comparisons show that the viscoelastic properties of the fractal composite differ qualitatively from the Hashin–Strikman one. The observed

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230

differences can be understood by noting that the hierarchical model considered takes into account clusters of various length scales and different parameters of their ‘‘resonances’’ which are formed in the fractal composite whereas the Hashin–Strikman composite is ‘‘uniform’’ in this aspect. The nature of the peaks and the way in which they are formed depending on the effective shear modulus m , the frequency o, and the concentration p can be understood by considering a single insert with negative shear modulus 0 0 m1 ¼ m ðm ¼ xK1 Þ immersed in a medium (stabilizing matrix) of positive 0 shear modulus m2 ¼ m2 . The shear modulus m of the composite with a single inclusion can be determined from [131–133] m ¼ m2 þ

pðm1  m2 Þ 1 þ ð1  pÞb2 ðm1  m2 Þ

ð397Þ

where b2 ¼

6ðK2 þ 2m2 Þ 5m2 ð3K2 þ 4m2 Þ

ð398Þ

It follows from Eq. (397) that when m1 ¼ m0 ðm0 ¼ xK1 Þ, then m ¼ m02 

pðxK1 þ m02 Þ 1  ð1  pÞb2 ðxK1 þ m02 Þ

ð399Þ

Finally, from the equation 1  ð1  pÞb2 ðxK1 þ m02 Þ ¼ 0

ð400Þ

one can determine the resonance parameters of the composite for which the peaks arise; that is, the parameters for which the external disturbance is in resonance with the inclusion parameters (being equal to its natural frequency). As the model studied constitutes a self-similar, chaotic system of clusters of various sizes, on the jth scale level, each cluster will have its own resonance parameters—that is, its own characteristic frequency, which produces the system of peaks (characteristic frequencies) in the dependence of the effective shear modulus m on the parameters of the composite. One should add that because the Hashin–Strikman composite is not very realistic and the Hashin–Strikman bounds are not particularly sharp, and since the probability function, Rðl; pÞ, used here is more or less arbitrary, the results obtained should be interpreted in a qualitative sense. We plan to study more realistic approximations in the future.

physical properties of fractal structures

231

Figure 63. Schematic for construction of a material with fractal structure.

Finally, we note that the materials with fractal structure exhibiting viscoelastic properties similar to those of the model described here can be manufactured. For example, at the first step (the lowest size level) one produces tablets—for example, of a polymer with required inclusions. At the next step, the tablets obtained at the preceding level are used as inclusions in larger tablets. The process is continued and the hierarchy shown in Fig. 63 is obtained [160]. Conclusions. It has been shown that a hierarchical ‘‘blob’’ model when used to study viscoelastic properties of an inhomogeneous fractal medium (the fractal composite) yields results which differ qualitatively from those obtained by applying the Hashin–Strikman approximation to an inhomogeneous medium (the Hashin–Strikman composite). In particular, studies of a fractal model composed of an elastic phase (which can be regarded as a stabilizing matrix) and a viscoelastic phase with a negative shear modulus prove that the effective shear modulus and the effective loss tangent calculated exhibit much more complex behavior (more singularities) than those of the standard Hashin–Strikman model. The new singularities observed in the fractal composite are interpreted as resonances originating in (mesoscopic) clusters of various length scales which are described by different (mesoscopic) resonance parameters. Such clusters are taken into account by the hierarchical model, whereas they are neglected completely in the standard Hashin–Strikman approximation.

vitaly v. novikov

232 4.

Appendix. Fractal Model of Shear Stress Relaxation

The linear viscoelastic connection between stress (response) and velocity of strain (stimulus) can be written [132] ðt deðtÞ dt sðtÞ ¼ m1 eðtÞ þ ðm0  m1 Þ fs ðt; tÞ dt

ð401Þ

0

We shall suppose that (in order to generate the initial iteration): e ¼ e0 ZðtÞ;

de ¼ e0 dðtÞ dt

ð402Þ

where ZðtÞ is the Heaviside unit function and dðtÞ is the Dirac delta function. For a standard linear medium (the Zener model) taking into account Eq. (401) we have [131] s ¼ 2mðtÞe0

ð403Þ

  t mðtÞ ¼ m1 þ ðm0  m1 Þ exp  te

ð404Þ

where

where m1 ; m0 are relaxed (t ! 1) and nonrelaxed (t ¼ 0) values of the shear modulus, correspondingly; te is the relaxation time for constant deformation of the solid. Therefore the probability of f ðtÞ to change from the initial state (t ¼ 0) to a random state mðtÞ is given by the formula f0 ðtÞ ¼

  m0  mðtÞ t ¼ exp  m0  m1 ts

ð405Þ

In accordance with the Arrhenius formula, the relaxation time is t ¼ t0 exp½Q=kT

ð406Þ

where Q is the energy barrier between the initial and final states, T is the temperature, k is Boltzmann’s constant, and t0 is constant. According to the fractal model and the iteration method of calculation of the shear modulus, the system will consecutively change from the initial to final

physical properties of fractal structures

233

mð0Þ ! mð1Þ ! . . . ! mðiÞ ! mðiþ1Þ ! . . . ! mðnÞ

ð407Þ

state via

each of these changes is described by the probability of a change fi ðtÞ from the i-state to the (i þ 1) one.The function fi ðtÞ is defined by an exponential relaxation time ti given by Eq. (406) and the energy barrier height Qi . According to Eq. (404), the chain of inequalities can be written tð0Þ < tð1Þ < . . . < tðiÞ < tðiþ1Þ < . . . < tðnÞ ¼ t

ð408Þ

The lower t0 and the upper t limits exist for the set of relaxation times ftðiÞ g. Therefore, in accordance with the fractal model, the set of relaxation times ftðiÞ g satisfies the self-similarity criterion and is bounded by the lower and upper asymptotic limits. The hierarchical chain of changes from the initial state (t ¼ 0) to the final one (t ! 1) can be compared to a Cayley tree [25,26] (see Fig. 64). Here, the knots of the Cayley tree will correspond to static ensembles a and b which correspond to the dots in ultrametric space divided by the distance lab .

U1 n=0

l1

U2

l2

U3

n=1

n=2

l3

U4

n=3

l4

Figure 64. Schematic of a self-similar structure potential energy landscape and of the Cayley tree.

234

vitaly v. novikov

The value of lab is defined by the number of steps over the levels of the Cayley tree up to the mutual knot in Fig. 64 and it yields the extent of a hierarchical link. Therefore, both the barrier height, Qab , and the relaxation time, tab , are connected with functions of the distance lab in ultrametric space, that is, Qab ¼ Qðlab Þ; tab ¼ tðlab Þ

ð409Þ

Just as in deletion of clusters the height of the barriers dividing the clusters increases and a monotonic increasing Qðlab Þ dependence is assumed. So, according to the fractal model, the collection of parallel relaxation channels is acting independently. The probability of changes between channels a and b is defined by the formula fab ðtÞ ¼ exp½t=tab ; tab ¼ t0 exp½Qab =kT

ð410Þ

where Qab is the height of the energy barrier dividing the channels. The parallel action of different relaxation channels is only possible under conditions of hierarchical co-subordination of the corresponding collection of static ensembles. The hierarchical co-subordination means that the parallel net of channels of the next level having relaxation time tðiÞ does not act until channels with the given relaxation time tðiþ1Þ > tðiÞ act. Thus, the fastest processes take place first; they correspond to surmounting the barriers of minimum height Qab . Here, static ensembles merge with each other, and the system attains a higher hierarchical level of the Cayley tree. Therefore, static ensembles a; b can combine to form clusters. Each of the clusters is characterized by the maximum height, Qab , of the barrier dividing each cluster from the others. The hierarchical co-subordination of this kind results in the deceleration of relaxation leading to the transformation of Debye’s process into more slowly decaying dependencies. Thus, according to the model of relaxation described, the dependence Qab ðlÞ and ja ðlÞ on the distance in ultrametric space can be defined as Qab ðlÞ Q ln l;

ja ðlÞ ldf

ð411Þ

where df is the fractal dimension. The distribution function jðlÞ of the conditions corresponding to different points in ultrametric space reveals the degree of hierarchical linking of an inhomogeneous medium. According to Eqs. (406) and (410) and taking into consideration Eq. (411), the dependence of the relaxation time tab ðlÞ on the distance l in ultrametric

physical properties of fractal structures

235

space is tab ðlÞ ¼ t0 lQ=kT

ð412Þ

The total probability can be expressed as 1 ð

f ðtÞ ¼

jðlÞfab ðt; lÞ dl

ð413Þ

0

Taking into account that  fab ðt; lÞ ¼ exp 

 t tab ðlÞ

ð414Þ

we have 1 ð

f ðtÞ ¼

 jðlÞ exp 

 t dl tab ðlÞ

ð415Þ

0

which describes the relaxation due to all collections of relaxation channels, where jðlÞ is the probability of realization of a given static ensemble. Taking into account formulas (411), (412), and (415), we obtain a non-Debye relaxation law, namely, f ðtÞ tg

ð416Þ

1  df Q=kT

ð417Þ

where g¼

df is the fractal dimension of the set of relaxation times (df < 1). Thus, even simple estimations of the influence of hierarchical structure on the relaxation process yield a stress characteristic leading to anomalous relaxation of the form (416). G.

Relaxation and Diffusion Processes 1.

Non-Debye Relaxation

Anomalous (nonexponential) relaxations have long been and still are a favorite topic in the physics of inhomogeneous media [179–204]. Broadly speaking, one

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236

may refer to three general relaxation laws encountered in experimental studies of complex systems: (i) stretched exponential [179,180,190],

 b  t f ðtÞ exp  ; t

0 < b < 1; t > t

ð418Þ

(ii) exponential-logarithmic [181–183], h  t i f ðtÞ exp B  lna t

ð419Þ

(iii) algebraic decay [186], f ðtÞ

 t a t

ð420Þ

where the a; b; t, and B are the appropriate fitting parameters. Currently, there seems to be no quantitative microscopic theory for the cited laws [184,185,193]; moreover, sometimes even the possibility of such a theory is denied [191–193]. The main argument is that a spatial inhomogeneity (e.g., a random distribution of impurities within a matrix, or of interatomic spacings in amorphous semiconductors) will necessarily result in an extremely broad range of microscopic transition rates. Hence, a spatial disorder is expected to induce a temporal energetic disorder. Another approach to the problem of anomalous relaxations uses fractal concepts [187–189,200–203]. Here the problem is analyzed using the mathematical language of fractional derivatives [194,200–203] based on the previously mentioned Riemann–Liouville fractional differentiation operator [205–208], ðt 1 d D ½ f ðtÞ ¼  ðt  tÞa  f ðtÞ dt ð1  aÞ dt a

ð421Þ

c

where ðxÞ is the gamma function In spite of the reasonable success of the latter approach, use of the fractional derivative as represented by Eq. (421) renders difficult the interpretation of differentiation procedures (e.g., the nonzero value of a fractional derivative of a constant), as well as their relevance to the assumed fractal ensemble. One may also note that so far fractional derivatives have been analyzed in essentially

physical properties of fractal structures

237

phenomenological terms; moreover, the evolution equations based on fractional derivatives have been constructed more by intuition (guessed), rather than obtained from first principles. The fractional derivative technique is used for the description of diverse physical phenomena (e.g., Refs. 208–215). Apparently, Blumen et al. [189] were the first to use fractal concepts in the analysis of anomalous relaxation. The same problem was treated in Refs. 190,194,200–203, again using the fractional derivative approach. An excellent review of the use of fractional derivative operators for the analysis of various physical phenomena can be found in Ref. 208. Yet, however, there seems to be little understanding of the relationship between the fractional derivative operator and/or differential equations derived therefrom (which are used for the description of various transport phenomena, such as transport of a quantum particle through a potential barrier in fractal structures, or transmission of electromagnetic waves through a medium with a fractal-like profile of dielectric permittivity, etc.), and the fractal dimension of a medium. In this context, attempts to construct fractional derivatives and to clarify their relevance to the assumed fractal ensemble are believed be still relevant for the treatment of anomalous relaxations. In our previous papers [215], the analysis of the classical problem of polarization of an inhomogeneous medium permitted us to establish the relationship between anomalous relaxation and the dimension of a temporal fractal ensemble which characterizes a nonequilibrium state of a medium. Thus, the main aim of the present work is the further extension and generalization of these results in the context of a fractal model. Dielectric Relaxation. The potential of fractional derivatives in the context of anomalous relaxation will become evident, and the relationship between the exponents b and a in Eqs. (418)–(420) and the fractal dimension df will be established, in the subsequent treatment of the classical problem of polarization PðtÞ of a dielectric medium (which is, in fact, equivalent to the general problem of relaxation of the internal parameters of a nonequilibrium phase). Assume that PðtÞ contains two contributions [185], PðtÞ ¼ P0 þ P1 ðtÞ

ð422Þ

where the first qualitity, the static polarization ðP0 ¼ w0 EÞ, varies exactly (at least, with negligibly small retardation) as the applied field E, while the timedependent quantity, P1 ðtÞ, is retarded. Let P ¼ w1 E be the upper limit (at fixed E ); then, the larger the amplitude (w1 E  PðtÞ), the greater the instantaneous rate of approach P1 ðtÞ to this limit. Hence, the corresponding relaxation equation

vitaly v. novikov

238 may be written as

dP1 ðtÞ 1 ¼  ðwE  P1 ðtÞÞ; dt t w ¼ w1  w0

ð423Þ

where t is the relaxation time. Thus h   t i PðtÞ ¼ P0 þ P1 ðtÞ ¼ w0 þ w 1  exp  E t

ð424Þ

(for a constant field E), and PðoÞ ¼ P0 þ P1 ðoÞ ¼ ½w0 þ w=ð1  iotÞ  E

ð425Þ

(for an alternating field E ¼ E0 eiot ). Therefore, the dielectric permittivity of a medium may be defined, finally in linear response, as [185] e ¼ e1 þ

e0  e1 1 þ iot

ð426Þ

where e1 ¼ lim e; e0 ¼ ejo¼0 . o!1 The next issue to concern us will be anomalous relaxation in which the ‘‘smearing out’’ of a relaxation spectrum (i.e., the deviation of complex susceptibility from its Debye form) is associated with the concept of a relaxation time distribution. As is well known, this concept implies an assembly of dipoles with a continuous distribution of relaxation times of Eq. (379). Consider Fro¨hlich’s relaxation model [221] which is based on the relaxation time distribution concept. It is usually assumed that the relaxators are homogeneously distributed along the height of potential barrier U; however, this assumption is not strictly correct. As can be seen from Fig. 65, many shallower minima may exist between two main minima; therefore, the system is involved in a continuous chain of transitions r1 ! r2 ! r3 ! . . . ! rk ! rn between adjacent minima (Fig. 65). Let Sl ðtÞ be the probability of transition from the lth minimum into the (l þ 1)th one, so that SðtÞ ¼ e where the relaxation time is

ð427Þ

l

 tl ¼ t0 exp

where Ql is the barrier height.

tt

Ql kT

 ð428Þ

physical properties of fractal structures

239

U

Q1

r1 r2 r3

Figure 65.

rk

rn

r

Schematic of the potential barrier landscape.

Implicit in the relaxation time distribution concept is the assumption of comparable magnitudes of barrier heights Ql (Fig. 65); hence, the characteristic times tl ¼ tlþ1  tl of transitions over the barriers will be also of comparable magnitudes. Therefore, during a time t  ntl we have for n barriers with a probability 1  S(t) so that SðtÞ ¼

n Y

Sl ;

l¼1

  tl Sl ¼ exp  tl

ð429Þ

Assuming tl ¼ nt , it becomes clear that the dispersion of the intervals tl may be neglected in the limit of n  1; as a result, the Eq. (429) will regain its Debye form with the mean relaxation time hti defined as hti1 ¼

n 1X t1 n l¼1 l

ð430Þ

Thus, the chain of transitions considered above is effectively reduced to exponential, Debye-like relaxation with the mean relaxation time hti. In other words, the concept of a relaxation time distribution implies Debye-like relaxation of a system. However, it is evident that the relaxation will become nonexponential, should a system be characterized by a complex susceptibility of, say, Cole–Cole type.

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240

Thus, it can be concluded that the relaxation time distribution concept applies to Debye-like relaxation (even though its frequency dependence may be smeared-out), whereas it becomes inapplicable for still slower relaxation patterns. In the latter situation, the distribution of relaxation times over a selfsimilar, fractal ensemble seems a physically more reasonable assumption. As is well known, the fractality of geometrical objects implies their non-integer dimension; however, a more exact definition of the fractal concept with respect to the ensemble of relaxation times is in order. As proved by Nigmatullin [200–203], fractional derivatives in time in Newton’s equations imply that the interactions between a system and an external field are not continuous but occur at discrete time intervals. In this context, the fractality of an ensemble of relaxation times simply means that the relaxation is not a single process with a unique relaxation time, rather it is a series of successive relaxation events with different relaxation times. Let us consider now the nonequilibrium state of a fractal-like medium assuming that this nonequilibrium state is characterized by many events such that a subsequent event is separated by a certain time interval ti from a previous event. In this case, some intervals will be eliminated from a continuous process of system evolution by a definite law. Assume that such a process is caused by a temporal fractal state of dimensionality df ; the corresponding relaxation equation can be written as Da ½P1 ðtÞ ¼

1 ðwE  P1 ðtÞÞ ta

ð431Þ

and rearranged as ½1 þ ðtDÞa P1 ðtÞ ¼ wE

ð432Þ

The latter Eq. (432) can be solved using the Laplace transform [205–208], thus ½1 þ ðtsÞa P1 ðsÞ ¼

wE ; p

1 ð P1 ðsÞ ¼ est P1 ðtÞ dt

ð433Þ

0

yielding P1 ðsÞ ¼

wE 1 s 1 þ ðtsÞa

ð434Þ

physical properties of fractal structures

241

1 X 1 ðstÞa ¼ ¼ ð1Þn ðstÞaðnþ1Þ 1 þ ðtsÞa 1 þ ðstÞa n¼0

ð435Þ

since

the solution of Eq. (434) in the time domain will have the following form:  aðnþ1Þ 1 X ð1Þn  tt ð436Þ P1 ðtÞ ¼ wE ½aðn þ 1Þ þ 1 n¼0 where ðxÞ is the gamma function. Therefore, "  aðnþ1Þ # 1 X ð1Þn  tt PðtÞ ¼ P0 þ P1 ðtÞ ¼ w0 þ w E ½aðn þ 1Þ þ 1 n¼0 When a ¼ 1 in Eq. (437), Eq. (424) is recovered so that "  ðnþ1Þ # 1 X ð1Þn  tt PðtÞ ¼ w0 þ w E ½n þ 2 n¼0 h   t i E ¼ w0 þ w 1  exp  t

ð437Þ

ð438Þ

(in the derivation, the standard Eq. (439) has been used): 1 X

ðzÞn ¼ expðzÞ;  ½ ð n þ 1Þ  n¼0

z¼

t t

ð439Þ

Thus, the crossover from a strictly exponential to an anomalous relaxation pattern can be associated with the change of a continuous distribution of relaxation times (a ¼ 1) into a fractal-like one (0 < a ¼ df < 1). Thus, the solution of Eq. (437) in the time domain is PðtÞ ¼ w0 E 1 

1 X ð1Þn  n¼0

 t an ! t

½an þ 1

  ¼ w0 E 1  Ma;1 ðzÞ

ð440Þ

where Ma;1 ðzÞ the Mittag–Leffler function is Ma;g ðzÞ ¼ In our case, g ¼ 1.

1 X

zn ; ½an þ g n¼0

z¼

 t a t

ð441Þ

vitaly v. novikov

242

It is convenient to use Fox functions (generalized Mellin–Barnes integras) when solving equations with fractional derivatives because Laplace and Fourier transformations for Fox functions may be expressed via Fox functions with given parameters. The connection of Mittag–Leffler functions with Fox functions is as follows [216,217]: 

ð0; 1Þ 1;1 z Ma;g ðzÞ ¼ H1;2 ð0; 1Þ; ð1  g; aÞ

ð442Þ

Then Eq. (440) becomes 

1;1 PðtÞ ¼ w0 E 1  H1;2 z

 ð0; 1Þ ð0; 1Þ; ð1  g; aÞ

ð443Þ

If a ¼ 1, then from Eq. (443) we obtain PðtÞ ¼ w0 E 1 

1 X ð1Þn n¼0

 t n ! t

½ n þ 1

  t  ¼ w0 E 1  exp  t

ð444Þ

If a 6¼ 1 then according to Eq. (443) it follows that PðtÞ 1 

 t a t

;

t  t

!1

ð445Þ

and PðtÞ

 t a t

;

t  t

!0

ð446Þ

Thus, the solution of an equation with fractional derivatives (431) describes relaxation in dielectrics having a power law dependence in asymptotic limits such as Eqs. (445) and (446). It follows from Eq. (447) that

PðtÞ  w0 þ

 w  t a E ða þ 1Þ t

ð447Þ

which can be compared with Eqs. (418)–(420). For an alternating field, Fourier transformation of Eq. (431) yields (s ¼ io)

PðioÞ  w0 þ

 w E ð1 þ iotÞa

ð448Þ

physical properties of fractal structures

243

and the dielectric permittivity is e ¼ e1 þ

e0  e1 1 þ ðiotÞa

ð449Þ

Equation (449) describes a frequency dependence of the Cole–Cole type. The real Re eðoÞ and imaginary Im eðoÞ parts of the total dielectric permittivity in Eq. (449) are, respectively, 2

3 h pai ð1  ZÞ 1 þ ðotÞa cos 6 7 2 ReeðoÞ ¼ e0 4Z þ 5; pa a 2a þ ðotÞ 1 þ 2  ðotÞ  cos 2 2 3 h pai a ðZ  1Þ 1 þ ðotÞ sin 6 2 7 ImeðoÞ ¼ e0 4Z þ 5 pa a þ ðotÞ2a 1 þ 2ðotÞ cos 2

ð450Þ

Therefore, the dielectric loss tangent is 2 6 tan d ¼ ðZ  1Þ4

3 ðotÞ

a

7 5 pa 2a þ ðotÞ 1 þ 2  ðotÞ  cos 2 a

ð451Þ

where Z ¼ ee10 . Equations (450) and (451), respectively, have been used to construct plots of the real, Re eðoÞ=e0 (Fig. 66b), and of the imaginary, Im eðoÞ=e0 (Fig. 66c), parts of the complex dielectric permittivity, as well as tan d(Fig. 66a) as a function of log ot for a medium with Z ¼ ee10 ¼ 10. As is easily verified, the relaxation spectrum pattern strongly depends on the dimension of a temporal fractal ensemble a ¼ df . Now we will try to summarize more of our results. Instead of (3.210), we consider yet another operator of fractional differentiation.

ðt

a

  v þD Þ ¼ ðt Þ DaðvnÞ n n¼0 a v

1 X

  v where is the binomial coefficient. n

a n

ð452Þ

vitaly v. novikov

244 (a)

(b)

1 0.8 0.6 0.4 α 0.2

–2 0 log10 (wt)

2

Re

( ε(w) ε 0

(

4 3 tan(d) 2 1 0

10 7.5 5 2.5 0 –2 log10 (wt)

0

1 0.8 0.6 0.4 α 0.2 2

0

0 (c)

(

(

Im

ε(w) ε0

10 7.5 5 2.5 0

1 0.8 0.6 0.4 α 0.2

–2

0 log10 (wt)

2

0

Figure 66. Dispersion dependence of tan(d) (a), Re eðoÞ=e0 (b), Im eðoÞ=e0 (c) for different values of the parameter a.

By Eq. (452), the complex susceptibility may be written as ðta þ Da Þv ½w expðiotÞ ¼

w 0 E0 expðiotÞ tav

ð453Þ

The solution of Eq. (453) yields the standard definition of complex susceptibility, wðioÞ ¼ w1 þ

w 0  w1 v ð1 þ ðiotÞa Þ

ð454Þ

which is identical to the empirical Havriliak–Negami law [184]. Here the dielectric permittivity will be eðioÞ ¼ e1 þ

e0  e1 v ð1 þ ðioÞa Þ

ð455Þ

physical properties of fractal structures

245

with real and imaginary parts, 13 ap sin 6 C7 B 2 cos4v  arctan@ A5 ap a cos þ ðotÞ 2 ; Re½eðioÞ ¼ e1 þ ðe0  e1 Þ ap 2a 1 þ ðotÞ þ2ðotÞa  cos 2 0 132 ap sin 6 B C7 2 sin4v  arctan@ A5 ap a þ ðotÞ cos 2 Im½eðioÞ ¼ ðe0  e1 Þ ap 2a 1 þ ðotÞ þ2ðotÞa  cos 2 2

0

ð456Þ

and 2

tan d¼

0

13 ap sin 6 B C7 sin4v  arctan@ ap 2 A5 a cos þ ðotÞ 2 2 0

13 ap h i sin e1 ap 6 C7 B þ cos4v  arctan@ ap 2 1þ ðotÞ2aþ2ðotÞa  cos A5 a 2 e0  e1 cos þ ðotÞ 2 ð457Þ

The relaxation equation for an initially polarized dielectric is ðta þ Da Þv PðtÞ ¼ 0

ð458Þ

with solution

PðtÞ ¼ P0 tav1

1 X n¼0

 t an

ð1Þn

vðv þ 1Þ    ðv þ n  1Þ t  n! ðaðn þ vÞ þ 1Þ

ð459Þ

where P0 is the initial polarization (this solution also diverges as t ! 1). The case of a dielectric without initial polarization (when the field is switched on at t ¼ 0) is described by Eq. (460), namely, ðta þ Da Þv PðtÞ ¼

w0 E tav

ð460Þ

vitaly v. novikov

246 U

∆U2 ∆U1 ∆U0

r Figure 67.

Schematic of the potential barrier landscape under the action of an external electrical

field.

with solution  t aðnþvÞ vðv þ 1Þ    ðv þ n  1Þ t ð1Þn PðtÞ ¼ w0 E n!  ð a ð n þ v Þ þ 1Þ n¼0 1 X

ð461Þ

Fractal Model. We consider in more detail the issue of relaxation times relevant to our fractal model of anomalous relaxation. As appears from the potential energy landscape for a system under the action of an external electrical field (Fig. 67), the energy differences between minima separated by energy maxima of different levels of self-similarity diminish, the larger the number of a selfsimilarity level. In view of the standard definition of a relaxation time, t eU=kT , one may write the following chain of inequalities, t0 > t1 > t2 > . . . > ti > tiþ1

ð462Þ

It is easy to see that these inequalities meet the model requirement of the difference between relaxation times in so far as both the width and the height of the energy maxima are assumed to decrease, the larger the number of a selfsimilarity level (Fig. 65). Note, however, the existence of the upper limit for an ensemble of relaxation times; that is, the self-similar process of growing complexity of the potential energy landscape is halted at a certain level N < 1. Thus, the proposed fractal relaxation model satisfies the criterion of selfsimilarity; moreover, its validity is restricted by the asymptotic lower and upper

physical properties of fractal structures

247

limits. Let us now analyze the physical meaning of a self-similar potential energy landscape. Assume that a system evolves by traversing a succession of potential barriers, each next one of greater height. In this context, initial relaxation processes with short relaxation times are assumed to be followed by those with ever increasing relaxation times. Now consider a relaxing ensemble of N < 1 particles. Let this system consist of smaller subsystems (clusters), each of which, in turn, consists of still smaller subsystems (subclusters), and so on. In principle, this kind of tessellation could be repeated down to infinitesimal scales; however, as mentioned above, the accepted model requires that such a self-similar increase of system complexity should halt at a certain level. In other words, the relaxation at the (n þ 1)st level would not set in until a certain fraction of particles at the previous nth level would have relaxed (here it is implicitly assumed that the enumeration of relaxation levels starts at the lower limit of self-similarity for a subcluster comprising a minimum number of relaxing particles, n ¼ 1, and attains the upper limit of self-similarity for a cluster comprising all smaller subclusters, n ¼ N, where N is the total number of hierarchical levels. Let oq be the probability of existence of each relaxation level corresponding to the kth statistical ensemble; then, the probability for a system to pass from the qth to the pth relaxation level during a time t may be defined as   t Sqp ðtÞ ¼ exp  tqp

ð463Þ

where tqp - is a relaxation time defined as tqp

  Qqp ¼ t0 exp kT

ð464Þ

and Qqp is the barrier height between levels q and p. The probability for a system to attain the level n  N after time t will be SðtÞ ¼

n X

oq op Sqp ðtÞ

ð465Þ

q¼1;p¼1

that is, the function SðtÞ is assumed to account for contributions from all available relaxation channels. Therefore, the relaxation process may be specified, provided the functions oq and Qqp are known. We remark that the self-similar potential energy landscape (Fig. 68) resembles a Cayley tree (Fig. 68), provided each minimum at a certain self-

vitaly v. novikov

248 U1

·

r1

n1

· · · ·

r2

n2

r3

n3

U2

U3

Figure 68.

Schematic of a self-similar potential energy landscape and of the Cayley tree.

similarity level of the former is associated with a branch of the same number on the latter [223]. The coordinates of branches on the Cayley tree comprise the ultrametric space; the metric of this space is specified by the interbranch distances which are defined as the numbers of steps between the branches and a common origin (for example, the distances between branches a and b and between a and c in Fig. 6b are unity and two, respectively). Figure 68 indicates that the statistical ensembles fq; pg may merge into clusters, each of which is characterized by the maximum barrier height Qqp separating this particular cluster from its neighbor. In view of the correspondence between the ensembles fq; pg and the branches of the Cayley tree referred to above, the former may be also characterized by points fq; pg in the ultrametric space separated by the distances lqp . In this context, the barrier heights Qqp , as well as the corresponding relaxation times tqp are functions of distances lqp in the ultrametric space. Insofar as elimination of clusters from this space may be achieved by increase of the corresponding barrier heights, one may conclude that QðlÞ should be a smoothly increasing function. It follows from the above analysis that parallel contributions of various relaxation channels may be possible only via hierarchical subordination of the corresponding series of statistical ensembles. Now, the smallest statistical ensembles (subclusters) merge, and the system passes on to a higher hierarchical level of the Cayley tree (Fig. 68). After passing over barriers of higher height, Qqp , the newly created clusters merge again into larger entities (superclusters) corresponding to the next hierarchical level, and so on. It is this kind of hierarchical subordination which is believed to be the main cause of the

physical properties of fractal structures

249

critical slowing down of the relaxation process which manifests itself as the transformation of exponential, Debye-like behavior into a slower, nonexponential decay. Within the framework of phenomenological approach, consider possible patterns of temporal dependence SðtÞ at variable distribution upper bounds oðlÞ and QðlÞ. Assume that the descending tails of the probability distribution may be approximated as   l ow ðlÞ exp  ; l0

os ðlÞ ldf

ð466Þ

where the first and the second functions apply to weakly hierarchical and to strongly hierarchical systems, respectively (here l0 and D are positive parameters). The reason is that the former exponential function ow ðlÞ decays at distances l l0 and, therefore, links only a limited number of hierarchical levels, while the latter, slowly changing power dependence os ðlÞ accounts, in effect, for the entire set of levels available. The landscape barrier heights QðlÞ will be approximated by three major types of functions, namely, Ql ðlÞ ¼ Q ln l;

Qp ðlÞ ¼ Qla ;

Qe ðlÞ ¼ Qel

ð467Þ

where Q is the characteristic barrier height, and a ¼ const > 0. The asymptotics at t ! 1 derived by the saddle-point method on substitution of Eq. (467) into eqs. (464) and (463), and of the result obtained using Eq. (466) and eq. (465), are shown in Table XI. It can be seen that all relaxation laws derived in this fashion are non-Debye-like, the weakest slowing down corresponding to a logarithmic growth of landscape heights in weakly hierarchical systems (i.e., the Kohlrausch–Williams–Watts stretched exponential law [179,180]). The descreasing function SðtÞ transforms into a power law as the hierarchical links become stronger and the increase of peaks on the

TABLE XI Asymptotics of the Correlator SðtÞ at t ! 1 SðtÞ

Ww Ws ðlÞ

Ql ðlÞ

   b exp  tt0 ;...;

 1 Q : b¼ 1þ kT tg ; g ¼

df T Q

Qp ðlÞ

 1=a  t exp  kT Q ln t0 

Q kT

ln tt0

df =a

Qe ðlÞ 

kT Q

ln tt0

1=l0

h  idf t ln kT Q ln t0

250

vitaly v. novikov

landscape follows a power law. The alternative cases of exponential and powerlaw increases of the barrier heights in weakly and strongly hierarchical systems, respectively, would correspond to a logarithmic relaxation law, as described elsewhere [185]. Finally, a double-logarithmic slowing down (i.e., the virtual arrest) of the relaxation process is expected for strongly hierarchical systems characterized by exponential growth of barrier heights. It is instructive now to discuss the relevance of these model predictions to the structural features of inhomogeneous media. Thus we define a statistical ensemble as a set of particles in a state of constant motion. The model of hierarchical subordination was then constructed by selecting groups of ensembles from the entire statistical set available ; hence, a similar tessellation procedure should be applied to the ensemble of particles comprising an inhomogeneous medium. A rule for selecting particular clusters and subclusters in the latter should be developed. In simple terms, solid bodies may be characterized by two main features, by the pattern of their mutual arrangement (packing) of particles, and by the pattern of interparticle interactions. These features are complementary, rather than independent; nevertheless, it is the former which will be used as a criterion for partitioning the entire inhomogeneous medium system into smaller subsystems. According to current concepts, an inhomogeneous medium may be considered as a structureless body at large length scales (i.e., above the characteristic correlation length x), whereas regions of a short-range order are assumed to exist at smaller scales (below x). In this context, it is the regions of short-range order which will be identified as the primitive (1st level) clusters; a new set of primitive clusters will be defined as the 2nd level clusters, and so on. Thus, the nth level cluster corresponding to the statistical ensemble of nth hierarchical level may be constructed using such a process of self-similar increase in complexity. The ‘‘blob’’ model based on these concepts has been introduced elsewhere [196]. It is now possible to establish a correspondence between the functions oðlÞ and QðlÞ and the accepted model of short-range hierarchy order. From Fig. 69, the primitive cluster comprises 7 particles, the next one at the 2-nd level 72 ¼ 49 particles, and the lth level cluster 7l particles (here l is the level number). Obviously, for a level l comprising N ¼ 7l particles, the level number may be defined as l ¼ a  lnðnÞ; a

1

¼ ln 7

ð468Þ

In the general case, l  ln N; therefore, eq. (466) may be rewritten as ow ðNÞ  N df ;

os ðNÞ  ðln N Þdf

ð469Þ

physical properties of fractal structures

Figure 69.

251

Schematic of the self-similar structure of a dielectric medium.

The results obtained imply a rather small probability of large-size clusters comprising many particles for a weakly hierarchical system; the reverse is true for a strongly hierarchical system. Thus, the structural implications of the concept of strongly and weakly hierarchical systems become more transparent. The physical meaning of the function accounting for the increase of potential barrier height may be clarified by considerating the microscopic kinetics of dielectric relaxation of a hierarchical structure. Assume that the initial polarization is induced in the latter, and that single particle and clusters of particles interact through dipole and multiple interactions, respectively. The relaxation processes commence after the field is switched off at t ¼ 0. Initially, the relaxation sets in at the primitive, 1st level, insofar as the elementary dipoles can easily cross the potential barrier created by their neighbors. In contrast, for the 2nd-level clusters the barrier heights created by neighboring clusters with preferential orientation of the majority of the dipoles are so high, that no such relaxation can occur. Therefore, relaxation at the 2nd level may set in only after completion of relaxation by the majority of the dipoles at the primitive level. In other words, it is only after sufficient weakening of the multipole correlations of a given cluster with its neighbors that its transition into a depolarized state becomes possible. Such self-similar processes occur in succession at the next higher level, and so on. Thus, the form of the function QðlÞ depends not only on the number of dipoles in a cluster but on the form of the multipole potential and the temperature. Finally, in view of Eq. (468), Eq. (467) can be rewritten as Ql ðNÞ ¼ Q ln½ln N ;

Qp ðNÞ ¼ Qðln N Þa ;

Qe ðNÞ ¼ QN

ð470Þ

252

vitaly v. novikov

The model considerations outlined above permit one to clarify the results presented in Table XI. For example, from the explicit definition of the Kohlrausch–Williams–Watts stretched exponent on the barrier height

 Q 1 b¼ 1þ kT

ð471Þ

it can be Inferred that b ! 1 for T  0; that is, for sufficiently high temperatures the anomalous relaxation becomes Debye-like. Physically, this effect may be associated with the increase of the fluctuation density of dipole reorientations; as a result, the relaxation of all available dipoles has already occurred at the 1st level, so that the entire chain of remaining parallel relaxation channels becomes ineffective. In like manner, it becomes easy to predict the pattern of anomalous, nonexponential relaxation at times, shorter than the relaxation time t1 at the lowest (i.e., 1-st) self-similarity level. This level may be considered as the primitive one (in a sense that it cannot be further tesselated into subclusters); hence, the relaxation should be of a classical, Debye-like type, PðtÞ  et t

ð472Þ

whatever the pattern of nonclassical relaxation at longer times. Conclusions. The power of the fractional derivative technique has been demonstrated using as example the derivation of three known patterns of anomalous, non-exponential dielectric relaxation of an inhomogeneous medium in the frequency domain. It is explicitly assumed that the fractional derivative is related to the dimension of a temporal fractal ensemble (in the sense that the relaxation times are distributed over a self-similar fractal system). The proposed fractal model of microstructure of inhomogeneous media exhibiting nonexponential dielectric relaxation is constructed by selecting groups of hierarchically subordinated ensembles (subclusters, clusters, superclusters, etc.) from the entire statistical set available. In this context, the regions of short-range order are identified as the primitive (1st-level) clusters; a set of primitive clusters are defined as 2nd-level clusters, and so on. Different relaxation functions are derived assuming that the actual (physical) ensemble of relaxation times is confined between the upper and lower limits of self-similarity. In this respect, the temporal fractal differs from a geometrical fractal (e.g., Cantor dust) for which only an upper limit (i.e., the initial segment before its subdivision) is assumed to exist. It is predicted that at times, shorter than the relaxation time at the lowest (primitive) self-similarity level the

physical properties of fractal structures

253

relaxation should be of a classical, Debye-like type, whatever the pattern of nonclassical relaxation at longer times. The methods and models used when describing dielectric relaxation in the previous chapter can also be used to describe diffusion of a Brownian particle. We mention that diffusion has been a focus of great attention for scientists for more than 100 years and has a vast bibliography. Recently, excellent reviews have been published [222–238]. We will only cite a few [239–251]. 2.

Anomalous Diffusion

Brownian Motion. The incessant haphazard motion of microscopic particles in a colloidal suspension is called the Brownian motion, and the particles themselves Brownian particles. The Brownian motion was first experimentally discovered and investigated by the botanist Robert Brown (1773–1858) in 1827, who observed pollen suspended in water through a microscope. The pollen particles moved randomly, and the average value of the movements in unit time did not change for the same parameter values of the fluid—for example, its temperature. Once the temperature increased, the Brownian motion became more intense. To describe the Brownian motion of a particle in a fluid, Langevin assumed [250,251] that a random force, x; affecting the particle having its origin in the particles of the fluid exists and that its autocorrelation function is hxðt1 Þxðt2 Þi ¼ 2kTrdðt1  t2 Þ meaning that x is Gaussian white noise. x must also obey [247,250] Wick’s theorem (Isserlis’s theorem) for the averages over the realizations of the trajectories of the particles. The equation of motion of a Brownian particle in the direction of the chosen axis, X, can then be written as mx00 þ rx0 ¼ x

ð473Þ

where m is the mass of the Brownian particle, r is the viscous drag coefficient of the Brownian particle resulting from the fluid. Now multipling Eq. (473) by x and taking account of the equation xx00 ¼

dðxx0 Þ  x02 dt

ð474Þ

we obtain m

dðxx0 Þ  mx02 þ rxx0 ¼ xx dt

ð475Þ

vitaly v. novikov

254

Averaging Eq. (475) over a large number of Brownian particles, we have m

  dh xx0 i  m x02 þ r h xx0 i ¼ h xxi dt

ð476Þ

Due to the haphazard motion of the Brownian particle, it may be assumed that the coordinate, x, and the force, x, are statistically independent, and the average value of their product is equal to zero: h xxi ¼ 0: Moreover, one may write assuming that the velocity distribution has reached equilibrium instantaneously 1  02  1 m x ¼ kT 2 2

ð477Þ

Then Eq. (476) becomes m

d h xx0 i þ r h xx0 i ¼ kT dt

ð478Þ

If at time t ¼ 0 the Brownian particle is at a point with coordinate x ¼ 0, then h xx0 ijt¼0 ¼ 0: Hence, the solution of Eq. (478) can be written as h xx0 i ¼

 t  kT  1  exp  r t

ð479Þ

where the inertial relaxation time, t ¼ r=m, is small compared with the time of observation of the Brownian particle. For t >> t, which corresponds to the stationary Brownian motion (i.e., the Maxwellian distribution of velocities has been attained by the Brownian particle), we have h xx0 i ¼

kT r

ð480Þ

Using the equation 1 dðx2 Þ 2 dt

ð481Þ

  d x2 kT ¼2 dt r

ð482Þ

xx0 ¼ we may transform formula (480) into

physical properties of fractal structures Integrating (482) with the initial condition mean-square displacement is  2 x ¼ 2Dt;

255

 2 x jt¼0 ¼ 0 shows that the

t>0

ð483Þ

where the diffusion coefficient of the Brownian particle is D¼

kT r

ð484Þ

Formula (483) was first obtained by Albert Einstein (1879–1955) in 1905 and bears his name. Independently of Einstein, the theory of the Brownian motion was developed by Marian von Smoluchowski (1872–1917) in 1905–1906. The expression obtained by him agrees with formula (483) with a constant multiplier equal to one. Thus, it follows from formula (483) that the mean-square distance traversed by a Brownian particle is proportional to the observation time. In accordance with formula (484), the mean-square distance traversed by a Brownian particle in equal periods of time increases with temperature T and decreases with the viscosity r. The above formulae were experimentally checked by Perrin in 1908. He measured the distances traversed by Brownian particles for equal periods of time with a microscope. Based on his experiments and formulae (483) and (484), Perrin was able to define the Boltzmann constant, k, and calculated the value of Avogadro’s number NA, both closely approximating their values obtained by other methods. Diffusion (Smoluchowski) Equation for Brownian Particles. The concentration pðx; tÞ of free Brownian particles at time t is described by the diffusion equation [250] qpðx; tÞ q2 pðx; tÞ ¼D qt qx2

ð485Þ

D is the diffusion coefficient. The diffusion equation (485), which is usually known as the Smoluchowski equation (a particular form of the Fokker–Planck equation), can be obtained using the equation of continuity in one dimension qpðx; tÞ qjðx; tÞ ¼ qt qx

ð486Þ

256

vitaly v. novikov

and the Fick law jðx; tÞ ¼ D

qpðx; tÞ qx

ð487Þ

If at t ¼ 0 the particle is at the origin, x ¼ 0, then the initial concentration is pðx; 0Þ ¼ dðxÞ

ð488Þ

and the solution of Eq. (485) is then   1 x2 pðx; tÞ ¼ pffiffiffiffiffiffiffiffi exp  ; 4Dt 2 pDt

t>0

ð489Þ

or pðx; tÞ ¼ Ct2  f ðx; Ct2 Þ 1

1

ð490Þ

where C ¼ p1ffiffiDffi, and the function

    1 z 2 f ðzÞ ¼ pffiffiffi exp  2 2 p

ð491Þ

is the centered Gaussian distribution. N(0,2). According to Eq. (491), the variance of the distribution is  2 x ¼ 2Dt

ð492Þ

The diffusion equation can also be written as the forced equation qpðx; tÞ q2 pðx; tÞ ¼D þ dðxÞdðtÞ qt qx2

ð493Þ

where now of course the initial conditions are zero. This form is useful in the generalization of Eq. (485) to fractional diffusion. The investigation of the diffusion equation (485) began when Louis Bachellier (Jules Poincare’s student) wrote his thesis in 1900. It was called ‘‘The Theory of Speculations’’ and was devoted to the evolution of the stock market. Many of the most famous scientists have contributed to our knowledge of diffusion processes, amongst them Einstein, Langevin, Smoluchowski, Fokker, Planck, Levy, and others.

physical properties of fractal structures

257

The investigation of Brownian particle motion led to the development of the mathematical theory of random processes which has been widely adopted. A great contribution to the mathematical theory of Brownian motion was made by Wiener and Kolmogorov; for example, Wiener proved that the trajectory of Brownian motion is almost everywhere continuous but nowhere differentiable, Kolmogorov introduced the concept of forward and backward Fokker-Planck equations. As far as contemporary research is concerned in recent years, many works devoted to anomalous diffusion have appeared [231–242]. The main difference of anomalous diffusion from normal diffusion is that the variance of the distribution obeys:  2 1 ð494Þ H 6¼ x  tH ; 2 To describe anomalous diffusion, some additional information about the diffusion process is needed, for example:  specific physical models for the jump processes;  the imposition of certain conditions, for example, the self-similarity condition, according to which pðx; tÞ ¼ tH Fðx; tH Þ

ð495Þ

where H is not equal to 1=2, and f ðxÞ is not a Gaussian distribution. We consider the main results which describe anomalous diffusion. Anomalous Diffusion. The standard random walk process underlying the Brownian motion assumes that the walker executes a step of fixed length in a random direction at each tick of a system clock [226]. After a very large number of steps the associated random variable—namely the position of the walker a time t after he departed from the origin—will be a gaussian random variable by virtue of the central limit theorem [226] and so leads to the normal diffusion discussed above. A generalization of this process is the continuous time random walk introduced by Montroll and Weiss [231]. Such a walk is defined to be passing [231] an alternation of steps and pauses with both step length and pausing (waiting) times instead of being fixed in same way being governed by a step length probability distribution and a pausing time probability distribution. Thus the step length and the pausing time are random variables which are not necessarily independent although we shall assume that they are here to the diffusion limit of a large number of steps, anomalous diffusion will occur either if no average pausing time or no mean-square step length exists or if both do not exist. We consider the CTRW using the terminology of renewal theory [236].

258

vitaly v. novikov

Consider a sequence of independent identically distributed time random variables, T1 ; T2 ; . . . ; the probability that any single one of the T s satisfies t  T  t þ dt will be denoted by qðtÞdt where qðtÞ is the pdf of the time between successive steps or the pausing time density. The time intervals TðnÞ at which the nth step is taken are given by the random sum TðnÞ ¼

n X

Tj ;

Tð0Þ ¼ 0

ð496Þ

j¼1

and are called renewal (because the jump process rests) times, fTðnÞg is the set of random variables constituting the renewal process, the individual Tj are called pausing (expectation) times since Eq. (496) is a sum of independent random variables a number of relations may be simplified when expressed in terms of characteristic functions as we shall do later. Now let us suppose that in a time interval ½0; t the random walker executes n steps, and then the displacement of the walker is the random sum XðtÞ ¼

NðtÞ X

Rj

ð497Þ

j¼1

where the step lengths Rj are independent random variables each Rj having the same step length pdf pðxÞ. As an illustrative example, suppose NðtÞ is a Poisson process; then the probability that walker has moved exactly n steps after an elapsed time t is Pð N ð t Þ ¼ nÞ ¼

ðmtÞn expðmtÞ n!

ð498Þ

where m is the mean rate at which steps occur, the pdf pðxÞ of the random sum XðtÞ (i.e., the displacement of the walker at time t) is then pð x; tÞ ¼

1 X ðmtÞn n¼0

n!

expðmtÞpn ðxÞ

ð499Þ

pn ðxÞ is the pdf for the position of the random walker at the step n, p0 ðxÞ means that the walker was definitely at zero at time zero and stays there until t ¼ T1 (the time at which the first jump occurs) p1 ðxÞ ¼ pðxÞ (the jump length pdf) for the position of the walker at step one, and pn ðxÞ ¼ pn ðxÞ (since the jump lengths are independent identically distributed random variables is the pdf for the position of the walker at step n).

physical properties of fractal structures

259

In passing we remark that Feller [218] replaces the process fX1 ¼ R1 , X2 ¼ X1 þ R2 , X3 ¼ X2 þ R3 ; . . .g with independent increments Rj by the Markov chain fX1 ; X2 ; X3 ; . . .g and calls this process pseudo-Poissonian. Subordinate Processes. The pdf of the pseudo-Poissonian process is defined as pð x; tÞ ¼

1 X

Wn0 ðtÞpð x; nÞ ¼ hpð x; N ðtÞÞi

ð500Þ

n¼0

where Wn0 ðtÞ ¼

ðmtÞn expðmtÞ n!

ð501Þ

and pð x; nÞ ¼ pn ðxÞ

ð502Þ

Equation (502) is interpreted as the pdf of a process that occurs at integer times n (the operational time). Feller calls the random variable NðtÞ the randomized operational time. The operational times need not necessarily be discrete and need not be distributed according to the Poisson law, which has been only used for illustration purposes. For example, suppose pðx; tÞ is the transition probability for a Markov process fXðtÞg, and wðy; tÞ dy is the randomized operational time distribution denoted now by fTðtÞg concentrated on the positive semiaxis; then the pdf pðx; tÞ for the process fXðtÞg is 1 ð

pð x; yÞwðy; tÞdy

pð x; tÞ ¼

ð503Þ

0

We call this process a subordinate process. If fXðtÞg is a Markov process with continuous transition probabilities and fTðtÞg a process with non-negative independent increments, then fXðTðtÞÞg is also a Markov process. Thus, this process is subordinated to fXðtÞg with operational time fTðtÞg. The process fTðtÞg is called a directing (controlling) process. When the process fXðtÞg has independent increments, we again arrive at the above formula for pðx; tÞ. In particular, if fXðtÞg is Brownian motion with

vitaly v. novikov

260 transition probability

 2 x pð x; tÞ ¼ ð2ptÞ1=2 exp  2t

and the directing process is the Smirnov–Le´vy process  2 t t wðy; tÞ ¼ pffiffiffiffiffiffiffiffiffi3ffi exp  2y 2py

ð504Þ

ð505Þ

then the subordinate process pðx; tÞ has the Cauchy distribution t pð x; tÞ ¼ 2p

1 ð

 2  x þ t2 t y2 exp  dy ¼ pð t 2 þ x 2 Þ 2y

ð506Þ

0

Note that a renewal process with transition probability qðtÞ , which is not necessarily a process with independent increments, can also be chosen as the controlling process fTðtÞg. Thus, assuming that the random variables Rj and that their pausing time density qðtÞ is arbitrary in the system under examination, we obtain pðx; tÞ ¼

1 X

Wn0 ðtÞpn ðxÞ

ð507Þ

n¼0

where Wn ðtÞ ¼ PðN ðtÞ ¼ nÞ ¼ PðN ðtÞ  nÞ  PðN ðtÞ  n þ 1Þ n

¼ Q ðtÞ  Q

ðnþ1Þ

n

ð508Þ



ðtÞ ¼ Q ½1  Q 

Hence [cf. Eq. 507] of Ref. 244 becomes 1 h i X Qn ðtÞ  Qðnþ1Þ ðtÞ pn ð xÞ pð x; tÞ ¼

ð509Þ

n¼0

Such models of the one dimensional random walk of a particle with expectation times distributed independently according to the same pausing time law qðtÞ and independent increments (both from each other and from the expectation times) distributed with equal density pðxÞ are, as we have seen, are called Continuous-Time Random Walks. The estimation of pðx; tÞ namely the pdf associated with the position XðtÞ of the random walker at time t in the subordinate process, in the above situation is called the Montroll–Weiss problem. Its solution may be obtained by Fourier

physical properties of fractal structures

261

transformation over coordinates and Laplace transformation over time that is by using the characteristic functions: 1 ð

pðk; sÞ ¼

1 ð

dx expðikx  stÞpð x; tÞ

dt 0

ð510Þ

1

where Fourier transformation over space variables is defined by 1 ð

expðikxÞf ð xÞdx

F f f ð xÞg ¼

ð511Þ

1

f ðkÞ ¼ F f f ð xÞg ¼ heikx i

ð512Þ

and Laplace transformation over time variables is defined by 1 ð

L f f ðt Þg ¼

expðstÞ f ðtÞdt

ð513Þ

f ðsÞ ¼ Lf f ðtÞg ¼ hest i

ð514Þ

0

Knowing by the properties of the characteristic function of independent identically distributed random variables that 1 ð

expðikxÞpn ð xÞdx ¼ ½pðkÞn

ð515Þ

1

We have like manner in terms of the Laplace transform qðsÞ of the pausing time pdf qðtÞ 1 ð

1 ð

n

expðstÞQ ðtÞdt ¼ 0

ðt

dt expðstÞ dsqn ðsÞ

0 1 ð

¼

0 1 ð

ds s

0

¼

1 s

dt expðstÞqn ðsÞ

1 ð

0

1 expðstÞqn ðtÞdt ¼ ½qðsÞn s

ð516Þ

vitaly v. novikov

262

again by the properties of the moment generating function of independent identically distribution random variables. Thus, we obtain the characteristic function of the subordinate process, namely, pðk; sÞ ¼

1   1X 1  qn ð s Þ qn ðsÞ  qnþ1 ðsÞ pn ðkÞ ¼ s n¼0 s½1  qn ðsÞpn ðkÞ

ð517Þ

Hence using the Bromwich integral (complex inversion formula for the Laplace transform) and the inverse Fourier transform we have

pð x; tÞ ¼

cþi1 ð

1 ð

1

dk

2

ð2pÞ i

1

ds

1  qn ð s Þ expðikx þ stÞ s½1  qn ðsÞpn ðkÞ

ð518Þ

ci1

According to Eq. (517), we have pðk; sÞ ¼

1  qn ð s Þ s½1  qn ðsÞpn ðkÞ

ð519Þ

The estimation of the subordinate density pðx; tÞ using formula (518) demands that the form of the distributions qðtÞ and pðxÞ should be given. Their estimation is a rather complicated task in the general case. Let us consider, however, the asymptotics of qðtÞ and pðxÞ as t ! 1. In order to obtain an evolution equation describing anomalous diffusion, we will use the characteristic function of pðx; tÞ. To calculate the asymptotics of the process under study, we use the stable densities gðx; b; yÞ and gðx; a; 1Þ, 0 < b  2; a  1 as the functions pðxÞ and qðtÞ which are defined below. The behavior of the characteristic function,  pðk; sÞ for small values of the arguments (i.e., at large x and t), is according to Tauberian theorems in terms of the functions just mentioned [218,239] pðkÞ ¼ gðk; b; y ¼ 1Þ




 ibp ibp b b sign k  1  jkj exp  sign k ; ¼ exp jkj exp  2 2

k!0 ð520Þ

and qðsÞ ¼ gðis; a; 1Þ ¼ expðsa Þ  1  sa ;

s!0

ð521Þ

physical properties of fractal structures

263

In accordance with (519), we have  pðk; sÞ½1  qn ðsÞpn ðkÞ ¼

1  qn ðsÞ s

ð522Þ

Taking Eqs. (520) and (521) into account, we obtain

 ibp sa pðk; sÞ ¼ jkjb exp  signk pðk; sÞ þ sa1 2

ð523Þ

On inverse Fourier transformation over k we have sa pð x; sÞ ¼ 

qb pð x; sÞ þ sa1 qxb

ð524Þ

Here it is supposed, that the Fourier transform of a fractional derivative qb pð x; sÞ qxb

ð525Þ

is (

qb pð x; sÞ F qxb

) ¼ ðikÞb pðk; sÞ

ð526Þ

so that qb pð x; sÞ 1 ¼ b qx 2p

1 ð

ðikÞb pðk; sÞ expðikxÞdk

ð527Þ

1

The product sa pð x; sÞ and the function sa1 are the Laplace transforms of the Riemann–Liouville fractional derivative ðt qa pð x; tÞ 1 q pð x; tÞ ¼ dt qta ð1  aÞ qt ðt  tÞa

ð528Þ

0

and the generalized function ta dð x Þ ð1  aÞ

ð529Þ

vitaly v. novikov

264 respectively, that is,


qa pð x; tÞ L qta




 ¼ s pð x; sÞ; a

ta L dð x Þ ð1  aÞ

 ¼ sa1

ð530Þ

Hence, the distribution pðx; tÞ, as t ! 1, satisfies the fractional partial differential equation qa pð x; tÞ qb pð x; tÞ ta dð x Þ ¼ D þ ab qta qxb ð1  aÞ

ð531Þ

which is called the anomalous diffusion equation. Here the scale of the variables x and t is chosen in a special way, and Dab is a positive constant (the anomalous diffusion coefficient). From (531) it follows that the diffusion equation for b ¼ 2 is qa pð x; tÞ q2 pð x; tÞ ta dð x Þ ¼ Dab þ a 2 qt qx ð1  aÞ

ð532Þ

Various aspects of such fractional differential equations have been studied in [205–208]. We shall now give an example of the solution of such an equation. 3.

Distribution Function of a Brownian Particle with Memory

We consider the pdf of the displacements of a Brownian particle in a process characterized by an equation like (532). The normal Fokker - Planck equation would now be (here the diffusion coefficient is denoted by B) qW ð x; tÞ q2 W ð x; tÞ ¼B qt qx2

ð533Þ

If we include anomalous diffusion of the pausing times the corresponding anomalous diffusion equation is [237] qa W ð x; tÞ dð x Þ q2 W ð x; tÞ ¼ B  qta ta ð1  aÞ qx2

ð534Þ

The boundary conditions for this equation are Wðx; tÞjt¼1 ¼ 0

ð535Þ

The initial condition as in the normal Brownian motion is Wðx; tÞjt¼0 ¼ dð xÞ

ð536Þ

physical properties of fractal structures

265

We apply Fourier transformation to Eq. (534). Hence 1 ð

W ðo; tÞ ¼

W ð x; tÞ expðioxÞdx

ð537Þ

1

so that qa W ðo; tÞ 1 þ Bo2 W ðo; tÞ ¼  a qta t ð1  aÞ

ð538Þ

According to the properties of fractional derivatives, from [227] the initial condition of Eq. (538) must be qa1 W ðo; tÞ jt¼0 ¼ b qta1

ð539Þ

where b is a constant. The following clear explanation of this condition can be given. The solution of Eq. (538) can be represented as the series W ðo; tÞ ¼

1 X

An ðoÞtnða1Þ

ð540Þ

n¼1

and only in this way by using the inverse Fourier transform of (539) can we obtain the initial condition for Eq. (534), namely, qa1 W ð x; tÞ jt¼0 ¼ bdðxÞ qta1

ð541Þ

The solution of Eq. (538) is, using the results of Ref. 217,     W ðo; tÞ ¼ bta1 Ma;a o2 Bta þ Ma;1 o2 Bta

ð542Þ

where Ma;b ðzÞ is the generalized Mittag–Leffler function [217] Ma;b ðzÞ ¼

1 X

zk ðak þ bÞ k¼0

ð543Þ

We now express Mittag–Leffler function in terms of the Fox function [216] and using the formulae from Eq. (543) apply the inverse Fourier transformation

vitaly v. novikov

266

to Eq. (542). Thus, the distribution function Wðx; tÞ is # "  rffiffiffi   p a1 2;0 x2 a2 ; a   W ð x; tÞ ¼ b t 2 H1;2 a B 4Bt 2 ð0; 1Þ 12 ; 1 "  #   a 2 1  ; a 1 x 2;0 2 1  þ pffiffiffiffiffiffiffiffiffiffi H1;2 a a 2 4Bt ð0; 1Þ 2 pBt 2;1

ð544Þ

Now since Wðx; tÞ is a pdf we have 1 ð

W ð x; tÞdx ¼ 1

ð545Þ

1

From Eq. (544) we obtain 1 ð

W ð x; tÞdx ¼ 1

1 2;0 W ð x; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi H1;2 2 pBta



bta1 2p þ1 ðaÞ

ð546Þ

   x2 1  a2 ; a a 4Bt 2 ð0; 1Þ

 1  ;1

ð547Þ

   1  a2 ; a ð0; 1Þ

# 1  2;1

ð548Þ

2

Hence b ¼ 0 so that finally 1 2;0 W ð x; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi H1;2 2 pBta

"

x2 a 4Bt 2

For a ¼ 1, we have classical diffusion. Otherwise the dependence on time is a power law one. In order to see this consider the asymptotic behavior as t ! 1: 1 lim W ð x; tÞ  pffiffiffiffia t!1 t

ð549Þ

and as t ! 0  2 x exp  a t pffiffiffiffi lim W ð x; tÞ  t!0 ta

ð550Þ

physical properties of fractal structures

267

Formula (3.328) shows that the diffusion is anomalous, that is a power low dependence on time exists. When a ¼ 1, Eq. (550) becomes Gaussian thus normal diffusion. All the above results are obtained assuming that the Maxwillian distribution of velocities is reached instantaneously by the ensemble of Brownian particles. In other words, the inertia of the particles is ignored. 4.

Inertial Effects of a Brownian Particle

We consider the Brownian motion when the assumption that the ensemble of Brownian particles is instantaneously thermalized is abandoned, in terms of simple models. Random Walks on a One-Dimensional Lattice. We consider [249, 251] stable random walks on a one-dimensional lattice. Here the particle moves at random, and the direction is defined by the direction of the previous step. Each step is only carried out to the nearest neighbor. The mathematical definition of stability demands that at any time and position on the lattice of the wandering particle, two previous coordinates and the direction of the previous step be known. To describe the random walk process, we consider two probabilities, ð1Þ ð2Þ ð1Þ fpn g and fpn g, where pn is the probability to be at place j at step ð2Þ n from place j  1 at the previous step. pn is the same probability but from place j þ 1. ð2Þ pnð1Þ ¼ pð1Þ n ð j  1Þp11 þ pn ð j  1Þp12 ; ð2Þ pnð2Þ ¼ pð1Þ n ð j þ 1Þp21 þ pn ð j þ 1Þp22

ð551Þ

where p11 ¼ p22 p0 is the probability of the transition when moving in the same direction as that at the previous step, and p12 ¼ p21 q ¼ 1  p0 is the probability of the transition in the opposite direction. For the conventional random walk, p0 ¼ 1=2. Equation (551) is an example [250,251] of a persistent random walk. By correctly scaling p0 , a diffusion equation for the persistent random walk can be obtained from the recurrence relations of Eq. (551). One of the possible variants of scaling is examined when changing to continuous variables x and t. To do this, we define x and t for all j and n: x ¼ jx;

t ¼ nt

ð552Þ

In the limit t; x ! 0 lim

t;x!0

x ¼v t

ð553Þ

vitaly v. novikov

268

In order to construct partial differential equations, we can scale the probability p0 as p0 ¼ 1 

t 2t

ð554Þ

where t ¼ mg is now the time between particle collisions. If t ! 0, then p0 ¼ 1=2, and we have the usual random walk process (i.e., initial effects are ignored). If t 6¼ 0, then we have a process with memory, i.e. the particle ‘‘remembers’’ its state at the previous step (the position and direction of motion). Stability embodies the fact that p0 differs but little from 1. Equations (552) and (553) are now substituted into the recurrence equations (551) yielding  t ð1Þ t ð2Þ p ð x  x; tÞ; p ð x  x; tÞ þ 2t 2t   t ð2Þ t ð1Þ p ð x þ x; tÞ pð2Þ ð x; t þ tÞ ¼ 1  p ð x þ x; tÞ þ 2t 2t pð1Þ ð x; t þ tÞ ¼



1

ð555Þ

Knowing that qpð1Þ ð x; tÞ qpð1Þ ð x; tÞ t þ x; qt qx qpð2Þ ð x; tÞ qpð2Þ ð x; tÞ pð2Þ ð x; t þ tÞ  pð2Þ ð x þ x; tÞ ¼ t  x qt qx

pð1Þ ð x; t þ tÞ  pð1Þ ð x  x; tÞ ¼

ð556Þ

and letting x and t ! 0, we obtain  qpð1Þ ð x; tÞ qpð1Þ ð x; tÞ 1  ð2Þ ¼ v þ p ð x; tÞ  pð1Þ ð x; tÞ ; qt qx 2t  qpð2Þ ð x; tÞ qpð2Þ ð x; tÞ 1  ð1Þ ¼v þ p ð x; tÞ  pð2Þ ð x; tÞ qt qx 2t

ð557Þ

From Eqs. (557) noting that the total probability density function pðx; tÞ is defined as pð x; tÞ ¼ pð1Þ ð x; tÞ þ pð2Þ ð x; tÞ

ð558Þ

we obtain the diffusion equation q2 pð x; tÞ 1 qpð x; tÞ q2 pð x; tÞ ¼ D þ ; qt2 t qt qx2

D ¼ v2 t

ð559Þ

physical properties of fractal structures

269

From Eqs. (557), the diffusive particle flow, jðx; tÞ, is defined. jð x; tÞ ¼ vpð1Þ ð x; tÞ  pð2Þ ð x; tÞ

ð560Þ

To do this, the second equation is subtracted from the first one:     q pð1Þ ð x; tÞ  pð2Þ ð x; tÞ q pð1Þ ð x; tÞ þ pð2Þ ð x; tÞ ¼v qt qx  1  ð1Þ  p ð x; tÞ  pð2Þ ð x; tÞ t

ð561Þ

Therefore, the equation for diffusive particle flow reads jð x; tÞ ¼ D

qpð x; tÞ qjð x; tÞ t ; qx qt

D ¼ v2 t ¼

kT g

ð562Þ

Equation (559) coincides with the Maxwell–Cattaneo equation [254]. Diffusion Equation with Fractional Derivatives. In normal as well as in anomalous diffusion the quantity lim

t;x!0

x t

ð563Þ

is indeterminate. However, consider the quantity

lim

t;x!0

x ¼ va ta

ð564Þ

which in general exists. In this connection, we replace t with ðtÞa in Eqs. (553) and (554); that is, we will measure time not in units t, but ðtÞa . Making the substitution

lim

t;x!0

x x ¼ v ) lim ¼ va ; 0 < a  1; t;x!0 ta t   t 1 t a ) p0 ¼ 1  p0 ¼ 1  2t 2 t

ð565Þ

vitaly v. novikov

270

we arrive at fractional derivatives in the time variable. In fact, Eqs. (557) and (559) can be rewritten as     1 t a ð1Þ ð1Þ p ð x; t þ tÞ ¼ 1  p ð x  x; tÞ 2 t  a 1 t pð2Þ ð x  x; tÞ; þ 2 t ð566Þ     1 t a ð2Þ ð2Þ p ð x; t þ tÞ ¼ 1  p ð x þ x; tÞ 2 t   1 t a ð1Þ þ p ð x þ x; tÞ 2 t Assuming that (see 5.Appendix) qa pð1Þ ð x; tÞ a t qta qpð1Þ ð x; tÞ x; þ qx qa pð2Þ ð x; tÞ a pð2Þ ð x; t þ tÞ  pð2Þ ð x þ x; tÞ ¼ t qta qpð2Þ ð x; tÞ x  qx pð1Þ ð x; t þ tÞ  pð1Þ ð x  x; tÞ ¼

ð567Þ

as x and ta ! 0. We obtain    qa pð1Þ ð x; tÞ qpð1Þ ð x; tÞ 1 1 a  ð2Þ ð1Þ þ ¼ v p ð x; t Þ  p ð x; t Þ ; a qta qx 2 t    qa pð2Þ ð x; tÞ qpð2Þ ð x; tÞ 1 1 a  ð1Þ þ ¼ va p ð x; tÞ  pð2Þ ð x; tÞ a qt qx 2 t

ð568Þ

Then the diffusion equation with fractional derivatives allowing for inertial effects is  a a 2 q2a pð x; tÞ 1 q pð x; tÞ 2 q pð x; tÞ þ ¼ v a qt2a t qta qx2

ð569Þ

The diffusion particle flow or probability current is jðx; tÞ ¼ Da

qpð x; tÞ qa jð x; tÞ  ta ; qx qta

Da ¼ v2a ta

ð570Þ

physical properties of fractal structures

271

The Solution of Equation (559). We remark that inertial effects are considerably influenced by three time intervals [225]:  Interval ð0; t1 Þ—ballistic mode  Interval ðt1 ; t2 Þ—intermediate mode  Interval ðt2 ; 1Þ—Einstein–Smoluchwski region. According to (562), the distinctive feature of the interval ðt1 ; t2 Þ is the deviation from the Fick law, that is, the Maxwell–Cattaneo law is correct rather than the Fick law (487) jðx; tÞ ¼ D

qpð x; tÞ qjð x; tÞ t qx qt

ð571Þ

combined with the continuity equation (571), this law leads to the equation for the pdf pðx; tÞ, namely, qpð x; tÞ q2 pð x; tÞ q2 pð x; tÞ þt ¼ D : qt qt2 qx2

ð572Þ

As we have already stated, Eq. (572) is called the telegraph equation.1 Applying Laplace transformation to Eq. (571), we obtain jð x; sÞ ¼ D

qpð x; sÞ  stjð x; sÞ qx

ð573Þ

or jð x; sÞ ¼ DðsÞ

qpð x; sÞ qx

ð574Þ

where the diffusion coefficient (memory function) is DðsÞ ¼ DkðsÞ; kðsÞ ¼

1 1 þ ts

ð575Þ

1 Editor’s Note: Severe criticisms have been concerning the use of the telegraph equation as an approximation to the configuration space distribution function yielded by the exact Fokker–Planck (Klein–Kramers) equation in phase space in order to describe inertial effects in the Brownian motion see H. Risken, The Fokker–Planck Equation, Springer-Verlag, Berlin, 1984, 1989, pp. 257–261; see also Refs. [244–246] Similar considerations apply to the fractional generalization.

vitaly v. novikov

272

Representing the diffusive flow as Eq. (574) allows one to draw an analogy between the diffusion of a particle including inertial effects and the frequency dependent dielectric and viscoelastic properties. On inversion  t 1 ð576Þ kðtÞ ¼ exp  t t Considering Eq. (574) as the Laplace transform of a convolution and noting Eq. (576), we obtain

 ðt D t  t0 q jð x; tÞ ¼ jð x; 0Þ  pð x; t0 Þdt0 exp  ð577Þ t qx t 0

or jð x; tÞ ¼ jð x; 0Þ  DkðtÞ 

q pð x; tÞ qx

ð578Þ

q where kðtÞ  qx pð x; tÞ is the convolution of the functions kðtÞ and  ðt  t  t0 q dt0 pð x; t0 Þ kðtÞ  rpð x; tÞ ¼ k qx t t

q qx pð x; tÞ:

ð579Þ

0

if the initial condition is pð x; 0Þ ¼ dð xÞ

ð580Þ

then, after Laplace and Fourier transformations, we obtain from (3.351) (cf Risken loc.cit.) pðo; sÞ ¼

1 þ st sð1 þ stÞ þ v2 o2

ð581Þ

Inverse transformation of Eq. (581) yields et=2t pð x; tÞ ¼ 2

(

dðhx  vtÞ þ dð xi þ vtÞþ 1 ðrÞ 1 þ 4vt I0 ðrÞ þ I2tr Zðvt  jxjÞ

where Zð xÞ is Heaviside’s step function pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 t 2  x2 r¼ 2et

) ð582Þ

ð583Þ

physical properties of fractal structures

273

I0 ðrÞ; I1 ðrÞ are modified Bessel functions of the first kind. The Laplace transformation of the variance is L



x 2 ðt Þ



¼

q2 pðo; sÞ t2 j ¼ o¼0 qo2 sðst þ 1Þ2

ð584Þ

so we obtain the Ornstein–Uhlenbeck result2 [247] 

nt o  t x2 ðtÞ ¼ 2v2 t2  1 þ et t

ð585Þ

From (3.364) it follows that:   If tt ! 0, then x2 ðtÞ t2 ballistic transport of particles.   If tt ! 1, then x2 ðtÞ t represents the Einstein results. Let us now consider the equation for diffusive particle flow with a fractional derivative (570). By Laplace transformation of equation (570), we obtain jð x; sÞ ¼ Da ðsÞ

qpð x; sÞ qx

ð586Þ

where on this occasion the s dependent diffusion coefficient is Da ðsÞ ¼ Dka ðsÞ;

k a ðsÞ ¼

1 1 þ ðtsÞa

ð587Þ

Knowing that 1 X 1 ðtsÞa ¼ ¼ ð1Þn ðtsÞaðnþ1Þ 1 þ ðtsÞa 1 þ ðtsÞa n¼0

ð588Þ

in the domain of the originals ka ðsÞ reads 0

 aðnþ1Þ1 1 n t  a 1 ð1Þ X 1B 1 C t Ma;a ðzÞ ka ðtÞ ¼ @ A ¼ ta1 t n¼0 t ½aðn þ 1Þ

2

ð589Þ

Editor’s Note: A peculiarity of the ‘‘telegraph equation,’’ when used to approximately include inertial effects, is that it yields the correct result for the variance while yielding a poorer approximation for the distribution function than the Smoluchowski equation [see Ref. 248 and Risken (loc.cit.).]

vitaly v. novikov

274

where Ma;g ðzÞ, the Mittag–Leffler function is Ma;g ðzÞ ¼

1 X

zn ; ½an þ g n¼0

 t a z¼ t

ð590Þ

In our case g ¼ 1: Moreover, Ma;g ðzÞ ¼

1;1 H1;2

 ð0; 1Þ z ð0; 1Þ ð1  g; aÞ

ð591Þ

Then Eq. (589) can be written in the time domain as ka ðtÞ ¼ t

a1

  a    a 1 t ð0; 1Þ 1;1 H1;2 t t ð0; 1Þ ð1  a; aÞ

ð592Þ

if a ¼ 1, then from Eq. (592) we obtain  n ! 1  t ð1Þn tt 1 X 1 kðtÞ ¼ ¼ exp  t n¼0 ½n þ 1 t t

ð593Þ

Conclusions. Equation (593) brings to an end our long discussion of anomalous diffusion. Throughout its course we have seen that the characteristic feature of random inhomogeneous structures and the physical processes taking place in them is their hierarchy causing in turn anomalous behavior of their physical properties on the macrolevel, namely, anomalous large spatial fluctuations of local electric and elastic fields. The fractal concept has proved to the helpful in describing such systems. In this connection our attention has been focused on using the fractal concept to make predictions concerning the physical properties of inhomogeneous media with a random structure. We remark that numerous other examples of fractal behavior than those treated here appear in the literature. We should mention resistance capacitance transmission lines [240] and fractal models for the alternating current response at a rough interface between materials of very dissimilar conductivities [241,242] and how a resistance capacitance line may be used as a semiintegrator [243]. Finally we have tried to demonstrate how rather complex phenomena may be described in unified fashion using simple fractal models and we have hopefully composed the text in such a manner that it could serve as an introduction to the subject for the beginner in the field.

physical properties of fractal structures 5.

275

Appendix. Derivative of Fractal Functions.

In general, functions for which the total increment, h f ðxÞ ¼ f ðx þ xÞ  f ðxÞ can be represented as



h f ðxÞ ¼ A½ xh þaðxÞ½ xh ;

ð594Þ 

lim aðxÞ ¼ 0

x!0

ð595Þ

(i) h ¼ 1; 0: f ðxÞ belongs to the classical set of differentiable functions. (ii) h ¼ 6 1(Hoelder index): f ðxÞ belongs to the set of functions for which not the classical derivative but only the fractional derivative exists d h f ðxÞ h f ðxÞ ¼ lim x!0 ½xh dxh

ð596Þ

Wiener’s process (i.e., Brownian motion) and Kolmogorov’s turbulence (i.e., a nonsmooth vector field) may be cited as examples of phenomena which can be described by continuous, nowhere differentiable functions (fractal functions). The displacement yðtÞ of a Brownian particle in the former (Wiener’s) process is defined as jyðt þ tÞ  yðtÞj ½ta

ð597Þ

whereas the singular velocity of the latter phenomenon (Kolmogorov’s turbulent flow) is characterized by [219] h½vp i ½tp=3 ;

ð598Þ

where v ¼ vðx þ xÞ  vðxÞ is the difference of velocities between two points separated by distance x. Assume that a function f ðxÞ is defined on a fractal ensemble
f , of dimension 0 < df < 1. Let the function f ðxÞ (hereafter referred to as a fractal function) be continuous through
f , be self-similar at different scales, and have no tangent at any point of its trajectory. It is assumed that f ðxÞ ¼ 0 if x < 0, and jf ðxÞj < 1. Let us divide a segment ½ x; x0  in such a manner that the length of each qth fragment at the nth scale level is n xðnÞ q ¼ x  ðx0  xÞ

ð599Þ

where x < 1 is the scaling factor (i.e., the index of similarity of the ensemble
f , ).

vitaly v. novikov

276

The number of dividing points of the segment ½ x; x0  at the nth step is therefore mn ¼ 1; 2; . . . ; jnþ1

ð600Þ

where j is the number of blocks (i.e., the branching index) involved in the construction of the fractal unit cell ( j ¼ 2 for Cantor’s ensemble). Let the unit scale at the nth step be xa , h

ia 1 xðnÞ ¼ ðx0  xÞa q Nn

ð601Þ

where N1 ¼ j1 ; . . .; Nn ¼ jn (that is, Nn ¼ jn determines the number of fragments at the nth scale level). This definition of the unit scale for the segment ½ x; x0  allows one to associate each point (element) of a fractal ensemble with a point of an ultrametric space which can be represented by the Cayley tree. ðnÞ ðnÞ It follows from Eq. (601) that lim xq ¼ 0; hence xq , is an infinitesimal n!1 ðnÞ quantity. From now on, the increment of the function argument xq at the nth ðnÞ step will be denoted by x (that is, x ¼ xq ), while the corresponding coordinates of the dividing points will be defined as xq ¼ x0  qxðnÞ q ¼ x0  qx

ð602Þ

where q ¼ 0; 1; 2; . . . ; jnþ1 . Recognition of the fractal dimension as df ¼ a further implies ðx0  xÞ x ¼  n ; 1 x

 ðx0  xÞa ½xa ¼ ; Nn  na 1 ¼ jn ¼ N n x

ð603Þ

Consider an increment, a f ðxÞ ¼ f ðx0 Þ  f ðx0  xÞ; then the qth increment q f ðxÞ will be determined via binomial coefficients with alternating signs m a f ðx0 Þ ¼

m X ½1q Cmq ð f ðx0  qxÞÞ;

m! ; ¼ q!ðm  qÞ!

Cmq ð604Þ

q¼0

m¼j

nþ1

physical properties of fractal structures

277

and the function f ðxÞ in the vicinity of point x0 will be f ðxÞ ¼ ð1  a Þm f ðx0 Þ

ð605Þ

Using Eqs. (599)–(605), one can derive an analogue of the Taylor series for a function f ðxÞ f ðxÞ ¼

m X

aq ðx0  xÞaq

ð606Þ

q¼0

q

j where aq ¼ q!  f ðaqÞ ðx0 Þ, and f ðaqÞ ðx0 Þ defines the fractional derivative of qth order of the fractal function f ðxÞ at the point x ¼ x0 as

qa f ðx0 Þ a q x!0 ð½x Þ

f ðaqÞ ðx0 Þ ¼ lim

ð607Þ

The coefficients of the series (606) depend both on the fractional derivative of qth order of the fractal function f ðxÞ at the point x ¼ x0 and on the branching index j of the fractal ensemble for which the function f ðxÞ is specified.

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CHAPTER 8 FRACTIONAL ROTATIONAL DIFFUSION AND ANOMALOUS DIELECTRIC RELAXATION IN DIPOLE SYSTEMS WILLIAM T. COFFEY Department of Electronic and Electrical Engineering, School of Engineering, Trinity College, Dublin 2, Ireland YURI P. KALMYKOV Laboratoire de Mathe´matiques et Physique des Syste`mes, Universite´ de Perpignan, 66860 Perpignan Cedex, France SERGEY V. TITOV Institute of Radio Engineering and Electronics of the Russian Academy of Sciences, Fryazino, Moscow Region, 141190, Russian Federation

CONTENTS I. Introduction II. Microscopic Models for Dielectric Relaxation in Disordered Systems A. Continuous-Time Random Walk Model B. Fractional Diffusion Equation for the Cole–Cole Behavior C. Anomalous Dielectric Relaxation in the Context of the Debye Noninertial Rotational Diffusion Model D. Fractional Diffusion Equation for the Cole–Davidson and Havriliak–Negami Behavior E. Fundamental Solution of the Fractional Smoluchowski Equation III. Fractional Noninertial Rotational Diffusion in a Potential A. Anomalous Diffusion and Dielectric Relaxation in a Double-well Periodic Potential B. Fractional Rotational Diffusion in a Uniform DC External Field C. Fractional Rotational Diffusion in a Bistable Potential with Nonequivalent Wells

Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

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1. Matrix Continued Fraction Solution 2. Bimodal Approximation IV. Inertial Effects in Anomalous Dielectric Relaxation A. Metzler and Klafter’s Form of the Fractional Klein–Kramers Equation B. Barkai and Silbey’s Form of the Fractional Klein–Kramers Equation C. Inertial Effects in Anomalous Dielectric Relaxation of Linear and Symmetrical Top Molecules 1. Rotators in Space 2. Symmetric Top Molecules D. Inertial Effects in Anomalous Dielectric Relaxation in a Periodic Potential E. Fractional Langevin Equation V. Conclusions Appendix I: Calculation of Inverse Fourier Transforms Appendix II: Exact Continued Fraction Solution for Longitudinal and Transverse Responses Appendix III: Dynamic Kerr-Effect Response: Linear Molecules Appendix IV: Ordinary Continued Fraction Solution for Spherical Top Molecules Appendix V: Kerr-Effect Response Acknowledgments References

I.

INTRODUCTION

One of the most striking features of the dielectric relaxation of disordered materials such as glass-forming liquids, amorphous polymers, and so on, is the failure of the Debye [1] theory of dielectric relaxation based on the Einstein theory of Brownian motion [2] to describe adequately the low-frequency spectrum, where the relaxation behavior may deviate considerably from the exponential (Debye) pattern and is characterized by a broad distribution of relaxation times. The relaxation process in such disordered systems is characterized by the temporally nonlocal behavior arising from the energetic disorder that produces obstacles or traps that delay the motion of the particle and introduce memory effects into the motion. Such behavior has been given the title anomalous dielectric relaxation and was first systematically described in the pioneering article [3] by Cole and Cole in 1941 on dielectric relaxation in polar liquids. These and subsequent investigators have proposed [4,5] (see also Ref. 6) various empirical formulas describing the departure from the Debye behavior. In specific terms, the normal Debye relaxation process is characterized by a complex susceptibility wðoÞ ¼ w0 ðoÞ
iw00 ðoÞ of the form wðoÞ ¼

w0 1 þ iot

ð1Þ

where w0 is the static susceptibility and t is a characteristic relaxation time known in the present context as the Debye relaxation time. Equation (1) adequately describes the low-frequency behavior of the observed complex susceptibility of many simple polar liquids.

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Equation (1) may be derived using a variety of microscopic models of the relaxation process. In the derivation of Eq. (1), Debye [1] used the theory of the Brownian motion developed by Einstein and Smoluchowski. Einstein’s theory of Brownian motion [2] is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows. Consider a twodimensional lattice; then, in discrete time steps of length
t, the random walker is assumed to jump to one of its nearest-neighbor sites, displayed, for example [7], on a square lattice with lattice constant
x, the direction being random. Such a process, which is local both in space and time, can be modeled [7] in the one-dimensional analogue by the master equation Wj ðt þ
tÞ ¼ ½Wj
1 ðtÞ þ Wjþ1 ðtÞ=2

ð2Þ

Here the index j denotes the position of the random walker on the underlying one-dimensional lattice; j þ 1, j
1 are the adjacent lattice sites. Wj ðt þ
tÞ is the probability for the random walker to be at site j at time t þ
t given that it was at sites j  1 at time t. In the continuum limit
t ! 0,
x ! 0, expansion of Wj in a Taylor series in
t and
x leads to the diffusion equation qW q2 W ¼D 2 qt qx

ð3Þ

for the transition probability function Wðx; tjx0 ; t0 Þ, where x0 ¼ xðt0 Þ (i.e., at t ¼ t0 the particle was at x0 ) and the diffusion coefficient D is defined as D¼

ð
xÞ2
x!0;
t!0 2
t lim

ð4Þ

Equation (3) in the presence of an external potential VðxÞ (e.g., the gravitational field of the earth) becomes the Smoluchowski equation [8,9]
 qW q qW W qV ¼D þ qt qx qx kB T qx

ð5Þ

where kB T is the thermal energy. For the noninertial translational Brownian motion of a particle in space, Eq. (5) can be written as [8,9]
 qW W ¼ D div grad W þ grad V qt kB T

ð6Þ

Debye extended the foregoing arguments in order to establish the Smoluchowski equation [Eq. (5)] for the rotational Brownian motion of a dipolar particle about a

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diameter in suspension (planar rotation). If f is an angular coordinate and if f2 is the mean-square displacement in time
t due to thermal agitation, then the underlying rotational diffusion equation is [2,8,9]
 qW q qW W qV ¼D þ qt qf qf kB T qf

ð7Þ

where Wðf; tjf0 ; t0 Þ is the transition probability function and D ¼ f2 =ð2
tÞ is the rotational diffusion coefficient, which determines the Debye relaxation time t ¼ 1=D ¼
=kB T for rotation about a fixed axis (
is the viscous drag coefficient of a dipole). The corresponding rotational diffusion equation for the noninertial rotational Brownian motion of a linear or spherical top molecule in space is given by [8,9]   qW 1 1 2 ¼ r  ðWrVÞ þ r W qt 2t kB T

ð8Þ

where r2 and r are the Laplacian and gradient on the surface of the unit sphere, Wð#; j; tj#0 ; j0 ; t0 Þ represents the transition probability density function of the orientations of the dipoles on the surface of the sphere (the orientation of the dipole moment vector l is described by the polar angle # and azimuth j), and t ¼
=2kB T is the Debye relaxation time for rotation in space. Equation (8) is the Smoluchowski equation, which is an approximate Fokker–Planck equation [7] in the space of angular coordinates for the distribution function of the orientations of the dipoles on the surface of the unit sphere when the influence of the inertia of the molecules on the relaxation process is ignored. We remark that in the context of dielectric relaxation, the rotational diffusion equation for the distribution function W (#, j, t) of the orientations of the dipoles derived by Debye [1] is simply a more general case of Eq. (3), where the tips of the unit vectors specifying the dipole orientations execute a discrete time random walk on the surface of the unit sphere. The discrete time random walk on the surface of the unit sphere then leads directly to the Debye equation [Eq. (1)]. The behavior predicted by Debye’s modification of Einstein’s theory is substantially in accordance with the experimental evidence for simple polar liquids, supporting the hypothesis that, in such liquids, the underlying random processes are local, both in space and time. We shall term this model the first Debye model. This model applies when one has (1) a dilute solution of dipolar molecules in a non polar liquid; (2) axially symmetric molecules; and (3) isotropy of the liquid, even on an atomic scale in the time average over a time interval small compared with the Debye relaxation time t.

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The Debye model then yields the mean dipole moment in the direction of the applied field and the complex susceptibility wðoÞ, Eq. (1). The Smoluchowski equation on which the Debye model is based applies [10] to strong dissipative coupling to the bath so that the first Debye model always contains the assumption that the dipolar molecule is bound so strongly to the surrounding molecules that large jumps of the dipole direction are extremely unlikely. This behavior according to Fro¨hlich [10] may be true in a number of cases, but others may exist in which the opposite (large jumps) is much more likely. A dipolar molecule will then [10] make many jumps over the potential barrier separating it from another dipole direction during the time required for an appreciable change in direction by viscous flow. Clearly, such behavior holds for solids where flow may be considered as entirely absent; however, it may also be expected where the viscosity is so high that flow is practically negligible. In liquids, it might also happen that the motion which prevails is different for different kinds of dissolved molecules. Moreover, both large and small jump transitions may exist simultaneously. The above observations lead us to the second microscopic model considered by Debye [1] (and much extended by Fro¨hlich [10]), which is a Poisson-like process, where relaxation occurs due to the crossing by large jumps of rare members of an assembly of dipoles over an internal potential barrier in a solid due to the shuttling action of thermal agitation. This microscopic model also produces a relaxation spectrum of the form of Eq. (1); however, the overbarrier relaxation time has Arrhenius-like behavior because it depends exponentially on the height of the potential barrier. The Debye–Fro¨hlich model also constitutes a rotational Brownian motion model based on the Fokker–Planck equation, as is apparent by considering a continuous distribution of orientations [8] and a double- (multi-) well potential rather than the discrete orientation approximation treated by Debye and Fro¨hlich. It should be noted that if a continuous distribution of orientations is used, then the prefactor of the exponential in the overbarrier relaxation time depends strongly on the dissipative coupling to the heat bath and the shape of the potential, as emphasized by Kramers [11] in his famous study of the escape of particles over potential barriers due to the shuttling action of thermal agitation. Moreover, the use of the Fokker–Planck equation allows one to account for the contribution of the fast decays in the wells of the potential to the relaxation process. The Debye–Fro¨hlich model is also very useful as a picture of the solidstate-like process of reversal of the magnetization in fine single-domain ferromagnetic nanoparticles possessing an internal potential barrier due to their inherent magnetocrystalline anisotropy [12–14]. In this context, taking into account the intrinsic differences between dielectric and magnetic relaxation, the model is known as the Ne´el–Brown model [12–14] of magnetic relaxation. If, on the other hand, the model is applied to the dielectric relaxation of nematic liquid crystals, it is known as the Maier–Saupe model [15].

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Thus the Debye equation [Eq. (1)] may be satisfactorily explained in terms of the thermal fluctuations of an assembly of dipoles embedded in a heat bath giving rise to rotational Brownian motion described by the Fokker–Planck or Langevin equations. The advantage of a formulation in terms of the Brownian motion is that the kinetic equations of that theory may be used to extend the Debye calculation to more complicated situations [8] involving the inertial effects of the molecules and interactions between the molecules. Moreover, the microscopic mechanisms underlying the Debye behavior may be clearly understood in terms of the diffusion limit of a discrete time random walk on the surface of the unit sphere. Returning to anomalous dielectric relaxation, it appears that a significant amount of experimental data on disordered systems supports the following empirical expressions for dielectric loss spectra, namely, the Cole–Cole equation wðoÞ ¼

w0 ; 1 þ ðiotÞs

0 < s1

ð9Þ

w0 ; ð1 þ iotÞn

0 < n1

ð10Þ

the Cole–Davidson equation wðoÞ ¼

and the Havriliak–Negami equation wðoÞ ¼

w0 ; ð1 þ ðiotÞs Þn

0 < s  1;

0 < n1

ð11Þ

which is a combination of the Cole–Cole and Cole–Davidson equations [6]. Each of the above equations by use of the superposition principle exhibits a broad distribution of relaxation times [6,10]. In the notation of Fro¨hlich [10], we have wðoÞ ¼ w0

1 ð

f ðT 0 ÞdT 0 1 þ ioT 0

1 ð

with

0

f ðT 0 ÞdT 0 ¼ 1

ð12Þ

0

where f ðT 0 Þ is the relaxation time distribution function.This equation is the mathematical expression of the idea that the dielectric behaves as a collection of individual components each being described by a Debye equation with relaxation time T 0 . One may show that for the Debye equation [Eq. (1)] we obtain fD ðT 0 Þ ¼ dðT 0
tÞ

ð13Þ

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(dðxÞ is the Dirac delta function); thus only one relaxation mechanism is involved as is obvious by definition, while for the Cole–Cole Eq. (9) [6] fCC ðT 0 Þ ¼

sin ps pT 0 ½ðT 0 =tÞs þ ðT 0 =tÞ
s þ 2 cos ps

ð14Þ

for the Davidson–Cole Eq. (10) [6] fDC ðT 0 Þ ¼

8
tÞ

ð15Þ

and for the Havriliak–Negami Eq. (11) [6] 
    sinps  ðT =tÞ sin v arctan s ðT 0 =tÞ þ cos ps  0

fHN ðT 0 Þ ¼

vs 

pT½ðT 0 =tÞ2s þ 2ðT 0 =tÞs cos ps þ 1v=2

ð16Þ

Thus it is apparent that the anomalous relaxation behavior may be characterized by a superposition of an infinite number of Debye-like relaxation mechanisms with a distribution of relaxation times given by Eqs. (14)–(16). In the Cole–Cole, Cole–Davidson, and Havriliak–Negami relaxation mechanisms unlike the Debye equation, where the distribution function is a d function, the relaxation time distribution (first evaluated for the Cole–Cole relaxation by Cole and Cole [3]) exhibits long-time tails typical of Le´vy probability distributions [7]. This observation has been formalized for Cole–Cole relaxation by Glo¨ckle and Nonnenmacher [16]. Returning to Eqs. (9)–(11), the Cole–Cole parameter s is a broadening parameter as the curve of w00 ðoÞ versus o broadens as s is reduced. On the other hand, the Cole–Davidson parameter n in Eqs. (10) and (11) is a skewing parameter, because in the Cole–Cole plot of w00 ðoÞ versus w0 ðoÞ the circular arc characteristic of the Debye equation is shifted toward the lowfrequency end of the spectrum [6]. An explanation of this behavior is the n-fold degeneracy induced in the Debye equation (1) by the Cole–Davidson parameter n, causing the simple pole
1=t of the Debye equation to become a branch point of order n—that is, a n-fold degenerate eigenvalue. As far as the physical mechanism underlying the Cole–Cole equation is concerned, we first remark that Eq. (9) arises from the diffusion limit of a continuous-time random walk (CTRW) [17] (see Section II.A). In this context one should recall that the Einstein theory of the Brownian motion relies on the diffusion limit of a discrete time random walk. Here the random walker makes a jump of a fixed mean-square length in a fixed time, so that the only random

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variable is the direction of the walker, leading automatically by means of the central limit theorem (in the limit of a large sequence of jumps) to the Wiener process describing the Brownian motion [8]. The CTRW, on the other hand, was introduced by Montroll and Weiss [17] as a way of rendering time continuous in a random walk without necessarily appealing to the diffusion limit. In the most general case of the CTRW, the random walker may jump an arbitrary length in arbitrary time. However, the jump length and jump time random variables are not statistically independent [18–21]. In other words a given jump length is penalized by a time cost, and vice versa. A simple case of the CTRW arises when one assumes that the jump length and jump time random variables are decoupled and that the jump length variances are always finite (so that the central limit theorem applies in the limit of a large sequence of jump lengths [8]); however, the jump times may be arbitrarily long so that they obey a Le´vy distribution with its characteristic long tail [18–22]. Thus the jump length distribution ultimately becomes Gaussian with finite jump length variance, while the mean waiting time between jumps diverges on account of the underlying Le´vy waiting time distribution. Such walks, which possess a discrete hierarchy of time scales, not all of which have the same probability of occurrence, are known as fractal time random walks [19]. In the limit of a large sequence of jump times, they give rise to a fractional Fokker–Planck equation in configuration space [7,18]. If this equation is now adapted to rotational Brownian motion as used by Debye [1] for the normal Fokker–Planck equation in his first model, then the Cole–Cole equation (9) automatically follows [8,23]. Inertial effects have also been included in the model [24]. The second model of Debye or the Debye–Fro¨hlich model may also be generalized to fractional diffusion [8,25] (including inertial effects [26]). Moreover, it has been shown [25] that the Cole–Cole equation arises naturally from the solution of a fractional Fokker–Planck equation in the configuration space of orientations derived from the diffusion limit of a CTRW. The broadening of the dielectric loss curve characteristic of the Cole–Cole spectrum may then be easily explained on a microscopic level by means of the relation [8,24] lp;s ¼ lp t1
s

ð17Þ

between the eigenvalues lp;s and lp of the fractional and normal configuration space Fokker–Planck equations, respectively. Here the relaxation behavior appears [8] as a superposition of Cole–Cole equations if the inertial effects are ignored. The fractal time random walk picture, whereby a particle is trapped in a given configuration for an arbitrarily long period before executing a jump [18], immediately suggests that the Cole–Cole parameter s (here the fractal dimension of the set of waiting times between jumps) arises from the anisotropy

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of the material on a microscopic scale. Thus assumption 3 underpinning the Debye theory breaks down for Cole–Cole relaxation. The microscopic anisotropy gives rise to a distribution of microscopic potential barrier heights [23] which in turn, because the individual jump probabilities constitute a hierarchy of Poisson processes [19], give rise to a hierarchy of relaxation times not all of which have the same probability of occurrence. Such models are usually known as random activation energy models (see Ref. 20, p. 280). They appear to be consistent with the concept of a distribution of microscopic Debye-like mechanisms embodied in Eq. (12) and the Le´vy-like behavior [Ref. 6, Eqs. (3.104) and (3.105)] of the various relaxation time distributions and with the breakdown in anomalous relaxation of Einstein’s ansatz [8,18,21] that in Brownian motion the random walker executes a discrete jump of finite meansquare length in an average time t. Our purpose is to demonstrate how it is possible to describe the anomalous dielectric relaxation from microscopic models of the underlying processes. Moreover, we shall illustrate how the effects of the inertia of the molecules and an external potential arising from crystalline anisotropy or indeed any other mechanism could be included. II.

MICROSCOPIC MODELS FOR DIELECTRIC RELAXATION IN DISORDERED SYSTEMS

The Cole–Cole equation can be derived from a kinetic equation based on the concept of a continuous-time random walk—that is, a walk with a long-tailed distribution of waiting times between the elementary jumps. It is also apparent [27] that the method may be extended to both the Cole–Davidson and the Havriliak–Negami equations using an extension of an approach proposed by Nigmatullin and Ryabov [28]. There are, however, certain mathematical and conceptual difficulties associated with such a fractional diffusion equation approach. The first of these is the justification of truncation of the generalized Kramers–Moyal expansion at the second term in the space derivative in order to obtain fractional probability density diffusion equations for the orientation distribution functions. This question is easily answered in the theory of the Brownian motion because the underlying processes are Gaussian; thus it is possible to express the higher-order even moments of the distribution function in terms of powers of the second moment while the odd moments are zero. This is the crucial factor that allows one to truncate the Kramers–Moyal expansion at the second derivative in the spatial derivatives. An important corollary to the above statement is the following: Since one may express the higher-order moments in terms of powers of the second-order moment, one may also generate the hierarchy of differential recurrence relations describing the time behavior of the statistical averages by averaging the underlying Langevin equation over its

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realizations using Isserlis’s theorem also known as Wick’s theorem. It is not yet apparent how to do this for fractional diffusion processes due to the lack of appropriate interpretation rules (Itoˆ–Stratonovich) and the absence of an analogue of Isserlis’s theorem [8]. Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters s and n in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole–Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers–Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). A.

Continuous-Time Random Walk Model

The theory of the Brownian motion, which we have described, is distinguished by a characteristic feature—namely, the concept of a collision rate—which is the inverse of the time interval between successive collision events of the Brownian particle with its surroundings; we recall the words of Einstein [2]. We introduce a time interval
t in our discussion, which is to be very small compared with the observed interval of time, but, nevertheless of such a magnitude that the movements executed by a particle in two consecutive intervals of time
t are to be considered as mutually independent phenomena.

This concept which is based on a random walk with a well-defined characteristic waiting time (thus called a discrete-time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included (see Note 8 of Ref. 2, due to Fu¨rth), we obtain the Klein–Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion. The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. Einstein’s approach, ignoring inertial effects always leads to a mean-square displacement proportional to jtj. A generalization of this is of the form jtjs ,

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where the case s > 1 is referred to as superdiffusion or enhanced diffusion while s < 1 is referred to as subdiffusion. We shall now indicate how these behaviors may be interpreted in terms of CTRW. We have mentioned that the concept of a CTRW was introduced by Montroll and Weiss in 1965 [17,29] as a way to render time continuous in a random walk without an appeal to the diffusion or continuum limit. Thus Einstein’s assumption of a discrete time
t in which a jump of mean-square length h
2 i is executed is abandoned. Instead, a jump probability distribution function cðx; tÞ is introduced in which in general the jump length and time between jumps are coupled random variables, meaning that a jump of a certain length involves a time cost and vice versa. The jump distribution is the probability density that a random walker executes a jump from x to x þ dx in a time interval dt having remained at some site for a waiting time t. We may determine from cðx; tÞ both the jump length probability density function lðxÞ ¼

1 ð

cðx; tÞ dt

ð18Þ

0

and the waiting time probability density function wðtÞ ¼

1 ð

cðx; tÞ dx

ð19Þ


1

Here lðxÞ dx yields the probability of a jump length L in the interval x ! x þ dx, and wðtÞ dt yields the probability of a waiting time TW in the interval t ! t þ dt. Moreover, unlike in Brownian motion where the introduction of the quantities
x and
t defines a physical length and a physical time scale, the second moment of the jump length distribution diverges and so does the first moment of the jump time. Thus both jump times and jump lengths exhibit chaotic behavior. Hence it is impossible to attribute underlying physical scales to such processes. This may be interpreted as the scale invariance which is typical of self-similarity and fractal behavior. The word fractal—coming from the Latin fractus, meaning broken—is used to describe dilation invariant objects which exhibit irregularities and chaotic behavior at any given scale. Specific examples are clouds and the coastline of islands. The self-similar behavior is of course constrained by certain boundaries such as in Brownian motion the selfsimilar behavior is limited to distances above the mean free path of a molecule, and so on. An important consequence of the lack of physical scales is that referring to the temporal behavior of such systems, all global characteristic

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times such as the mean first passage time (i.e., the average time at which a process reaches a predetermined level for the first time), the integral relaxation time, and so on, will diverge [8]. In analyzing the complex susceptibility in such systems, therefore, one should not use the concept of characteristic times; in constrast, the physically meaningful quantities are the frequencies of maximum absorption. The characteristic times of the normal diffusion process are merely parameters in anomalous diffusion processes. An example of divergent characteristic times is the divergence of the relaxation time of strongly interacting magnetic nanoparticle systems in the vicinity of a spin glass phase transition. Despite all these difficulties, it is, however, possible to analyze distribution functions having divergent moments of the kind we have mentioned. The underlying limiting distribution function is not Gaussian, it is a stable or Le´vy distribution, which is marked by the presence of long-range inverse power law tails in the distribution function which may lead to divergence of even the lowest-order moments. The tails prevent convergence to the Gaussian distribution if they pertain to a sequence of random variables, however, not the existence of a limiting distribution. It is in general very difficult to treat CTRW problems when the jump time and jump length distributions are coupled. Thus we shall assume for the most part that the jump time and jump length are independent random variables so that the jump probability factorizes and we have the decoupled (separable) form cðx; tÞ ¼ wðtÞlðxÞ

ð20Þ

cðx; tÞ ¼ pðxjtÞwðtÞ

ð21Þ

cðx; tÞ ¼ pðtjxÞlðxÞ

ð22Þ

If they are coupled, we have

or

that is, a jump of a certain length involves a time cost or, on the other hand, in a given time span, the walker can only travel a maximum distance [7]. We remark that if the jump length distance is also a Le´vy process, the meansquare displacement does not exist which has led to conceptual difficulties in applying this process to dielectric relaxation. Using these simplifications, one can identify two specialized forms of a continuous time random walk: 1. The first is the long rests or fractal time random walk where the mean waiting time diverges; however, the second moment of the jump length distribution remains finite. The fractal time random walk always leads to

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subdiffusion because the random walker always risks being trapped at some site for an arbitrarily long time before he can advance a distance equal to the finite variance of the jump length distribution. 2. The other uncoupled case with finite mean waiting time and divergent jump length variance is called the long-jump or Le´vy flight model. A famous example of the Le´vy flight model is the Weierstrass random walk. Here all steps take the same average time and the distribution of step lengths exhibits a Le´vy-type inverse power decay law for the largest jumps, which of course leads to the divergent second moment. Such a walk leads to enhanced diffusion and ultimately turbulence as the overall displacement is dominated by the largest jumps without any time cost; that is, jumps of arbitrary length all take the same time if the jump length is a Le´vy process. 3. Both divergences of the moments associated with the random walks mentioned above may be avoided by considering nonseparable CTRWs or Le´vy walks where, in contrast to Le´vy flights, a Le´vy walker does not simply jump an arbitrary length in the same time [7,19] but instead has to move with a given velocity from his starting point. For the simplest case of constant velocity [7,19] we see that large jumps require a longer time than shorter ones and the overall consequence is finite mean-square displacement for all values of s. Such walks are important in the context of probability density diffusion equations for the phase-space distribution function that is the generalization of the Klein–Kramers equation to fractional diffusion. If a purely fractal time random walk is used to generalize the Klein–Kramers equation for rotational diffusion to anomalous diffusion, then a nonphysical divergence of the absorption coefficient occurs at very high frequencies [8]. If, however, a Le´vy walk is used which appears to be at the root of the generalization of the Klein–Kramers equation proposed by Barkai and Silbey [30], then the undesirable divergence of the absorption coefficient is removed [8]). The fact that the temporal occurrence of the motion events performed by the random walker is so broadly distributed that no characteristic waiting time exists has been exploited by a number of investigators [7,19,31] in order to generalize the various diffusion equations of Brownian dynamics to explain anomalous relaxation phenomena. The resulting diffusion equations are called fractional diffusion equations because in general they will involve fractional derivatives of the probability density with respect to the time. For example, in fractional noninertial diffusion in a potential, the diffusion Eq. (5) becomes [7,31]  2
 qW q W qV s 1
s q W ¼ D 0 Dt þ qt qx2 qx kB T qx

ð23Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

where s is the anomalous exponent, the fractional derivative 0 Dt1
s is given by (the Riemann–Liouville definition) [7,32,33] 1
s 0 Dt

¼

q
s 0D qt t

ð24Þ

in terms of the convolution (recall Cauchy’s integral formula)
s 0 Dt Wðx; tÞ

ðt 1 Wðx; t0 Þdt0 ¼ ðsÞ ðt
t0 Þ1
s

ð25Þ

0

where  (z) denotes the gamma function. Equation (23) with 0 < s < 1 describes slow diffusion or subdiffusion and with 1 < s < 2 describes enhanced diffusion (s ¼ 2 defines the ballistic limit); normal diffusion occurs when s ¼ 1 [7,31]. The fractional derivative is a type of memory function with a slowly decaying power law kernel in time. Such behavior arises from random torques with an anomalous waiting time distribution—that is, from a fractal time random walk with t as the intertrapping time. The derivation of fractional diffusion equations such as Eq. (23) hinges on the observation (cf. Ref. 33, p. 118) that fractional diffusion is equivalent to a CTRW with waiting time density wðtÞ given by a generalized Mittag–Leffler function (see below and also Refs. 7 and 31). The fact that wðtÞ is given by a generalized Mittag–Leffler function amounts to assuming an asymptotic (longtime) power law form for the waiting time probability distribution function, namely considering slow diffusion, wðtÞ  As ts t
1
s ;

ð0 < s < 1Þ

ð26Þ

(As is a constant). The characteristic (mean) waiting time hTW i ¼

1 ð

twðtÞ dt

ð27Þ

0

then always tends to 1 except in the limit s ! 1 (the classical Brownian motion), where wðtÞ ¼ dðt
tÞ, so that hTW i ¼ t. A famous example [19] of a distribution function with a long-time tail like Eq. (26) is the Cauchy distribution wðtÞ ¼

a 1 p a2 þ t 2

ð28Þ

fractional rotational diffusion

299

with infinite second moment. This distribution is just one example of a whole class of distributions which if applied to a sum of random variables do not converge to the Gaussian distribution as the number of random variables tends to infinity. Thus, the central limit theorem, on which the theory of Brownian motion is based, is not obeyed because the long-time tails preclude convergence to the Gaussian distribution. Nevertheless, limiting (now called Le´vy [19], Chapter 4) distributions may exist. The divergence of the waiting time associated with the long-time-tailed nature of the waiting time probability distribution function Eq. (26) is according to Metzler and Klafter [7,31] a manifestation of the self-similar nature of the waiting time process. This has prompted many investigators [7] to use in the present context the term fractal time processes to describe anomalous relaxation. Returning to the fractional diffusion equation [Eq. (23)], that equation will now follow from Eq. (26) and CTRW theory because (Ref. 33, p. 118) the integral equation for the probability density f ðx; tÞ for a continuous-time random walker to be in a position x at time t starting from x ¼ 0 at t ¼ 0 with waiting time density given by Eq. (26) is equivalent in the diffusion limit to the fractional diffusion equation Eq. (23). We remark that postulating wðtÞ as a generalized Mittag–Leffler function with long-time behavior given by Eq. (26) so that fractional diffusion may be described as a CTRW is (just as the postulate of the existence of a discrete time
t, the duration of an elementary jump in the Einstein theory of the Brownian movement) equivalent to a stosszahlansatz for the Boltzmann equation. That equation must of necessity underpin the entire theory. In other words, the transition probability or ‘‘mechanism’’ of the fractional diffusion process is the CTRW. Below, we shall see in detail how the introduction of a waiting time density of the form of Eq. (26) allows one to generalize the Fokker–Planck equation of normal diffusion to fractional diffusion. We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye–Fro¨hlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye–Fro¨hlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34–36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker–Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics.

300

william t. coffey, yuri p. kalmykov and sergey v. titov B.

Fractional Diffusion Equation for the Cole–Cole Behavior

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate f ) and a set of discrete orientations on the unit circle with fixed angular spacing
. We note that
may not necessarily be fixed; for example, if we have a Gaussian distribution of jumps, the standard deviation of
serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. Following the procedure suggested in [7,37] for the translational motion, we first denote individual discrete orientations by f. . . 1; 0; 1; . . .g. A typical dipole is supposed to have orientation n ¼ 0 at time t ¼ 0. The dipole having oriented to the site n at time t is fixed in that orientation that is trapped, for some random time. The random waiting times after which changes in orientation take place are denoted by fti g; i ¼ 1; 2; . . .; these times are assumed to be independent identically distributed random variables with the probability density function wðtÞ. Thus the situation is unlike that in the Einstein theory where t is fixed. Note that an exponential distribution of t so that the mean waiting time is finite also leads in the diffusion limit to Einstein’s result. It follows therefore that a random walk where a mean waiting time exists and a finite jump length variance exists will always lead to Einstein’s result. (This behavior again stems from the central limit theorem.) We assume that  the probability density function wðtÞ is independent of the orientation of the dipole at time t, that is, independent of n,  the dipole when at orientation specified by n reorients only to its nearestneighbor sites, that is, n  1 as in Einstein’s theory,  the probability of orienting from site n to n þ 1 is AðnÞ and the probability of orienting from site n to n
1 is BðnÞ, where AðnÞ and BðnÞ obey the normalization condition AðnÞ þ BðnÞ ¼ 1

ð29Þ

and are independent of the time. The probability GðtÞ that the dipole has survived in a given orientation–that is, at a given site for a time t—is ðt

1 ð

0

t

GðtÞ ¼ 1
wðtÞ dt ¼

wðtÞ dt

ð30Þ

fractional rotational diffusion

301

Using the Laplace transform of the survival probability GðtÞ at a site, we have ~ ~ ðsÞ GðsÞ ¼ s
1 ½1
w

ð31Þ

Here a tilde denotes the Laplace transform, namely, ~f ðsÞ ¼

1 ð

f ðtÞe
st dt

ð32Þ

0

Now the waiting times are identically distributed random variables. Hence on introducing Qi ðtÞ, the probability that the dipole has changed i times in orientation ~ i ðsÞ in the time interval ð0; tÞ, we will have for the Laplace transform Q ~ wi ðsÞ ¼ s
1 ½1
w ~ i ðsÞ ¼ GðsÞ~ ~ ðsÞ~ wi ðsÞ Q

ð33Þ

Following Ref. 37, let us now introduce Wðn; tÞ, which is the probability of finding the dipole in discrete orientation n at time t. Let us further introduce pi ðnÞ, which is the probability that the dipole has orientation n after i changes in orientation; then summing over all the orientation changes, we have ~ Wðn; sÞ ¼

1 X

~ ðsÞ pi ðnÞQi ðsÞ ¼ s
1 ½1
w

i¼0

1 X

pi ðnÞ~ wi ðsÞ

ð34Þ

i¼0

Since only nearest neighbors are involved, now the evolution of pi ðnÞ is determined by the discrete time (i) and space (n) equation piþ1 ðnÞ ¼ Aðn
1Þpi ðn
1Þ þ Bðn þ 1Þpi ðn þ 1Þ

ð35Þ

The continuous distribution of orientations, f, is obtained by the replacement of pi ðnÞ by pi ðfÞ where pi ðfÞ df is the probability of finding the dipole after the ith jump in the angle f ! f þ df. In like manner Aðn
1Þ; Bðn þ 1Þ ! Aðf

Þ; Bðf þ
Þ. Thus Eq. (35) becomes piþ1 ðfÞ ¼ Aðf

Þpi ðf

Þ þ Bðf þ
Þpi ðf þ
Þ

ð36Þ

On expanding Aðf

Þpi ðf

Þ and Bðf þ
Þpi ðf þ
Þ in Taylor series, we have the following in the continuum limit with regard to terms in
2: Aðf

Þpi ðf

Þ ¼ AðfÞpi ðfÞ



q
2 q2 ½AðfÞpi ðfÞ þ ½AðfÞpi ðfÞ ð37Þ qf 2 qf2

Bðf þ
Þpi ðf þ
Þ ¼ BðfÞpi ðfÞ þ


q
2 q2 ½BðfÞpi ðfÞ þ ½BðfÞpi ðfÞ ð38Þ qf 2 qf2

302

william t. coffey, yuri p. kalmykov and sergey v. titov

Moreover, by the principle of detailed balance, if the system is close to thermal equilibrium at temperature T [37], then we have AðfÞ
BðfÞ ’



qVðfÞ 2kB T qf

ð39Þ

where VðfÞ is the external potential energy—for example, due to the electric field acting on the system. Such a requirement on AðfÞ and BðfÞ guarantees that the system relaxes to the equilibrium Boltzmann distribution [37]. Equation (36) then becomes  
2 q qpi ðfÞ pi ðfÞ qV piþ1 ðfÞ ¼ pi ðfÞ þ þ kB T qf 2 qf qf

ð40Þ

Now replacing n by f, we may rewrite Eq. (34) as ~ ðsÞ ~ ðsÞ 1
w 1
w ~ p0 ðfÞ þ Wðf; sÞ ¼ s s

1 X

pi ðfÞ~ wi ðsÞ

ð41Þ

i¼1

or according to Eq. (40) ~ ðsÞ ~ ðsÞ 1
w 1
w ~ dðfÞ þ Wðf;sÞ ¼ s s
  1  X
2 q q pi
1 ðfÞ qV ~ i ðsÞ ð42Þ  pi
1 ðfÞ þ pi
1 ðfÞ þ þ  w kB T qf 2 qf qf i¼1 We may now eliminate the summation in Eq. (42) by noting that according to Eq. (34) ~ ðsÞ 1
w ~ Wðf; sÞ~ wðsÞ ¼ s

1 X

pi ðfÞ~ wiþ1 ðsÞ

ð43Þ

i¼0

on change of the summation variable to i ¼ j
1. Thus ~ ðsÞ 1
w ~ dðfÞ Wðf; sÞ ¼ s 
  ~ Wðf; sÞ qV
2 q q ~ ~ ~ ðsÞ Wðf; sÞ þ þw Wðf; sÞ þ þ  kB T qf 2 qf qf

ð44Þ

fractional rotational diffusion

303

We now explicitly consider the waiting time distribution. First we reiterate that the Einstein theory of the Brownian motion relies on the central limit theorem that a sum of independent identically distributed random variables (the sum of the elementary displacements of the Brownian particle) SN ¼

 N X

Xi

i¼0

 ! 1, provided that the first and becomes a Gaussian distribution in the limit N second moments of Xi do not diverge. However, there are famous exceptions; for example, for the Cauchy distribution, f ðxÞ ¼

a 1 p a2 þ x2

ð45Þ

the second moment of this distribution is infinite. The Cauchy distribution is just one example of a whole class of distributions which possess long inverse power law tails—for example, pðxÞ ¼

1

ð46Þ

jxj1þs

 ! 1, but The tails prevent [19] convergence to the Gaussian distribution for N not the existence of a limiting distribution. These distributions as we have seen are called stable distributions. If the concept of a Le´vy distribution is applied to an assembly of temporal random variables such as the fti g of the present chapter, then wðtÞ is a long-tailed probability density function with long-time asymptotic behavior [7,37], wðtÞ 

sAs ; ð1
sÞt1þs

0 < s  1;

t!1

ð47Þ

The restriction to 0 < s < 1 ensures that the first moment of the waiting-time distribution is divergent as is usual in a CTRW (corresponding to chaotic behavior of the waiting times). Moreover, in the s domain, the long-time behavior is manifested in the small s expansion [7,37] ~ ðsÞ ¼ 1
As ss þ c1 ðAs ss Þ2 þ    w

ð48Þ

Thus it is obvious by the properties of the Laplace transform that the second term corresponds to the asymptotic behavior given by Eq. (47). If we now

304

william t. coffey, yuri p. kalmykov and sergey v. titov

substitute Eq. (48) into Eq. (44), on multiplying across by s and simplifying, we have ~ sÞ
s
1 ð1
c1 As ss þ   ÞdðfÞ þ ð1
c1 As ss þ   ÞWðf; 
  2 ~

s q q ~ Wðf; sÞ qV 2 s Wðf; sÞ þ þ  ¼ 0 þ ðs
As þ c1 As s þ   Þ qf qf kB T qf 2As ð49Þ If introducing the limiting procedure As ! 0,
! 0 with
2
!0 2As

t
s ¼ lim

ð50Þ

As !0

finite, Eq. (49) becomes ~
s
1 dðfÞ þ Wðf; sÞ ¼ ðtsÞ
s


 ~ Wðf; sÞ qV q q ~ Wðf; sÞ þ kB T qf qf qf

ð51Þ

or, on inversion to the time domain, becomes qWðf; tÞ ¼ t1
s 0 D1
s LFP Wðf; tÞ t qt

ð52Þ


 1 q2 Wðf; tÞ 1 q qV Wðf; tÞ LFP Wðf; tÞ ¼ þ t kB T qf qf qf2

ð53Þ

where

is, in effect, the Fokker–Planck operator for normal rotational diffusion about a fixed axis. The time t has the meaning of the intertrapping time scale which is identified with the Debye relaxation time t ¼
=kB T (
is friction coefficient), is defined in terms of the Riemann–Liouville fractional and the operator 0 D1
s t integral definition, Eq. (24) and (25), namely, 1
s 0 Dt

q ¼ 0 D
s ; qt t


s 0 Dt f ðtÞ

ðt 1 f ðt0 Þ dt0 ¼ ðsÞ ðt
t0 Þ1
s

ð54Þ

0

Equation (54) means [31] that Eq. (52) now contains a slowly decaying (s < 1) memory function with a power law kernel so that the process is no longer Markovian and thus depends on the history of the system. Equation (52) stems

fractional rotational diffusion

305

from a fractal-waiting-time or long-rests model, where the system is jammed in a particular configuration for an arbitrary long interval. Such behavior stems in turn from assuming random forces with an anomalous waiting-time distribution. Equation (52) for fractional rotational diffusion has the same mathematical form as that for fractional translation diffusion Eq. (23) derived by Metzler and Klafter [7,31]. Thus the solution of Eq. (52) may be obtained [7,31] in similar manner using generalized Mittag–Leffler or Fox functions [38]. The fractional diffusion equation [Eq. (52)] for the time evolution of the probability density function Wðf; tÞ in configuration space is then the same as that previously derived [31] for a particle of one translational degree of freedom; however, rotational quantities replace translational ones (see, for example, Eq. (19) of Ref. 32). The advantage of using the CTRW formalism is that it is now possible to gain some insight into the meaning of the parameter s. It is the order of the fractional derivative in the fractional diffusion equation describing the continuum limit of a random walk with a chaotic set of waiting times (often known as a fractal-time random walk). However, a more physical and useful definition of s is as the fractal dimension of the set of waiting times which is the scaling of the waiting-time segments in the random walk with magnification. The parameter s thus measures the statistical self-similarity (or how the whole looks like its parts [19]) of the waiting time segments. In order to construct such an entity in practice, a whole discrete hierarchy of time scales is needed. For example, a fractal time Poisson process [19] with a waiting-time distribution assumes the typical form of the Le´vy stable distribution. This is explicitly discussed in Ref. 19 where a formula for s is given and is also discussed in Ref. 23 where the fractal time process is essentially generated by considering jumps over the wells of a chaotic potential barrier landscape (random activated energy modes). The microscopic picture presented in Refs. 19 and 23 appears to completely support the commonly used experimental representation of the anomalous behavior as a distribution of Debye-like relaxation mechanisms with a continuous relaxation time distribution function. C.

Anomalous Dielectric Relaxation in the Context of the Debye Model of Noninertial Rotational Diffusion

In his work on dielectric relaxation of an assembly of noninteracting dipolar molecules, Debye [1] considered two models: (a) an assembly of fixed axis rotators, each having permanent dipole moment l and subjected to Brownian motion torques having their origin in the background or heat bath, and (b) the same assembly, but the restriction to fixed axis rotation is removed. Nevertheless, the results in both instances are qualitatively the same, since in each case the time-dependent distribution function, if inertial effects are disregarded [39–41], depends only on a single space coordinate. In case (a) this is the azimuthal angle f while in case (b) it is the polar angle # (the colatitude). Thus in the calculations

306

william t. coffey, yuri p. kalmykov and sergey v. titov

which follow which are identical to those in Chapter V of the Debye book [1] with the fractional diffusion equation in configuration space (here the space of polar angles) replacing the Smoluchowski equation used by him, we shall mainly confine ourselves to fixed axis rotation and merely allude to the corresponding result for rotation in space. In general, an assembly of fixed axis rotators qualitatively reproduces the principal features of dielectric relaxation of dipolar molecules in space while allowing considerable mathematical simplification of the problem. Again in the spirit of the Debye calculations [1] we shall consider two separate cases of dielectric relaxation, namely (a) the response following the sudden removal of a constant field (after effect response) and (b) the response to an alternating current (ac) field which has been applied for a long time so that a steady state has been attained. An advantage of the two separate calculations is that they explicitly demonstrate that the concepts of linear response theory will hold (as is to be expected) for relaxation processes in fractal structures. Before proceeding, we remark that just as in the Einstein theory of the Brownian motion of which the Debye theory is a rotational version, the characteristic microscopic time scale is a time interval t1 so long that the motion of the particle at time t is independent of its motion at time t  t1 , but small compared to the observation time intervals [17]. It is also supposed that during the time t1 , which is the mean of the time intervals between collision events, any external nonstochastic forces which may be applied to the system do not alter. In the fractal waiting time picture, however, the concept of collision rate does not hold and the mean of the time intervals between collision events diverges. In the fixed axis rotation model of dielectric relaxation of polar molecules a typical member of the assembly is a rigid dipole of moment m rotating about a fixed axis through its center. The dipole is specified by the angular coordinate f (the azimuth) so that it constitutes a system of 1 (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function Wðf; tÞ in configuration space is given by Eq. (52) which we write here as
 qWðf; tÞ q qWðf; tÞ Wðf; tÞ qVðf; tÞ ¼ t
s 0 D1
s þ t qt qf qf kB T qf

ð55Þ

Here Vðf; tÞ ¼
mFðtÞ cos f is the potential arising from an external applied electric field FðtÞ. Here, just as with the translational diffusion equation treated in Ref. 7, we consider subdiffusion, 0 < s < 1 phenomena only. Here, the internal field effects are ignored, which means that the effects of long-range torques due to the interaction between the average moments and the Maxwell fields are not taken into account. Such effects may be discounted for dilute systems in first approximation. Thus, the results obtained here are relevant to situations where dipole–dipole interactions have been eliminated by extrapolation of data to infinite dilution.

fractional rotational diffusion

307

As we have mentioned, we shall consider two classes of solution of Eq. (55): The first is the aftereffect solution following the removal of the constant field. In the case of the aftereffect solution of the fractional diffusion equation, a uniform field F, having been applied to the assembly of dipoles at a time t ¼
1 so that equilibrium conditions prevail by the time t ¼ 0, is switched off at t ¼ 0. In addition, it is supposed that the field is weak (mF kB T); thus for t > 0, Eq. (55) becomes qWðf; tÞ q2 Wðf; tÞ ¼ t
s 0 D1
s t qt qf2

ð56Þ

which must be solved subject to the initial condition Wðf; 0Þ ¼ Ce

mF kB T

cos f

  1 mF 1þ cos f

2p kB T

ð57Þ

where C and 2p are the normalizing constants. Just as in normal diffusion, the form of the initial condition Eq. (57) suggests that the time-dependent solution should be of the form   1 mF 1 þ gðtÞ cos f Wðf; tÞ ¼ 2p kB T

ð58Þ

Substitution of Eq. (58) into Eq. (56) then yields the following fractional differential equation for the function gðtÞ: d gðtÞ ¼
t
s 0 D1
s gðtÞ t dt

ð59Þ

The solution of this fractional relaxation equation [31] is gðtÞ ¼ Es ½
ðt=tÞs 

ð60Þ

where Es ðzÞ is the Mittag–Leffler function defined by Es ðzÞ ¼

1 X n¼0

zn ð1 þ s nÞ

ð61Þ

The Mittag–Leffler function interpolates between the initial stretched exponential form [7,31]
 ðt=tÞs Es ½
ðt=tÞs   exp
ð1 þ sÞ

ð62Þ

308

william t. coffey, yuri p. kalmykov and sergey v. titov

and the long-time inverse power-law behavior Es ½
ðt=tÞs   ½ðt=tÞs ð1
sÞ
1

ð63Þ

The Debye result for gðtÞ corresponds to s ¼ 1, namely, E1 ð
t=tÞ ¼ e
t=t

ð64Þ

that is, the exponential function is a special case of the Mittag–Leffler function [31]. The Mittag–Leffler function can be expressed in terms of the Fox H-function [38] [see Eq. (A1.8) in Appendix I]. We may now calculate the mean dipole moment. The mean dipole moment due to orientation alone is given at any time t > 0 (f is a unit vector in the direction of field F) by hl  fiðtÞ ¼

2ðp

m cos fWðf; tÞ df

ð65Þ

0

so that with Eq. (58) we have hl  fiðtÞ ¼

m2 F Es ½
ðt=tÞs  2kB T

ð66Þ

in contrast to the Debye result embodied in Eq. (64). A practically much more important result is the behavior of the system in a periodic field FðtÞ ¼ Feiot so that the fractional diffusion equation [Eq. (55)] for the distribution function becomes 

 qWðf; tÞ q mF q2 Wðf; tÞ iot ¼ t
s 0 D1
s sin fe Wðf; tÞ þ t qt qf kB T qf2

ð67Þ

Following Debye, let us try as a solution   1 mF iot 1 þ BðoÞ cos fe Wðf; tÞ ¼ 2p kB T

ð68Þ

where BðoÞ is a constant to be determined. Substitution of Eq. (68) into Eq. (67) yields f½1
BðoÞeiot g ioBðoÞeiot ¼ t
s 0 D1
s t

ð69Þ

fractional rotational diffusion

309

Equation (69) may be further simplified if we recall the integration theorem of Laplace transformation as generalized to fractional calculus, [31], namely, L

f0 Dt1
s f ðtÞg

¼



s1
s~f ðsÞ
0 D
s t f ðtÞj t¼0; 1
s~ s f ðsÞ;

0<s 0;

Wðx; 0Þ ¼ W0 ðxÞ

ð87Þ

where LFP is the Fokker–Planck operator. Equation (87) can equivalently be rewritten as an equation for an impulse response so that the initial condition appears as the amplitude of a forcing function dðtÞ, [50], namely,  qWðx; tÞ  ¼ LFP Wðx; tÞ þ dðtÞW0 ðxÞ qt

ð88Þ

 where Wðx; tÞ ¼ yðtÞWðx; tÞ, yðtÞ is the Heaviside unit step function, and dðtÞ is the Dirac delta function. The fractional analog of Eq. (88), [i.e., Eq. (52)] can now be rewritten as  qWðx; tÞ  ¼ t1
s 0 D1
s LFP Wðx; tÞ þ dðtÞW0 ðxÞ t qt

ð89Þ

where the fractional derivative is given by Eq. (54). Equation (89) may be written in an equivalent form as [7,20]   ts 0 Dst Wðx; tÞ ¼ tLFP Wðx; tÞ þ

ðt=tÞ
s W0 ðxÞ ð1
sÞ

ð90Þ

314

william t. coffey, yuri p. kalmykov and sergey v. titov

The time behavior of the right-hand side of Eq. (90) indicates that the initial state W0 ðxÞ decays slowly with a long-time tail unlike the exponential decay of normal diffusion, which is an indication of the fractal time character of the process. Equations (89) and (90) are fractional analogs of the conventional Fokker–Planck equation [Eq. (88)] giving rise to the Cole–Cole anomalous behavior. Another approach to fractionalize the Fokker–Planck equation incorporating Cole–Davidson behavior can now be given by extending a hypothesis of Nigmatullin and Ryabov [28]. They noted that the ordinary first-order differential equation describing an exponential decay d f ðtÞ þ 0 f ðtÞ ¼ 0 dt with initial condition f ðtÞjt¼0 ¼ f ð0Þ may be written as an equation for an impulse response, namely, e
0 t

d 0 t e yðtÞ f ðtÞ ¼ f ð0ÞdðtÞ dt

ð91Þ

which in turn may be written as [recalling that the derivative of the step function is the Dirac delta function, qt yð
tÞ ¼
dðtÞ] e
0 t

 d  0 t e ½yðtÞf ðtÞ þ yð
tÞf ð0Þ ¼ yð
tÞ0 f ð0Þ dt

ð92Þ

In order to obtain in heuristic fashion a fractional analog of Eq. (92), one may simply replace the ordinary derivative by a fractional derivative [28] so that Eq. (92) becomes e0
0 t Dnt fe0 t ½yðtÞf ðtÞ þ yð
tÞf ð0Þg ¼ yð
tÞ0 f ð0Þ

ð93Þ

In the particular application to dielectric relaxation, f ðtÞ is the aftereffect function following the removal of a constant field [8]. The solution of Eq. (93) rendered in the frequency domain yields the Cole–Davidson equation [Eq. (10)] [28]. The approach of Nigmatullin and Ryabov [28] is, however, entirely phenomenological because no underlying kinetic equation is involved. Nevertheless, their method may also be applied to the Fokker–Planck equation [Eq. (88)] so that a kinetic equation and thus a microscopic model is involved. Indeed, we can rewrite the normal Fokker–Planck equation [Eq. (88)] as an

fractional rotational diffusion

315

equation for an impulse response in the form of the right-hand side of Eq. (91), namely, eLFP t

q
LFP t  ½e Wðx; tÞ ¼ dðtÞW0 ðxÞ qt

ð94Þ

Equation (94) assumes the form of Eq. (92), namely, eLFP t

q
LFP t  e ½Wðx; tÞ þ yð
tÞW0 ðxÞ þ yð
tÞLFP W0 ðxÞ ¼ 0 qt

ð95Þ

For the purpose of using a kinetic equation incorporating the Cole–Davidson mechanism according to the heuristic procedure of Nigmatullin and Ryabov [28], we may replace the partial time derivative in Eq. (95) by a fractional time derivative 0 Dnt . Thus Eq. (95) becomes [cf. Eq. (93)]  tÞ þ yð
tÞW0 ðxÞ þ yð
tÞLFP W0 ðxÞ ¼ 0 tn
1 eLFP t 0 Dnt e
LFP t ½Wðx;

ð96Þ

Next we recall the cumulant operator expansion for the operator e
B AeB [28], namely, e


B



½A; B; B ½A; B þ þ  Ae ¼ A þ 2! 1! B

where the commutator ½A; B is defined as ½A; B ¼ AB
BA. This expansion allows one to represent Eq. (96) as the series of operators: t

n
1




n 0 Dt

n

 ½0 Dt ; LFP t; LFP t ½0 Dnt ; LFP t  þ yð
tÞW0  þ þ    ½W
1! 2!

ð97Þ

þ yð
tÞLFP W0 ¼ 0 This series can be simplified by using the relationship between fractional derivatives [32]: n 0 Dt

½t f ðtÞ
t 0 Dnt ½ f ðtÞ ¼ n 0 Dn
1 ½ f ðtÞ t

Thus Eq. (97) reduces to tn
1




n 0 Dt


 nLFP 0 Dtn
1 nðn
1ÞL2FP 0 Dtn
2  þ yð
tÞW0  þ þ    ½W 1! 2!

þ yð
tÞLFP W0 ¼ 0

ð98Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

which in turn assumes the shifted fractional differential operator form, namely,  tÞ þ yð
tÞW0 ðxÞ þ t yð
tÞLFP W0 ðxÞ ¼ 0 ðt0 D1t
tLFP Þn ½Wðx;

ð99Þ

Here we have recalled the binomial expansion ða þ bÞn ¼

1 X ð
1Þn ð
nÞn n
n n a b n! n¼0

ð100Þ

where ðaÞn ¼ ðn þ aÞ=ðaÞ is a Pochhammer symbol [51]. Equation (99) represents the generalization (by heuristic reasoning) of the normal Fokker– Planck equation to anomalous diffusion governed by a Cole–Davidson relaxation mechanism. In like manner, combining the ideas embodied in the fractional diffusion Eq. (90) describing Cole–Cole relaxation and Eq. (96) describing Cole– Davidson relaxation, we may introduce the fractional kinetic equation  ½ts 0 Dst
tLFP n ½Wðx; tÞ þ yð
tÞW0 ðxÞ þ t yð
tÞLFP W0 ðxÞ ¼ 0

ð101Þ

For two particular cases n ¼ 1, 0 < s < 1 and s ¼ 1, 0 < n < 1, Eq. (101) reduces to Eqs. (90) and (99), respectively. Equation (101) represents a fractional generalization of the normal Fokker– Planck equation incorporating the Havriliak–Negami relaxation mechanism. Thus in any model described by a Fokker–Planck equation the effect of the Havriliak–Negami mechanism on the normal diffusion response may be included by solving Eq. (101). It allows one to generalize both the original Debye model and the Debye–Fro¨hlich model to include this relaxation mechanism. In the particular case of the first Debye model, namely, an assembly of noninteracting dipoles, Eq. (101) will yield the simple Havriliak– Negami Eq. (11) as demonstrated below. E.

Fundamental Solution of the Fractional Smoluchowski Equation

In order to demonstrate how the anomalous relaxation behavior described by the hitherto empirical Eqs. (9)–(11) may be obtained from our fractional generalizations of the Fokker–Planck equation in configuration space (in effect, fractional Smoluchowski equations), Eq. (101), we first consider the fractional rotational motion of a fixed axis rotator [1], which for the normal diffusion is the first Debye model (see Section II.C). The orientation of the dipole is specified by the angular coordinate f (the azimuth) constituting a system of one rotational degree of freedom. Electrical interactions between the dipoles are ignored.

fractional rotational diffusion

317

The normal Fokker–Planck equation for the time evolution of the probability density function Wðf; tÞ on the unit circle in configuration space is then qWðf; tÞ ¼ LFP Wðf; tÞ; qt

Wðf; 0Þ ¼ W0 ðfÞ

ð102Þ

where the Fokker–Planck operator is defined by Eq. (53) [see also Eq. (7)]. In the absence of external fields the Fokker–Planck operator is given by LFP ¼ t
1

q2 qf2

ð103Þ

Equation (101) thus becomes  tÞ þ yð
tÞW0 ðfÞ þ t yð
tÞLFP W0 ðfÞ ¼ 0 ½ts 0 Dst
tLFP n ½Wðf;

ð104Þ

Here we wish to obtain the aftereffect solution for an assembly of fixed axis rotators. Recalling that for rotation, the probability density function must be  periodic in f, we expand Wðf; tÞ in the Fourier series:  Wðf; tÞ ¼ yðtÞ

1 X

fp ðtÞeipf

ð105Þ

p¼
1

Moreover, W must be real so that the Fourier coefficient fp ðtÞ, which is the characteristic function of W must satisfy f
p ðtÞ ¼ fp ðtÞ, where the asterisk denotes the complex conjugate. Substitution of Eq. (105) into Eq. (104) now yields 1 X ð
1Þn ð
nÞn 2n sðn
nÞ sðn
nÞ p t Dt ½yðtÞfp ðtÞ þ fp ð0Þyð
tÞ ¼ p2 yð
tÞfp ð0Þ n! n¼0

ð106Þ Here we have noted Eq. (100) and q2 ipf e ¼
t
1 p2 eipf qf2 1 X  LnFP Wðf; ð
1Þn fp ðtÞp2n eipf tÞ ¼ yðtÞt
n LFP eipf ¼ t
1

ð107Þ ð108Þ

p¼
1

W0 ðfÞ ¼

1 X p¼
1

fp ð0Þeipf

ð109Þ

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william t. coffey, yuri p. kalmykov and sergey v. titov

By using the integration theorem of two-sided Fourier transformation generalized to fractional calculus [32], namely, Ff0 Dat yðtÞf ðtÞg ¼ ðioÞa~f ðoÞ

ð110Þ

where ~f ðoÞ ¼

1 ð

e


iot

yðtÞf ðtÞ dt ¼


1

1 ð

e
iot f ðtÞ dt

ð111Þ

0

with inverse 1 f ðtÞ ¼ 2p

1 ð

eiot~f ðoÞ dt

ð112Þ


1

we have from Eq. (106)  X 1 ð
1Þn ð
nÞn 2n p2 ~fp ðoÞ
fp ð0Þ p ðiotÞsðn
nÞ ¼
fp ð0Þ io n¼0 n! io

ð113Þ

Using Eq. (100), Eq. (113) can be further rearranged as ½ fp ð0Þ
io~fp ðoÞ½ p2 þ ðiotÞs n ¼ p2 fp ð0Þ

ð114Þ

Here we have also recalled that Dtn
n yð
tÞ ¼
Dn
n
1 dðtÞ. We remark that t posing the problem of the step of solution in terms of an impulse response in the time domain with the initial conditions regarded as the amplitude of the impulse has enabled us to obtain the solution in a very simple way using the generalized Fourier integration theorem [Eq. (110)]. This method, unlike the corresponding generalized Laplace transform integration theorem [32], does not involve the initial conditions. The same result may, however, be obtained using the Laplace transform if we rearrange Eq. (101) as an equation for the time-dependent part of the distribution function only so that initial conditions are not involved. Inverting Eq. (114), we have for the inverse Fourier transform over o (see Appendix I) 
  fp ðtÞ p2ð1
nÞ 1;1 2 ð1; 1Þ s ¼1
H p ðt=tÞ  ðn; 1Þ; ð0; sÞ fp ð0Þ ðnÞ 1;2

ð115Þ

319

fractional rotational diffusion

1;1 where H1;2 is the Fox H function [48]. If the initial distribution function is a delta function, namely, W0 ðfÞ ¼ dðf
f0 Þ, the solution (Green’s function) is given by

Wðf; tjf0 ; 0Þ 
 1   ð1; 1Þ 1 X p2ð1
nÞ 1;1 2 s H eipðf
f0 Þ ; 1
p ðt=tÞ  ¼ 2p p¼
1 ðnÞ 1;2 ðn; 1Þ; ð0; sÞ

t 0 ð116Þ

Here we have noted that [50] dðfÞ ¼

1 1 X eipf 2p p¼
1

For two particular cases n ¼ 1, 0 < s < 1 (Cole–Cole relaxation mechanism) and s ¼ 1, 0 < n < 1 (Cole–Davidson relaxation mechanism), Eq. (116) can be considerably simplified. Here Eq. (116) becomes, respectively (see Appendix I), Wðf; tjf0 ; 0Þ ¼

1 1 X Es ½
p2 ðt=tÞs eipðf
f0 Þ ; 2p p¼
1

t 0

ð117Þ

and  1  1 X p2ð1
nÞ 2 gðn; p t=tÞ eipðf
f0 Þ ; 1
Wðf; tjf0 ; 0Þ ¼ 2p p¼
1 ðnÞ

t 0 ð118Þ

where Es ðzÞ and gða; zÞ are the Mittag–Leffler and incomplete Gamma functions, respectively. The Mittag–Leffler function is given by Eq. (61) and the incomplete Gamma function [51] is defined as ðz

gða; zÞ ¼ ta
1 e
t dt ¼ 0

1 1X ð
zÞnþa a n¼0 ða þ nÞn!

ð119Þ

For s ¼ 1, Eq. (116) yields the Green function for normal rotational diffusion of a planar rotator, namely, Wðf; tjf0 ; 0Þ ¼

1 1 X 2 eipðf
f0 Þ
p t=t ; 2p p¼
1

t 0

ð120Þ

320

william t. coffey, yuri p. kalmykov and sergey v. titov

which, we emphasize, is periodic in f. For comparison, the Green function Wðx; tjx0 ; 0Þ for translational diffusion of a free particle along the x axis [8] is given by 1 Wðx; tjx0 ; 0Þ ¼ 2p

1 ð

eixðx
x0 Þ
x

2

Dt=2

dx ð121Þ


1

2 1 ¼ pffiffiffiffiffiffiffiffiffiffi e
ðx
x0 Þ =ð4DtÞ ; 4pDt


1 < x < 1;

t 0

As one can see, the continuous x in Eq. (121) replaces the discrete p of the rotational case. In order to calculate dielectric response functions, we suppose that a uniform field F (having been applied to the assembly of dipoles at a time t ¼
1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, we suppose that the field is weak (i.e., mF kB T, which is the linear response condition [8]). Thus the initial distribution function Wðf; 0Þ is the Boltzmann distribution function and is given by Eq. (57). Now one can readily obtain the corresponding aftereffect solution  1 mF 1þ f1 ðtÞ cos f Wðf; tÞ ¼ 2p kB T

ð122Þ

where the aftereffect function f1 ðtÞ is given by Eq. (115) for p ¼ 1 with f1 ð0Þ ¼ 1, namely, 
  1 ð1; 1Þ 1;1 ð123Þ H1;2 f1 ðtÞ ¼ 1
ðt=tÞs  ðn; 1Þ; ð0; sÞ ðnÞ The function f1 ðtÞ has initially ðt tÞ stretched exponential form sn

f1 ðtÞ  e
ðt=tÞ

=ð1þnsÞ

ð124Þ

In contrast, at long times ðt tÞ, it has the inverse power law behavior f1 ðtÞ 

ðn þ 1Þðt=tÞ
s ðnÞð1
sÞ

ð125Þ

The behavior of f1 ðtÞ is shown in Figs. 1 and 2 for various values of s and n; the asymptotes [Eqs. (124) and (125)] are also shown in these figures for

321

fractional rotational diffusion

1"

2"

3"

100 3 2 f1(t) / f1(0)

1' 1

10−1 σ = 0.5

1,1',1": ν =0.2 2,2',2": ν =0.5 3,3',3": ν =1.0

2'

10−2

10−2

10−4

100

3'

102

t/τ Figure 1. f1 ðtÞ=f1 ð0Þ as a function of t=t for s ¼ 0:5 and different values of n [Eq. (A1.4): solid line] with the short [Eq. (124): dotted lines] and long [Eq. (125): dashed lines] time asymptotes.

comparison. For pure Cole–Davidson relaxation, where s ¼ 1 (see Fig. 2), the asymptote f1 ðtÞ 

e
t=t ðt=tÞn
1 ðnÞ

ð126Þ

must be used instead of Eq. (125) because this is the correct asymptote for the incomplete Gamma function. Thus the decay of f1 ðtÞ is essentially exponential in the limit of long times in Cole–Davidson relaxation. Thus it is apparent from Eqs. (125) and (126) that the origin of the long-time tail is the parameter s, which is ultimately due to the trapping effects inherent in the fractal time random walk—that is, the microscopic disorder. As far as the short-time behavior [Eq. (124)] is concerned, the role of n is to enhance the stretched exponential behavior for given s. Equation (122) allows us readily to evaluate the polarization PðtÞ, namely, PðtÞ ¼ N0 m

2ðp

0

cos fWðf; tÞ df ¼ w0 Ff1 ðtÞ

ð127Þ

322

william t. coffey, yuri p. kalmykov and sergey v. titov 2"

1" 10

0

3"

3

f1(t) / f1(0)

2 1 1' 10

−1

2' ν = 0.5

3'

1,1',1": σ = 0.2 2,2',2": σ = 0.5 3,3',3": σ = 1.0 10

−2

10

−4

10

−2

10

0

10

2

t /τ f1 ðtÞ=f1 ð0Þ as a function of t=t for n ¼ 0:5 and different values of s [Eq. (A1.4): solid line] with the short [Eq. (124): dotted lines] and long [Eqs. (125) (for s ¼ 0:2 and 0.5) and (126) (for s ¼ 1): dashed lines] time asymptotes.

Figure 2.

and the complex dielectric susceptibility, which is defined by linear response theory as [8] wðoÞ ¼ 1
io~f1 ðioÞ w0

ð128Þ

with w0 given by Eq. (76). Thus we have the Havriliak–Negami equation [Eq. (11)]. We remark that Eq. (11) can be obtained by extracting f1 ðtÞ from the Green function, Eq. (116), and noting the linear response theory relation [8] f1 ðtÞ ¼

hcos fð0Þ cos fðtÞi0 hcos2 fð0Þi0

ð129Þ

relating the aftereffect function to the equilibrium dipole moment autocorrelation function. Here h. . .i0 means the equilibrium statistical averages over the

fractional rotational diffusion

323

equilibrium distribution function Weq ðfÞ. The autocorrelation function from Eq. (129) is defined as hcos fð0Þ cos fðtÞi0 ¼

2ðp 2ðp

cos f0 cos fWeq ðfÞWðf; tjf0 ; 0Þ dfdf0

ð130Þ

0 0

As Weq ðfÞ ¼ 1=ð2pÞ for rotation in a plane, one can readily verify that Eqs. (123), (128), (129), and (130) yield Eq. (11). So far we have considered the planar rotator model. However, the above equations can readily be generalized to rotation in space. Here, the space coordinate is the polar angle # and the Fokker–Planck operator for normal rotational diffusion assumes the form [8] [cf. Eq. (80)] 
 1 q qVð#; tÞ qWð#; tÞ LFP Wð#; tÞ ¼ sin # Wð#; tÞ þ 2t sin # q# q# q#

ð131Þ

so that Eq. (101) now becomes  tÞ þ yð
tÞW0 ð#Þ þ t yð
tÞLFP W0 ð#Þ ¼ 0 ½ts 0 Dst
tLFP n ½Wð#;

ð132Þ

 By expanding Wð#; tÞ in the Fourier series  Wð#; tÞ ¼ yðtÞ

1 X

fp ðtÞPp ðcos #Þ

p¼0

(Pn ðzÞ are the Legendre polynomials [51] which now constitute the appropriate basis set), Eq. (132) may be solved to yield the corresponding results for rotation in space, namely, the aftereffect function [Eq. (123)] and the complex susceptibility [Eq. (11)], with t and w0 from Eqs. (81) and (84), respectively. Apparently as in normal diffusion, the results differ from the corresponding two-dimensional analogs only by a factor 2=3 in w0 and the appropriate definition of the Debye relaxation time. Thus we have demonstrated how the empirical Havriliak–Negami equation [Eq. (11)] can be obtained from a microscopic model, namely, the fractional Fokker–Planck equation [Eq. (101)] applied to noninteracting rotators. This model can explain the anomalous relaxation of complex dipolar systems, where the anomalous exponents s and n differ from unity (corresponding to the classical Debye theory of dielectric relaxation); that is, the relaxation process is characterized by a broad distribution of relaxation times. Hence, the empirical Havriliak–Negami equation of anomalous dielectric relaxation which has been

324

william t. coffey, yuri p. kalmykov and sergey v. titov

extensively used to analyze the experimental dielectric loss of disordered glassy systems (see, e.g., Refs. 52 and 53) may be formulated in terms of a microscopic model just as the original Debye equation. A complete understanding of the Havriliak–Negami mechanism requires an understanding of the microscopic origin of the parameters s and n. As far as s is concerned, this fractional exponent arises naturally from the diffusion limit of a fractal time walk and may be construed as arising from a chaotic set [23] of microscopic potential barrier heights; that is, s has its origin in random activation energy models [20]. Further evidence for the random activation energy model concept as the generator of the Cole–Cole mechanism is provided by the recent experimental results of Fannin and Giannitsis [54]. They successfully applied the Cole–Cole equation to the analysis of complex magnetic susceptibility data on ferrofluids with a distribution of particle sizes giving rise naturally to a distribution of Arrhenius-like (Ne´el) microscopic relaxation times. Here the volume of the particles appears explicitly in the argument of the experimental relaxation time. The particle size distribution then automatically leads to a hierarchy of Ne´el relaxation times. The Cole–Cole parameter s is thus a measure of the particle size distribution. Thus our understanding of the parameter s may be said to be reasonably complete. However, the same cannot be said concerning our understanding of the Cole–Davidson parameter n (although some reasons for its origin have been advanced by Nigmatulin and Ryabov [28] in their phenomenological treatment of the Cole–Davidson relaxation process). Here, unlike the Cole–Cole relaxation mechanism based on a fractal time random walk, it is not clear how the kinetic equation for the Cole–Davidson relaxation mechanism and its extension for the Havriliak–Negami relaxation may be derived from anything other than replacing the partial time derivative in the Fokker–Planck equation by a fractional partial time derivative of order n. This represents a gap in our understanding of the Cole–Davidson process. Despite these reservations, the model kinetic equations (96) and (101) are important because they allow one to incorporate the Cole–Davidson and Havriliak– Negami mechanisms into the existing microscopic theory of Debye and Cole– Cole relaxation. The kinetic equation or fractional Fokker–Planck equation [Eq. (101)] that we have proposed may be applied to a system with Havriliak–Negami behavior in the presence of an external potential. The advantage of being in possession of a kinetic equation incorporating the Havriliak–Negami mechanism then becomes apparent because it is now possible to study the effect of the anomalous behavior on fundamental parameters associated with the Brownian motion in a potential such as the Kramers escape rate. Moreover, it is possible to generalize the Debye–Fro¨hlich model of relaxation over a potential barrier to incorporate the Havriliak–Negami mechanism. We remark that in the Cole–Davidson and Havriliak–Negami mechanisms, both of which

fractional rotational diffusion

325

involve branch points in the fractional diffusion operator, the analysis of the aftereffect solution has been greatly facilitated by writing the appropriate kinetic equation as an equation for an impulse response using the properties of the unit step and delta functions. The advantage of such a formulation of the problem is that it avoids the difficulties associated with the inherent dependence of fractional derivatives on initial conditions. These difficulties are completely eliminated by considering the impulse response, which allows us to solve the problem using two-sided Fourier transforms so that the initial conditions are involved only as a forcing function. Finally we remark that in the context of stochastic resonance [8,55], Eqs. (9)–(11) for the complex susceptibility and their extensions to diffusion in a potential may be regarded as transfer functions [56] (Fourier transforms of the true impulse response of the system) whence the spectral density may be easily calculated using the Wiener–Khinchin theorem [8] so that the effect of anomalous diffusion on the stochastic resonance may be ascertained. III.

FRACTIONAL NONINERTIAL ROTATIONAL DIFFUSION IN A POTENTIAL

Anomalous rotational diffusion in a potential may be treated by using the fractional equivalent of the diffusion equation in a potential [7]. This diffusion equation allows one to include explicitly in Fro¨hlich’s model as generalized to fractional dynamics (i) the influence of the dissipative coupling to the heat bath on the Arrhenius (overbarrier) process and (ii) the influence of the fast (high-frequency) intrawell relaxation modes on the relaxation process. The fractional translational diffusion in a potential is discussed in detail in Refs. 7 and 31. Here, just as the fractional translational diffusion treated in Refs. 7 and 31, we consider fractional rotational subdiffusion ð0 < s < 1Þ phenomena only. For simplicity, we first consider rotation about fixed axis in a potential V0 ðfÞ. We suppose that a uniform field F1 (having been applied to the assembly of dipoles at a time t ¼
1 so that equilibrium conditions prevail by the time t ¼ 0) is switched off at t ¼ 0. In addition, we suppose that the field is weak (i.e., mF1 kB T, which is the linear response condition). The underlying fractional rotational diffusion equation is Eq. (52) with Vðf; tÞ ¼ V0 ðfÞ
mF1 ðtÞ cos f

ð133Þ

and the Fokker–Planck operator LFP is defined as LFP ¼ L0FP þ Lext

ð134Þ

326

william t. coffey, yuri p. kalmykov and sergey v. titov

where L0FP W ¼


 1 q2 1 q qV0 W W þ t qf2 kB T qf qf

Lext W ¼

and

mF1 ðtÞ q ðsin fWÞ tkB T qf ð135Þ

The formal solution of the fractional rotational diffusion equation is obtained just as that of Eq. (135) from the Sturm–Liouville representation [7,57] Wðf; tÞ ¼ W0 ðfÞ þ

1 X

p ðfÞfp ðtÞ

ð136Þ

p¼1

with the initial (equilibrium) distribution function Wðf; 0Þ ¼ Ce
½V0 ðfÞ
mF1 ð0Þ cos f=ðkB TÞ ¼ W0 ðfÞ½1 þ x1 ðcos f
hcos fi0 Þ þ Oðx21 Þ

ð137Þ

where W0 ðfÞ ¼ e
V0 =ðkB TÞ =Z is the equilibrium distribution function in the absence of the external field F1 , which satisfies L0FP W0 ðfÞ ¼ 0, x1 ¼ mF1 =ðkB TÞ is the dimensionless field parameter, Z is the partition function given by Z¼

2ðp

e
V0 ðfÞ=ðkB TÞ df

ð138Þ

0

and h i0 means the equilibrium statistical average over W0 ðfÞ. Here, remembering the fact that fp must be periodic in f, the decay modes fp ðtÞ of the fractional diffusion equation obey the equation d fp ðtÞ ¼
lp;s0 D1
s fp ðtÞ t dt

ð139Þ

The eigenvalues lp;s of the operator occurring in Eq. (52) 
  1 q 1 qV q2 þ ts qf kB T qf qf2 may also be expressed in terms of the eigenvalues lp of the Fokker–Planck operator L0FP for normal diffusion in the absence of the electric field, namely, L0FP p ðfÞ ¼
lp p ðfÞ

ð140Þ

lp;s ¼ lp t1
s

ð141Þ

so that

fractional rotational diffusion

327

The solutions of Eq. (139) are the Mittag–Leffler functions Es ðzÞ [7,31], namely, fp ðtÞ ¼ Es ð
lp;s ts Þ

ð142Þ

Equation (141) exemplifies how the eigenvalues of the normal distribution process are altered, in this case reduced, by the nonlocal character of the anomalous diffusion process. The eigenvalues of the nonlocal process are related to their Brownian counterparts by the prefactor t1
s. This scaling effect is significant in the context of escape of particles over potential barriers. There, the smallest nonvanishing eigenvalue l1 of the Fokker–Planck equation, written for the Brownian motion in a potential, yields in the high barrier limit, the Kramers escape rate [58,59]. Since we consider the anomalous diffusion analog of the overdamped Brownian motion, the scaling Eq. (141) shows that the overdamped Kramers escape rate for normal diffusion   l1  e

V=ðkB TÞ (
V is the barrier height) is slowed by the factor t1
s so that l1;s  t1
s l1 ¼ t1
s 

ð143Þ

Therefore, in the present context, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag–Leffler functions governing the highly nonexponential relaxation behavior of the system. The Sturm–Liouville representation (136) is a formal solution as a knowledge of all eigenfunctions p ðfÞ, and corresponding eigenvalues lp is required. However, this representation is very useful because it allows one to obtain a formal solution for the longitudinal complex susceptibility wðoÞ ¼ w0 ðoÞ
iw00 ðoÞ. According to linear response theory [60], wðoÞ is defined as wðoÞ ¼ 1
io w0

1 ð

e
iot Cs ðtÞ dt

ð144Þ

0

where Cs ðtÞ ¼

hcos fiðtÞ
hcos fi0 X ¼ cp Es ½
tlp ðt=tÞs  hcos fið0Þ
hcos fi0 p

is the normalised dipole relaxation function CðtÞ, 2ðp

cp ¼

ðcos f
hcos fi0 Þp ðfÞ df

0

Xð

2p

p

0

ðcos f
hcos fi0 Þp ðfÞ df

ð145Þ

328

william t. coffey, yuri p. kalmykov and sergey v. titov

P

p cp ¼ 1, and the angular brackets h iðtÞ denotes the statistical averages over Wðf; tÞ, namely,

hcos fiðtÞ ¼

2ðp

ð146Þ

cos fWðf; tÞ df 0

Noting the Laplace transform of the Mittag–Leffler function Eq. (79), we have from Eqs. (144) and (145) wðoÞ X cp ¼ s w0 1 þ ðiotÞ =ðtlp Þ p

ð147Þ

The susceptibility from Eq. (147) may simply be evaluated in the low- ðo ! 0Þ and high- ðo ! 1Þ frequency limits. We obtain from Eq. (147) accordingly for o ! 0 and o ! 1 wðoÞ tint ðiotÞs þ   

1
w0 t wðoÞ t  þ  w0 ðiotÞs tef

ð148Þ ð149Þ

where the parameters tint and tef are defined as tint ¼

X

cp =lp

and

tef ¼ 1=

p

X

cp lp

ð150Þ

p

For normal diffusion, s ¼ 1, these parameters correspond to the integral relaxation P time tint [the area under the corresponding relaxation function C1 ðtÞ ¼ p cp e
lp t ] tint ¼

1 ð

C1 ðtÞ dt

ð151Þ

0

and the effective relaxation time tef [which gives precise information on the initial decay of the relaxation function C1 ðtÞ] tef ¼
1=C_ 1 ð0Þ

ð152Þ

fractional rotational diffusion

329

It is of importance that for one-dimensional rotational Brownian motion in a potential, tint and tef defined by Eq. (151) may be expressed in closed form for an arbitrary potential V0 ðfÞ Ref. 8, Chapter 4), namely, tef ¼ t

1 þ hcos 2fi0
2hcos fi20 1
hcos 2fi0

ð153Þ

and (Ref. 8, Chapter 2)

tint ¼

2p ð

t Zðhcos2 fi0
hcos fi20 Þ

0

2f 32 ð eV0 ðfÞ=kT 4 ðcos x
hcos fi0 Þe
V0 ðxÞ=kT dx5 df 0

ð154Þ We note that the characteristic times tint and tef do not exist in anomalous diffusion ðs 6¼ 1Þ. This is obvious from the properties of the Mittag–Leffler function [see Eqs. (62) and (63)]. However, we shall now demonstrate that the above time constants may also be used to characterize the dynamic susceptibility in anomalous relaxation. At low frequencies, ignoring the contributions of all decay modes in Eq. (147), save the slowest one, and noting Eq. (143), we will have for the complex dielectric susceptibility wðoÞ wðoÞ 1 1 ¼ ffi s w0 1 þ ðiotÞ =ðtl1 Þ 1 þ ðio=oc Þs

ð155Þ

where the characteristic frequency oc (here the imaginary part of wðoÞ has a maximum) may thus be defined as oc ¼ t
1 ðtl1 Þ1=s t
1 ðt Þ1=s

ð156Þ

The characteristic time tc of the relaxation process may be defined as
1=s tc ¼ o
1 c ¼ tðtl1 Þ

ð157Þ

Equation (156) illustrates how anomalous relaxation influences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. Therefore, in anomalous relaxation the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag–Leffler function governing the relaxation behavior of the system. We remark that the slowest decay mode, the parameters of which are determined by Eq. (156), will

330

william t. coffey, yuri p. kalmykov and sergey v. titov

be highly nonexponential in character because its behavior will be dictated by the Mittag–Leffler function, Eq. (142), even though the Arrhenius character [albeit stretched according to Eq. (156)] of the activation process is preserved. Equation (155) may be used to evaluate the complex susceptibility in the low-frequency range ðo  oc Þ only. In order to describe the dielectric spectra at all frequencies, we can generalize the method of Coffey et al. [8], Chapter 2, Section 2.13 developed for normal diffusion in a potential. According to this method, a knowledge of the characteristic times 1=l1 , tint , and tef allows one to give a simple analytical description of the relaxation behavior of the system in the context of a two-mode approximation. In order to accomplish this generalization for anomalous relaxation [8,25], let us now suppose that the relaxation function CðtÞ from Eq. (145), which in general comprises an infinite number of Mittag–Leffler functions, may be approximated by two Mittag– Leffler functions only: Cs ðtÞ
1 Es ½
ðt=tÞs tl1  þ ð1

1 ÞEs ½
ðt=tÞs t=tW 

ð158Þ

In turn, the spectrum of the longitudinal susceptibility wðoÞ may essentially be approximated by a sum of two Cole–Cole mechanisms, namely, wðoÞ
1 1

1 ¼ þ w0 1 þ ðio=oc Þs 1 þ ðio=oW Þs

ð159Þ

where the characteristic frequencies oc and oW are given by oc ¼ t
1 ðtl1 Þ1=s ;

oW ¼ t
1 ðt=tW Þ1=s

ð160Þ

Here, we implicitly suppose that the contribution of the high-frequency modes may be approximated by a single mode with characteristic frequency oW . The high-frequency band is due to ‘‘intrawell’’ modes corresponding to the eigenvalues lk ðk 6¼ 1Þ. These near-degenerate ‘‘intrawell’’ modes are indistinguishable in the frequency spectrum of w00 ðoÞ, appearing merely as a single high-frequency band. The low-frequency band is due to the slowest (overbarrier) relaxation mode; the characteristic frequency oc and the halfwidth of this band are determined by l1. Thus, the anomalous low-frequency behavior is dominated by the barrier crossing mode as in the normal diffusion. The parameters
1 and tW in Eqs. (159) and (160) are determined from the condition that the approximate Eq. (159) must obey the exact asymptotic Eqs. (148) and (149) yielding tint ¼
1 =l1 þ tW ð1

1 Þ


1 and t
1 ef ¼
1 l1 þ tW ð1

1 Þ

331

fractional rotational diffusion

It follows that
1 and tW may readily be evaluated from the above equations yielding [8] tint =tef
1 l1 tint
2 þ 1=ðl1 tef Þ l1 tint
1 tW ¼ l1
1=tef


1 ¼

ð161Þ ð162Þ

Equation (159), which involves the integral relaxation time tint , the effective relaxation time tef , and the smallest nonvanishing eigenvalue l1 , correctly predicts wðoÞ both at low ðo ! 0Þ and high ðo ! 1Þ frequencies. Moreover, for a particular form of the potential V, wðoÞ may be determined in the entire frequency range 0  o < 1 as we shall presently see for a double-well periodic potential representing the internal field due to neighboring molecules. A.

Anomalous Diffusion and Dielectric Relaxation in a Double-Well Periodic Potential

Here, we shall present both the exact and approximate solution for the anomalous dielectric relaxation of an assembly of fixed axis dipoles rotating in a double-well potential: V0 ðfÞ ¼ V0 sin2 f

ð163Þ

This potential has two potential minima on the sites at f ¼ 0 and f ¼ p as well as two energy barriers located at f ¼ p=2 and f ¼ 3p=2. This model has been treated in detail for normal diffusion in Refs. 8, 61, and 62. Here we consider the fractional Fokker–Planck equation [Eq. (55)] for a fixed axis rotator with dipole moment m moving in a potential [Eq. (163)]. We calculate wðoÞ by converting the solution of the fractional diffusion Eq. (55) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,62]. By expanding the distribution function Wðf; tÞ in Fourier series Wðf; tÞ ¼

1 X

eipf cp ðtÞ

p¼
1

we have from Eq. (55) the differential-recurrence equation: f_p ðtÞ ¼ t
s 0 D1
s fxV p½fp
2 ðtÞ
fpþ2 ðtÞ
p2 fp ðtÞg; t where xV ¼ V0 =ð2kB TÞ is the barrier height parameter and fp ðtÞ ¼ Re½cp ðtÞ=ð2pÞ ¼ hcos pfiðtÞ

t 0

ð164Þ

332

william t. coffey, yuri p. kalmykov and sergey v. titov

Applying the integration theorem of Laplace transformation generalized to fractional calculus, Eq. (70), we have from Eq. (164)

st~fp ðsÞ
fp ð0Þ ¼ ðstÞ1
s fxV p ~fp
2 ðsÞ
~fpþ2 ðsÞ
p2~fp ðsÞg

ð165Þ

where ~f ðsÞ denotes the Laplace transform of f ðtÞ. The exact solution of the threeterm recurrence equation [Eq. (165)] can be obtained, just as normal diffusion, in continued fraction form (see Refs. 8 and 62 for details). The initial values fp ð0Þ can be obtained as follows. At time t ¼ 0, the steady field FðtÞ is switched off. Thus 2ðp

f2pþ1 ð0Þ ¼ hcos½ð2p þ 1Þfið0Þ ffi

cos½ð2p þ 1ÞfexV cos 2f ½1 þ x1 cos fdf

0 2ðp

exV cos 2f ½1 þ x1 cos fdf

0

ð166Þ where x1 ¼ mF1 =ðkB TÞ and we have supposed that the external field is weak so that x1 1. We also have [51] ez cos 2y ¼

1 X

Im ðzÞei2my

ð167Þ

m¼
1

where Im ðzÞ is the modified Bessel function of the first kind of order m [51]. Whence, using the orthogonality property of the circular functions and noting that Im ðzÞ ¼ I
m ðzÞ, we have 8 2p ð 1 x1 < X Im ðxV Þei2mf ½ei2ðpþ1Þf þ e
i2ðpþ1Þf df f2pþ1 ð0Þ ¼ 4 : m¼0 0

922p 3
1 1 = ðX i2mf i2pf
i2pf i2mf þ Im ðxV Þe ½e þe df 4 Im ðxV Þe df5 ; m¼0 m¼0 0 0   x1 I
ðpþ1Þ ðxV Þ þ Ipþ1 ðxV Þ I
p ðxV Þ þ Ip ðxV Þ þ ¼ I0 ðxV Þ I0 ðxV Þ 4 x1 ½Ipþ1 ðxV Þ þ Ip ðxV Þ ð168Þ ¼ 2I0 ðxV Þ 2ðp

1 X

fractional rotational diffusion

333

On noting the initial values fp ð0Þ, we have " # p 1 X ~f1 ðsÞ tðstÞs
1 ð
1Þp Ipþ1 ðxV Þ þ Ip ðxV Þ Y ¼ 1þ S2kþ1 ðsÞ f1 ð0Þ ðstÞs þ 1
xV þ xV pS3 ðsÞ 2p þ 1 I1 ðxV Þ þ I0 ðxV Þ k¼1 p¼1 ð169Þ with successive convergents being calculated from the continued fraction Sp ðsÞ ¼ ¼

xV p ðstÞs þ p2 þ xV pSpþ2 ðsÞ xV p : xV 2 pðp þ 2Þ s 2 ðstÞ þ p þ xV 2 ðp þ 2Þðp þ 4Þ ðstÞs þ ðp þ 2Þ2 þ ðstÞs þ ðp þ 4Þ2 þ   

~ s ðioÞ ¼ ~f1 ðioÞ=f1 ð0Þ, we may Thus, by setting s ¼ io and by noting that C calculate the complex susceptibility w (o) from Eq. (144), where the static susceptibility w0 [62] is w0 ¼

m2 N0 m2 N0 I1 ðxV Þ þ I0 ðxV Þ hcos2 fi0 ¼ 2I0 ðxV Þ kB T kB T

(N0 is the number of dipoles per unit volume). The ease of calculation of wðoÞ from Eq. (169) represents the chief advantage of the continued fraction in comparison to the Sturm–Liouville method as applied to anomalous diffusion. Now, we shall demonstrate that the characteristic times of the normal diffusion process, namely, the inverse of the smallest nonvanishing eigenvalue 1=l1 , the integral and effective relaxation times tint and tef obtained in [8,62,63], also allow us to evaluate the dielectric response of the system for anomalous diffusion using the two-mode approximation just as normal diffusion (Ref. 8, Section 2.13). Here, we can use known equations for tint , tef , and l1 for the normal diffusion in the potential Eq. (163) [8,62,63]; these equations are tint

ðp

pffiffiffiffiffiffiffiffi  te2xV 2xV sin f df ¼ e
xV cos 2f erf 2 4B½I1 ðxV Þ þ I0 ðxV Þ 0

I0 ðxV Þ þ I1 ðxV Þ tef ¼ t I0 ðxV Þ
I1 ðxV Þ " #
1 1 X p ð
1Þp 2 l1 t ¼ I ðx Þ 1
e
2xV p¼0 2p þ 1 pþ1=2 V

334

william t. coffey, yuri p. kalmykov and sergey v. titov

where erf(z) is the error function [51]. We recall that in the high barrier limit 2xV =8xV and tef  4xV t [8,62,63], yielding (xV 1), we have l
1 1  tint  tpe simple asymptotic equations for the characteristic frequencies oc and oW from Eqs. (160), namely, oc  ð8xV =pÞ1=s e
2xV =s =t

ð170Þ

oW  ð8xV Þ1=s =t

ð171Þ

and

Thus oc and oW depend not only on the barrier height (as in normal diffusion) but also on the anomalous exponent s which substantially modifies the dielectric loss spectra. The characteristic frequency oc and oW given by Eqs. (160) are shown in Fig. 3 as a function of s and xV . ^ðoÞ ¼ wðoÞ=w0 from the exact Calculations of the normalized susceptibility w continued fraction solution, Eq. (169), and the approximate bimodal approximation, Eqs. (159)–(162), are shown in Figs. 4 and 5. Here, the lowand high-frequency asymptotes, Eqs. (148) and (149), are also presented, demonstrating that the bimodal approximation obeys the exact asymptotic equations, Eqs. (148) and (149). Two bands, which appear in the dielectric loss ^00 ðoÞ at 2xV 1 (in the high barrier limit), reach a maximum at spectrum of w the characteristic frequencies oc and oW given by Eqs. (170) and (171), respectively. Apparently, the agreement between the exact continued fraction calculations and the approximate equation Eq. (159), is very good [the maximum relative deviation between the corresponding curves, which appears at o  t
1 , does not exceed a few (3–5) percent]. Similar (or even better) agreement exists for all values of xV and s. Such a good accuracy of the bimodal approximation is due to the fact that the infinite number of highfrequency ‘‘intrawell’’ modes (these near degenerate modes are indistinguishable appearing merely as a single Cole–Cole high-frequency band in the dielectric loss spectrum) may be approximated effectively by a single mode. Thus, one may conclude that Eq. (159) accurately describes the behavior of wðoÞ for all frequencies of interest and for all values of the barrier height (xV ) and anomalous exponent (s) parameters. We remark that the bimodal approximation works extremely well both for anomalous ðs 6¼ 1Þ and normal ðs ¼ 1Þ cases (various applications for the normal diffusion in a potential are given in Ref. 8). Thus the anomalous relaxation in a double-well potential is effectively determined by the bimodal approximation, Eq. (159); the characteristic times of the normal diffusion process—namely, the inverse of the smallest nonvanishing eigenvalue, the integral, and effective relaxation times—appear as time

335

fractional rotational diffusion

0 –5 –10 log10[tw c] –15

0 1

1 2

0.8 B

3 4

0.4

0.6 s

50.2

6 4 2

0.2 0

0.4

log10[tw w]

0

1

0.6 s

3

0.8

2 B

4

1 Figure 3.

Characteristic frequencies oc and oW as functions of s and xV .

parameters. Moreover, the simple asymptotic equations, Eqs. (170) and (171), allow one to easily evaluate the characteristic frequencies of the dielectric loss spectrum in terms of the physical model parameters xV and s. Thus Fro¨hlich’s model of relaxation over a potential barrier based on the concept of normal diffusion may be generalized to anomalous diffusion in disordered energyscapes giving rise to temporally nonlocal behavior [7]. The results obtained may be regarded as a generalization of the solution for the normal Brownian motion in a

336

william t. coffey, yuri p. kalmykov and sergey v. titov

(a)

100

χ^ '(ω)

10−2

1

2

3 σ = 0.5

1: ξ V = 0.01 −4

2: ξ V = 2.5

10

3: ξ V = 5.0 10−12

−8

0

10

4

10

10

8

10 (b)

10 0 χ^''(ω)

−4

10

3

2

1

−2

10

−4

10

−12

10

−8

10

−4

10

0

ωτ

10

4

10

8

10

Figure 4.

The real (a) and imaginary (b) parts of the normalized complex susceptibility evaluated from the exact continued fraction solution [Eqs. (169): solid lines] for s ¼ 0:5 and various values of xV and compared with those calculated from the approximate Eq. (159) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (148) and (149), respectively.

cosine periodic potential [8,62] to fractional dynamics (giving rise to anomalous diffusion). It should be mentioned that if one is interested only in low-frequency ðo  oc Þ part of the dielectric spectrum, one may use a more simple singlemode (Cole–Cole) equation (155) [or Eq. (9) with t ¼ 1=oc ] for the normalized complex susceptibility. The characteristic frequencies oc is given by Eq. (160). The divergence of all the global characteristic times for anomalous diffusion—as defined in their conventional sense (which is a natural consequence of the underlying Le´vy distribution), rendering them useless as a measure of the relaxation behavior—signifies the importance of characteristic times for such processes in terms of the frequency-domain representation of

337

fractional rotational diffusion

10

0

χ^ '(ω )

3

2

ξ V = 5.0

−2

10

1: σ = 1.00 2: σ = 0.75

−4

10

3: σ = 0.50 10

χ^ ''( ω )

(a)

1

10

0

10

−2

10

−4

−12

10

−8

10

3

10

−12

Figure 5.

10

−8

−4

2

10

10

0

10

4

ωτ

8

(b)

1

−4

10

10

0

10

4

10

8

The same as in Fig. 4 for xV ¼ 5 and various values of s.

such processes. As shown in this section, the anomalous diffusion process in the periodic potential will have two associated characteristic times, namely, the inverse of the overbarrier frequency oc and the inverse of the well frequency o W . Moreover, we have the novel and surprising result that the anomalous dielectric relaxation at any frequency is governed by the characteristic times l
1 1 , tint , and tef of the normal diffusion processes. Thus, the difficulties associated with divergent global characteristic times in anomalous diffusion may be avoided. In conclusion, we remark that throughout this discussion, we have confined ourselves to an interpretation of Cole–Cole behavior. Phenomenological fractional diffusion equations in the time domain describing other types of anomalous relaxation such as Cole–Davidson or Havriliak–Negami behavior can be treated in a similar manner. As further applications of the approach developed, we consider in Sections III.C and III.D two problems of fractional rotational diffusion in space, namely,

338

william t. coffey, yuri p. kalmykov and sergey v. titov

(i) the fractional rotational diffusion in a dc electric field and (ii) the fractional rotational diffusion in a uniaxial potential with a superimposed dc electric field (i.e., a double-well potential with nonequivalent wells). B.

Fractional Rotational Diffusion in a Uniform DC External Field

The application of a strong direct current (dc) electric field F0 to a polar fluid comprised of dipolar molecules results in a transition from free thermal rotation of the molecules to partial orientation with hindered rotation. This change in the character of the molecular motion under the influence of the field has a marked effect on the dielectric properties of the fluid insofar as dispersion and absorption of electromagnetic waves will be observed at the characteristic frequencies of rotation of the molecule in the field F0 . A similar effect arises in magnetic relaxation of ferrofluids subjected to a strong dc magnetic field H0 . The similarity of the problems of dielectric relaxation of a polar fluid and magnetic relaxation of a ferrofluid is not surprising because, from a physical point of view, the rotational Brownian motion of magnetic particles (magnetic dipoles) in a constant magnetic field H0 is similar to that of polar molecules (electric dipoles) in a constant electric field F0 . Orientational relaxation of Brownian particles in the context of the normal rotational diffusion in the presence of a dc field has been treated in detail, for example, in Refs. 8 and 64–71. In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F1 applied parallel and perpendicular to the bias field F0 may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker–Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. Let us suppose that the uniform dc field F0 is directed along the Z axis of the laboratory coordinate system and that a small probing field F1 , having been applied to the assembly of dipoles in the distant past ðt ¼
1Þ so that

fractional rotational diffusion

339

equilibrium conditions are attained by the time t ¼ 0, is switched off at t ¼ 0. In addition it is supposed that the field F1 is weak (i.e., mF1 kB T, which is the linear response condition; m is the permanent dipole moment of a molecule). Here, the appropriate fractional rotational diffusion equation [8] is qW ¼ t1
s 0 D1
s LFP W t qt

ð172Þ

where LFP is the Fokker–Planck operator for normal noninertial rotational diffusion defined by Eq. (8), namely,   1 1 r2 W þ r  ðWrVÞ 2t kT  

 1 1 q qW W qV 1 q qW W qV ¼ sin # þ þ þ 2 2t sin # q# q# kB T q# sin # qj qj kB T qj

LFP W ¼

ð173Þ # and j are the polar and azimuthal angles, respectively, and V is the orientational potential energy of the particle in the field. The characteristic relaxation time t for the normal diffusion is given by Eq. (81). For the longitudinal and transverse responses, one has Vk ð#; tÞ ¼
mF0 cos #
mF1 ðtÞ cos #

ð174Þ

V? ð#; j; tÞ ¼
mF0 cos #
mF1 ðtÞ cos j sin #

ð175Þ

and

respectively. According to linear response theory [8], the longitudinal and transverse components of the complex dielectric susceptibility wg ðoÞ ¼ w0g ðoÞ
iw00g ðoÞ, ðg ¼ k; ?Þ are defined as wg ðoÞ ¼ 1
io wg

1 ð

e
iot Cg ðtÞ dt

ð176Þ

hcos #iðtÞ
hcos #i0 hcos #ið0Þ
hcos #i0

ð177Þ

0

where Ck ðtÞ ¼

340

william t. coffey, yuri p. kalmykov and sergey v. titov

and C? ðtÞ ¼

hcos j sin #iðtÞ hcos j sin #ið0Þ

ð178Þ

are the normalized relaxation functions, wk ¼

m2 N0 ð1 þ x
2
coth2 xÞ and kB T

w? ¼

m2 N0 ðcoth x
1=xÞ kB Tx

are the components of the static susceptibility tensor, x ¼ mF0 =ðkB TÞ is the dimensionless field parameter, h. . .iðtÞ denotes the statistical averages over the assembly of rotators in the presence of a small probing ac electric field, and h. . .i0 means the equilibrium statistical averages. The formal step-off transient solution of Eq. (172) for t 0 is obtained from the Sturm–Liouville representation Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X

gp ð#; jÞEs ð
lgp;s ts Þ

ð179Þ

p¼1

with the initial (equilibrium) distribution function

Wð#; j; 0Þ ¼ Ce
Vg ð#;j;0Þ=kT ¼ W0 ð#Þ 1 þ x1 ðug
hug i0 Þ þ Oðx21 Þ

ð180Þ

where W0 ð#Þ ¼ ex cos # =Z is the equilibrium distribution function, Z is the partition function given by

Z¼

ð1

exx dx ¼ 2 sinh x=x

ð181Þ


1

and the angular brackets h. . .i0 mean the equilibrium statistical average over W0. The Sturm–Liouville representation is very useful as it allows one readily to obtain a solution for the relaxation functions Cg ðtÞ. According to Eqs. (176)–(179), Cg ðtÞ ¼

X p

cgp Es ð
lgp;s ts Þ

cgp wg ðoÞ X ¼ p 1 þ ðiotÞs =ðtlg Þ wg p

ð182Þ ð183Þ

341

fractional rotational diffusion

P (where p cp ¼ 1). In the low- ðo ! 0Þ and high- ðo ! 1Þ frequency limits, the susceptibility tensor components may readily be evaluated in terms of the integral relaxation time tint and the effective relaxation time tgef . We have from Eq. (183) wg ðoÞ tgint

1
ðiotÞs þ    w0g ð0Þ t

ð184Þ

wg ðoÞ t  þ  0 wg ð0Þ ðiotÞs tgef

ð185Þ

for o ! 0, and

for o ! 1, where tgint ¼

X

cg =lgp p p

and

tgef ¼ 1=

X

ð186Þ

cg lg p p p

The relaxation times tgint and tgef were obtained in the context of the normal rotational diffusion model in Refs. 8, 65, and 67 and are given by (in our notation) k tint

tx ¼
2 ð1 þ x
coth2 xÞ sinh x

ð1
1

exz ½z
coth x þ e
xð1þzÞ ð1 þ coth xÞ2 dz 1
z2 ð187Þ

k tef

t

¼

x
xLðxÞ
2 LðxÞ

ð188Þ

for the longitudinal response and t? int

¼

t

1 X

xI3=2 ðxÞ n¼1

ð
1Þ

nþ1

n ð2n þ 1ÞInþ1=2 ðxÞ Y ~S? ð0Þj s¼1 k nðn þ 1Þ k¼1

t? ef ¼ 2t

LðxÞ x
LðxÞ

! ð189Þ ð190Þ

for the transverse response, where LðxÞ ¼ coth x
1=x is known as the Langevin function, Inþ1=2 ðxÞ are the spherical Bessel functions [51], and the continued fraction ~ S? k ð0Þjs¼1 is defined in Appendix II, Eq. (A2.9). The behavior of the relaxation times tgint and tgef is very similar: They are very close to each other and decrease with increasing x (see Fig. 6).

342

william t. coffey, yuri p. kalmykov and sergey v. titov 1.0

1: τint / τ : τef / τ ⊥

2: τint / τ ⊥

Relaxation Times

: τef / τ

0.5

2 1

0.0 0

5

ξ

10

15

Integral (tgint : solid lines) and effective (tgef : filled circles and asterisks) relaxation times vs. x for normal rotational diffusion in a dc bias field. Equations (187)–(190) have been used in the calculation.

Figure 6.

According to Eq. (147), each of the infinite number of relaxation modes (corresponding to the eigenvalues lgk ) gives a contribution to the spectra wg ðoÞ. However, as we shall see, these near-degenerate individual modes are indistinguishable in the frequency spectrum of wg ðoÞ appearing merely as a single band. This is due to the fact that in contrast to the rotational diffusion in a double-well potential treated above, for the single-well potential under consideration, the overbarrier relaxation mode is not involved in the relaxation process. Thus noting that tgint tgef for all values of x (see Fig. 6), the spectrum of wg ðoÞ may be approximated by the Cole–Cole equation wg ðoÞ 1

0 wg ð0Þ 1 þ ðio=ogc Þs

ð191Þ

ogc ¼ t
1 ðt=tgef Þ1=s

ð192Þ

where

is the characteristic frequency at which the loss spectrum a00g ðoÞ attains its maximum. Noting the low-temperature behavior of the effective relaxation

343

fractional rotational diffusion

5 4 3 log [twc] 2 10 1

0

0.2 2 0.4 s

4 0.6

6 0.8

x

8 1 10

Frequency okc as a function of x and s.

Figure 7. k

times, namely, tef  t=x and t? ef  2t=x at x 1 [8,67], one can readily obtain from Eqs. (188), (190), and (192) okc  t
1 x1=s

1=s
1 and o? at x 1 c  t ð2xÞ

ð193Þ

The frequencies okc and o? c as functions of x and s are plotted in Figs. 7 and 8. In the time domain, the single-mode approximation, equation Eq. (191), is equivalent to assuming that the relaxation function Cg ðtÞ as determined by the exact equation, Eq. (182) (which in general comprises an infinite number of Mittag–Leffler functions), may be approximated by one Mittag–Leffler function only, namely " Cg ðtÞ Es

t t s
g tef t

# ð194Þ

Equation (191) correctly predicts wg ðoÞ both at low ðo ! 0Þ and high ðo ! 1Þ frequencies; moreover, wg ðoÞ may be determined in the entire frequency range as one shall presently see. In order to estimate the accuracy of the approximate Eq. (191), the longitudinal and transverse components of the complex susceptibility are evaluated by converting the problem of solving the

344

william t. coffey, yuri p. kalmykov and sergey v. titov

4 3 log10 [twc] 2

1

0

0.2 2 0.4 s

4 0.6

6 0.8

x

8 1 10

Figure 8.

Frequency o? c as a function of x and s.

fractional diffusion Eq. (172) with V given by Eq. (174) into the calculation of successive convergents of a differential-recurrence relation just as normal diffusion [8,67]. Just as a double-well potential, the exact solutions for the longitudinal and transverse responses are given in terms of ordinary continued fractions in Appendix II. The results of the calculation of the normalized ½m2 N0 =ðkB TÞ ¼ 1 loss spectra w00k ðoÞ and w00? ðoÞ from these exact continued fraction solutions and the approximate Eqs. (191) are shown in Figs. 9–12; here, the low- and high-frequency asymptotes, Eqs. (148) and (149), are also presented. Apparently as x increases, the spectra shift to higher frequencies in accordance with Eq. (193); simultaneously, the half-width of the spectra increases with decreasing s. Furthermore, the agreement between the exact continued fraction calculations and the approximate Eq. (191) is good [the maximum relative deviation between the corresponding curves does not exceed a few (3–5) percent]. Similar (or even better) agreement exists for all values of x. The accuracy of the single-mode approximation is because for F1 ¼ 0, the potentials (174) and (175) are single-well potentials so that the long-lived mode due to overbarrier relaxation (as in multiwell potentials with two or more metastable states) [63] does not exist. Thus the infinite number of high-frequency ‘‘intrawell’’ modes (these near-degenerate modes are indistinguishable appearing merely as a single high-frequency band in the

345

fractional rotational diffusion

10

−1

σ = 0.5

1 - ξ = 0.01 2- ξ =5 3 - ξ = 10

χ||''(ω)

1

10

2

−3

3

10

−5 −4

−2

10

10

10

0

ωτ

10

2

10

4

10

6

Figure 9. Dielectric loss spectra w00k ðoÞ evaluated from the exact continued fraction solution [Eqs. (176) and (A2.3): solid lines] for s ¼ 0:5 and various values of x, and compared with those calculated from the approximate Eq. (191) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (184), (187), and (185), (188), respectively.

ξ = 5.0

10−1

χ ''( ω ) ||

1 - σ = 1.00 2 - σ = 0.75 3 - σ = 0.50 10−3

3 2 1

10−5 −4

−2

10

0

10

4

10

Figure 10.

The same as in Fig. 9 for x = 5 and various values of s.

ωτ

10

2

10

10

6

346

william t. coffey, yuri p. kalmykov and sergey v. titov

10

σ = 0.5 1 - ξ = 0.01 2- ξ =5 3 - ξ = 10

−1

χ⊥''(ω)

1 10

2

−2

3

10

−3

10

−4

10

−2

10

0

ωτ

10

2

10

4

10

6

Figure 11. Dielectric loss spectra w00? ðoÞ evaluated from the exact continued fraction solution [Eqs. (176) and (A2.8): solid lines] for s ¼ 0:5 and various values of x, and compared with those calculated from the approximate Eq. (191) (asterisks). The low- (dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (184), (189), and (185), (190), respectively.

χ⊥''(ω )

ξ = 5.0 10

−1

10

−2

1 - σ = 1.00 2 - σ = 0.75 3 - σ = 0.50

3 2 10

−3

10

1

−4

Figure 12.

10

−2

10

0

ωτ

10

2

10

4

The same as in Fig. 11 for x ¼ 5 and various values of s.

10

6

fractional rotational diffusion

347

dielectric loss spectrum) may be effectively approximated by a single mode. Thus one may conclude that Eq. (191) accurately describes the behavior of wk ðoÞ and w? ðoÞ for all frequencies of interest and for all values of the field strength (x) and anomalous exponent (s) parameters. Hence the generalized Debye model can explain the anomalous relaxation of complex dipolar systems where the anomalous exponent s differs from unity (corresponding to the classical Debye theory of dielectric relaxation); that is, the relaxation process is characterized by a broad distribution of relaxation times. In particular, the theory may be applied to dilute suspensions of fine magnetic particles (ferrofluids) by a simple change of notation. Experiments on the magnetization induced by a weak ac field superimposed on a strong dc magnetic field may be realized in practice in a ferrofluid because a large value of x can be achieved with a moderate constant magnetic field due to the large value of the magnetic dipole moment m (104–105 Bohr magnetons) of singledomain particles. As observed by Fannin and Giannitsis [54,73], with increasing x, both the magnetic loss spectra and the relaxation times for ferrofluids in a strong dc magnetic field decrease compared with those in the isotropic case. The anomalous relaxation behavior naturally appears in ferrofluids due to the broad distribution of particle volume v (for fine particles, the magnetic moment and the Debye relaxation time strongly depend on v) [74]. The results obtained may be regarded as a generalization of the solution for the normal Brownian motion in a dc bias field [8,67] to fractional dynamics (giving rise to anomalous diffusion). We remark that the single-mode approximation also works extremely well for normal diffusion ðs ¼ 1Þ in a dc external field. C.

Fractional Rotational Diffusion in a Bistable Potential with Nonequivalent Wells

In this section, we shall demonstrate how to evaluate the linear response of an assembly of noninteracting polar Brownian particles in a uniaxial potential
Kðn  eÞ2 (K is the anisotropy constant, e ¼ l=m, and n is the unit vector in the direction of the easy axis) with a superimposed strong dc electric field F0 . In order to retain axial symmetry, we suppose that the field F0 and axis of the uniaxial anisotropy potential n are directed along the Z axis of the laboratory coordinate system. The total potential is an asymmetric double-well potential, which can be written in a dimensionless form as V0 ð#Þ=ðkB TÞ ¼
xV cos2 #
x cos #

ð195Þ

where xV ¼ K=ðkB TÞand x ¼ mF0 =ðkB TÞ are the barrier height and bias field parameters, respectively. The potential (195) is a bistable potential with two nonequivalent wells (see Fig. 13); for xV ¼ 0, it reduces to that considered in the

348

william t. coffey, yuri p. kalmykov and sergey v. titov ϑ

0

π

0

β [V0 (ϑ )−V0 (ϑ max )]

1

2 1

2 −5 3

−10

Figure 13.

4

ξV = 2

1: ξ = 0 2: ξ = ξV 3: ξ = 2 ξV 4: ξ = 3 ξV

Potential V0 ð#Þ=kB T ¼
xV cos2 #
x cos # for xV ¼ 2 and various values of x.

previous section. This model has been treated in Refs. 74–80 in the context of the normal diffusion. A very interesting feature of the asymmetric double-well potential is that at values of the field which are much less than that required to destroy the double-well structure of the potential, the integral relaxation time (which is the area under the curve of the decay function of the electric polarization) switches from the exponential (Arrhenius-like) behavior characteristic of an overbarrier relaxation process to simple algebraic behavior characteristic of the relaxation process in a well [79]. This constant field effect, which is due to the depletion for high-bias fields of the population of the shallower well of the potential, has been described in detail in Ref. 8, Section 1.20. In terms of the complex susceptibility, the effect of the bias field is then to destroy the overbarrier contribution to the susceptibility so that only the highfrequency contribution due to the intrawell modes remains. The dielectric response in the context of anomalous rotational diffusion in the potential Eq. (195) may be calculated 81 in the same manner as normal rotational diffusion (Ref. 8, Chapter 8). Here we present (following [81]) both exact and approximate solutions of fractional Fokker–Planck equation for this particular bistable asymmetric potential. The exact solution of the problem reduces to the solution of infinite hierarchies of differential-recurrence relations for the corresponding relaxation functions. The longitudinal and transverse components of the

fractional rotational diffusion

349

susceptibility tensor may be calculated exactly by matrix continued fractions from the Laplace transform of these relaxation functions using linear response theory. We shall also demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times, allow one to evaluate the dielectric response for anomalous diffusion in the context of the two-mode approximation. It will be shown how this procedure yields a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The longitudinal component of the susceptibility tensor in general exhibits a very broad band low-frequency peak arising from the overbarrier relaxation, superimposed on which is a weaker high-frequency wing. However, if the field F0 is sufficiently strong, the lowfrequency peak as explained earlier can be made to disappear due to depletion of the dipoles in the shallowest well of the potential. We suppose that a small probing field F1 , having been applied to the assembly of dipoles in the distant past ðt ¼
1Þ so that equilibrium conditions have been attained at time t ¼ 0, is switched off at t ¼ 0. Our starting point is the fractional Smoluchowski equation (172) for the evolution of the probability density function Wð#; j; tÞ for normal diffusion of dipole moment orientations on the unit sphere in configuration space (# and j are the polar and azimuthal angles of the dipole, respectively), where the Fokker– Planck operator LFP for normal rotational diffusion in Eq. (8) is given by LFP ¼ L0FP þ Lext , where L0FP W ¼

  1 1 r2 W þ r  ðWrV0 Þ ; 2t kB T

Lext W ¼

1 r  ðWrVg Þ 2tkB T ð196Þ

and the potential energy Vg of a dipole due to the perturbing field F1 is Vg ð#; j; tÞ ¼
mF1 ðtÞug ð#; jÞ

ð197Þ

Here g ¼ k for the longitudinal response and g ¼ ? for the transverse response with uk ¼ cos # and u? ¼ cos j sin #. Our objective is to ascertain how anomalous diffusion modifies the dielectric relaxation in a bistable potential with two nonequivalent wells, Eq. (195). The formal step-off transient solution of Eq. (172) for t 0 is obtained from the Sturm–Liouville representation, Eq. (179), with the initial (equilibrium) distribution function Wð#; j; 0Þ ¼ Ce
½V0 ð#ÞþVg ð#;j;0Þ=ðkB TÞ ¼ W0 ð#Þ½1 þ x1 ðug
hug i0 Þ þ Oðx21 Þ ð198Þ

350

william t. coffey, yuri p. kalmykov and sergey v. titov

where W0 ð#Þ ¼ e
V0 =ðkB TÞ =Z is the equilibrium distribution function in the absence of the external field E1 , which satisfies L0FP W0 ð#Þ ¼ 0, Z is the partition function given by [8] Z¼

ð1

2

exxþxV x dx ¼


1

rffiffiffiffiffiffiffiffi o pffiffiffiffiffi pffiffiffiffiffi p
xV h2 n e erfi½ xV ð1 þ hÞ þ erfi½ xV ð1
hÞ 4xV ð199Þ

Ðx 2 erfiðxÞ ¼ p2ffiffip et dt is the error function of imaginary argument, h ¼ x=ð2xV Þ is 0

the reduced field (or asymmetry) parameter, and x1 ¼ mF1 =ðkB TÞ. The longitudinal and transverse normalized relaxation functions Cg ðtÞ and components of the complex dielectric susceptibility wg ðoÞ are given by Cg ðtÞ ¼

hug iðtÞ
hug i0 X g ¼ c E ½
lgp tðt=tÞs  p p s hug ið0Þ
hug i0 cgp wg ðoÞ X ¼ p 1 þ ðiotÞs =ðtlg Þ wg p

ð200Þ ð201Þ

where lgp;s are expressed in terms of the eigenvalues lgp of the operator L0FP for the normal diffusion, namely, L0FP gp ð#; jÞ ¼
lgp gp ð#; jÞ so that lgp;s ¼ lgp t1
s , and wk ¼

m2 N0 m2 N0 ðhcos2 #i0
hcos #i20 Þ; w? ¼ hcos2 j sin2 #i0 kB T kB T

are the components of the static susceptibility tensor. 1.

Matrix Continued Fraction Solution

As we have already mentioned, it is difficult to evaluate dielectric parameters from Eq. (201) because a knowledge of all the eigenvalues lgk and corresponding amplitudes cgk is required. A more simple (from the computational point of view) solution can be given in terms of matrix continued fractions. The general transient response solution of Eq. (172) for t 0, one can be sought in the form Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X l 1 X ð2l þ 1Þðl
mÞ! g cl;m ðtÞeimj Pm l ðcos #Þ 4p l¼0 m¼
l ðl þ mÞ!

ð202Þ

fractional rotational diffusion

351

where the Pm l ðxÞ are the associated Legendre functions and the equilibrium distribution function may be expanded in the Legendre polynomials Pl ðxÞ, namely, W0 ð#Þ ¼

1 1 X ð2l þ 1ÞGl Pl ðcos #Þ 4p l¼0

ð203Þ

and Gl ¼ hPl ðcos #Þi0 . By substituting Eq. (202) into Eq. (172), we have the fractional differential recurrence relations for the longitudinal and transverse k relaxation functions fl ðtÞ and fl? ðtÞ, namely,   d k xV lðl þ 1Þ lðl þ 1Þ k
s 1
s f ðtÞ ¼ t 0 Dt
fl ðtÞ dt l ð2l
1Þð2l þ 3Þ 2 x xV ðl
1Þ k k k þ ½f ðtÞ
flþ1 ðtÞ þ f ðtÞ 2ð2l þ 1Þ l
1 ð2l
1Þð2l þ 1Þ l
2 xV ðl þ 2Þ k flþ2 ðtÞ ð204Þ
ð2l þ 1Þð2l þ 3Þ   d ? lðl þ 1Þ
3 lðl þ 1Þ ?
s 1
s f ðtÞ ¼ t 0 Dt
fl ðtÞ xV dt l ð2l
1Þð2l þ 3Þ 2

x ? ? þ ðtÞ
l2 flþ1 ðtÞ ðl þ 1Þ2 fl
1 2ð2l þ 1Þ xV lðl þ 1Þ2 xV l2 ðl þ 1Þ ? ? flþ2 fl
2 ðtÞ
ðtÞ þ ð2l þ 1Þð2l þ 3Þ ð2l
1Þð2l þ 1Þ ð205Þ where

k fl ðtÞ

and fl? ðtÞ are defined by k

k

fl ðtÞ ¼ cl;0 ðtÞ ¼ hPl ðcos #ÞiðtÞ
hPl i0

ð206Þ

fl? ðtÞ

ð207Þ

¼

Re½c? l;1 ðtÞ

¼

hcos jP1l ðcos #ÞiðtÞ

so that the relaxation function Cg ðtÞ describing the decay of the electric polarization is Cg ðtÞ ¼ f1g ðtÞ=f1g ð0Þ. We now present the solution of Eqs. (204) and (205) in terms of matrix continued fractions. The advantage of posing the problem in this way is that exact formulae in terms of such continued fractions may be written for the Laplace transform of the aftereffect function, the relaxation time, and the complex susceptibility. The starting point of the calculation is Eqs. (204) and (205) written as the matrix differential recurrence relation g g g gþ g ½Qg
C_ gl ðtÞ ¼ t
s 0 D1
s t l Cl
1 ðtÞ þ Ql Cl ðtÞ þ Ql Clþ1 ðtÞ

ð208Þ

352

william t. coffey, yuri p. kalmykov and sergey v. titov

Here the vectors Cgl ðtÞ are defined as

0

Cgl ðtÞ ¼ @

g f2l
1 ðtÞ

f2lg ðtÞ

1 A

ð209Þ

g where the 2  2 matrices Qg l ; Ql in Eq. (208) are

0

1 4xV ðl
1Þð2l
1Þl xð2l
1Þl B ð4l
1Þð4l
3Þ C 4l
1 B C k
C Ql ¼ B B C @ 2xV ð2l
1Þð2l þ 1Þl A 0 ð4l þ 1Þð4l
1Þ 1 0   2xV
xlð2l
1Þ
1 lð2l
1Þ C B 4l
1 ð4l
3Þð4l þ 1Þ C B k C B Ql ¼ B  C A @ xlð2l þ 1Þ 2xV lð2l þ 1Þ
1 4l þ 1 ð4l þ 3Þð4l
1Þ 1 0
2xV ð2l þ 1Þð2l
1Þl 0 C B ð4l
1Þð4l þ 1Þ C B kþ C B Ql ¼ B C @
xlð2l þ 1Þ
4xV ðl þ 1Þð2l þ 1Þl A 4l þ 1 ð4l þ 1Þð4l þ 3Þ 0 1 4xV l2 ð2l
1Þ 2xl2 B ð4l
1Þð4l
3Þ C 4l
1 B C B C Q?
¼ l B C @ 2xV lð2l þ 1Þ2 A 0 ð4l
1Þð4l þ 1Þ 1 0 2lð2l
1Þ
3 xð2l
1Þ2
C B xV ð4l
3Þð4l þ 1Þ
lð2l
1Þ 2ð4l
1Þ C B ? C B Ql ¼ B C 2 A @ xð2l þ 1Þ 2lð2l þ 1Þ
3
lð2l þ 1Þ xV ð4l
1Þð4l þ 3Þ 2ð4l þ 1Þ 1 0 2xV lð2l
1Þ2 0 C B
ð4l
1Þð4l þ 1Þ C B ?þ C Ql ¼ B C B @ 2xl2 4xV l 2 ð2l þ 1Þ A

ð4l þ 1Þð4l þ 3Þ 4l þ 1

353

fractional rotational diffusion

The three-term matrix recurrence relation, Eq. (208), may now be solved for the ~ g ðoÞ in terms of matrix continued fractions to yield [8] Fourier–Laplace transform C 1 " ! # 1 l X Y ~ g ðoÞ ¼ tðiotÞs
1 Dg ðoÞ Cg ð0Þþ Qgþ Dg ðoÞ Cg ð0Þ ð210Þ C 1

1

1

k
1

l¼2

k

l

k¼2

where Dgk ðoÞ is the 2  2 matrix continued fraction defined as g g

1 Dgk ðoÞ ¼ ðiotÞs I
Qgk
Qgþ k Dkþ1 ðoÞQkþ1

ð211Þ

the tilde denotes the one-sided Fourier transform, namely, ~ g ðoÞ C l

¼

1 ð

e
iot Cgl ðtÞ dt

0

and the initial value vectors Cgl ð0Þ ¼




g ð0Þ f2l
1 f2lg ð0Þ



ð212Þ

can be also evaluated in terms of continued fractions. The initial value vectors Cgl ð0Þ in Eq. (212) may be determined using matrix continued fractions because the initial values flg ð0Þ are expressed just as in normal diffusion in terms of the equilibrium averages (Ref. 8, Chapter 8)

k fl ð0Þ ¼ x1 hcos #Plþ1 i0
hcos #i0 hPl i0   lþ1 l hPlþ1 i0 þ hPl
1 i0
hP1 i0 hPl i0 ¼ x1 2l þ 1 2l þ 1 lðl þ 1Þ ½hPl
1 i0
hPlþ1 i0  fl? ð0Þ ¼ x1 hcos j sin #Pl i0 ¼ x1 2ð2l þ 1Þ and, in turn, the equilibrium quantities hPl i0 satisfy the set of equations (Ref. 8, Chapter 8)


 k
hP2l
3 i0 k hP2l
1 i0 kþ hP2lþ1 i0 Ql þ Ql þ Ql ¼0 ð213Þ hP2l
2 i0 hP2l i0 hP2lþ2 i0 The solution of Eq. (213) is then given by


hP2l
1 ðcos #Þi0 hP2l ðcos #Þi0



k

k

k

¼ Sl ð0ÞSl
1 ð0Þ . . . S1 ð0Þ


 0 1

ð214Þ

354

william t. coffey, yuri p. kalmykov and sergey v. titov k

k

k


k

where Sl ð0Þ ¼ Dl ð0ÞQl and Dl ð0Þ is given by Eq. (211) at s ¼ 0. Thus the initial conditions Cgl ð0Þ using matrix continued fractions are 1 0 1 2l

 2l
1 i
hP 1 0 6 0 B hP2l
1 i0 hP2l
3 i0 k 4l
1 C C B @ A þ Cl ð0Þ ¼ x1 6 4l
1 A hP i 4 @ 2l hP2l
2 i0 2l 0 0 0
hP1 i0 4l þ 1 0 1 3 0 0
hP i  0 2lþ1 A 5 þ @ 2l þ 1 0 hP2lþ2 i0 4l þ 1 82 13 0 0 1 2l > >
hP i > :
hP1 i0 4l þ 1 4l þ 1 0 19
 2l
1 = 0 0 k k A ð0Þ . . . S ð0Þ S þ@ 4l
1 1 l
1 ; 1 0 0 2

0

and 0 1 1 lð2l
1Þ lð2l
1Þ
hP i  B 0
4l
1 C 2l
3 0 C þB 4l
1 A @ A lð2l þ 1Þ hP2l
2 i0 0 0 4l þ 1 3 0 1

 0 0 hP2l
1 i0 hP2lþ1 i0 7 A 7 þ @ lð2l þ 1Þ 0
hP2l i0 hP2lþ2 i0 5 4l þ 1 82 0 13 0 1 lð2l
1Þ > > 0
0 0 0 lð2l þ 1Þ
> : 0 4l þ 1 4l þ 1 0 19
 lð2l
1Þ = 0 0 k k @ A S ð0Þ . . . S1 ð0Þ þ 4l
1 ; l
1 1 0 0

2 0 6 6@ 0 C? l ð0Þ ¼ x1 4 0

Equations (210) and (211) constitute the exact solution of our problem formulated in terms of matrix continued fractions. Having determined the ~ g ðioÞ ¼ ~f g ðioÞ=f g ð0Þ, one may Laplace transform ~flg ðoÞ and noting that C 1 1 calculate the susceptibility wg ðoÞ from Eq. (201).

355

fractional rotational diffusion 2.

Bimodal Approximation

The matrix continued fraction method we have just described yields the exact solutions for the complex susceptibility for all values of the thermal and anisotropy energies. Consequently, that method is an indispensable tool in estimating the accuracy of approximate solutions for typical parameters of the system. For example, in normal diffusion, the Kramers escape rate provides a close approximation to the smallest nonvanishing eigenvalue, the inverse of which provides an approximation to the longest relaxation time in a system where barrier crossing is involved as well as the relaxation time for bistable potentials with equivalent wells ½8. The matrix continued fraction method then allows one to determine the range of system parameters (e.g., barrier height, friction, etc.) in which the approximate solution for the longest relaxation time provided by the Kramers method is valid. Nevertheless, in practical applications such as the analysis of experimental results, the matrix continued fraction method is of very limited use since the dependence of the susceptibility tensor on the model parameters (anisotropy constants) is not obvious from this method. Thus it is desirable to try to obtain simple approximate formulas describing the dynamical behavior. This has been accomplished in previous sections for similar models in the context of the two-mode approximation. For the model under consideration, the corresponding longitudinal complex susceptibility wk ðoÞ can be effectively described then by the sum of two Cole–Cole spectra, namely, wk ðoÞ
1 1

1 ¼ þ k s k wk 1 þ ðio=oc Þ 1 þ ðio=oW Þs

ð215Þ

k

where the characteristic frequencies okc and oW are given by k

k

okc ¼ t
1 ðtl1 Þ1=s ;

oW ¼ t
1 ðt=tW Þ1=s

ð216Þ

k

and l1 is the smallest nonvanishing eigenvalue of the Fokker–Planck operator L0FP for the normal rotational diffusion defined by Eq. (196). The parameters
1 and tW are defined in terms of the characteristic times of the normal diffusion k k (the integral relaxation time tint , the effective relaxation time tef , and the inverse k of the smallest non-vanishing eigenvalue 1=l1 ) k


1 ¼

k

k k

tint =tef
1 k k

k k

l1 tint
2 þ 1=ðl1 tef Þ

;

tW ¼

l1 tint
1 k

k

l1
1=tef

ð217Þ

356

william t. coffey, yuri p. kalmykov and sergey v. titov

Furthermore, for the transverse response, noting that ? ? t? int ffi tef ffi l1

ð218Þ

for all values of the model parameters, the spectrum of w? ðoÞ may be approximated by the single Cole–Cole equation w? ðoÞ 1 ¼ s w? 1 þ ðio=o? cÞ

ð219Þ

where the characteristic frequency o? is given by
1 ? 1=s o? c ¼ t ðt=tef Þ

ð220Þ

In the foregoing equations tgint , tgef , and lg1 are calculated as follows. The k effective relaxation times tef and tgef are given in terms of equilibrium averages hP1 i0 and hP2 i0 only (Ref. 8, Chapter 8): 2hP2 i0 þ 1
3hP1 i20 1
hP2 i0 1
hP2 i0 t? ef ¼ t 1 þ hP2 i0 =2 k

tef ¼ t

ð221Þ ð222Þ

where exV sinh x x
xV Z 2xV
 3exV x 3x2 3 1 coshx
sinhx þ 2

hP2 i0 ¼ 2xV 4x 2 2xV Z 8xV V

hP1 i0 ¼

ð223Þ ð224Þ

the partition function Z is given by Eq. (199). The integral relaxation time tgint can be evaluated numerically from the continued fraction solution ~ g ð0Þ=Cg ð0Þ at s ¼ 1. Moreover, the longitudinal integral relaxation tgint ¼ C k time tint is given in an exact integral form [8] as k tint

ð1

2

ðz

32

eV0 ðzÞ=kT  4 ðz0
hcos #i0 Þe
V0 ðz Þ=kT dz05 ¼ dz 1
z2 Z hcos2 #i0
hcos #i20
1
1 2t

0

ð225Þ

fractional rotational diffusion

357

k

where z ¼ cos #. The smallest eigenvalue l1 can also be evaluated numerically as described in detail in Ref. 8, Chapter 8. In the low-temperature limit, 2xV þ x 1,
1 ; 1

1 , and tW in Eq. (159) may be evaluated from the simple equations [79] wk
1 

4ð1
h2 Þ ½ð1 þ hÞe
2xV h þ ð1
hÞe2xV h 2

wk ð1

1 Þ  ½2xV ð1 þ hÞ
2   5 þ h
1 tW  t 2xV ð1 þ hÞ
1þh

ð226Þ ð227Þ ð228Þ

k

Moreover, l1 can be approximated by Brown’s formula ½76 k

l1 ¼

3=2

2 2 xV ð1
h2 Þ ½ð1 þ hÞe
xV ð1þhÞ þ ð1
hÞe
xV ð1
hÞ  1=2 tp

ð229Þ

For small values of xV , this formula is inadequate. However, we can then use an k approximate equation for the effective relaxation time tef because for k k xV 1 l1 1=tef [8]. Also, for small values of x, one can use the exact Taylor series expansion ([8], Section 8.3.2) 2 48 2 32 3 15552 4 k xV
xV þ x l1 t ¼ 1
x V þ 875 21875  58953125 V
5 1 1 686 2 2 1 4 þ x
x x
x þ  þ 10 875 V 84375 V 7000 k

k

k

ð230Þ

Selected numerical values of tint , tef , 1=l1 , tW , and t? ef are presented in Table I of Ref. [81]. Equations (228) and (229) allow one to readily estimate the asymptotic k behavior of the characteristic frequencies okc and oW in Eq. (160). k The frequencies okc and oW as functions of xV (i.e., the inverse dimensionless temperature) at a fixed value of h ¼ 0:2 are plotted in Figs. 14 and 15 for various values of s. As one can see in these figures, the fractional exponent k s strongly influences the temperature dependence of okc and oW . The results of the calculation of the real w0g ðoÞ and imaginary w00g ðoÞ parts of susceptibility from both the exact continued fraction solutions and the approximate Eqs. (159) and (219) for typical values of the model parameters are shown in Figs. 16–19. The agreement between the exact continued fraction calculations and the approximate Eqs. (159) and (219) is very good [the maximum relative deviation between the corresponding curves does not exceed a few (3–5) percent]. Similar

358

william t. coffey, yuri p. kalmykov and sergey v. titov

10

0

1: σ = 1.0 2: σ = 0.8 3: σ = 0.5 4: σ = 0.2

h = 0.2 −1

10

−2

τωc

||

10

1

10

−3

2

4

10

−4

10

−5

3 (a)

0

5

10

ξV

15

(b)

10 0

4 10 −1

τωc

||

3

σ = 0.8

10 −2

10

1: h = 0.0 2: h = 0.2 3: h = 0.4 4: h = 0.6

−3

0

1

5

2

10

ξV k

15

okc [solid lines: Eq. (160) with exact values of l1 ] versus xV (inverse temperature) for (a) h ¼ 0:2 and various values of s and (b) s ¼ 0:8 and various values of h. Dashed lines: Eq. (160) k with l1 from Eq. (229).

Figure 14.

359

fractional rotational diffusion 10

6

10

4

10

2

1: σ = 1.0 2: σ = 0.8 3: σ = 0.5 4: σ = 0.2

τωW

||

4

3 2 1 h = 0.2

10

0

0

5

10 ξV

15

20

Figure 15. okW versus xV for h ¼ 0:2 and various values of s [solid lines: Eqs. (216) and (217) k

k

k

with exact values of l1 , tint , and tef ].

(or even better) agreement is obtained for all values of x and s. Thus one may conclude that Eqs. (159) and (219) accurately describe the behavior of wg ðoÞ for all frequencies of interest and for all values of the field strength (x), anisotropy constants (xV ), and anomalous exponent (s) parameters. We remark that h ¼ 0 corresponds to the symmetric bistable potential while h ¼ 1 is the value of h at which the bistable structure of the potential—that is, the nucleation field— disappears. Furthermore, for normal diffusion in the asymmetric bistable potential given by Eq. (195), it may be shown that the shallower of the two potential wells becomes completely depopulated at h 0:17 so that the relaxation is no longer dominated by the low-frequency peak generated by the overbarrier process ½79. All that effectively remains is the high-frequency absorption peak due to the well process. The behavior is reminiscent of a phase transition controlled by the asymmetry parameter h. With a simple change of notation the present theory may be used to set the Gilroy and Philips model ½82 of structural relaxation processes in amorphous materials and Dyre and Olsen’s minimal model for beta relaxation in viscous liquids ½83 in the framework of the general theory of stochastic processes. Moreover, the formulation of the theory in terms of kinetic equations as the

χ|| '(ω)

360

william t. coffey, yuri p. kalmykov and sergey v. titov 10 0

1

10

−1

2

10

−2

σ = 0.5

ξV = 10

1: ξ = 0 2: ξ = 2 3: ξ = 5 4: ξ = 10

3 10

−3

10

−4

4

10

10

−1

10

−3

10

−5

−1 0

10

−6

−2

10 ωτ

10

2

10

6

1

χ||'' (ω)

2

ξV = 10 σ = 0.5

4 10

1: ξ = 0 2: ξ = 2 3: ξ = 5 4: ξ = 10

3

−7

10

−1 0

−6

10

−2

10 ωτ

10

2

6

10

w0k ðoÞ and w00k ðoÞ versus ot evaluated from the exact matrix continued fraction solution (solid lines) for s ¼ 0:5, xV ¼ 10 and various values of x and compared with those calculated from the approximate Eq. (215) (filled circles) and with the low-(dotted lines) and highfrequency (dashed lines) asymptotes Eqs. (184) and (185), respectively.

Figure 16.

361

fractional rotational diffusion

σ = 0.5 −1

10

ξ=2

1 2

χ||' (ω)

3 4

−3

10

1: ξV = 1 2: ξV = 5 3: ξV = 10 4: ξV = 15

−5

10

10

10

−1 2

10

−8

10

4

3

−4

ωτ

10

0

2

10

4

10 8

1: ξV = 1

1

2: ξV = 5

−2

3: ξV = 10

χ ''|| (ω )

4: ξV = 15

10

−4

σ = 0.5 ξ=2

10

−6

10 −12

Figure 17.

10 −8

10 −4

ωτ

10 0

10 4

108

The same as in Fig. 16 for s ¼ 0:5, x = 2 and various values of xV .

diffusion limit of, in general, fractal time random walks allows anomalous diffusion effects to be incorporated into both of these models. In the context of dielectric relaxation, we remark that the area of applicability of these results is restricted to the low-frequency range, as defined

362

william t. coffey, yuri p. kalmykov and sergey v. titov 10−1 1

σ = 0.5 ξV = 10

2 3

χ⊥' (ω)

10

−2

10−3

1: ξ = 0 2: ξ = 7 3: ξ = 15 10−4 10−4

10−2

10 0

ωτ

10 2

10 4

10 6

10 2

10 4

10 6

10 −1 σ = 0.5 ξV = 10

10

−2

χ ''⊥ (ω)

1 10 −3

3 2 1: ξ = 0 2: ξ = 7 3: ξ = 15

10 −4

10 −5

10−4

10−2

10 0

ωτ

w0? ðoÞ and w00? ðoÞ versus ot evaluated from the exact matrix continued fraction solution (solid lines) for s ¼ 0:5, xV ¼ 10 and various values of x and compared with those calculated from the approximate Eq. (219) (filled circles) and with the low- (dotted lines) and highfrequency (dashed lines) asymptotes Eqs. (184) and (185), respectively.

Figure 18.

363

fractional rotational diffusion 10

0

σ = 0.5 ξ=2

1 10

2

−1

χ⊥' (ω)

3 10

−2

1: ξV = 1 10

−3

10

−4

2: ξV = 5 3: ξV = 10

10

10

−4

10

−2

10

0

ωτ

10

2

10

σ = 0.5 ξ=2

−1

4

10

6

1: ξV = 1 2: ξV = 5 3: ξV = 10

1

−2

2

χ''⊥ (ω)

10

3

10

−3

10

−4 −4

10

Figure 19.

−2

10

0

10

ωτ

10

2

4

10

6

10

The same as in Fig. 18 for s ¼ 0:5, x ¼ 2 and various values of xV .

by the inequality otg  1 ðg ¼ k; ?Þ, because the theory does not include the effects of molecular inertia. A consistent treatment of inertial effects must be carried out using the appropriate inertial kinetic equation for the evolution of the probability density function in phase space ½8.

364

william t. coffey, yuri p. kalmykov and sergey v. titov IV.

INERTIAL EFFECTS IN ANOMALOUS DIELECTRIC RELAXATION

We shall now discuss the role played by inertial effects in anomalous relaxation on the basis of various generalizations of the inertial Fokker–Planck equation for the motion of a Brownian particle in a potential (or Klein–Kramers equation) to anomalous diffusion. In order to give a physically meaningful description of the anomalous dielectric behavior at high frequencies, inertial effects must be included just as in normal relaxation ½43. As already mentioned in Section II.C, the omission of inertial effects in the relaxation process gives rise, in the context of dielectric relaxation, to the phenomenon of infinite dielectric absorption at high frequencies ½43. Inclusion of them on the other hand yields the return to transparency at high frequencies (in the far-infrared region). In practice, models of the inertial rotational Brownian motion ½43 are frequently used in studying orientational relaxation in liquids in order to compare spectra obtained by various probe techniques such as dielectric relaxation, the dynamic Kerr effect, infrared absorption, Raman scattering, and so on, with the corresponding theoretical spectra ½8; 43; 84. In the context of Brownian motion, inertial effects may be included either by averaging the inertial Langevin equation or by constructing a kinetic equation for the evolution of the probability density function in phase space. The resulting differential recurrence relation may then be solved by various methods to yield the complex susceptibility ½8. Coffey and others ½8; 24 have recently shown how inertial effects may be included in the fractional dynamics of an assembly of noninteracting rotators (rotating either in a plane or in space). They have also demonstrated that this generalized model can reproduce nonexponential (Cole–Cole type) anomalous dielectric relaxation behavior at low frequencies, and how the unphysical high-frequency behavior of the absorption coefficient due to the neglect of inertia may be removed in fractional relaxation just as in inertia corrected Debye relaxation. The inclusion of inertial effects in the fractional dynamics just as in the conventional Brownian dynamics thus ensures the desired return to optical transparency at high frequencies. Moreover, Gordon’s sum rule ½24 for the integral dipolar absorption is satisfied. The starting point of the solution outlined in Refs. 8 and 24 is a generalization of the Klein– Kramers equation to fractional dynamics. The various generalizations of the Klein–Kramers equation which have been proposed have been fully described by Metzler [85], and so we shall merely list them here. Moreover, we shall select the particular generalization of the Klein–Kramers equation which provides a physically acceptable description of the far-infrared absorption in dipolar fluids.

365

fractional rotational diffusion A.

Metzler and Klafter’s Form of the Fractional Klein–Kramers Equation

Metzler and Klafter ½86 and Metzler ½85 have proposed a fractional Klein– Kramers equation (FKKE), which, according to them, corresponds to a multiple trapping picture where the tagged particle executes translational Brownian motion in accordance with the Langevin equation: m€xðtÞ ¼
mb_xðtÞ


q V½xðtÞ; t þ FðtÞ qx

ð231Þ

where FðtÞ is the usual white noise driving force. The particle then gets successively immobilized in traps whose mean distance apart is
¼ ðkB T=mÞt

ð232Þ

where t is the mean time between successive trapping events. The time spans spent in the traps are governed by the waiting-time probability density function wðtÞ  Aa t
1
a ;

0 1Þ, however, this frequency does not exhibit such behavior. We remark that a very high-frequency band is visible in all the figures. This band is due to the fast inertial librations of the dipoles in the potential wells. This band corresponds to the THz (far-infrared) range of frequencies and is usually associated with the Poley absorption [24]. For xV 1, the characteristic

408

william t. coffey, yuri p. kalmykov and sergey v. titov 10

0

10

−1

10

−2

10

−3

α = 0.5

3

4

−χ''(ηω)

γ ' = 0.4

1

2

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0

10

4 - ξV = 5.0

−4 −9

−7

10

−5

10

10

−3

ηω

10

−1

1

10

1.0

10

1 2

0.5

3

χ'(ηω )

4

0.0

1 - ξV = 0.1 2 - ξV = 1.0 3 - ξV = 3.0 α = 0.5

4 - ξV = 5.0 −0.5

10

−9

10

−7

Figure 38.

γ ' = 0.4

10

−5

ηω

10

−3

The same as in Fig. 36 for a ¼ 0:5.

10

−1

10

1

409

fractional rotational diffusion 10

0

α = 1.5 −1

10

ξV = 3

6 5

−χ''(ηω)

4 3

−2

10

2

123456-

1 −3

10

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

−4

10

10−9

10−7

10−5

10−3

ηω

1.0

10−1

101

10−1

101

1 3 5

2

4

6 χ '(ηω )

0.5 1 - γ ' = 0.01 2 - γ ' = 0.1 0.0

3- γ '=1 4 - γ ' = 10 5 - γ ' = 100 6 - γ ' = 1000

10−9

10−7

α = 1.5

10−5 ηω

ξV = 3

10−3

Figure 39. Real and imaginary parts of the complex susceptibility wðZoÞ versus Zo for a ¼ 1:5 and xV ¼ 3 and various values of the friction parameter g0.

410

william t. coffey, yuri p. kalmykov and sergey v. titov

10−1

4

5 4

5

6 −χ''(ηω)

6 1

2

10−3

123456-

10−5 10−7

3

2

3 γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

10−5

1 α =1

10−3

ξV = 3 10−1

ηω

α =1

101

ξV = 3

1.0

5

6

1

4 2

3

χ'(ηω)

0.5

0.0

–0.5 −7 10

123456-

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

−5

10

Figure 40.

−3

10 ηω

−1

10

1

10

The same as in Fig. 39 for a ¼ 1.

pffiffiffiffiffiffiffiffiffiffi frequency of librations oL increases as  xV =I (this frequency is weakly dependent on a). As far as the behavior as a function of g0 is concerned, the amplitude of the high-frequency band decreases progressively with increasing g0 for small inertial effects g0 1, as one would intuitively expect. On the other

411

fractional rotational diffusion 5

6

100

4 2

10−2

3

−χ''(ηω)

1

10−4

1 - γ ' = 0.01 4 - γ ' = 10 2 - γ ' = 0.1 5 - γ ' = 100 6 - γ ' = 1000 3- γ'=1

10−6

10−8

10−6

α = 0.5

10−4 ηω

ξV = 3

10−2

100

1.5 α = 0.5

ξV = 3

χ'(ηω)

1.0

2

3

0.5

6

5

123456-

γ ' = 0.01 γ ' = 0.1 γ'=1 γ ' = 10 γ ' = 100 γ ' = 1000

4

1 0.0

–0.5 10−8

10−6 Figure 41.

10−4 ηω

10−2

The same as in Fig. 39 for a ¼ 0:5.

100

412

william t. coffey, yuri p. kalmykov and sergey v. titov

hand, for large inertial effects g0 1, a fine structure appears in the highfrequency part of the spectra (due to resonances at high harmonic frequencies of the almost free motion in the (anharmonic) cosine potential) again in accordance with intuition. We further remark that the high-frequency ^00 ðoÞ is entirely determined by the inertia of system. ðo oL Þ behavior of w Moreover, just as in the normal Brownian dynamics, the inertial effects ^00 ðoÞ at high frequencies. It can be shown that the produce a rapid fall-off of w present fractional model satisfies the Gordon sum rule for the dipole integral absorption of rotators in a plane [see Eq. (85)]. For a ¼ 1, the anomalous rotational diffusion solution coincides with that for normal rotational diffusion. Finally, it is apparent that between the low-frequency and very highfrequency bands, at some values of model parameters, a third band exists in the dielectric loss spectra (see, e.g., Fig. 30). This band is due to the high-frequency relaxation modes of the dipoles in the potential wells (without crossing the potential barrier) which will always exist in the spectra even in the noninertial limit (see Section III.B). Such relaxation modes are generally termed the intrawell modes. The characteristic frequency of this band depends on the barrier height xV and the anomalous exponent a. In Fig. 42, a comparison of experimental data for 10% v/v solution of a probe molecule CH2Cl2 in glassy decalin at 110 K [98] with the theoretical dielectric loss spectrum e00 ðoÞ  ðe0
e1 Þw00 ðoÞ=w0 calculated from Eqs. (320) and (331) is shown. The reduced moment of inertia Ir used in the calculation is defined by Ir
1 ¼ Ib
1 þ Ic
1, where Ib and Ic are the principal moments of inertia about molecular axes perpendicular to the principal axis a along which the dipole moment vector is directed. For the CH2Cl2 molecule, Ir ¼ 0:24  10
38 g  cm2 [98]. The use of the reduced moment Ir allows one to obtain the correct value for the dipolar integral absorption for twodimensional models. The phenomenological model parameters xV , g0 , and a were adjusted by using the best fit of experimental data. It is known that in order to describe the low-frequency dielectric relaxation in such organic glasses, one must consider anomalous diffusion and relaxation [98]. The highfrequency Poley absorption is also observed in molecular glasses in the farinfrared region (e.g., Ref. 98). Figure 42 indicates that our generalized Fro¨hlich model explains qualitatively the main features of the whole broadband (0-THz) dielectric loss spectrum of the CH2Cl2/decalin solution in contrast to the normal diffusion in a periodic potential (curve 2), which is incapable of explaining the anomalous dielectric relaxation behavior at low frequencies. One can also see in Fig. 33 that the low-frequency part of the loss ^00 ðoÞ, which may be approximated by the Cole–Cole equation [Eq. spectrum w (9)] with s ¼ 2
a and t ¼ 1=oR , is also explained by the generalized Fro¨hlich model.

413

fractional rotational diffusion 100

ε''

10−2

3

10−4

1 2 10−6

10−8 10

1

10

3

10

5

7

10 f [Hz]

10

9

10

11

10

13

Figure 42. Broad-band dielectric loss spectrum of 10% v/v solution of probe molecule CH2Cl2 in glassy decalin at 110 K. Filled circles are the experimental data [98]. Curve 1 is the best fit for the anomalous diffusion in the double-well cosine potential (a ¼ 1:5, xV ¼ 8, and g ¼ 0:003); curve 2 is the best fit for the normal diffusion (a ¼ 1, xV ¼ 7, and g ¼ 0:001) in the double-well cosine potential. Dashed line (curve 3) is the Cole–Cole equation [Eq. (9)] with s ¼ 2
a.

We remark that all the above results are obtained by using the Barkai– Silbey [30] fractional form of the Klein–Kramers equation for the evolution of the probability distribution function in phase space. In that equation, the fractional derivative, or memory term, acts only on the right-hand side—that is, on the diffusion or dissipative term. Hence the form of the Liouville operator, or convective derivative, is preserved [cf. the right-hand side of Eq. (317)] so that Eq. (317) has the conventional form of a Boltzmann equation for the single-particle distribution function. We reiterate that the preservation of the Liouville operator is equivalent to stating that the Newtonian form of the equations of motion underlying the Klein–Kramers equation is preserved. Thus the high-frequency behavior is entirely controlled by the inertia of the system and does not depend on the anomalous exponent. Consequently, the fundamental sum rule, Eq. (85), for the dipole integral absorption of singleaxis rotators is satisfied, ensuring a return to transparency at high frequencies

414

william t. coffey, yuri p. kalmykov and sergey v. titov

as demanded on physical grounds. We also remark that a general characteristic of the systems we have treated is that they are nonlocal both in space and time and thus give rise to anomalous diffusion. The generalized Fro¨hlich model we have outlined incorporates both resonance and relaxation behavior and thus may simultaneously explain both the anomalous relaxation (low-frequency) and far infrared absorption spectra of complex dipolar systems. E.

Fractional Langevin Equation

In previous sections, we have treated anomalous relaxation in the context of the fractional Fokker–Planck equation. As far as the Langevin equation treatment of anomalous relaxation is concerned, we proceed first by noting that Lutz [47] has introduced the following fractional Langevin equation for the translational Brownian motion in a potential V: m

d vðtÞ þ mga 0 Dt1
a vðtÞ þ qx V½xðtÞ; t ¼ lðtÞ dt

ð333Þ

where vðtÞ ¼ x_ ðtÞ is the velocity of the particle, m is the mass of the particle, ga is vðtÞ and lðtÞ are, respectively, the the friction coefficient, and mga 0 D1
a t generalized frictional and random forces with the properties lðtÞ ¼ 0;

lðt0 ÞlðtÞ ¼

mkB Tga jt
t0 ja
2 ða
1Þ

ð334Þ

(the parameter a corresponds to 2
a as used in Ref. 47). The overbar means the statistical average over an ensemble of particles starting at the instant t with the same sharp values of the velocity and the position. The fractional derivative 0 D1
a t in Eq. (333) has the form of a memory function so that Eq. (333) may be regarded as a generalized Langevin equation (a didactic account of the generalized Langevin equation is given by Mazo [9]): ðt d m vðtÞ þ Ka ðt
t0 Þvðt0 Þ dt0 þ qx V½xðtÞ; t ¼ lðtÞ dt

ð335Þ

0

The memory function Ka ðtÞ is given (in accordance with the fluctuation dissipation theorem) by Ka ðtÞ ¼

1 lð0ÞlðtÞ kB T

ð336Þ

fractional rotational diffusion

415

Lutz [47] also supposed that the random force lðtÞ is Gaussian. Equation (335) may also describe non-Gaussian processes. However, in that case, the higherorder moments lðt1 Þlðt2 Þ . . . lðtn Þ may not be expressed in terms of lðtÞ and lðt0 ÞlðtÞ. The formal exact solution of Eq. (335) for a free Brownian particle ðV ¼ 0Þ may readily be obtained using Laplace transforms [47]. We have ~vðsÞ ¼

~ lðsÞ vð0Þ 1 þ s þ s1
a ga m s þ s1
a ga

ð337Þ

so that, noting the Laplace transform of the Mittag–Leffler function, Eq. (79), we obtain

vðtÞ ¼ vð0ÞEa ð
t ga Þ þ m a


1

ðt

Ea ð
ðt
t0 Þa ga Þlðt0 Þ dt0

ð338Þ

0

and ðt

xðtÞ ¼ xð0Þ þ vðt0 Þ dt0 ¼ xð0Þ þ vð0Þ t Ea;2 ð
ga ta Þ 0

þ m
1

ðt ðt00

ð339Þ Ea ð
ðt
t0 Þa ga Þlðt0 Þ dt0 dt00

0 0

Here we have noted that ðt

Ea ð
ga t0a Þ dt0 ¼ t Ea;2 ð
ga ta Þ

0

where Ea;b ðzÞ is the generalized Mittag–Leffler function defined by [89] Ea;b ðzÞ ¼ In particular, Ea;1 ðzÞ ¼ Ea ðzÞ.

1 X

zk ; ðb þ kaÞ k¼0

a; b > 0

ð340Þ

416

william t. coffey, yuri p. kalmykov and sergey v. titov

One can also obtain the first-order statistical moments. We have for the mean displacement [47] xðtÞ ¼ xð0Þ þ vð0Þt Ea;2 ð
ga ta Þ

ð341Þ

and for the first moment of the velocity vðtÞ ¼ vð0ÞEa;1 ð
ga ta Þ

ð342Þ

Moreover, we have from Eq. (342) the equilibrium velocity correlation function cv ðtÞ [cf. Eq. (264) for the angular velocity correlation function] cv ðtÞ ¼ hvð0ÞvðtÞi0 ¼ ðkB T=mÞEa;1 ð
ga ta Þ

ð343Þ

since for the Maxwell–Boltzmann distribution hv2 ð0Þi0 ¼ kB T=m. Noting that x2 ðtÞ is given by ðt

x ðtÞ ¼ x ð0Þ þ 2 xðt0 Þvðt0 Þ dt0 2

2

ð344Þ

0

we may obtain from Eqs. (338) and (339) [30] 2kB T hx2 ðtÞi0
hx ð0Þi0 ¼ m 2

¼2

ðt ðt0

Ea ð
ga ðt0
t00 Þa Þ dt00 dt0

0 0 ðt

kB T 2kB T 2 t Ea;3 ð
ga ta Þ ðt
t0 ÞEa ð
ga ðt
t0 Þa Þ dt0 ¼ m m 0

ð345Þ Lutz also compared his results with those predicted by the fractional Klein– Kramers equation for the probability density function f ðx; v; tÞ in phase space for the inertia-corrected one-dimensional translational Brownian motion in a potential V of Barkai and Silbey [30], which in the present context is
 qf qf 1 qV qf q k B T q2 f 1
a þv
¼ 0 D t ga ðvf Þ þ qt qx m qx qv qv m qv2

ð346Þ

Here x and v ¼ x_ are the position and the velocity of the particle, respectively. Lutz showed that Eqs. (343) and (345) can be obtained in the context of both

fractional rotational diffusion

417

(Langevin and Fokker–Planck) methods. However, the two methods apparently predict different equations for the second moment of the velocity, namely, 2 v2 ðtÞ ¼ ðv2 ð0Þ
kB T=mÞEa;1 ð
ga ta Þ þ kB T=m

ð347Þ

by the Langevin method and v2 ðtÞ ¼ ðv2 ð0Þ
kB T=mÞEa;1 ð
2ga ta Þ þ kB T=m

ð348Þ

by the Fokker–Planck method. Likewise, each approach apparently predicts different results for all higher-order moments (e.g., vn ðtÞ for n 2). Thus, it has been concluded [47] that the fractional equations, Eqs. (346) and (333), describe fundamentally different stochastic processes, although they share striking common features. The above results are indicative of a wider problem, which will be encountered in all attempts to calculate higher-order statistical moments from the generalized Langevin equation. Namely, the fact that a knowledge of the first two moments of the random force is insufficient to calculate higher moments. In other words, the advantage conferred by Isserlis’s theorem (see Ref. 8, Chapter 1, Section 1.3) in the calculation of statistical moments for Markovian Gaussian processes is entirely lost when memory effects are taken into account. This is particularly important in the context of the averaging procedure for the construction of differential-recurrence equations from the Langevin equation, which we have used throughout the book, because it is no longer apparent how the general term of the hierarchy of the averages may be calculated. Thus, at the present time, it is not clear how our procedure may be extended to Langevin equations with a memory term and thus extended to fractional Langevin equations. Similar arguments will of course apply to fractional Fokker–Planck equations such as the Barkai–Silbey or Metzler–Klafter equations, since an analog of the Isserlis theorem is needed in order to justify truncation of the fractural Kramers– Moyal expansion. The general nature of the problems that are encountered in identifying a generalized Langevin equation with a Fokker–Planck equation possessing a memory kernel have been succinctly discussed by Mazo in Chapters 10 and 11 of Ref. 9. For example, taking as a dynamical variable the momentum p of a particle and retaining his notation, we have the generalized Langevin equation ðt dp ¼
Kðt
t0 Þpðt0 Þ dt0 þ FðtÞ dt 0

ð349Þ

418

william t. coffey, yuri p. kalmykov and sergey v. titov

He demonstrates that this equation cannot, in general, be identified with the Fokker–Planck equation [ f ¼ f ðx; p; tÞ]
 ðt qf p qf q p 0 q þ ¼ Kðt
t Þ þ f ðx; p; t0 Þ dt0 qt m qx qp qp mkB T

ð350Þ

0

In view of these difficulties, it appears then that the best course to adopt at the present time is merely to regard the right-hand side of a fractional Klein– Kramers equation as a stosszahlansatz for the Boltzmann equation of which, for example, the Barkai–Silbey equation is simply a special case. We remark that there is nothing unusual about this hypothesis, because there are many examples of collision operators (kinetic models) in statistical mechanics, where it is possible to have a well-behaved right-hand side of the Boltzmann equation, where no corresponding Langevin equation exists, because one cannot separate the stochastic forces into systematic and random parts. Indeed, the classical theory of the Brownian motion is rather particular insofar as a Newtonian-like equation of motion (e.g., the Langevin equation) underlies the dynamical process. There exist many examples of collision operators where a dynamical (Langevin) equation is not defined—for example, the Van Vleck–Weisskopf model [88], the Bhatnagar–Gross–Krook model [39,100], and so on. These kinetic models yield physically acceptable results for the observed variables (such as the complex susceptibility). Therefore, at the present stage of development, it appears to us that the best way forward is to regard kinetic equations such as the Barkai–Silbey equation, as belonging to that particular class of kinetic (collision) models which takes into account long-memory effects. Collision models are, in general, described by a Boltzmann equation such as [39,40,88,100] qf qf qv qf df þv þ ¼ qt qx qx qv dt

ð351Þ

where the right-hand side represents the disturbance of the streaming motion of the distribution function due to collisions. In particular, Sack [40] and Gross [39] (considering rotation about a fixed axis) have shown, by means of an expansion of the distribution function in Fourier series in the angular variable f, have shown how differential-recurrence relations for four distinct collision mechanisms may be obtained. These, in turn, may be solved [25,49] using continued fraction methods to yield the complex susceptibility. We may summarize by stating that as far as progress using the generalized Langevin equation is concerned, the main problem is the lack of a stochastic integral formalism (analogous to the Wiener integrals for normal diffusion), which

fractional rotational diffusion

419

would allow one to calculate the statistical averages needed for the construction of the hierarchy of differential-recurrence equations from the Langevin equation. V.

CONCLUSIONS

By constructing the appropriate probability density diffusion equations, we have demonstrated how conventional Brownian motion solutions for dielectric relaxation may be generalized to fractional dynamics, thereby providing one with a reasonably well-grounded framework for treating the rotational diffusion in disordered fractal systems. Such systems are, in general, governed by a joint probability distribution for two random variables, which are, in general, vectorvalued. The two random variables in question are the waiting time and the jump length. Thus the situation is radically different from that in a discrete-time random walk where the random walker makes a step of fixed mean-square length in a discrete time
t so that considering a one-dimensional random walk, the only variable is the direction of the walker. Instead, in the continuous-time random walk, both the jump length and the time duration between steps are random variables. In general, it is very difficult to treat such walks if these two random variables are not independent. Thus apart from the sections in which inertial effects are considered, we have for the most part concentrated on random walks where one has a Le´vy distribution of waiting times and in general a Gaussian jump length distribution so that the variance of the jump length is finite. The Le´vy distribution of jump times has a convincing physical origin insofar as it is easy to visualize such a chaotic distribution of waiting times as resulting from random activation energies that naturally arise from a chaotic distribution of potential barrier heights that is microscopic disorder. In particular, in the diffusion limit of such fractal time random walks, we have shown how probability density diffusion equations may be constructed in substantially the same manner as those pertaining to normal diffusion. The advantage of such a formulation is that one may just as in normal diffusion easily introduce an external potential. Thus, because of the simple scaling relation that exists between the eigenvalues of the fractional probability density diffusion equation and those of the normal diffusion equation, one may predict the effect of anomalous diffusion on important parameters of the relaxing system such as the Kramers escape rate, and so on. Moreover, just as in normal diffusion, one may derive simple formulas for the complex polarizability, and so on. These predict accurately the effect of anomalous diffusion on the interwell (overbarrier) and intrawell relaxation processes. The overall conclusion is that one may analyze systems governed by such fractal time random walks (of which the Cole–Cole relaxation is the most important example) in a manner almost as simple as the Debye relaxation, which arises from normal diffusion. However, one should remark that in order to obtain diffusion equations for fractal time random walks

420

william t. coffey, yuri p. kalmykov and sergey v. titov

yielding the Cole–Cole relaxation, it is necessary to truncate the fractional analog of the Kramers–Moyal expansion for normal diffusion. Such a truncation may be rigorously justified for the normal diffusion since Isserlis’s theorem allows one to express all moments of the transition probability [101] in terms of powers of the second moment. Thus one may rigorously justify for normal diffusion the truncation of the Kramers–Moyal expansion at the second moment. One cannot do this at present for the diffusion limit of fractal time random walks because of the lack of a fractional diffusion equivalent of Isserlis’s theorem. In other words, one requires the generalization of Isserlis’s theorem (which pertains to Gaussian white noise) to fractal time Le´vy processes. One of the most important consequences of this is that it is impossible at present to establish a correspondence between the diffusion equation for the fractal time random walk and a Langevin equation. Without an analog of Isserlis’s theorem, one cannot by averaging the appropriate Langevin equation over its realizations generate the same hierarchy of differential recurrence relations for the relaxation functions as that which arises from the fractional diffusion equation. In spite of these difficulties, however, the fractal time random walk for the most part provides a reasonable theoretical model for the Cole–Cole process. The Cole–Davidson and Havriliak–Negami processes are, however, more difficult to justify because the fractional diffusion equations governing these processes are essentially generated by a purely mathematical device involving the replacement of ordinary differential operators by fractional ones as detailed in the text. The merit of such an approach is that because of the existence of a kinetic equation, albeit generated by a purely mathematical transformation, one may again incorporate a potential into systems governed by these relaxation processes. Thus one may predict the relaxation behavior in a manner similar to that used for the fractal time random walk. However, unlike in the Cole–Cole process, it is not readily apparent from the underlying physics of the problem how one may justify on physical grounds the purely mathematical replacement of the normal diffusion operator by a fractional diffusion operator in order to generate kinetic equations. We have summarized the present state of the theory when the inertial effects are ignored. Initially, in studying these we considered a diffusion equation in configuration angular velocity space in which the fractional diffusion operator acts on both the Liouville and dissipative terms. Thus the resulting diffusion equation no longer has the form of a Boltzmann equation. Now allowing the fractional diffusion operator to act on the Liouville term destroys the Hamiltonian character of the noiseless motion so that Hamilton’s canonical equations no longer apply. The result of this is that at high frequencies where the relaxation behavior is controlled by the inertia of the system a nonacceptable divergence of the absorption coefficient is obtained. This behavior suggests that the fractional dynamics should be included in the dissipative term only. In this way the form of the Boltzmann equation is retained. Moreover, the noiseless

fractional rotational diffusion

421

motion is still governed by Hamilton’s equations. Thus the absorption coefficient no longer diverges at very high frequencies. We remark that allowing the fractional derivatives to act only on the dissipative term, causes the resulting diffusion equation (first given by Barkai and Silbey [30]) to be nonseparable in the space and time variables. This appears to be consistent with a coupled Le´vy walk picture where any given jump length involves a time cost and vice versa. Although such a diffusion equation fully incorporates inertial effects and produces physically meaning results insofar as the Cole–Cole relaxation behavior is reproduced at low frequencies and the absorption coefficient returns to zero at high frequencies, much work still remain to be done in order to provide a rigorous justification for such inertial kinetic equations. Moreover, the question still remains how such equations may arise from generalized Langevin equations. APPENDIX I: CALCULATION OF INVERSE FOURIER TRANSFORMS For simplicity, we calculate fp ðtÞ for p ¼ 1. Commencing with Eq. (114) for p ¼ 1, we have ð ~f1 ðioÞ 1 1 1 ¼
¼ e
io t f1 ðtÞ dt f1 ð0Þ io ioð1 þ ðiotÞs Þn f1 ð0Þ 1

ðA1:1Þ

0

The inverse Laplace transformation yields f1 ðtÞ 1 ¼ f1 ð0Þ 2pi

gþi1 ð

  du ut 1 e 1
u ð1 þ ðutÞs Þn

ðA1:2Þ

g
i1

The part inside the brackets in Eq. (A1.2) can be rearranged as follows: 1 1
n ¼ ð1 þ ðutÞ
s Þ ð1 þ ðutÞs Þn ðutÞsn Using Eq. (100), we have (Ref. 51, Eqs. (6.1.22) and (15.1.1)) ð1 þ ðutÞ
s Þ
n ¼ 1
nðutÞ
s þ ¼

1 X ðnÞ n¼0

n!

n

nðn þ 1Þ nðn þ 1Þðn þ 2Þ ðutÞ
2s
ðutÞ
3s þ    2! 3!

ð
ðutÞ
s Þn

ðA1:3Þ

422

william t. coffey, yuri p. kalmykov and sergey v. titov

This equation is then substituted back into Eq. (A1.2) to get f1 ðtÞ 1 ¼ f1 ð0Þ 2pi

" # 1 X du ut ðnÞn
sn
s n e 1
ðutÞ ð
ðutÞ Þ u n! n¼0

gþi1 ð g
i1

¼1
t


sn

gþi1 ð 1 X ðnÞn ð
t
s Þn 1 du ut
sn
sn e u 2pi u n! n¼0 g
i1

¼ 1
ðt=tÞsn

1 X ðnÞ n¼0

ð
ðt=tÞs Þn n! ð1 þ sn þ snÞ n

ðA1:4Þ

Finally f1 ðtÞ ~ s; 1 þ sn;
ðt=tÞs Þ ¼ 1
ðt=tÞsn fðn; f1 ð0Þ

ðA1:5Þ

~ is a generalization of a Wright function [99] containing an extra where f Pochhammer symbol. Equation (A1.5) can be also expressed in terms of the Fox 1;1 , [102], namely, H function H1;2 
  f1 ðtÞ 1 ð1; 1Þ 1;1 H1;2 ¼1
ðt=tÞs  ðn; 1Þ; ð0; sÞ f1 ð0Þ ðnÞ

ðA1:6Þ

For p 6¼ 1, the corresponding equation is 1 fp ðtÞ p2 ðt=tÞsn X ð
1Þn ðn þ nÞðp2 ðt=tÞs Þn ¼1
fp ð0Þ ðnÞ n¼0 ð1 þ sn þ snÞn! 
  ð1; 1Þ p2ð1
nÞ 1;1 2 H1;2 p ðt=tÞs  ¼1
ðnÞ ðn; 1Þ; ð0; sÞ

ðA1:7Þ

For n ¼ 1 and s ¼ 1, the function fp ðtÞ from Eq. (A1.7) reduces, respectively, to 
 X 1  ð0; 1Þ fp ðtÞ ð
p2 ðt=tÞs Þn 1;1 s 2 ¼ H1;2 p ðt=tÞ  ¼ fp ð0Þ ð1 þ nsÞ ð0; 1Þ; ð0; sÞ n¼0 s

¼ Es ð
p ðt=tÞ Þ 2

ðA1:8Þ

423

fractional rotational diffusion and


 fp ðtÞ p2ð1
nÞ 1;1 2  ð1; 1Þ ¼1
H1;2 p t=t fp ð0Þ ðnÞ ðn; 1Þ; ð0; 1Þ nX n 1 2 p ðt=tÞ ð
1Þ ðn þ nÞðp2 ðt=tÞÞn p2ð1
nÞ ¼1
gðn; p2 t=tÞ ¼1
ðnÞ n¼0 ð1 þ n þ nÞn! ðnÞ ðA1:9Þ the Mittag–Leffler function Es ðzÞ and the incomplete Gamma function gða; zÞ are defined accordingly by Eqs. (61) and (119); for p ¼ 1, the corresponding equations have been obtained in Ref. 102. Here we recalled that the Fox H m;n can be expressed as a series [48] function Hp;q m Q ðbj
ðbi þ kÞBj =Bi Þ   X m 1 X j¼1; j6¼i  ða1 ; A1 Þ; . . . ðap ; Ap Þ m;n  ¼ Hp;q z q Q ðb1 ; B1 Þ; . . . ðbq ; Bq Þ i¼1 k¼0 ð1
bj þ ðbi þ kÞBj =Bi Þ j¼1þm n Q



ð1
aj þ ðbi þ kÞAj =Bi Þ

j¼1 p Q

ðaj
ðbi þ kÞAj =Bi Þ

ð
1Þk zðbi þkÞ=Bi k!Bi

j¼1þn

if
¼

q P

Bj


j¼1

p P

Aj 0 and Bk ðbj þ lÞ 6¼ Bj ðbk þ sÞ; j 6¼ k; j; k ¼ 1; 2; . . . ; m;

j¼1

l; s ¼ 1; 2; . . .; and n Q ð1
aj
ð1
ai þ kÞAj =Ai Þ   X m 1 X j¼1; j6¼i  ða1 ; A1 Þ; . . . ðap ; Ap Þ m;n  Hp;q z ¼ p Q ðb1 ; B1 Þ; . . . ðbq ; Bq Þ i¼1 k¼0 ðaj þ ð1
ai þ kÞAj =Ai Þ j¼1þn m Q



ðbj þ ð1
ai þ kÞBj =Ai Þ

j¼1 q Q

ð1
bj
ð1
ai þ kÞBj =Ai Þ

ð
1Þk z
ð1
ai þkÞ=Ai k!Ai

j¼1þm

if
¼

q P j¼1

Bj


p P

Aj  0 and Ak ð1
aj þ lÞ 6¼ Aj ð1
ak þ sÞ; j 6¼ k; j; k ¼

j¼1

1; 2; . . . ; n; l; s ¼ 1; 2; . . .; Ai ; Bj > 0; i ¼ 1; 2; . . . ; p; j ¼ 1; 2; . . . ; q. Nearly

424

william t. coffey, yuri p. kalmykov and sergey v. titov

all special functions appearing in applied mathematics can be expressed in terms of the Fox H function [48]. Solutions of Eq. (90), which governs the Cole–Cole mechanism, have been given in Refs. 22 and 101. Here we give details of the aftereffect solution for an assembly of fixed axis rotators. Thus we expand the  probability density function Wðf; tÞ in the Fourier series, Eq. (105). By substituting Eq. (105) into Eq. (90), applying the Fourier transformation and using Eq. (110), we have ½ðioÞs þ p2 =ts ~fp ðoÞ ¼ ðioÞs
1 fp ð0Þ or s
1 ~fp ðoÞ ¼ tðiotÞ fp ð0Þ ðiotÞs þ p2

ðA1:10Þ

Noting that the Laplace transform of the Mittag–Leffler function is given by Eq. (79), one has Eq. (A1.8) [3,4]. The Green function is given by Eq. (117). In like manner for the Cole–Davidson mechanism, substitution of Eq. (105) into Eq. (99) yields 1 X ð
1Þn ðn
nÞ 2n n
n n
n p t 0 Dt ½yðtÞfp ðtÞ þ fp ð0Þyð
tÞ ¼ yð
tÞp2 fp ð0Þ ðn þ 1Þð
nÞ n¼0

ðA1:11Þ yð
tÞ ¼
Dn
n
1 dðtÞ and using Eq. (110), we have Recalling that Dn
n t t  X 1 ð
1Þn ðn
nÞ 2n p2 ~fp ðoÞ
fp ð0Þ p ðiotÞn
n ¼
fp ð0Þ io n¼0 ðn þ 1Þð
nÞ io or ½fp ð0Þ
io~fp ðoÞðp2 þ iotÞn ¼ p2 fp ð0Þ Using the known relation 1 ð

gðn; ctÞe
iot dt ¼

0

one can readily obtain Eq. (A1.9).

ðnÞ ioð1 þ io=cÞn

ðA1:12Þ

425

fractional rotational diffusion

APPENDIX II: EXACT CONTINUED FRACTION SOLUTION FOR LONGITUDINAL AND TRANSVERSE RESPONSES The complex susceptibility components wg ðoÞ can be evaluated from Eq. (147) by calculation of the eigenvalues lgk for normal rotational diffusion (see Section III.C). However, wg ðoÞ may be much more effectively calculated by using the continued fraction method (see Ref. 103 for detail). Let us first evaluate the longitudinal response. By expanding the distribution function Wð#; tÞ in a Fourier series (here W is independent of j) Wð#; tÞ ¼ W0 ð#Þ þ

1 X

ðn þ 1=2ÞPn ðcos #Þ fn ðtÞ

n¼0

one has from Eq. (172) a differential-recurrence equation just as for normal diffusion [8]: þ f_n ðtÞ ¼ t
s 0 D1
s ½q
t n fn
1 ðtÞ þ qn fn ðtÞ þ qn fnþ1 ðtÞ

ðA2:1Þ

where W0 ð#Þ ¼ expðx cos #Þ=Z is the equilibrium distribution function, the Pn ðzÞ are the Legendre polynomials [51], and fn ðtÞ ¼ hPn iðtÞ
hPn i0 are the þ relaxation functions, so that Ck ðtÞ ¼ f1 ðtÞ=f1 ð0Þ and qn ; q
n ; qn are defined as qn ¼


nðn þ 1Þ ; 2

q
n ¼

xnðn þ 1Þ ; 2ð2n þ 1Þ

qþ n ¼


xnðn þ 1Þ 2ð2n þ 1Þ

Applying the integration theorem of one-sided Fourier transformation generalized to fractional calculus, we have from Eq. (A2.1) þ~ ~ ~ iot~fn ðioÞ
fn ð0Þ ¼ ðiotÞ1
s ½q
n fn
1 ðioÞ þ qn fn ðioÞ þ qn fnþ1 ðioÞ

ðA2:2Þ

where ~f ðsÞ denotes the Laplace transform, defined by Eq. (71). The three-term recurrence Eq. (A2.2) can be solved exactly for the Fourier– Laplace transform ~f1 ðioÞ in terms of ordinary continued fractions to yield s
1 X 1 n ~ 2n þ 1 Y ~Sk ðioÞ ~ k ðioÞ ¼ f1 ðioÞ ¼ 2tðiotÞ C ð
1Þnþ1 fn ð0Þ f1 ð0Þ nðn þ 1Þ k¼1 k x f1 ð0Þ n¼1

where the continued fraction Skn ðsÞ is defined by the recurrence equation k

s
1 þ Skn ðioÞ ¼ q
n ½ðiotÞ
qn
qn Snþ1 ðioÞ  
1 x 2ðiotÞs x ~k 1þ Snþ1 ðioÞ þ ¼ 2n þ 1 nðn þ 1Þ 2n þ 1

ðA2:3Þ

426

william t. coffey, yuri p. kalmykov and sergey v. titov

The initial values fn ð0Þ are evaluated just as normal diffusion [8,67]:

fn ð0Þ ¼ x1 hP1 Pn i0
hP1 i0 hPn i0   nþ1 n ¼ x1 hPnþ1 i0 þ hPn
1 i0
hP1 i0 hPn i0 2n þ 1 2n þ 1

ðA2:4Þ

where hPn i0 ¼

Inþ1=2 ðxÞ I1=2 ðxÞ

ðA2:5Þ

x1 ¼ mF1 =ðkB TÞ and Ip ðzÞ is the modified Bessel function of the first kind [51]. Here we have used the relation ð2n þ 1ÞP1 Pn ¼ ðn þ 1ÞPnþ1 þ nPn
1 [51]. In particular, "

#   2 2 I5=2 ðxÞ 1 I3=2 ðxÞ 1 þ
2 f1 ð0Þ ¼ x1 ¼ x1 1 þ 2
coth2 x 3 I1=2 ðxÞ 3 I1=2 ðxÞ x The appropriate differential-recurrence equation for the transverse relaxation functions gn ðtÞ ¼ hcos jP1n ðcos #ÞiðtÞ

ðA2:6Þ

so that C? ðtÞ ¼ g1 ðtÞ=g1 ð0Þ can be obtained from Eq. (172) with V given by Eq. (175) just as for normal diffusion [8]. By expanding the distribution function W in a Fourier series (here W is dependent on j) Wð#; j; tÞ ¼ W0 ð#Þ þ

1 X l 1 X ð2l þ 1Þðl
mÞ! cl;m ðtÞeimj Pm l ðcos #Þ 4p l¼0 m¼
l ðl þ mÞ!

we obtain

d þ gn ðtÞ ¼ t
s 0 Dt1
s q
n gn
1 ðtÞ þ qn gn ðtÞ þ qn gnþ1 ðtÞ dt

ðA2:7Þ

where Pm n ðzÞ is the associated Legendre function [51], gn ðtÞ ¼ Re½cn;1 ðtÞ, and þ ; q qn ; q
n n are defined as qn ¼


nðn þ 1Þ ; 2

q
n ¼

xðn þ 1Þ2 ; 2ð2n þ 1Þ

qþ n ¼


xn2 2ð2n þ 1Þ

427

fractional rotational diffusion

Just as the longitudinal response, Eq. (A2.7) can be solved exactly for the Fourier–Laplace transform ~ g1 ðioÞ in terms of ordinary continued fractions to yield s
1 X 1

g1 ðioÞ 2tðiotÞ ~ ? ðioÞ ¼ ~ ¼ C g1 ð0Þ xg1 ð0Þ

ð
1Þnþ1

n¼1

2n þ 1 n2 ðn

þ 1Þ

2

gn ð0Þ

n Y

S? k ðioÞ

k¼1

ðA2:8Þ where the continued fraction Skn ðsÞ is defined by the following recurrence equation: q
n ? ðiotÞ
qn
qþ n Snþ1 ðioÞ  
1 s xðn þ 1Þ 2ðiotÞ xn ~S? ðioÞ þ1þ ¼ nð2n þ 1Þ nðn þ 1Þ ðn þ 1Þð2n þ 1Þ nþ1

S? n ðioÞ ¼

s

ðA2:9Þ

and the initial values gn ð0Þ are given by gn ð0Þ ¼ x1

nðn þ 1Þ nðn þ 1Þ Inþ1=2 ðxÞ hPn
1 i0
hPnþ1 i0 ¼ x1 I1=2 ðxÞ 2ð2n þ 1Þ 2x

Equations (A2.3) and (A2.8) are the exact solutions of the problem. They allow one to calculate the longitudinal and transverse components of the complex susceptibility from Eqs. (176)–(178). APPENDIX III. DYNAMIC KERR-EFFECT RESPONSE: LINEAR MOLECULES The physical quantity of interest from an experimental point of view and which is appropriate to Kerr effect relaxation is the electric birefringence function K(t) defined by [69] KðtÞ ¼ b2

2pN0 0 ðak
a0? ÞhP2 ðcos #ÞiðtÞ  n

where N0 denotes the number of molecules per unit volume, a0k and a0? are the components of the optical polarizability due to the electric field (optical frequency) of the light beam passing through the liquid medium, and n is the mean refractive index. The coefficient b2 depends on the particle depolarization factors and the dielectric susceptibility of the medium.

428

william t. coffey, yuri p. kalmykov and sergey v. titov

In the transient (step-off) Kerr-effect response, it is also possible to obtain from Eqs. (273)–(275) for l ¼ 2 the system of recurrence equations for the Laplace transforms of the corresponding relaxation functions c2;m n ðtÞ ðm ¼ 0; 1; 2Þ pertaining to that response, namely, 1 2;1 2;0 c ðsÞ
2~c2;1 ½Zs þ 2nx~c2;0 n
1 ðsÞ ¼ dn;0 Zc0 ð0Þ n ðsÞ
~ 2 n ½sZ þ ð2n þ 1Þx~c2;1 n ðsÞ þ

ðA3:1Þ

3ðn þ 1Þ 2;0 ~cnþ1 ðsÞ þ 6ðn þ 1Þ~c2;0 n ðsÞ 2

1 2;2
~c2;2 c ðsÞ ¼ 0 n ðsÞ
~ 4 nþ1

ðA3:2Þ

c2;1 c2;1 ½sZ þ 2nx~c2;2 n ðsÞ þ n~ n ðsÞ þ 4ðn þ 1Þ~ n
1 ðsÞ ¼ 0

ðA3:3Þ

Here, we have taken into account that all the c2;m n ð0Þ vanish with the exception 2 n ¼ 0 and m ¼ 0, namely, c2;0 0 ð0Þ ¼ x =15. This follows from the initial Maxwell–Boltzmann distribution, Eq. (278). Just as in the dielectric response, the solution of Eqs. (A3.1)–(A3.3) for g yÞiðsÞ can be obtained in terms of an infinite continued ~c2;0 ðsÞ ¼ hP2 ðcos 0

fraction, namely, ~c2;0 0 ðsÞ c2;0 0 ð0Þ

Z

¼ Zs þ

3 5
Zs þ x þ Zs þ 2x

b0 Zs þ a1


b1 Zs þ a2


b2

. Zs þ a3
. . ðA3:4Þ

where an ¼ ð2n þ 1Þx þ

4n þ 3 4n þ 5 þ 2nx þ Zs 2ðn þ 1Þx þ Zs

and bn ¼

16ðn þ 1Þðn þ 2Þ ½2ðn þ 1Þx þ Zs2

429

fractional rotational diffusion

The scalar recurrence Eqs. (A3.1)–(A3.3) may also be recast in the form of the matrix three-term recurrence relation
~ ~ n ðsÞ
Qþ C ~ ðZs
Qn ÞC n nþ1 ðsÞ
Qn Cn
1 ðsÞ ¼ d1;n ZC1 ð0Þ

ðA3:5Þ

~ 0 ðsÞ ¼ 0, where C 0

~c2;0 n
1 ðsÞ

1

B C ~ n ðsÞ ¼ B ~c2;1 ðsÞ C C @ n
1 A ~c2;2 n
1 ðsÞ 0 B C1 ð0Þ ¼ B @

c2;0 0 ð0Þ 0 0

1

ðA3:6Þ

C C A 0

0

B B Q
n ¼ @0 0 0

2

0

C 0C A

0
4n 0

1 ðA3:7Þ

0 0

0

1

B C B C Qþ n ¼ @
3n=2 0 1=4 A 0 0 0 0 1
2xðn
1Þ 1=2 0 B C C Qn ¼ B
6n
xð2n
1Þ 1 @ A 0
ðn
1Þ
2xðn
1Þ

ðA3:8Þ

ðA3:9Þ

The solution of Eq. (A3.5) is given by ~ 1 ðsÞ ¼ Z C

I ZsI
Q1


Qþ 1

I ZsI
Q2
Qþ 2

I ..

Q
3

Q
2

C1 ð0Þ

ZsI
Q3 . ðA3:10Þ þ where the column vector C1 ð0Þ and the matrices Q
n ; Qn ; and Qn are defined by Eqs. (A3.6)–(A3.9).

430

william t. coffey, yuri p. kalmykov and sergey v. titov

APPENDIX IV: ORDINARY CONTINUED FRACTION SOLUTION FOR SPHERICAL TOP MOLECULES A solution of the system of moment equations (295)–(297) can be obtained in terms of ordinary continued fractions for spherical top molecules (b0 ¼ b0z , and B ¼ 1) as follows. On substituting Eqs. (295) and (297) into Eq. (296) and introducing a new quantity an;k defined as an;k ¼ ~b1;1 n;2k =k!, we obtain  Zs þ ð2ðn þ kÞ þ 1ÞrðsÞ þ þ

2ðn þ kÞ þ 3 2½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ

 2ðn þ kÞ þ 2 an;k 2½Zs þ 2ðn þ kÞrðsÞ

2 nþ1 anþ1;k ¼
b1;0 ð0Þdnþk;0
s 0;0 4½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ


kþ1 an;kþ1 4½Zs þ ð2ðn þ kÞ þ 2ÞrðsÞ




4ðn þ 1Þ 4k
2 an
1;k
an;k
1 Zs þ 2ðn þ kÞrðsÞ Zs þ 2ðn þ kÞrðsÞ

ðA4:1Þ

Furthermore, for the series Sm ¼

m X

am
i;i

ðA4:2Þ

i¼0

where the summation is taken over the elements an;k with n þ k ¼ m, e.g., S0 ¼ a0;0 , S1 ¼ a1;0 þ a0;1 , S2 ¼ a2;0 þ a1;1 þ a0;2 , etc.), we have from Eqs. (A4.1) and (A4.2) 

 2m þ 3 2m þ 2 þ Zs þ ð2m þ 1ÞrðsÞ þ Sm 2½Zs þ 2ðm þ 1ÞrðsÞ 2½Zs þ 2mrðsÞ 2 mþ1 4m þ 6 Smþ1
Sm
1 ¼
b1;0 ð0Þdm;0
s 0;0 4½Zs þ 2ðm þ 1ÞrðsÞ Zs þ 2mrðsÞ ðA4:3Þ

or 2 1;0
ðZs
qm ÞSm
qþ m Smþ1
qm Sm
1 ¼
b0;0 ð0Þdm;0 s

ðA4:4Þ

fractional rotational diffusion

431

where qm ¼
ð2m þ 1ÞrðsÞ


2m þ 3 mþ1
2½Zs þ 2ðm þ 1ÞrðsÞ Zs þ 2mrðsÞ

mþ1 4½Zs þ 2ðm þ 1ÞrðsÞ 4m þ 6 q
m ¼
Zs þ 2mrðsÞ qþ m ¼


ðA4:5Þ ðA4:6Þ ðA4:7Þ

The continued fraction solution of the three-term recurrence equation, Eq. (A4.4), is S0 ¼


2 Zs

Zs
q0


Zb1;0 0;0 ð0Þ
qþ 0 q1 Zs
q1


ðA4:8Þ
qþ 1 q2

. Zs
q2
. .

~1;0 is then given by The quantity b 0;0 ~ b1;0 0;0 ¼

b1;0 1 0;0 ð0Þ þ S0 s 2Zs

ðA4:9Þ

Taking into account Eqs. (A4.4)–(A4.9) and noting the equality
 1 1 2n þ 3
2n þ 3 ðn þ 1Þð2n þ 5Þ 1 1 2 nþ2
þ ¼ 2 n þ 2 n þ 2 2A 2A 2ðn þ 2ÞA þB Aþ A B we obtain ~ b1;0 0;0 ðsÞ b1;0 0;0 ð0Þ

¼

Z sZ þ Z2

ðA4:10Þ

where the infinite continued fraction Z2 is determined by the recurrence relation Zn ¼

n=2 1 nðn þ 3Þ þ sZ þ rðsÞðn
1Þ þ ðn þ 2Þ½sZ þ nrðsÞ 2ðn þ 2Þ½sZ þ nrðsÞ þ Znþ2  ðA4:11Þ

Equations (A4.10) and (A4.11) yield Eq. (303).

432

william t. coffey, yuri p. kalmykov and sergey v. titov APPENDIX V: KERR-EFFECT RESPONSE

One may also readily derive differential-recurrence equations for the statistical moments involving the associated Legendre functions of order 2 (l ¼ 2) pertaining 2;0 to the dynamic Kerr effect, namely, b2;m n;k ðtÞ [so that b0;0 ðtÞ ¼ hP2 ðcos #ÞiðtÞ]. These equations can be written as a system of algebraic recurrence relations in the frequency domain using Laplace transformation, namely, 1 ~2;1 2;0 ~2;1 ½Zs þ 2nrðsÞ þ krz ðsÞ=B~ b2;0 n;k ¼ Zc0;0 ð0Þdnþk;0 þ bn;k þ 2bn
1;k 2 ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ b2;1

ðA5:1Þ

n;k

3 ~2;0 1 ~2;2 ~2;2 b2;0 ¼
ðn þ 1Þ~ nþ1;k
6ðn þ 1Þbn;k þ bnþ1;k þ bn;k 2 4 1 pffiffiffi ~2;
1 ~2;
1 Þ Bðbn;kþ1 þ 2kb
n;k
1 2 ½Zs þ ð2n þ 1ÞrðsÞ þ krz ðsÞ=B~ b2;
1

ðA5:2Þ

n;k

1 2;
2 1 pffiffiffi ~2;1 bnþ1;k þ ~ Bðbn;kþ1 þ 2k~ ¼ ~ b2;
2 b2;1 n;k þ n;k
1 Þ 4 2 ~2;2 ½Zs þ 2nrðsÞ þ krz ðsÞ=Bb n;k pffiffiffi 2;
2 2;1 2;1 ¼
n~ bn;k
4ðn þ 1Þ~ bn
1;k
Bð~ bn;kþ1 þ 2k~b2;
2 n;k
1 Þ 2;
2 ½Zs þ 2nrðsÞ þ krz ðsÞ=B~ bn;k

~2;
1 ¼
n~ b2;
1 n;k
4ðn þ 1Þbn
1;k þ

ðA5:3Þ

ðA5:4Þ

pffiffiffi 2;2 Bð~ bn;kþ1 þ 2k~b2;2 n;k
1 Þ

ðA5:5Þ

Then the hierarchy of equations for ~ b2;m n;k ðsÞ Eqs. (A5.1)–(A5.5) can be transformed into the matrix three-term differential-recurrence equation
~ ~ n ðsÞ
Qþ C ~ ½ZsI5n
Qn ðsÞC n nþ1 ðsÞ
Qn Cn
1 ðsÞ ¼ dn;1 ZC1 ð0Þ

ðA5:6Þ

~ ðsÞ is comprised of the five subvectors where the supercolumn vector C n 0

~c2;0 n
1 ðsÞ

1

C B 2;1 B ~cn
1 ðsÞ C C B C ~ n ðsÞ ¼ B C C; B ~c2;
1 ðsÞ C B n
1 C B 2;2 @ ~cn
1 ðsÞ A ~c2;
2 n
1 ðsÞ

0

~b2;m ðsÞ n;0

1

C B 2;m C B ~b B n
1;1 ðsÞ C 2;m C B ~cn ðsÞ ¼ B C .. C B . A @ ~b2;m ðsÞ 0;n

ðA5:7Þ

433

fractional rotational diffusion þ and the supermatrices Q
n , Qn , and Qn are given by 1 0 0 q
0 0 0 n C B 0 0 C B 0 0
p
n C B
B 0 0 0 C Q
n ¼ B 0 pn C C B 0 0
2p
@ 0 v
n n A

0 0

0

B þ B 3qn B B 0 Qþ ¼ n B B @ 0 0 0

v
n

0

q0n ðsÞ

2p
n

0

0

0

0

0 pþ n


pþ n 0

vþ n 0

0 0

0 0

0 2pþ n

In =2

0

0

C C C C C þC
2pn A 0 0 vþ n

0

B 0 In B 6ðrn
In Þ q1n ðsÞ B 1 B Qn ðsÞ ¼ B 0 0 qn ðsÞ 0 B 0 0 rn 0 qn ðsÞ @ 0 0 0 rn Here the submatrices p n, submatrices v n are 0 n 0  B0 n
1  B. .. .. B. v
. . n ¼
4B . @0 0  0

0

1

1

0

C C C C C C 0 A q0n ðsÞ 0 In

M q n , rn , and qn (s) are defined above, and the

1 0 0C .. C .C C 2A

 0

0

1 B 0 1B vþ n ¼ @ .. 4 . 0

;

0 1 .. . 0

 0  0 . .. . ..  1

nðn
1Þ

1 0 0C .. C .A 0

nðnþ1Þ

~ 1 ðsÞ is then given by the The exact solution, for the Laplace transform C matrix continued fraction, namely, ~ 1 ðsÞ ¼ Z C

I5 ZsI5
Q1
Qþ 1

I10 ZsI10
Q2
Qþ 2

I15 ..

Q
3

Q
2

C1 ð0Þ

ZsI15
Q3 . ðA5:8Þ

434

william t. coffey, yuri p. kalmykov and sergey v. titov

with initial conditions 1 x2 =15 B 0 C C B C C1 ð0Þ ¼ B B 0 C @ 0 A 0 0

and

Cn ð0Þ ¼ 0

for all n 2

As an example the results of numerical calculations for linear ðIz ¼ 0Þ molecules based on the above matrix continued fraction solution have been compared with that of Ref. 67 presented in terms of ordinary continued fractions. The numerical calculations show that both matrix and ordinary continued fraction solutions yield the same results. We remark that evaluation of the Kerr effect response in the context of the fractional noninertial rotational diffusion model has been carried out by De´jardin and Jadzyn [104]. Acknowledgments The support of this work by INTAS (project 01–2341) and HEA Ireland (Programme for Research in Third Level Institutions, Nanomaterials Initiative) is gratefully acknowledged. WTC thanks the Queen’s University of Belfast for the award of a Distinguished Visitor Fellowship for the period 2002–2005. The Trinity College Dublin Trust is thanked for financial support.

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CHAPTER 9 FUNDAMENTALS OF LE´VY FLIGHT PROCESSES ALEKSEI V. CHECHKIN and VSEVOLOD Y. GONCHAR Institute for Theoretical Physics, National Science Center, Kharkov Institute for Physics and Technology, Kharkov 61108, Ukraine JOSEPH KLAFTER School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel RALF METZLER NORDITA—Nordic Institute for Theoretical Physics, DK-2100 Copenhagen Ø, Denmark

CONTENTS I. Introduction II. Definition and Basic Properties of Le´vy Flights A. The Langevin Equation with Le´vy Noise B. Fractional Fokker–Planck Equation 1. Rescaling of the Dynamical Equations C. Starting Equations in Fourier Space III. Confinement and Multimodality A. The Stationary Quartic Cauchy Oscillator B. Power-Law Asymptotics of Stationary Solutions for c
2, and Finite Variance for c > 2 C. Proof of Nonunimodality of Stationary Solution for c > 2 D. Formal Solution of Equation (38) E. Existence of a Bifurcation Time 1. Trimodal Transient State at c > 4 2. Phase Diagrams for n-Modal States F. Consequences IV. First Passage and Arrival Time Problems for Le´vy Flights A. First Arrival Time B. Sparre Anderson Universality Fractals, Diffusion, and Relaxation in Disordered Complex Systems: A Special Volume of Advances in Chemical Physics, Volume 133, Part B, edited by William T. Coffey and Yuri P. Kalmykov. Series editor Stuart A Rice. Copyright # 2006 John Wiley & Sons, Inc.

439

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aleksei v. chechkin et al.

C. Inconsistency of Method of Images V. Barrier Crossing of a Le´vy Flight A. Starting Equations B. Brownian Motion C. Numerical Solution D. Analytical Approximation for the Cauchy Case E. Discussion VI. Dissipative Nonlinearity A. Nonlinear Friction Term B. Dynamical Equation with Le´vy Noise and Dissipative Nonlinearity C. Asymptotic Behavior D. Numerical Solution of Quadratic and Quartic Nonlinearity E. Central Part of PðV; tÞ F. Discussion VII. Summary Acknowledgements References VIII. Appendix. Numerical Solution Methods A. Numerical Solution of the Fractional Fokker–Planck Equation [Eq. (38)] via the Gru¨nwald–Letnikov Method B. Numerical Solution of the Langevin Equation [Eq. (25)]

I.

INTRODUCTION

Random processes in the physical and related sciences have a long-standing history. Beginning with the description of the haphazard motion of dust particles seen against the sunlight in a dark hallway in the astonishing work of Titus Lucretius Carus [1], followed by Jan Ingenhousz’s record of jittery motion of charcoal on an alcohol surface [2] and Robert Brown’s account of zigzag motion of pollen particles [3], made quantitative by Adolf Fick’s introduction of the diffusion equation as a model for spatial spreading of epidemic diseases [6], and culminating with Albert Einstein’s theoretical description [4] and Jean Perrin’s experiments tracing the motion of small particles of putty [5], the idea of an effective stochastic motion of a particle in a surrounding heat bath has been a triumph of the statistical approach to complex systems. This is even more true in the present Einstein year celebrating 100 years after his groundbreaking work providing our present understanding of Brownian motion. In Fig. 1, we display a collection of typical trajectories collected by Perrin. Classical Brownian motion of a particle is distinguished by the linear growth of the mean-square displacement of its position coordinate x [9–11],1 hx2 ðtÞi ’ Dt 1

Editor’s note. The inertia of the particle is ignored.

ð1Þ

fundamentals of le´vy flight processes

441

Figure 1. Random walk traces recorded by Perrin [5]: Three trajectories obtained by tracing a small grain of putty at intervals of 30sec. Using Einstein’s relation between the macroscopic gas constant and the diffusion constant, Perrin found a quite accurate result for Avogadro’s number. Refined results were successively obtained by Westgren and Kappler [7,8].

and the Gaussian form

  1 x2 Pðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp  4Dt 4pDt

ð2Þ

of its probability density function (PDF) Pðx; tÞ to find the particle at position x at time t. This PDF satisfies the diffusion equation q q2 Pðx; tÞ ¼ D 2 Pðx; tÞ qt qx

ð3Þ

for natural boundary conditions Pðjxj ! 1; tÞ ¼ 0 and d function initial condition ÐPðx; 0Þ ¼ dðxÞ. If the particle moves in an external potential x VðxÞ ¼  Fðx0 Þdx0 , the force FðxÞ it experiences enters additively into the diffusion equation, and the diffusion equation [Eq. (3)] is the particular term of the Fokker–Planck equation known as the Smoluchwski equation [11, 12] q Pðx; tÞ ¼ qt



 q V 0 ðxÞ q2 þ D 2 Pðx; tÞ qx mZ qx

ð4Þ

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where m is the mass of the particle and Z the friction constant arising from existing from exchange of energy with the surrounding heat bath. This Fokker– Planck equation is a versatile instrument for the description of a stochastic process in external fields [12]. Requiring that the stationary solution defined by qPðx; tÞ=qt ¼ 0 is the equilibrium distribution,     V 0 ðxÞ ! V 0 ðxÞ Pst ðxÞ ¼ N exp  ð5Þ ¼ N exp  DmZ kB T where N is the normalization constant and kB T the thermal energy, one obtains the Einstein–Stokes relation D¼

kB T mZ

ð6Þ

for the diffusion constant. The second important relation connected with the Fokker–Planck equation [Eq. (4)] is the linear response 1 hx2 ðtÞiF¼0 hxðtÞiF0 ¼ F0 2 kB T

ð7Þ

between the first moment (drift) in presence of a constant force F0 and the variance in absence of that force, sometimes referred to as the second Einstein relation. The Fokker–Planck equation can be obtained phenomenologically following Fick’s approach by combining the continuity equation with the constitutive equation for the probability current j, q q Pðx; tÞ ¼  jðx; tÞ; qt qx

jðx; tÞ ¼ D

q Pðx; tÞ qx

Alternatively, that equation follows from the master equation [11]2 ðn o q Wðxjx0 ÞPðx0 ; tÞ  Wðx0 ; xÞPðx; tÞ dx0 Pðx; tÞ ¼ qt

ð8Þ

ð9Þ

by Taylor expansion of the transition probabilities W under specific conditions. The master equation is thus a balance equation for the ‘‘state’’ Pðx; tÞ, and as such is a representation of a Pearson random walk: The transition probabilities quantify jumps from position x0 to x and vice versa [11]. Finally, the Fokker–Planck equation emerges from the Langevin equation [13] (ignoring inertial effects)3: dxðtÞ FðxÞ ¼ þ
ðtÞ dt mZ 2 3

The differential form of the Chapman–Kolmogorov equation [11]. That is, we consider the overdamped case.

ð10Þ

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443

relating the velocity of a particle to the external force, plus an erratic, timefluctuating force
ðtÞ. This random force
ðtÞ is supposed to represent the many small impacts on the particle by its surroundings (or heat bath); It constitutes a measure of our ignorance about the microscopic details of the ‘‘bath’’ to which the particle is coupled. On the typical scale of measurements, the Langevin description is, however, very successful. The random force
ðtÞ is assumed independent of x, and it fluctuates very rapidly in comparison to the variations of xðtÞ. We quantify this by writing
ðtÞ ¼ 0;


ðtÞ
ðt0 Þ ¼ dðt  t0 Þ

ð11Þ

where noise strength  and overbars denotes bath particle averages. d denotes the Dirac-delta function,
ðtÞ is Gaussian, white noise which obeys Isserlis’s (Wick’s) theorem [13]. We will see below the differences which occur when the noise is no longer Gaussian. Gaussian diffusion is by no means ubiquitous, despite the appeal of the central limit theorem. Indeed, many systems exhibit deviations from the linear time dependence of Eq. (1). Often, a nonlinear scaling of the form [14–16] hx2 ðtÞi ’ Dta

ð12Þ

is observed, where the generalized diffusion coefficient now has the dimension cm2 =seca . One distinguishes subdiffusion (0 < a < 1) and sub-ballistic, enhanced diffusion (1 < a < 2). Subdiffusive phenomena include charge carrier transport in amorphous semiconductors [17], tracer diffusion in catchments [18], or the motion of inclusions in the cytoskeleton [19], just to name a few.4 In general, subdiffusion corresponds to situations where the normal diffusion is slowed down by trapping events [21–25]. Conversely, sub-ballistic, enhanced diffusion can stem from advection among random directional motions [26,27], from trapping of a wave-like process [28], or in Knudsen diffusion [29,30,31], among others. Trapping processes in the language of continuous time random walk theory are characterized by a waiting time drawn from a waiting time distribution cðtÞ, exhibiting a long tail, cðtÞ ’ t1b , where 0 < b < 1 [14,21,32]. Now no characteristic waiting time exists; and while this process endures longer and longer, waiting times may be drawn from this cðtÞ. The nonexistence of a characteristic waiting time alters the Markovian character of normal diffusion, giving rise to slowly decaying memory effects (‘semi-Markov’ character). Among other consequences, this causes the aging effects. From a probability theory point of view, such behavior corresponds to the limiting distribution of a sum of positive, independent identically distributed random 4

An extensive overview can be found in Ref. [20].

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variables with a diverging first moment, enforcing by the generalized central limit theorem a one-sided Le´vy stable density with characteristic function [14,33,34] 1 ð

cðuÞ ¼ LfcðtÞg 

b

eut cðtÞ dt ¼ eðt=tÞ ;

0 < b < 1:

ð13Þ

0

The above relation is valid also for b ¼ 1. Indeed, in that limit, we have cðuÞ ¼ et=t , whence cðtÞ ¼ dðt  tÞ. This sharp distribution of the waiting time is but one possible definition of a Markovian process. In the remainder of this review, we solely focus on processes with b ¼ 1. Apart from trapping, there also exist situations where, as far as ensemble average hi is concerned, the mean square displacement does not exist. This corresponds to a jump length distribution lðxÞ emerging from an Le´vy stable density for independent identically distributed random variables of the symmetric jump length x, whose second moment diverges. The characteristic function of this Le´vy stable density is [14,33,34] 1 ð

lðxÞeikx dx ¼ expðsa jkja Þ

lðkÞ ¼ FflðxÞg 

ð14Þ

1

for 0 < a  2. For a ¼ 2, one immediately recovers a Gaussian jump length distribution with finite variance s2 . Figure 2 describes the data points for the

Figure 2. The starting point of each step from Fig. 1 is shifted to the origin. This illustrates the continuum approach of the jump length distribution if only a large number of jumps is considered [5].

fundamentals of le´vy flight processes

445

Figure 3. Comparison of the trajectories of a Gaussian (left) and a Le´vy (right) process, the latter with index a ¼ 1:5. While both trajectories are statistically self-similar, the Le´vy walk trajectory possesses a fractal dimension, characterizing the island structure of clusters of smaller steps, connected by a long step. Both walks are drawn for the same number of steps (7000).

jump lengths collected by Perrin, which were then fitted to a Gaussian. Asymptotically for 0 < a < 2, relation (14) implies the long-tailed form lðxÞ ’ jxj1a

ð15Þ

During the random walk governed by lðxÞ with 0 < a < 2, longer and longer jump lengths occur, leading to a characteristic trajectory with fractal dimension a. Thus, processes with an underlying Le´vy stable jump length distribution are called Le´vy flights [35,36]. A comparison between the trajectory of a Gaussian and a Le´vy flight process is shown in Fig. 3, for the same number of steps. A distinct feature of the Le´vy flight is the hierarchical clustering of the trajectory. Le´vy-flight processes have been assigned to spreading of biological species [37–39], related to the high efficiency of a Le´vy flight as a search mechanism for exactly this exchange of long jumps and local exploration [40], in contrast to the locally oversampling (in one or two dimensions) of a Gaussian process. A number of trajectories monitored for the motion of spider monkeys are displayed in Fig. 4, along with the power-law motion length distribution for individual monkeys and the entire group [41]. Le´vy flights have been also used to model groundwater flow [42], which exhibits Le´vy stable features, these are implicated in plasma processes [43] and other turbulent phenomena, among many others, see, for instance [16,20]. It is worthwhile noting that a diverging kinetic energy has been reported for an ion in an optical lattice [44]. Le´vy flights are the central topic of this review. For a homogeneous environment the central relation of continuous time random walk theory is given by [14,45] Pðk; uÞ ¼

1  cðuÞ 1 u 1  cðuÞlðkÞ

ð16Þ

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aleksei v. chechkin et al.

Figure 4. Daily trajectories of adult female (a,b) and male (c) spider monkeys. In panel d, a zoom into the square of c is shown [41]. On the right, the step length distribution is demonstrated to approximately follow power-law statistics with exponent ag corresponding to Le´vy motion.

that is, the Fourier–Laplace transform of the propagator, which immediately produces in the limit ks ! 0 and ut ! 0 (i.e., long distance and long time limit, in comparison to s and t) the characteristic function   Pðk; tÞ ¼ exp  Djkja t

ð17Þ

with diffusion coefficient D ¼ sa =t with dimensions cma =sec. That is, the PDF Pðx; tÞ of such a Le´vy flight process is a Le´vy stable density. In particular, it decays like Pðx; tÞ ’ Dt=jxj1þa . Although the variance of Le´vy flights diverges, one can obtain by means of rescaling of fractional moments a relation that is formally equivalent to expression (1), namely [46] hjxj
i2=
’ Dt2=a

ð18Þ

where 0 <
< a for convergence. This scaling relation indicates that Le´vy flights are indeed move superdiffusively. We note here that instead of the decoupled jump length and waiting time distributions used in this continuous time random walk description of Le´vy flights, one can introduce a coupling between lðxÞ and cðtÞ, such that long jumps invoke a higher time cost than short jumps. Such a coupling therefore introduces a finite ‘‘velocity,’’ leading to the name Le´vy walks, compare [45,47,48]. These are non-Markovian processes, which we shall not consider any further.

fundamentals of le´vy flight processes

447

Continuous time random walk processes with decoupled lðxÞ and cðtÞ can be rephrased in terms of a generalized master equation [49]. This is also true for a general external force FðxÞ, where we obtain a relation of the type q Pðx; tÞ ¼ qt

1 ð

dx 1

0

ðt

dt0 Kðx; x0 ; t  t0 ÞPðx0 ; t0 Þ

ð19Þ

0

The kernel K determines the jump length dependence of the starting position x0 , as well as the waiting time. Only in the spatially homogeneous case, is Kðx;0 x; t  t0 Þ ¼ Kðx  x0 ; t  t0 Þ [50,51]. In continuous time random walk language, one needs to replace lðxÞ by ðx; x0 Þ [52]. A convenient way to formulate a dynamical equation for a Le´vy flight in an external potential is the space-fractional Fokker–Planck equation. Let us quickly review how this is established from the continuous time random walk. We will see below, how that equation also emerges from the alternative Langevin picture with Le´vy stable noise. Consider a homogeneous diffusion process, obeying relation (16). In the limit k ! 0 and u ! 0, we have lðkÞ  1  sa jkja and cðuÞ  1  ut, whence [52–55] uPðk; uÞ  1 ¼ Djkja Pðk; uÞ

ð20Þ

  From the differentiation theorem of Laplace transform, L f_ ðtÞ ¼ uPðuÞ Pðt ¼ 0Þ, we infer that the left-hand side in ðx; tÞ space corresponds to qPðx; tÞ=qt, with initial condition Pðx; 0Þ ¼ dðxÞ. Similarly in the Gaussian limit a ¼ 2, the right-hand side is Dq2 Pðx; tÞ=qx2 , so that we recover the standard diffusion equation. For general a, the right-hand side defines a fractional differential operator in the Riesz–Weyl sense (see below) and we find the fractional diffusion equation [52–56] q qa Pðx; tÞ ¼ D Pðx; tÞ qt qjxja

ð21Þ

where we interpret Ffqa gðxÞ=qjxja g ¼ jkja gðkÞ. The drift exerted by the external force FðxÞ should enter additively (as proved in Ref. 52), and we finally obtain the fractional Fokker–Planck equation for Le´vy flight processes, [52,54–56] q Pðx; tÞ ¼ qt



 q V 0 ðxÞ qa þD Pðx; tÞ qx mZ qjxja

ð22Þ

The fractional Fokker–Planck equation (22) which ignores inertial effects can be solved exactly for an harmonic potential (Ornstein–Uhlenbeck process),

448

aleksei v. chechkin et al.

giving rise to the restoring Hookean force FðxÞ ¼ mo2 x. In the space of wavenumbers k, the solution is [57]  i ZDjkja h ao2 t=Z Pðk; tÞ ¼ exp  1  e ð23Þ ao2 which is a Le´vy stable density with the same stable index a, but time-dependent width ZD=ðaoÞ  ½1  expðao2 t=ZÞ. In particular, the stationary solution Pst ðxÞ ¼ F

1





ZDjkja exp  ao2

 

ZD ao2 jxj1þa

ð24Þ

leads to an infinite variance. Thus, although the harmonic potential introduces a linear restoring force, the process never leaves the basin of attraction of the Le´vy stable density with index a, imposed by the external noise. In particular, due to the diverging variance, the Einstein–Stokes relation and the linear response found for standard diffusion,5 no longer hold. After addressing the Langevin and fractional Fokker–Planck formulations of Le´vy flight processes in some more detail, we will show that in the presence of steeper than harmonic external potentials, the situation changes drastically: The forced Le´vy process no longer leads to an Le´vy stable density but instead to a multimodal PDF with steeper asymptotics than any Le´vy stable density. Mutimodality of the PDF and a converging variance are just one result, which one would not expect at first glance. We will show that Le´vy flights in the presence of non-natural boundary conditions are incompatible with the method of images, leading to subtleties in the first passage and first arrival behaviour. Moreover, we will demonstrate how a driving Le´vy noise alters the standard Kramers barrier crossing problem, thereby preserving the exponential decay of the survival probability. Finally, we address the long-standing question of whether or not a Le´vy flight with a diverging variance (or diverging kinetic energy) exhibits pathological behavior. As we will show, within a proper framework, nonlinear dissipative effects will cause a truncation of the Le´vy stable nature; however, within a finite experimental window, Le´vy flights are a meaningful approximation to real systems. These questions touch on the most fundamental properties of a stochastic process, and the question of the thermodynamic interpretation of processes that leave the basin of attraction of standard Gaussian processes. Le´vy flights, despite having been studied for many decades, still leave numerous open questions. In the following we explore the new physics of Le´vy flight processes and demonstrate their subtle and the intruguing nature. 5

And, in generalized form, also for subdiffusive processes [58].

fundamentals of le´vy flight processes II.

449

DEFINITION AND BASIC PROPERTIES OF LE´VY FLIGHTS

In this section, we formulate the dynamical description of Le´vy flights using both a stochastic differential (Langevin) equation and the deterministic fractional Fokker–Planck equation. For the latter, we also discuss the corresponding form in the domain of wavenumbers, which is a convenient form for certain analytical manipulations in later sections. A.

The Langevin Equation with Le´vy Noise

Our starting point in the stochastic description is the overdamped Langevin equation [54,59]6 dx FðxÞ ¼ þ
a ðtÞ ð25Þ dt mZ where F ¼ dV=dx is an external force with potential VðxÞ, which we choose to be VðxÞ ¼

ajxjc c

ð26Þ

with amplitude a > 0 and exponent c
2 (for reasons that become clear below); as before, m is the particle mass, Z the friction coefficient, and
a ðtÞ represents a stationary white Le´vy noise with Le´vy index a (1  a  2). By white Le´vy noise
a ðtÞ we mean that the process tþt ð

LðtÞ ¼


a ðtÞ dt

ð27Þ

t

that is, the time integral over an increment t, is an a-stable process with stationary independent increments. Restricting ourselves to symmetric Le´vy stable distributions, this implies a characteristic function of the form pa ðk; tÞ ¼ expðDjkja tÞ

ð28Þ

The constant D in this description constitutes the intensity of the external noise. In Fig. 5 we show realizations of white Le´vy noises for various values of a. The sharply pronounced ‘spikes’, due to the long-tailed nature of the Le´vy stable distribution, are distinctly apparent in comparison to the Gaussian case a ¼ 2. 6

A more formal way of writing this Langevin equation is xðt þ dtÞ  xðtÞ ¼ 

1 dVðxÞ dt þ D1=a
a ðdtÞ mZ dx

450

aleksei v. chechkin et al.

Figure 5. Examples of white Le´vy noise with Le´vy index a ¼ 2; 1:7; 1:3; 1:0. The outliers are increasingly more pronounced the smaller the Le´vy index a becomes. Note the different scales on the ordinates.

B.

Fractional Fokker–Planck Equation

The Langevin equation [Eq. (25)] still defines a Markov process, and it is therefore fairly straightforward to show that the corresponding fluctuation-averaged (deterministic) description is given in terms of the space-fractional Fokker–Planck equation (22) [54,60]. In what follows, we solve it with d-initial condition Pðx; 0Þ ¼ dðxÞ

ð29Þ

The space-fractional derivative qa =qjxja occurring in the fractional Fokker– Planck equation (22) is called the Riesz fractional derivative. We have already seen that it is implicitly defined by  a q Pðx; tÞ F ð30Þ ¼ jkja Pðk; tÞ: qjxja The Riesz fractional derivative is defined explicitly, via the Weyl fractional operator ( Da Pðx;tÞþDa Pðx;tÞ da Pðx; tÞ ; a 6¼ 1  þ 2 cosðpa=2Þ ¼ ð31Þ d djxja  dx HPðx; tÞ; a¼1

fundamentals of le´vy flight processes

451

where we use the following abbreviations: ðDaþ PÞðx; tÞ

1 d2 ¼
ð2  aÞ dx2

ðx

Pðx; tÞ dx

1

ðx  xÞa1

ð32Þ

and ðDa PÞðx; tÞ

1 d2 ¼
ð2  aÞ dx2

1 ð

x

Pðx; tÞ dx ðx  xÞa1

ð33Þ

for the left and right Riemann–Liouville derivatives (1  a < 2), respectively, and [61] 1 ð 1 Pðx; tÞ dx ð34Þ ðHPÞðx; tÞ ¼ p xx 1

is the Hilbert transform. Note that the integral is to be interpreted as the Cauchy principal value. The definitions of qa =qjxja demonstrate the strongly nonlocal property of the space-fractional Fokker–Planck equation. 1.

Rescaling of the Dynamical Equations

Passing to dimensionless variables x0 ¼ x=x0 ;

t0 ¼ t=t0

ð35Þ

with  x0 ¼

 mDZ 1=ðc2þaÞ ; a

t0 ¼

xa0 D

ð36Þ

the initial equations take the form (we omit primes below) dx dV ¼ þ
a ðtÞ dt dx

ð37Þ

instead of the Langevin equation (25), and qPðx; tÞ q dV qa Pðx; tÞ ¼ Pðx; tÞ þ qt qx dx qjxja

ð38Þ

instead of the fractional Fokker–Planck equation (22); also, VðxÞ ¼ instead of Eq. (26).

jxjc c

ð39Þ

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aleksei v. chechkin et al. C.

Starting Equations in Fourier Space

For the PDF Pðx; tÞ and its Fourier image Pðk; tÞ ¼ FfPðx; tÞg, we use the notation Pðx; tÞ Pðk; tÞ;

ð40Þ

where the symbol denotes a Fourier transform pair. Since [62] Da Pðx; tÞ ðikÞa Pðk; tÞ

ð41Þ

HPðx; tÞ isignðkÞPðk; tÞ

ð42Þ

qa Pðx; tÞ jkja Pðk; tÞ qjxja

ð43Þ

and

we obtain

for all a. The transformed fractional Fokker–Planck equation [Eq. (38)] for the characteristic function then follows immediately: qPðk; tÞ þ jkja Pðk; tÞ ¼ V k Pðk; tÞ qt

ð44Þ

with the initial condition Pðk; t ¼ 0Þ ¼ 1

ð45Þ

Pðk ¼ 0; tÞ ¼ 1

ð46Þ

and the normalization

The external potential VðxÞ becomes the linear differential operator in k, 1 ð

V k Pðx; tÞ ¼

e 1

ikx

  q dV Pðx; tÞ dx qx dx

1 ð

eikx signðxÞjxjc1 Pðx; tÞ dx

¼ ik

ð47Þ

1

Next, by using the following inverse transforms ð ixÞa PðxÞ Da PðkÞ

ð48Þ

fundamentals of le´vy flight processes

453

iðsignðxÞPðxÞ HPðkÞ

ð49Þ

and

we obtain the explicit expression for the external potential operator, (

c1  k Pðk; tÞ; c 6¼ 3; 5; 7; . . . Dþ  Dc1  V k Pðk; tÞ ¼ 2 cosðpc=2Þ m d2m ð1Þ k dk2m HPðk; tÞ; c ¼ 3; 5; 7; . . .

ð50Þ

Note that for the even potential exponents c ¼ 2m þ 2 , m ¼ 0; 1; 2; . . . , we find the simplified expression V k ¼ ð1Þmþ1 k

q2mþ1 qk2mþ1

ð51Þ

in terms of conventional derivatives in k. We see that the force term can be written in terms of fractional derivatives in k-space, and therefore it is not straightforward to calculate even the stationary solution of the fractional Fokker–Planck equation [Eq. (38)] in the general case c 2 = N. In particular, in this latter case, the nonlocal equation [Eq. (38)] in x-space translates into a nonlocal equation in k-space, where the nonlocality shifts from the diffusion to the drift term. III.

CONFINEMENT AND MULTIMODALITY

In the preceding section, we discussed some elementary properties of the spacefractional Fokker–Planck equation for Le´vy flights; in particular, we highlighted in the domain of wave numbers k the spatially nonlocal character of Eq. (38), and its counterpart (44). For the particular case of the external harmonic potential corresponding to Eq. (26) with c ¼ 2, we found that the PDF does not leave the basin of attraction imposed by the external noise
a ðtÞ—that is, its stable index a. In this section, we determine the analytical solution of the fractional Fokker– Planck equation for general c
2. We start with the exactly solvable stationary quartic Cauchy oscillator, to demonstrate directly the occurring steep asymptotics and the bimodality, that we will then investigate in the general case. The findings collected in this section were first reported in Refs. 60, 63 and 64. A.

The Stationary Quartic Cauchy Oscillator

Let us first consider a stationary quartic potential with c ¼ 4 for the Cauchy– Le´vy flight with a ¼ 1 that is, the solution of the equation d 3 d x Pst ðxÞ þ Pst ðxÞ ¼ 0 dx djxj

ð52Þ

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aleksei v. chechkin et al.

or d 3 Pst ðkÞ ¼ signðkÞjkjPst ðkÞ dk3

ð53Þ

in the k domain. Its solution is   pffiffiffi  3jkj p 2 jkj  cos Pst ðkÞ ¼ pffiffiffi exp  2 2 6 3

ð54Þ

whose inverse Fourier transform results in the simple analytical form 1 : Pst ðxÞ ¼ pð1  x2 þ x4 Þ

ð55Þ

We observe surprisingly that the variance hx2 i ¼ 1

ð56Þ

of the solution (55) is finite, due to the long-tailed asymptotics Pst ðxÞ  x4 . In addition, aspffiffishown in Fig. 6, this solution has two global maxima at ffi xmax ¼ 1= 2 along with the local minimum at the origin (that is the position of the initial condition). These two distinct properties of Le´vy flights are a central theme of the remainder of this section.

0.45 0.4 0.35

fst(x)

0.3 0.25 0.2 0.15 0.1 0.05 0 –4

–3

–2

–1

0 x

1

2

3

4

Figure 6. Stationary PDF (55)pof ffiffiffiffiffiffiffithe ffi Cauchy-Le´vy flight in a quartic (c ¼ 4) potential. Two global maxima exist at xmax ¼ 1=2, and a local minimum at the origin also exists.

fundamentals of le´vy flight processes B.

455

Power-Law Asymptotics of Stationary Solutions for c
2, and Finite Variance for c > 2

We now derive the power-law asymptotics of the stationary PDF Pst ðxÞ for external potentials of the form (39) with general c
2. Thus, we note that as x ! þ1, it is reasonable to assume Da Pst Daþ Pst

ð57Þ

since the region of integration for the right-side Riemann–Liouville derivative Da Pst ðxÞ, ðx; 1Þ, is much smaller than the region of integration for the left-side derivative Daþ Pst ðxÞ, ð1; xÞ, in which the major portion of Pst ðxÞ is located. Thus, at large x we get for the stationary state,   ðx d dV 1 d2 Pst ðxÞ dx Pst ðxÞ  ffi0 2 dx dx 2 cosðpa=2Þ dx ðx  xÞa1

ð58Þ

1

This relation corresponds to the approximate equality x

c1

1 d Pst ðxÞ ffi 2 cosðpa=2Þ dx

ðx

Pst ðxÞ dx

1

ðx  xÞa1

ð59Þ

We are seeking asymptotic behaviors of Pst ðxÞ in the form PðxÞ  C1 =xm (x ! þ1, m > 0Þ. After integration of relation (59), we find 2C1 cosðpa=2Þ
ð2  aÞ mþc x ffi m þ c

ðx 1

Pst ðxÞ dx ðx  xÞa1

ð60Þ

The integral on the right-hand side can be approximated by 1 xa1

ðx Pst ðxÞ dx ffi 1

1 xa1

1 ð

Pst ðxÞ dx ¼ 1

1 xa1

ð61Þ

Thus, we may identify the powers of x and the prefactor, so that m¼aþc1

ð62Þ

and C1 ¼

sinðpa=2Þ
ðaÞ p

ð63Þ

456

aleksei v. chechkin et al.

By symmetry of the PDF we therefore recover the general asymptotic form Pst ðxÞ 

sinðpa=2Þ
ðaÞ ; pjxjm

x ! þ1

ð64Þ

for all c
2. This result is remarkable, for several reasons: (i) despite the approximations involved, the asymptotic form (64) for arbitrary c
2 corresponds exactly to previously obtained forms, such as the exact analytical result for the harmonic Le´vy flight (linear Le´vy oscillator), c ¼ 2 reported in Ref. 57; the result for the quartic Le´vy oscillator with c ¼ 4 discussed in Ref. 60 and 64; and the case of even power-law exponents c ¼ 2m þ 2 (m 2 N0 ) given in Ref. 60. It is also supported by the calculation in Ref. 65. (ii) The prefactor C1 is independent of the potential exponent c; in this sense, C1 is universal. (iii) For each value a of the Le´vy index a critical value ccr ¼ 4  a

ð65Þ

exists such that at c < ccr the variance hx2 i is infinite, whereas at c > ccr the variance is finite. (iv) We have found a fairly simple method for constructing stationary solutions for large x in the form of inverse power series. The qualitative consequence of the steep power-law asymptotics can be visualized by direct integration of the Langevin equation for white Le´vy noise, the latter being portrayed in Fig. 5. Typical results for the sample paths under the influence of an external potential (39) with increasing superharmonicity are shown in Fig. 7 in comparison to the Brownian case (i.e., white Gaussian noise). For growing exponent c, the long excursions typical of homogeneous Le´vy flights are increasingly suppressed. For all cases shown, however, the qualitative behavior of the noise under the influence of the external potential is different from the Brownian noise even in this case of strong confinement. In the same figure, we also show the curvature of the external potential. Additional investigations have shown that the maximum curvature is always very close to the positions of the two maxima, leading us to conjecture that they are in fact identical. C.

Proof of Nonunimodality of Stationary Solution for c > 2

In this subsection we demonstrate that the stationary solution of the kinetic equation (38) has a nonunimodal shape. For this purpose, we use an

fundamentals of le´vy flight processes

457

Figure 7. Left column: The potential energy functions V ¼ xc =c, (solid lines) and their curvatures (dotted lines) for different values of c: c ¼ 2 (linear oscillator), and c ¼ 4; 6; 8 (strongly non-linear oscillators). Middle column: Typical sample paths of Brownian oscillators, a ¼ 2, with the potential energy functions shown on the left. Right column: Typical sample paths of Le´vy oscillators, a ¼ 1. On increasing m the potential walls become steeper, and the flights become shorter; in this sense, they are confined.

alternative expression for the fractional Riesz derivative (compare, e.g., Ref. 62), d a PðxÞ sinðap=2Þ 
ð1 þ aÞ p djxja 1 ð Pðx þ xÞ  2PðxÞ þ Pðx  xÞ  dx x1þa 0

ð66Þ

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aleksei v. chechkin et al.

valid for 0 < a < 2. In the stationary state (qP=qt ¼ 0), we have from Eq. (38)  da P ðxÞ d  st sgnðxÞjxjc1 Pst ðxÞ þ ¼0 dx djxja Thus, it follows that at c > 2 (strict inequality) da Pst ðxÞ ¼0 djxja x¼0

ð67Þ

ð68Þ

or, from definition (66) and noting that Pst ðxÞ is an even function, 1 ð

dx

Pst ðxÞ  Pst ð0Þ ¼0 x1þa

ð69Þ

0

we can immediately obtain a proof of the nonunimodality of Pst , from the latter relation, which we produce in two steps: 1. If we assume that the stationary PDF Pst ðxÞ is unimodal, then due to the symmetry x ! x, it necessarily has one global maximum at x ¼ 0. Here the integrand in equation (69) must be negative, and therefore contradicts equation (69). Therefore, Pst ðxÞ is nonunimodal. 2. We can in addition exclude Pð0Þ ¼ 0, as now the integrand will be positive, which again contradicts Eq. (69). Since PðxÞ ! 0 at x ! 1, based on statements 1 and 2, one may conclude that the simplest situation is such that x0 > 0 exists with the property 1 ð

dx

PðxÞ  Pð0Þ 0 x1þa

ð71Þ

x0

and xð0

0

that is, the condition for a two-hump stationary PDF for all c > 2. At intermediate times, however, we will show that a trimodal state may also exist. If such bimodality occurs, it results from a bifurcation at a critical time t12 [64] when evolution commences (as usually assumed) from the delta function at the origin. A typical result is shown in Fig. 8, for the quartic case c ¼ 4 and

fundamentals of le´vy flight processes

459

Figure 8. Time evolution of the Le´vy flight-PDF in the presence of the superharmonic external potential [Eq. (26)] with c ¼ 4 (quartic Le´vy oscillator) and Le´vy index a ¼ 1:2, obtained from the numerical solution of the fractional Fokker–Planck equation, using the Gru¨nwald–Letnikov representation of the fractional Riesz derivative (full line). The initial condition is a d-function at the origin. The dashed lines indicate the corresponding Boltzmann distribution. The transition from one to two maxima is clearly seen. This picture of the time evolution is typical for 2 < c  4 (see below).

Le´vy index a ¼ 1:2: from an initial d-peak, eventually a bimodal distribution emerges. D.

Formal Solution of Equation (38)

Returning to the general case, we rewrite Eq. (44) in the equivalent integral form, ðt Pðk; tÞ ¼ pa ðk; tÞ þ dt pa ðk; t  tÞV k Pðk; tÞ 0

ð72Þ

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aleksei v. chechkin et al.

where pa ðk; tÞ ¼ expðjkja tÞ

ð73Þ

is the characteristic function of a free (homogeneous) Le´vy flight. This relation follows from equation (44) by formally treating it as a nonhomogeneous linear first-order differential equation, where V k plays the role of the nonhomogeneity. Then, Eq. (44) is obtained by variation of parameters. [Differentiate Eq. (72) to return to Eq. (44).] Equation (72) can be solved formally by iteration: Let f ð0Þ ðk; tÞ ¼ pa ðk; tÞ

ð74Þ

then f

ð1Þ

ðt

ðk; tÞ ¼ pa ðk; tÞ þ dtpa ðk; t  tÞV k f ð0Þ ðk; tÞ

ð75Þ

0

ðt

f ð2Þ ðk; tÞ ¼ pa ðk; tÞ þ dtpa ðk; t  tÞV k pa ðk; tÞ 0

ðt

ðt

þ dt dt0 pa ðk; t  tÞV k pa ðk; t  t0 ÞV k pa ðk; t0 Þ 0

ð76Þ

0

and so on. From the convolution, ðt

ðt

A  B ¼ dtAðt  tÞBðtÞ ¼ dtAðtÞBðt  tÞ 0

ð77Þ

0

using A  B  C ¼ ðA  BÞ  C ¼ A  ðB  CÞ

ð78Þ

we arrive at the formal solution Pðk; tÞ ¼

1 X

pa ðV k pa Þn

ð79Þ

n¼0

This procedure is analogous to perturbation theory, with V k P playing the role of the interaction term (see, for instance, Ref. 66, Chapter 16). Applying a Laplace Transformation, namely, 1 ð

Pðk; uÞ ¼

dt expðutÞPðk; tÞ 0

ð80Þ

fundamentals of le´vy flight processes

461

to Eq. (72), we obtain Pðk; uÞ ¼ pa ðk; uÞ þ pa ðk; uÞV k Pðk; uÞ

ð81Þ

where pa ðk; uÞ ¼

1 u þ ka

ð82Þ

is the Fourier–Laplace transform of the homogeneous Le´vy stable PDF. Thus, we obtain the equivalent of the solution (79) in ðk; uÞ-space: Pðk; uÞ ¼

1 X

½pa ðk; uÞV k n pa ðk; uÞ

ð83Þ

n¼0

This iterative construction scheme for the solution of the fractional Fokker– Planck equation will be useful below. E.

Existence of a Bifurcation Time

For the unimodal initial condition Pðx; 0Þ ¼ dðxÞ we now prove the existence of a finite bifurcation time t12 for the turnover from a unimodal to a bimodal PDF. At this time, the curvature at the origin will vanish; that is, it is a point of inflection: q2 PðxÞ qx2

¼0

ð84Þ

x¼0;t¼t12

Introducing 1 ð

JðtÞ ¼

dkk2 Pðk; tÞ

ð85Þ

0

Eq. (84) is equivalent to (note that the characteristic function is an even function) Jðt12 Þ ¼ 0

ð86Þ

The bifurcation can now be obtained from the iterative solution (83); we consider the specific case c ¼ 4. From the first-order approximation   1 1 P1 ðk; uÞ ¼ 1 þ Vk ð87Þ u þ ka u þ ka where Vk ¼ k

q3 qk3

ð88Þ

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aleksei v. chechkin et al.

Combining these two expressions, we have P1 ðk; uÞ ¼

1 ka2 þ aða  1Þð2  aÞ a uþk ðu þ ka Þ3 þ 6a2 ða  1Þ

k2a2 ðu þ ka Þ

 6a3 4

k3a2 ðu þ ka Þ5

or, on inverse Laplace transformation,  a3 ka t P1 ðk; tÞ ¼ e 1  t4 k3a2 þ a2 ða  1Þt3 k2a2 4 t2 a2 þ aða  1Þð2  aÞ k 2

ð89Þ

ð90Þ

The first approximation to the bifurcation time t12 is then determined via Eq. (85); that is, we calculate 1 ð

ð1Þ

dk k2 P1 ðk; t12 Þ ¼ 0

ð91Þ

0

to obtain ð1Þ t12

 ¼

a=ð2þaÞ 4
ð3=aÞ 3ð3  aÞ
ð1=aÞ

ð92Þ ð1Þ

In Fig. 9, we show the dependence of this first approximation t12 as a function of the Le´vy index a (dashed line), in comparison to the values determined from the numerical solution of the fractional Fokker–Planck equation (38) shown as the dotted line. The second-order iteration for the PDF, P2 ðk; tÞ, can be obtained with maple6, whence the second approximation for the bifurcation time is found by analogy with the above procedure. The result is displayed as the full line in Fig. 9. The two approximate results are in fact in surprisingly good agreement with the numerical result for the exact PDF. Note that the second approximation appears somewhat worse than the first; however, it contains the minimum in the a-dependence of the t12 behavior. 1.

Trimodal Transient State at c > 4.

we have already proved the existence of a bimodal stationary state for the quartic ðc ¼ 4Þ Le´vy oscillator. This bimodality emerges as a bifurcation at a critical time t12 , at which the curvature at the origin vanishes. This scenario is changed

fundamentals of le´vy flight processes

463

t12

1.0

0.5

0 1.0

1.5 a

2.0

Figure 9. Bifurcation time t12 versus Le´vy exponent a for external potential exponent c ¼ 4:0. Black dots: bifurcation time deduced from the numerical solution of the fractional Fokker–Planck equation [Eq. (38)] using the Gru¨nwald–Letnikov representation of the fractional Riesz derivative (see ð1Þ ð2Þ appendix). Dashed line: first approximation t12 ; solid line: second approximation t12 .

for c > 4, as displayed in Fig. 10: There exists a transient trimodal form of the PDF. Thus, there are obviously two time scales that are relevant: the critical time for the emergence of the two off-center maxima, which are characteristic of the stationary state; and a second one, which corresponds to the relaxing initial central hump—that is, the decaying initial distribution Pðx; 0Þ ¼ dðxÞ. The formation of the two off-center humps while the central one is still present, as detailed in Fig. 11. The existence of a transient trimodal state was found to be typical for all c > 4. 2.

Phase Diagrams for n-Modal States

The above findings can be set in the context of the purely bimodal case discussed earlier. A convenient way of displaying the n-modal character of the PDF in the presence of a superharmonic external potential of the type (39) is the phase diagram shown in Fig. 12. There, we summarize the findings that for 2 < c  4 the bifurcation occurs between the initial monomodal and the stationary bimodal PDF at a finite critical time, whereas for c > 4, a transient trimodal state exists. Moreover, we also include the shaded region, in which c is too small to ensure a finite variance. In Fig. 13, in complementary fashion the temporal domains of the n-modal states are graphed, and the solid lines separating these domains correspond to the critical time scales tcr ð¼ t12 ; t13 ; t32 Þ. Again, the transient nature of the trimodal state is distinctly apparent.

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aleksei v. chechkin et al.

Figure 10. Time evolution of the PDF governed by the fractional Fokker–Planck equation (38) in a superharmonic potential (26) with exponent c ¼ 5:5, and for Le´vy index a ¼ 1:2, obtained from numerical solution using the Gru¨nwald–Letnikov method explained in the appendix. Initial condition is Pðx; 0Þ ¼ dðxÞ. The dashed lines indicate the corresponding Boltzmann distribution. The transitions between 1 ! 3 ! 2 humps are clearly seen. This picture of time evolution is typical for c > 4. On a finer scale, we depict the transient trimodal state in Fig. 11.

F.

Consequences

By combining analytical and numerical results, we have discussed Le´vy flights in a superharmonic external potential of power c. Depending on the magnitude of this exponent c, different regimes could be demonstrated. Thus, for c ¼ 2, the character of the Le´vy noise imprinted on the process, is not altered by the external potential: The resulting PDF has Le´vy index a, the same as the noise itself, and will thus give rise to a diverging variance at all times. Conversely, for c > 2, the variance becomes finite if only c > ccr ¼ 4  a. Because the PDF no

465

fundamentals of le´vy flight processes

0.74

0.75

0.76

0.78

0.79

0.81

0.83

0.84

0.87

0.89

0.90

0.93

Figure 11. The transition 1 ! 3 ! 2 from Fig. 10 on a finer scale (c ¼ 5:5, a ¼ 1:2).

2.0

a

1

1

2

3

2

1.5

1.0 2

4

C

6

8

Figure 12. ðc; aÞ map showing different regimes of the PDF. The region with infinite variance is shaded. The region c < 4 covers transitions from 1 to 2 humps during the time evolution. For c > 4, a transition from 1 to 3, and then from 3 to 2 humps occurs. In both cases, the stationary PDF exhibits 2 maxima. Compare Fig. 13.

466

aleksei v. chechkin et al.

1.0 2

tcr

3

0.5 1

0 2

4

C

6

8

ðc; tÞ map showing states of the PDF with different number of humps and the transitions between these. Region 1: The PDF has 1 hump. Region 2: The PDF exhibits 2 humps. Region 3: Three humps occur. At c < 4, there is only one transition 1 ! 2, whereas for c > 4, there occur two transitions, 1 ! 3 and 3 ! 2.

Figure 13.

longer belongs to the set of Le´vy stable PDFs and acquires an inverse power-law asymptotic behavior with power m ¼ a þ c  1. Obviously, moments of higher order will still diverge. Apart from the finite variance, the PDF is distinguished by the observation that it bifurcates from the initial monomodal to a stationary bimodal state. If c > 4, there exists a transient trimodal state. This interesting behavior of the PDF both during relaxation and under stationary conditions, depending on a competition between Le´vy noise and steepness of the potential is in contrast to the universal approach to the Boltzmann equilibrium, solely defined by the external potential, encountered in classical diffusion. One may demand the exact kinetic reason for the occurrence of the multiple humps. Now the nontransient humps seem to coincide with the positions of maximum curvature of the external potential, which at these points changes almost abruptly for larger c from a rather flat to a very steep slope. Thus one may conclude that the random walker, which is driven towards these flanks by the anomalously strong Le´vy diffusivity, is thwarted, thus the PDF accumulates close to these points. Apart from this rudimentary explanation, we do not yet have a more intuitive argument for the existence of the humps and their bifurcations, we also remark that other systems exist where multimodality occurs, for instance, in the transverse fluctuations of a grafted semiflexible polymer [67]. We will later return to the issue of finite variance in the discussion of the velocity distribution of a Le´vy flight. The different regimes for c > 2 can be classified in terms of critical quantities, in particular, the bifurcation time(s) tcr ð¼ t12 ; t13 ; t32 Þ and the critical

fundamentals of le´vy flight processes

467

external potential exponent ccr . Le´vy flights in superharmonic potentials can then be conveniently represented by phase diagrams on the ðc; aÞ and ðc; tcr Þ plains. The numerical solution of both the fractional Fokker–Planck equation in terms of the Gru¨nwald–Letnikov scheme used to find a discretized approximation to the fractional Riesz operator exhibits reliable convergence, as corroborated by direct solution of the corresponding Langevin equation. Our findings have underlined the statement that the properties of Le´vy flights, in particular under nontrivial boundary conditions or in an external potential are not fully understood. The general difficulty, which hampers a straightforward investigation as in the regular Gaussian or the subdiffusive cases, is connected with the strong spatial correlations associated with such problems, manifested in the integrodifferential nature of the Riesz fractional operator. Thus it is not easy to determine the stationary solution of the process. We expect, since diverging fluctuations appear to be relevant in physical systems, that many hitherto unknown properties of Le´vy flights remain to be discovered. Some of these features are discussed in the following sections. IV.

FIRST PASSAGE AND ARRIVAL TIME PROBLEMS FOR LE´VY FLIGHTS

The first passage time density (FPTD) is of particular interest in random processes [14,68–70]. For Le´vy flights, the first passage time density was determined by the method of images in a finite domain in reference [71], and by similar methods in reference [72]. These methods lead to results for the first passage time density in the semi-infinite domain, whose long-time behavior explicitly depends on the Le´vy index a. In contrast, a theorem due to Sparre Andersen proves that for any discrete-time random walk process starting at x0 6¼ 0 with each step chosen from a continuous, symmetric but otherwise arbitrary distribution, the first passage time density asymptotically decays as  n3=2 with the number n of steps [70,73,74], being fully independent of the index of the Le´vy flight—that is, universal. In the case of a Markov process, the continuous time analogue of the Sparre Andersen result reads [69,70] pðtÞ  t3=2

ð93Þ

The analogous universality was proved by Frisch and Frisch for the special case in which an absorbing boundary is placed at the source of the Le´vy flight at t > 0 [75], and numerically corroborated by Zumofen and Klafter [76]. In the following, we demonstrate that the method of images is generally inconsistent with the universality of the first passage time density, and therefore cannot be applied to solve first passage time density-problems for Le´vy flights. We also

468

aleksei v. chechkin et al.

show that for Le´vy flights the first passage time density differs from the PDF for first arrival. The discussion will be restricted to the case 1 < a < 2 [77]. A.

First Arrival Time

By incorporating in the fractional diffusion equation (21) a d-sink of strength pfa ðtÞ, we obtain the diffusion-reaction equation for the non-normalized density function f ðx; tÞ, q qa f ðx; tÞ ¼ D f ðx; tÞ  pfa ðtÞdðxÞ qt qjxja

ð94Þ

from which by integration over all space, we may define the quantity d pfa ðtÞ ¼  dt

1 ð

f ðx; tÞ dx

ð95Þ

1

that is, pfa ðtÞ is the negative time derivative of the survival probability. By definition of the sink term, pfa ðtÞ is the PDF of first arrival: once a random walker arrives at the sink, it is annihilated. By solving equation (94) by standard methods (determining the homogeneous and inhomogeneous solutions), it is straightforward to calculate the solution f in terms of the propagator P of the fractional diffusion

equation (21) with initial condition Pðx; 0Þ ¼ dðx  x0 Þ yielding f ðx; tÞ ¼ eikx0 þ pðuÞ =ðu þ Djkja Þ, whence pfa ðtÞ satisfies the chain rule (pfa implicitly depending on x0 ) ðt Pðx0 ; tÞ ¼ pfa ðtÞPð0; t  tÞ dt

ð96Þ

0

which corresponds to the m domain relation pfa ðuÞ ¼ Pðx0 ; uÞ=Pð0; uÞ. Equation (96) is well known and for any sufficiently well-behaved continuum diffusion process is commonly used as a definition of the first passage time density [14,70]. pffiffiffiffiffiffiffiffiffiffi For Gaussian processes with propagator Pðx; tÞ ¼ 1= 4pDt expðx2 =½4DtÞ, one obtains by direct integration of the diffusion equation with appropriate boundary condition the first passage time density [70]   x0 x2 pðtÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp  0 4D 4pDt3

ð97Þ

including the asymptotic behaviour pðtÞ  t3=2 for t  x20 =ð4DÞ. In this Gaussian case, the quantity pfa ðtÞ is equivalent to the first passage time density. From a random walk perspective, this occurs because individual steps all have the same increment, and the jump length statistics therefore ensure that the

469

fundamentals of le´vy flight processes

walker cannot hop across the sink in a long jump without actually hitting the sink and being absorbed. The behaviour is very different for Le´vy jump length statistics: There, the particle can easily cross the sink in a long jump. Thus, before eventually being absorbed, it can pass by the sink location many times, and therefore the statistics of the first arrivalÐ will be different from those of the 1 1 first passage. In fact, with Pðx; uÞ ¼ ð2pÞ1 1 eikx ðu þ Djkja Þ dk , we find 1 ð ð1  cos kx0 Þ=ðu þ Dka Þdk pfa ðuÞ ¼ 1 

0

ð98Þ

1 ð

1=ðu þ Dka Þdk Since 1 ð

Ð1 0

0 a 1

ðu þ Dk Þ dk ¼ pu

1=a1

=ðaD1=a sinðp=aÞÞ and

1  cos kx0
ðð2  aÞ sinðpð2  aÞ=2Þxa1 0 ;  u þ Dka ða  1ÞD

for u ! 0; a > 1

0

we obtain the limiting form 11=a 1þ1=a ~ pfa ðuÞ  1  xa1 ðaÞ D 0 u

ð99Þ

~ ðaÞ ¼ a
ð2  aÞ sinðpð2  aÞ=2Þ sinðp=aÞ=ða  1Þ. We note that the where  same result may be obtained using the exact expressions for Pðx0 ; uÞ and Pð0; uÞ in terms of Fox H-functions and their series expansions [78]. The inverse Laplace transform of the small u-behavior (99) can be obtained by completing (99) to an exponential, and then computing the Laplace inversion using the 1;0 identity ez ¼ H0;1 ½zjð0; 1Þ in terms of the Fox H-function [78], for which the exact Laplace inversion can be performed [79]. Finally, series expansion of this result leads to the long-t form xa1 0 ð100Þ pfa ðtÞ  CðaÞ 11=a D t21=a 1ÞÞ. with CðaÞ ¼ a
ð2  aÞ
ð2  1=aÞ sinðp½2  a=2Þ sin2 ðp=aÞ=ðp2 ðap ffiffiffiffiffiffiffiffiffiffiffiffi Clearly, in the Gaussian limit, the required asymptotic form pðtÞ  x0 = 4pDt3 for the first passage time density is consistently recovered, whereas in the general case the result (100) is slower than in the universal first passage time density behavior embodied in Eq. (93), as it should be since the d-trap used in equation (94) to define the first arrival for Le´vy flights is weaker than the absorbing wall used to properly define the first passage time density. For Le´vy flights, the PDF for first arrival thus scales like (100) (i.e., it explicitly depends on the index a of the underlying Le´vy process), and, as shown below, it differs from the corresponding first passage time density.

470

aleksei v. chechkin et al.

log t*p(t)

α=1.2, x0=0.0, w=0.3 α=1.2, x0=0.3, w=0.3 α=1.2, x0=1.0, w=0.3 α=1.2, x0=10.0, w=1.0 α=1.8, x0=10.0, w=0.25 t-2+1/α , α=1.2 t-2+1/α, α=1.8 t-3/2

0.01 10

100

1000

10000

log t Figure 14. First arrival PDF for a ¼ 1:2 demonstrating the t2þ1=a scaling, for optimal trap width w ¼ 0:3. For comparison, we show the same scaling for a ¼ 1:8, and the power-law t3=2 corresponding to the first passage time density. The behavior for large w ¼ 1:0 shows a shift of the decay toward the 3=2 slope. Note that the ordinate is lg tpðtÞ. Note also that for the initial condition x0 ¼ 0:0, the trap is activated after the first step, consistent with Ref. [76].

Before calculating this first passage time density, we first demonstrate the validity of Eq. (100) by means of a simulation the results of which are shown in Fig. 14. Random jumps with Le´vy flight jump length statistics are performed, and a particle is removed when it enters a certain interval of width w around the sink; in our simulations we found an optimum value w  0:3. As seen in Fig. 14 (note that we plot lg tpðtÞ!) and for analogous results not shown here, relation (100) is satisfied for 1 < a < 2 , whereas for larger w, the slope increases. B.

Sparre Anderson Universality

To corroborate the validity of the Sparre Anderson universality, we simulate a Le´vy flight in the presence of an absorbing wall—that is, random jumps with Le´vy flight jump length statistics exist along the right semi-axis—and a particle is removed when it jumps across the origin to the left semi-axis. The results of such a detailed random walk study are displayed in Figs. 15 and 16. The expected universal t3=2 scaling is confirmed for various initial positions x0 and

471

fundamentals of le´vy flight processes

log t*p(t)

0.1

x_0=0.10 x_0=1.00 x_0=10.0 x_0=100.0 t**(-1.5) t**(-1.5) t**(-1.5) t**(-1.5) First arrival Images method

0.01

0.001

1

10

100

1000

10000

log t Figure 15. Numerical results for the first process time density process on the semi-infinite domain, for an Le´vy flight with Le´vy index a ¼ 1:2. Note abscissa, is tpðtÞ. For all initial conditions x0 ¼ 0:10 1.00, 10.0, and 100.0 the universal slope 3=2 in the log10 –log10 plot is clearly reproduced, and it is significantly different from the two slopes predicted by the method of images and the direct definition of the first process time density.

Le´vy stable indices a. Clearly, the scaling for the first arrival as well as the image method–first passage time density derived below are significantly different. The following qualitative argument may be made in favor of the observed universality of the Le´vy flight–first passage time density: The long-time decay is expected to be governed by short-distance jump events, corresponding to the central region of very small jump lengths for the Le´vy stable jump length distribution. However, in this region the distribution function is, apart from a prefactor, indistinguishable from the Gaussian distribution, and therefore the long-time behavior should in fact be the same for any continuous jump length distribution lðxÞ. In fact, the universal law (93) can only be modified in the presence of non-Markov effects such as broad waiting time processes or spatiotemporally coupled walks [45,46,70,80,81]. In terms of the special case covered by the theorem of Frisch and Frisch [75], in which the absorbing boundary coincides with the initial position, we can understand the general situation for finite x0 > 0, as in the long-time limit, the distance x0 becomes negligible in comparison to the diffusion length hjxðtÞji  t1=a :

472

aleksei v. chechkin et al.

log t*p(t)

alpha=2.0 alpha=1.5 alpha=1.0 alpha=0.6 t**(-1.5) ditto ditto ditto

0.01

0.001 100

1000 log t

10000

Figure 16. Same as in Fig. 15, for a ¼ 2:0, 1.5, 1.0, and 0.6, and for the initial condition x0 ¼ 10:0. Again, the universal  t3=2 behavior is obtained.

therefore the asymptotic behavior is necessarily governed by the same universality. C.

Inconsistency of Method of Images

We now demonstrate that the method of images produces a result, which is neither consistent with the universal behavior of the first passage time density (93) nor with the behavior of the PDF of first arrival (100), Given the initial condition dðx  x0 Þ, the solution fim ðx; tÞ for the absorbing boundary value problem with the analogous Dirichlet condition fim ð0; tÞ ¼ 0 according to the method of images is given in terms of the free propagator P by the difference [69,70] fim ðx; tÞ ¼ Pðx  x0 ; tÞ  Pðx þ x0 ; tÞ

ð101Þ

that is, a negative image solution originating at x0 balances the probability flux across the absorbing boundary. The corresponding pseudo-first passage time density is then calculated just as Eq. (95). For the image solution in the ðk; uÞ domain, we obtain fim ðk; uÞ ¼

2i sinðkx0 Þ u þ Djkja

ð102Þ

fundamentals of le´vy flight processes

473

for a process which starts at x0 > 0 and occurs in the right half-space. In u space, the image method–first passage time density becomes 1 ð

pim ðuÞ ¼ 1  u

1 ð

dx 1

0

dk ikx 2i sin kx0 e 2p u þ Djkja

ð103Þ

After some transformations, we have 2 pim ðuÞ ¼ 1  p

1 ð

 sin xs1=a x0 =D1=a dx xð1 þ xa Þ

ð104Þ

0

In the limit of small Ðs, this expression reduces to pim ðuÞ  1  ðaÞx0 D1=a u1=a , 1 with ðaÞ ¼ ð2=pÞ 0 ð1 þ xa Þ1 dx ¼ 2=ða sinðp=aÞÞ. In like manner, we find the long-t form pim ðtÞ  2
ð1=aÞ

x0 1=a paD t1þ1=a

ð105Þ

for the image method–first passage timepdensity. ffiffiffiffiffiffiffiffiffiffiffiffi In the Gaussian limit a ¼ 2, expression (105) produces pim ðtÞ  x0 = 4pDt3 , in accordance with Eq. (97). Conversely, for general 1 < a < 2, pðtÞ according to Eq. (105) would decay faster than  t3=2 . The failure of the method of images is closely related to the strongly nonlocal character of Le´vy flights. Under such conditions, the random variable x  x0 is no longer independent of x þ x0 , so that the method of images is not appropriate. The proper dynamical formulation of a Le´vy flight on the semi-infinite interval with an absorbing boundary condition at x ¼ 0, and thus the determination of the first passage time density, has to ensure that in terms of the random walk picture jumps across the sink are forbidden. This objective can be consistently achieved by setting f ðx; tÞ  0 on the left semi-axis, i.e., actually removing the particle when it crosses the point x ¼ 0. This procedure formally corresponds to the modified dynamical equation qf ðx; tÞ D q2 ¼ qt k qx2

1 ð

0

f ðx0 ; tÞ jx  x0 ja1

dx0 

q2 F ðx; tÞ qx2

ð106Þ

in which the fractional integral is confined to the semi-infinite interval. Here, we have written pa k ¼ 2
ð2  aÞ cos ð107Þ 2

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After Laplace transformation and integrating over x twice, one obtains 1 ð

Kðx  x0 ; uÞ f ðx0 ; uÞ dx0 ¼ ðx  x0 Þðx  x0 Þ  xpðuÞ  F ð0; uÞ

ð108Þ

0

where pðtÞ is the FPTD and the kernel Kðx; uÞ ¼ uxðxÞ  ðkjxja1 Þ. This equation is formally a Wiener–Hopf equation of the first kind [82]. After some manipulations similar to those applied in Ref. 76, we arrive at the asymptotic expression pðuÞ ’ 1  Cu1=2 ;

where C ¼ const

ð109Þ

in accordance with the expected universal behavior (93) and with the findings of reference [76]. Thus, the dynamic equation (106) governs the first passage time density problem for Le´vy flights. We note that due to the truncation of the fractional integral it was not possible to modify the well-established Gru¨nwald– Letnikov scheme [61] to numerically solve Eq. (106) with enough computational efficiency to obtain the direct solution for f ðx; tÞ. V.

BARRIER CROSSING OF A LE´VY FLIGHT

The escape of a particle from a potential well is a generic problem investigated by Kramers [84] that is often used to model chemical reactions, nucleation processes, or the escape from a potential well 84. Keeping in mind that many stochastic processes do not obey the central limit theorem, the corresponding Kramers escape behavior will differ. For subdiffusion, the temporal evolution of the survival behavior is bound to change, as discussed in Ref. 85. Here, we address the question how the stable nature of Le´vy flight processes generalizes the barrier crossing behavior of the classical Kramers problem [86]. An interesting example is given by the a-stable noise-induced barrier crossing in long paleoclimatic time series [87]; another new application is the escape from traps in optical or plasma systems (see, for instance, Ref. 88). A.

Starting Equations

Here, we investigate barrier crossing processes in a reaction coordinate xðtÞ governed by a Langevin equation [Eq. (25)] with white Le´vy noise
a ðtÞ. Now, however, the external potential VðxÞ is chosen as the (typical) double-well shape a b VðxÞ ¼  x2 þ x4 2 4

ð110Þ

compare, for instance, Ref. 89. For convenience, we introduce dimensionless variables t ! t=t0 and x ! x=x0 with t0 ¼ mZ=a and x20 ¼ 1=ðbt0 Þ and

fundamentals of le´vy flight processes 1=a

1=a1

dimensionless noise strength D ! Dt0 =x0 (by
a ðt0 tÞ ! t0 that we have the stochastic equation  dxðtÞ

¼ x  x3 þ D1=a
a ðtÞ dt

475
a ðtÞ) [43], so

ð111Þ

Here, we restrict our discussion to 1  a < 2. B.

Brownian Motion

In normal Brownian motion corresponding to the limit a ¼ 2, the survival probability S of a particle whose motion at time t ¼ 0 which is initiated in one of the potential minima xmin ¼ 1, follows an exponential decay SðtÞ ¼ exp ðt=Tc Þ with mean escape time Tc , such that the probability density function pðtÞ ¼ dS=dt of the barrier crossing time t becomes pðtÞ ¼ Tc1 expðt=Tc Þ

ð112Þ

The mean crossing time (MCT) follows the exponential law Tc ¼ C expðh=DÞ

ð113Þ

where h is the barrier height (equal to 1/4 for the potential (110)) in rescaled variables, and the prefactor C includes details of the potential [84]. We want to determine how the presence of Le´vy stable noise modifies the laws (112) and (113). C.

Numerical Solution

The Langevin equation [Eq. (111)] was integrated numerically following the procedure developed in Ref. 90. Whence, we obtained the trajectories of the particle shown in Fig. 17. In the Brownian limit, we reproduce qualitatively the behavior found in Ref. 89. Accordingly, the fluctuations around the positions of the minima are localized in the sense that their width is clearly smaller than the distance between the minima and barrier. In contrast, for progressively smaller stable index a, characteristic spikes become visible, and the individual sojourn times in one of the potential wells decrease. In particular, we note that single spikes can be of the order of or larger than the distance between the two potential minima. From such single trajectories we determine the individual barrier crossing times as the time interval between a jump into one well across the zero line x ¼ 0 and the escape across x ¼ 0 back to the other well. In Fig. 18, we demonstrate that on average, the crossing times are distributed exponentially, and thus follow the same law (112) already known from the Brownian case. Such a result has been reported in a previous study of Kramers’ escape driven by Le´vy noise [91]. In fact, the exponential decay of the survival probability

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aleksei v. chechkin et al.

Figure 17.

Typical trajectories for different stable indexes a obtained from numerical integration of the Langevin equation [Eq. (111)]. The dashed lines represent the potential minima at 1. In the Brownian case a ¼ 2, previously reported behavior is recovered [89]. In the Le´vy stable case, occasional long jumps of the order of or larger than the separation of the minima can be observed. Note the different scales.

-7 -8

ln p(t)

-9 -10 -11 -12 -13 0

1000

2000

3000

4000

5000

t

Figure 18. Probability density function pðtÞ of barrier crossing times for a ¼ 1:0 and D ¼ 102:5  0:00316. The dashed line is a fit to Eq. (112) with mean crossing time Tc ¼ 1057:8 17:7.

477

fundamentals of le´vy flight processes

S observed in a Le´vy flight is not surprising, given the Markovian nature of the process. Due to the Le´vy stable properties of the noise
a , the Langevin equation [Eq. (111)] produces occasional long jumps, by which the particle can cross the barrier. Large enough values of the noise
a thus occur considerably more frequently than in the Brownian case with Gaussian noise (a ¼ 2), causing a lower mean crossing time. The numerical integration of the Langevin equation (111) was repeated for various stable indices a, and for a range of noise strengths D. From these simulations we obtain the detailed dependence of the mean crossing time Tc ða; DÞ on both of the parameters, a and D. As expected, for decreasing noise strength, the mean crossing time increases. For sufficiently large values of 1=D and fixed a, a power-law trend in the double-logarithmic plot is clearly visible. These power-law regions, for the investigated range of a are in very good agreement with the analytical form Tc ða; DÞ ¼

CðaÞ DmðaÞ

ð114Þ

over a large range of D. Equation (114) is the central result of this study. It is clear from Fig. 19, that this relation is appropriate for the entire a-range studied 5.5 5 4.5

log Tc

4 3.5 3

α = 2.00 α = 1.95 α = 1.90 α = 1.80 α = 1.60 α = 1.40 α = 1.20 α = 1.20

2.5 2 1.5 1

1.5

2

2.5 log 1/D

3

3.5

Figure 19. Escape time Tc as a function of noise strength D for various a. Above roughly lg 1=D ¼ 1:5, a power-law behavior is observed that corresponds to Eq. (114). The curve [Eq. (113)] for a ¼ 2:0 appears to represent a common envelope.

478

aleksei v. chechkin et al. 1.6

2

µ(α) fitted by 1+0.401 (α-1)+0.105 (α-1) log C(α)

1.14 1.12

1.4

1.1

µ(α)

1.06

1

log C(α)

1.2 1.08

1.04 0.8 1.02 0.6

1

0.98 0.4 1

1.2

1.4

α

1.6

1.8

2

Scaling exponent m as function of stable index a. The constant behavior mðaÞ  1 over the range 1  a / 1:6 is followed by an increase above 1.6, and it eventually shows an apparent divergence close to a ¼ 2, where Eq. (113) holds. Corresponding to the right ordinate, we also plot the decadic logarithm of the amplitude CðaÞ.

Figure 20.

in our simulations. For larger noise strength, we observe a breakdown of the power-law trend, and the curves seem to approach the mean crossing time behavior of the Brownian process (a ¼ 2) as a common envelope. A more thorough numerical analysis of this effect will be necessary in order to ascertain its exact nature. The main topic we want to focus on here is the behavior embodied in Eq. (114). We note from Fig. 19 that for a ranging roughly between the Cauchy case a ¼ 1 and the Holtsmark case a ¼ 3=2, the exponent m is almost constant; that is, the corresponding lines in the log–log plot are almost parallel. The behavior of both the scaling exponent m and the prefactor C as a function of the stable index a becomes clear in Fig. 20. There, we recognize a slow variation of m for values of a between 3/2 and slightly below 2, before a steeper rise in close vicinity of 2. This apparent divergence must be faster than any power, so that in the Gaussian noise limit a ¼ 2, the activation follows the exponential law (113) instead of the scaling form (114). The mðaÞ results are fitted with the parabola indicated in the plot where, for the analytical results derived below, we forced the fit function to pass through the point mð1Þ ¼ 1. D.

Analytical Approximation for the Cauchy Case

In the Cauchy limit a ¼ 1, we can find an approximate result for the mean crossing time as a function of noise strength D. To this end, we start with the

479

fundamentals of le´vy flight processes

rescaled fractional Fokker–Planck equation [20,46,54,57,71], corresponding to equation (111),  qPðx; tÞ q

qa ¼ x þ x3 Pðx; tÞ þ D Pðx; tÞ qt qx qjxja

ð115Þ

Rewriting Eq. (115) in continuity equation form qPðx; tÞ=qt þ qjðx; tÞqx ¼ 0, that is equivalent to qPðk; tÞqt ¼ ikjðk; tÞ in k space, we obtain for the flux the expression   q3 q a1 jðkÞ ¼  3  i þ i D signðkÞjkj Pðk; tÞ ð116Þ qk qk To obtain an approximate expression for the mean crossing time, we follow the standard steps [92] and for large values of 1=D make the constant flux approximation assuming that the flux across the barrier is a constant, j0 , corresponding to the existence of a stationary solution Pst ðxÞ. By integration of the continuity equation, it then follows that equation (112) is satisfied, and Tc ¼ 1=j0 . Due to the low Ð 0 noise strength, we also assume that for all relevant times the normalization 1 Pst ðxÞ ¼ 1 obtains. In this constant flux approximation, we obtain from equation (116) the relation d 3 Pst ðkÞ dPst ðkÞ  D signðkÞPst ðkÞ ¼ 2pij0 dðkÞ þ dk3 dk in the Cauchy case a ¼ 1. With the ansatz Pst ðkÞ ¼ C1 ez



k

ð117Þ 

þ C2 eðz Þ

k

for k > < 0,

3

we find the characteristic equation ðz Þ þ z  D ¼ 0 solved by the Cardan pffiffiffi

pffiffiffiffiffiffiffi expressions z ¼  12 ðu þ v Þ þ 12 i 3ðu  v Þ, with u3þ ¼ D 1 þ 1þ  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4=½27D2 Þ=2 ¼ v3 and v3þ ¼ D 1  1 þ 4=½27D2  =2 ¼ u3 . Matching the left and right solutions at k ¼ 0, requiring that Pst ðkÞ 2 R, and assuming that Pst ðkÞ in the constant flux approximation is far from the fully relaxed (t ! 1) solution, we obtain the shifted Cauchy form Pst ðkÞ ¼

j0
þ ; 2
þ
 ð x þ
 Þ2 þ
þ2

pffiffiffi 3 1 ð118Þ ;
þ ¼ ðuþ þ vþ Þ;
 ¼ ðuþ  vþ Þ 2 2 Ð0 With the normalization 1 Pst ðxÞ dx ¼ 1, we arrive at the mean crossing time   p 2
 Tc ¼ 1 þ arctan ð119Þ 4
þ
 p
þ

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aleksei v. chechkin et al.

For D 1,
þ  D=2 and
  1, so that Tc  p=D. In comparison with the numerical result corresponding to Fig. 18 with Tc ¼ 1057:8 for D ¼ 0:00316, we calculate from our approximation Tc  994:2, which is within 6% of the numerical result. This good agreement also corroborates the fact that the constant flux approximation appears to pertain to Le´vy flights. E.

Discussion

We observe from numerical simulations an exponential decrease of the survival probability SðtÞ in the potential well, at the bottom of which we initialize the process. Moreover, we find that the mean crossing time assumes the scaled form (114) with scaling exponent m being approximately constant in the range 1  a / 1:6, followed by an increase before the apparent divergence at a ¼ 2, that leads back to the exponential form of the Brownian case, Eq. (113). An analytic calculation in the Cauchy limit a ¼ 1 reproduces, consistently with the constant flux approximation commonly applied in the Brownian case, the scaling Tc  1=D, and, within a few percent error, the numerical value of the mean crossing time Tc . Employing scaling arguments, we can restore the dimensionality into expression (114) for the mean crossing time. From our model potential (110), where we absorb the friction factor pffiffiffiffiffiffiffiffimZ via a ! a=ðmZÞ and b ! b=ðmZÞ, we find that the minima are xmin ¼ a=b and the barrier height V ¼ a2 =ð4bÞ . In terms of the rescaled prefactors a and b with dimensions ½a ¼ sec1 and ½b ¼ sec1 cm2 , we can now reintroduce the dimensions via t0 ¼ 1=a and x20 ¼ b=a. In the domain where Tc  1=D (i.e., mðaÞ  1), we then have the scaling Tc 

xa0 ða=bÞa=2 jxmin ja ¼ ¼ D D D

ð120Þ

by analogy with the result reported in Ref. 91. However, we emphasiz two caveats based on our results: (i) The linear behavior in 1=D is not valid over the entire a-range. For larger values, a ’ 1:6, the scaling exponent mðaÞ assumes nontrivial values; then, the simple scaling used to establish Eq. (120) has to be modified. It is not immediately obvious how this should be done systematically. (ii) From relation (120) it cannot be concluded that the mean crossing time is independent of the barrier height V, despite the fact that Tc depends on the distance jxmin j from the barrier only. The latter statement is obvious from the expressions for xmin and V derived for our model potential: The location of the minima relative to the barrier is in fact coupled to the barrier height. Therefore, a random walker subject to Le´vy noise senses the potential barrier and does not simply move across it with the characteristic time given by the free mean-square displacement. Apparently, the activation for the mean crossing time as a function of noise strength D varies only as a power law instead of the standard exponential behaviour.

fundamentals of le´vy flight processes

481

The time dependence of the probability density dSðtÞ=dt for first barrier crossing time of a Le´vy flight process is exponential, just as the standard Brownian case. This can be understood qualitatively because the process is Markovian. From the governing dynamical equation (115), it is clear that the relaxation of modes is exponential, compare Ref. 46. For low noise strength D, the barrier crossing will be dominated by the slowest time-eigenmode ’ el1 t with eigenvalue l1 . This is indeed similar to the first passage time problem of Le´vy flights discussed in the previous section. VI.

DISSIPATIVE NONLINEARITY

The alleged ‘‘pathology’’ of Le´vy flights is related to their divergent variance, unless confined by a steeper than harmonic external potential. There indeed exist examples of processes where the diverging variance does not pose a problem: for example, diffusion in energy space [93], or the Le´vy flight in the chemical coordinate of diffusion along a polymer chain in solution, where Le´vy jump length statistics are invoked by intersegmental jumps, which are geometrically short in the embedding space [94]. Obviously however, for a particle with a finite mass moving in Euclidian space, the divergence of the variance is problematic.7 There are certain ways of overcoming this difficulty: (i) by a time cost through coupling between x and t, producing Le´vy walks [45,98], or (ii) by a cutoff in the Le´vy noise to prevent divergence [99,100]. While (i) appears a natural choice, it gives rise to a nonMarkov process. Conversely, (ii) corresponds to an ad hoc measure. A.

Nonlinear Friction Term

Here, we pursue an alternative, physical way of dealing with the divergence; namely, inclusion of nonlinear dissipative terms. They provide a mechanism, that naturally regularizes the Le´vy stable PDF PðV; TÞ of the velocity distribution. Dissipative nonlinear structures occur naturally for particles in a frictional environment at higher velocities [101]. A classical example is the Riccati equation MdvðtÞ=dt ¼ Mg  KvðtÞ2 for the motion of a particle of mass M in a gravitational field with acceleration g [102], autonomous oscillatory systems with a friction that is nonlinear in the velocity [101,103], or nonlinear corrections to the Stokes drag as well as drag in turbulent flows [104]. The occurrence of a non-constant friction coefficient gðVÞ leading to a nonlinear dissipative force 7 Note that in fact the regular diffusion equation includes a similar flaw, although less significant: Due to its parabolic nature, it features an infinite propagation speed; that is, even at very short times, there exists a finite value of Pðx; tÞ for large jxj. In that case, this can be removed by invoking the telegrapher’s (Cattaneo) equation [95–97]. (Editor’s note: For a critical discussion of this procedure, see Risken [12, p. 257 et seq.)

482

aleksei v. chechkin et al.

gðVÞV was highlighted in Klimontovich’s theory of nonlinear Brownian motion [105]. In what follows, we show that dissipative nonlinear structures regularize a stochastic process subject to Le´vy noise, leading to finite variance of velocity fluctuations and thus a well-defined kinetic energy. The velocity PDF PðV; tÞ associated with this process preserves the properties of the Le´vy process for smaller velocities; however, it decays faster than a Le´vy stable density and thus possesses a physical cutoff. In what follows, we start with the asymptotic behavior for large V and then address the remaining, central part of PðV; TÞ, that preserves the Le´vy stable density property. B.

Dynamical Equation with Le´vy Noise and Dissipative Nonlinearity

The Langevin equation for a random process in the velocity coordinate V is usually written as [59] dVðtÞ þ gðVÞVðtÞ ¼
a ðtÞ dt

ð121Þ

with the constant friction g0 ¼ gð0Þ.
a ðtÞ is the a-stable Le´vy noise defined in terms of a characteristic function (see Section I). The characteristic function of the velocity PDF PðV; tÞ, Pðk; tÞ  FfPðV; tÞg is then governed by the dynamical equation [59] qPðk; tÞ qPðk; tÞ ¼ g0 k  Djkja Pðk; tÞ qt qk

ð122Þ

This is exactly the V-space equivalent of the Le´vy flight in an external harmonic potential discussed in the introduction. Under stationary conditions the characteristic function assumes the form   Djkja Pst ðk; tÞ ¼ exp  ð123Þ g0 a So that the PDF PðV; tÞ converges toward a Le´vy stable density of index a. This stationary solution possesses, however, a diverging variance. To overcome the divergence of the variance hV 2 ðtÞi, we introduce into Eq. (121) the velocity-dependent dissipative nonlinear form gðVÞ for the friction coefficient [101,105]. We require gðVÞ to be symmetric in V [105], assuming the virial expansion up to order 2N gðVÞ ¼ g0 þ g2 V 2 þ    þ g2N V 2N

; g2N > 0

ð124Þ

The coefficients g2n are assumed to decrease rapidly with growing n (n 2 N). To determine the asymptotic behavior, it is sufficient to retain the highest power 2N.

fundamentals of le´vy flight processes

483

More generally, we will consider a power gn jVjn with n 2 Rþ and gn > 0. We will show that, despite the input driving Le´vy noise, the inclusion of the dissipative nonlinearity (124) ensures that the resulting process possesses a finite variance. To this end, we pass to the kinetic equation for PðV; tÞ, the fractional Fokker– Planck equation [20,46,54,60,64] qPðV; tÞ q qa PðV; tÞ ¼ ðVgðVÞPðV; tÞÞ þ D qt qV qjVja

ð125Þ

The nonlinear friction coefficient gðVÞ thereby takes on the role of a confining potential: while for g0 ¼ gð0Þ the drift term Vg0 , as mentioned before, is just the restoring force exerted by the harmonic Ornstein–Uhlenbeck potential, the next higher-order contribution g2 V 3 corresponds to a quartic potential, and so forth. The fractional operator qa =qjVja in Eq. (125) for the velocity coordinate for 1 < a < 2 is explicitly given by [20,64] d a PðVÞ d2 a ¼ k dV 2 djVj

1 ð

PðV 0 Þ

1

jV  V 0 ja1

dV 0

ð126Þ

by analogy with the x-domain operator (31), with k being defined in Eq. (107). C.

Asymptotic Behavior

To derive the asymptotic behavior of PðV; tÞ in the presence of a particular form of gðVÞ, it is sufficient to consider the highest power, say, gðVÞ  gn jVjn . In particular, to infer the behavior of the stationary PDF ÐPst ðVÞ for V ! 1, it is 1 reasonable to assume that we can truncate the integral 1 dV 0 in the fractional 0 operator (126) at the pole V ¼ V, since the domain of integration for the remaining left-side operator is much larger than the cutoff right-side domain. Moreover, the remaining integral over ð1; V also contains the major portion of the PDF. For V ! þ1, we find in the stationary state after integration over V, gn V

nþ1

d Pst ðVÞ ’ Dk dV

ðV 1

Pst ðV 0 Þ ðV  V 0 Þa1

dV 0

ð127Þ

We then use the ansatz Pst ðVÞ  C=jVjm , m > 0. With the approximation ÐV Ð1 ÐV 0 0 a1 dV 0  V 1a 1 Pst ðV 0 ÞdV 0  V 1a 1 Pst ðV 0 ÞdV 0 ¼ 1 Pst ðV Þ=ðV  V Þ V 1a we obtain the asymptotic form Pst ðVÞ ’

Ca D gn jVjm

;m ¼ a þ n þ 1

ð128Þ

484

aleksei v. chechkin et al. -2

α = 1.5; γ2 = 0.0; γ4 = 0.00; Ntra = 15,000 , N = 25,000 α = 1.5; γ2 = 0.1; γ4 = 0.00; Ntra = 30,000 , N = 25,000 α = 1.2; γ2 = 0.1; γ4 = 0.01; Ntra = 100,000, N = 50,000 Slope -1.5 Slope -3.5 Slope -5.2

ln V Pst(V)

-4

-6

-8

-10

-12 0.5

1

1.5

2

2.5

3 ln V

3.5

4

4.5

5

5.5

Figure 21.

Power-law asymptotics of the stationary PDF, ln–ln scale. We observe the expected scaling with exponent m from Eq. (128). In the graph, we also indicate the number Ntra of trajectories of individual length N simulated to produce the average PDF.

valid for V ! 1 due to symmetry. We conclude that for all n > ncr ¼ 2  a the variance hV 2 i is finite, and thus a dissipative nonlinearity whose highest power n exceeds the critical value ncr counterbalances the energy supplied by the Le´vy noise
a ðtÞ. D.

Numerical Solution of Quadratic and Quartic Nonlinearity

Let us consider dissipative nonlinearity up to the quartic order contribution, gðVÞ ¼ g0 þ g2 V 2 þ g4 V 4 . According to the previous result (128), the stationary PDF for the quadratic case with g2 > 0 and g4 ¼ 0 falls off like Pst ðVÞ  jVja3 , and thus 8a 2 ð0; 2Þ the variance hV 2 i is finite. Higher-order moments such as the fourth-order moment hV 4 i are, however, still infinite. In contrast, if g4 > 0, the fourth-order moment is finite. We investigate this behavior numerically by solving the Langevin equation (121); compare Ref. 64 for details. In Fig. 21 we show the asymptotic behavior of the stationary PDF Pst ðVÞ for three different sets of parameters. Clearly, in all three cases the predicted powerlaw decay is obtained, with exponents that, within the estimated error bars agree well with the predicted relation for m according to Eq. (128).8 8

From the scattering of the numerical data after repeated runs, see Fig. 7.

485

fundamentals of le´vy flight processes -2

γ2 = 0.0001, γ4 = 0 γ2 = 0, γ4 = 0.000001 Slope -1 Slope -3 Slope -5

-4

ln V Pst(V)

-6 -8 -10 -12 -14

1

2

3

4 ln V

5

6

7

Figure 22. Stationary PDF Pst ðVÞ for g0 ¼ 1:0 and (i) g2 ¼ 0:0001 and g4 ¼ 0, and (ii) g2 ¼ 0 and g4 ¼ 0:000001, with a ¼ 1:0. The lines indicate the slopes 1, 3, and 5.

E.

Central Part of PðV; tÞ

The nonlinear damping (124) mainly affects larger velocities, while smaller velocities (V 1) are mainly subject to the lowest-order friction gð0Þ. We therefore expect that in the central region close to V ¼ 0, the PDF PðV; TÞ preserves it Levy stable density character. This is demonstrated in Fig. 22, where the initial power-law decay of the Levy stable density eventually gives way to the steeper decay caused by the nonlinear friction term. In general, the PDF shows transitions between multiple power laws in the case when several higher-order friction terms are retained. The turnover point from the unaffected Levy stable density to steeper decay caused by nonlinear friction depends on the ratio g0 : g2n , where 2n is the next higher-order nonvanishing friction coefficient. In Fig. 23, we show the time evolution of the variance hV 2 ðtÞi for various combinations of Le´vy index a and magnitude g2 of the quadratic nonlinearity (g0 ¼ 1:0 and g4 ¼ 0:0). For all cases with finite g2 (g2 ¼ 0:1), we find convergence of the variance to a stationary value. For the two smaller a values (1.2 and 1.5), we observe some fluctuations; however, these are comparatively small with respect to the stationary value they oscillate around. For a ¼ 1:8, the fluctuations are hardly visible, and in fact the stationary value is practically the same as in the Gaussian case a ¼ 2:0. In contrast, the case with vanishing

486

aleksei v. chechkin et al. α = 2.0; γ2 = 0.0 α = 1.2; γ2 = 0.1 α = 1.5; γ2 = 0.1 α = 1.8; γ2 = 0.1 α = 1.2; γ2 = 0.0

1400

2



1000 1.5 800 600

1

2

1200

2

(α = 1.2, γ2 = 0)

2.5

1600

400 0.5 200 0

0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t Figure 23. Variance hV 2 ðtÞi as function of time t, with the quartic term set to zero, g4 ¼ 0 and g0 ¼ 1:0 for all cases. The variance is finite for the cases a ¼ 2:0; g2 ¼ 0:0; a ¼ 1:2; g2 ¼ 0:1; a ¼ 1:5; g2 ¼ 0:1; and a ¼ 1:8; g2 ¼ 0:1. These correspond to the left ordinate. For the case a ¼ 1:2; g2 ¼ 0:0, the variance diverges and strong fluctuations are visible; note the large values of this curve corresponding to the right ordinate.

g2 (and a ¼ 1:2) clearly shows large fluctuations requiring a right ordinate whose span is roughly two orders of magnitude larger than that of the left ordinate. Similarly, in Fig. 24, we show the fourth order moment hV 4 ðtÞi as a function of time. It is obvious that only for finite g4 (g4 ¼ 0:01 and a ¼ 1:8) the moment converges to a finite value that is quite close to the value for the Gaussian case (a ¼ 2:0) for which all moments converge. In contrast to this behavior, both examples with vanishing g4 exhibit large fluctuations. These are naturally much more pronounced for smaller Le´vy index (a ¼ 1:2, corresponding to the right ordinate). F.

Discussion

Strictly speaking, all naturally occurring power-laws in fractal or dynamic patterns are finite. Scale-free models nevertheless provide an efficient description of a wide variety of processes in complex systems [16,20,46,106]. This phenomenological fact is corroborated by the observation that the power-law properties of Le´vy processes persist strongly even in the presence of cutoffs [99]

487

fundamentals of le´vy flight processes α = 2.0; α = 1.8; α = 1.2; α = 1.8;

7

900

γ2 = 0.0; γ4 = 0.00 γ2 = 0.1; γ4 = 0.01 γ2 = 0.1; γ4 = 0.00 γ2 = 0.1; γ4 = 0.00

800 700

6

600 500 4 400 300

2

4

3



4



5

(α = 1.2, γ4 = 0.0)

8

200

1

100 0 0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

t

Figure 24. Fourth-order moment hV 4 ðtÞi as function of t, with g0 ¼ 1. hV 4 ðtÞi converges to a

finite value for the two cases a ¼ 2 (Gaussian) and a ¼ 1:8 with g4 ¼ 0:01. The other two examples with vanishing quartic contribution (g4 ¼ 0) show large fluctuations—that is, diverging hV 4 ðtÞi. Note that the case a ¼ 1:2 and g4 ¼ 0 corresponds to the right ordinate.

and, mathematically, by the existence of the generalized central limit theorem due to which Le´vy stable laws become fundamental [69]. A categorical question is whether in the presence of Le´vy noise, there exists a physical cause to remove the consequential divergences. A possible, physically reasonable answer is given by introducing a nonconstant friction coefficient gðVÞ, as occurs in various classical systems. Here, we present a concise derivation of the regularization of a stochastic process in velocity space driven by Le´vy stable noise, in the presence of dissipative nonlinearities. These dissipative nonlinearities remove the divergence of the kinetic energy from the measurable subsystem of the random walker. In idealized mathematical language, the surrounding heat bath provides an infinite amount of energy through the Le´vy noise, and the coupling via the nonlinear friction dissipates an infinite amount of energy into the bath, and thereby introduces a natural cutoff in the kinetic energy distribution of the random walker subsystem. Physically, such divergences are not expected, but correspond to the limiting behavior associated with large numbers in probability theory. In this section, we showed that both statements can be reconciled, and that Le´vy processes are indeed physical.

488

aleksei v. chechkin et al. VII.

SUMMARY

A hundred years after Einstein’s seminal work [4], the theory of stochastic processes has been put on solid physical and mathematical foundations, at the same time playing a prominent role in many branches of science [36,107–109]. Le´vy flights represent a widely used tool in the description of anomalous stochastic processes. By their mathematical definition, Le´vy flights are Markovian and their statistical limit distribution emerges from independent identically distributed random variables, by virtue of the central limit theorem. Despite this quite straightforward definition, Le´vy flights are less well understood than one might at first assume. This is due to their strongly nonlocal character in space, these long-range correlations spanning essentially the entire available geometry; as exemplified by the infinite range of the integration boundaries in the associated fractional operator. In this review, we have addressed some of the fundamental properties of random processes, these being the behaviour in external force fields, the first passage and arrival behaviour, as well as the Kramers-like escape over a potential barrier. We have examined the seemingly pathological nature of Le´vy flights and showed that dissipative non-linear mechanisms cause a natural cutoff in the PDF, so that with a finite experimental range the untruncated Le´vy flight still provides a good description. These investigations have been almost entirely based on fractional diffusion and Fokker–Planck equations with a fractional Riesz derivative and have turned out to be a convenient basis for mathematical manipulations, while at the same time being easy to interpret in the context of a dynamical approach. Acknowledgments We would like to thank Iddo Eliazar and Igor M. Sokolov for helpful discussions.

VIII.

APPENDIX. NUMERICAL SOLUTION METHODS

In this appendix, we briefly review the numerical techniques, which have been used in this work to determine the PDF from the fractional Fokker–Planck equation [Eq. (38)] and the Langevin equation [Eq. (37)]. A.

Numerical solution of the fractional Fokker–Planck equation [Eq. (38)] via the Gru¨nwald–Letnikov Method

From a mathematical point of view, the fractional Fokker–Planck equation [Eq. (38)] is an first-order partial differential equation in time, and of nonlocal, integrodifferential kind in the position coordinate x. It can be solved numerically via an efficient discretization scheme following Gru¨nwald and Letnikov [110–112].

fundamentals of le´vy flight processes

489

Let us designate the force component on the right-hand side of Eq. (38) as   q dV  P ð129Þ F ðx; tÞ  qx dx and the diffusion part as a  ðx; tÞ  q Pa D qjxj

ð130Þ

With these definitions, we can rewrite Eq. (38) in terms of a discretisation scheme as Pj;nþ1  Pj;n   j;n ¼ Fj;n þ D t

ð131Þ

where we encounter the term   Pjþ1;n  Pj1;n c2  Fj;n ¼ xj ðc  1ÞPj;n þ xj 2x

ð132Þ

which is the force component of the potential VðxÞ ¼ jxjc =c. Here, t and x are the finite increments in time and position, such that tn ¼ ndt and xj ¼ jx, for n ¼ 0; 1; . . . ; N and j ¼ 0; 1; . . . ; J, and Pj;n  Pðxj ; tn Þ. Due to the inversion symmetry of the kinetic equation (38), it is sufficient to solve it on the right semiaxis. In the evaluation of the numerical scheme, we define xJ such that the PDF in the stationary state is sufficiently small, say, 103 , as determined from the asymptotic form (64). In order to find a discrete time and position expression for the fractional Riesz derivative in Eq. (130), we employ the Gru¨nwald–Letnikov scheme [110–112], whence we obtain  j;n ¼  D

J X

 1 xq Pjþ1q;n þ Pj1þq;n a 2ðxÞ cosðpa=2Þ q¼0

where xq ¼ ð1Þq with

  a q

   a aða  1Þ . . . ða  q þ 1Þ=q!; ¼ 1; q

ð133Þ

ð134Þ

q>0 q