Contemporary Problems in Statistical Physics

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS Edited by George H. Weiss Division of Computer Research and Technology National Institutes of Health

51HJTL Society for Industrial and Applied Mathematics Philadelphia 1994

Cover Illustration: Two samples of the trajectory of a symmetrical bistable system driven by quasimonochromatic noise (Figure 2.12 of Chapter 2, Fluctuations in Nonlinear Systems Driven by Colored Noise, Mark Dykman and Katja Lindenberg, p. 94).

Library of Congress Cataloging-in-Publication Data Contemporary problems in statistical physics/ edited by George H. Weiss. p. cm. Includes bibliographical references. ISBN 0-89871-323-4 1. Statistical physics. I. Weiss, George H. (George Herbert), 1930QC174.8.C66 1994 530.13—dc20

94-11368

All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the Publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, Pennsylvania 19104-2688. Copyright © 1994 by the Society for Industrial and Applied Mathematics.

siamfr

is a registered trademark.

Contents

Contributors

xi

Preface

xiii

Introduction

xv

Chapter 1. Diffusion Kinetics in Microscopic Nonhomogeneous Systems Peter Clifford and Nicholas J. B. Green 1.1. Introduction 1.2. Models of molecular motion 1.2.1. Molecular dynamics 1.2.2. Langeviri equations 1.2.3. Stochastic differential equations 1.2.4. Singular diffusions 1.3. First-passage times for diffusions 1.3.1. Geminate recombination 1.3.2. Simplifications 1.3.3. Boundary conditions 1.3.4. The backward equation 1.3.5. A general method 1.3.6. Zero intcrparticle force 1.3.7. Other cases 1.3.8. Many particles in one dimension 1.3.9. Other models of reaction 1.4. First-passage times for integrated diffusions 1.4.1. Return times and impact velocities 1.4.2. Hitting times from return times 1.4.3. Escape probabilities 1.5. Computer simulation methods 1.5.1. Particle motion 1.5.2. Absorbing boundary 1.5.3. Reflecting boundary v

1 1 4 4 6 7 7 8 8 9 10 10 11 12 12 13 15 17 18 18 20 22 22 22 24

vi 1.5.4. Radiation boundary 1.6. Independent pairs 1.6.1. Independently and identically distributed distances 1.6.2. The Smoluchowski theory 1.7. The independent reaction times approximation 1.7.1. The IRT simulation method 1.7.2. Analytical formulation of the method 1.7.3. Competition between scavenging and geminate recombination Chapter 2. Fluctuations in Nonlinear Systems Driven by Colored Noise Mark Dykman and Katja Lindenberg 2.1. Introduction 2.2. Spectral density of fluctuations and statistical distribution near a stable state 2.2.1. Spectral densities of fluctuations in nonthermal systems 2.2.2. Statistical distribution near the maximum 2.2.3. Spectral density of fluctuations in thermal equilibrium 2.3. Large fluctuations: Methods of the optimal path 2.3.1. Variational problem for the optimal path 2.3.2. Variational equations and their analysis in limiting cases 2.3.3. Activation energy for noise of small correlation time. Comparison to other approaches 2.3.4. Statistical distribution for noise with large correlation time 2.4. Fluctuations induced by quasi-monochromatic noise 2.4.1. Double-adiabatic approximation for a QMN-driven system 2.4.2. Quasi-singularity of the activation energy: Breakdown of the adiabatic approximation 2.5. Pre-history problem 2.5.1. General expression for the pre-history probabilify density 2.5.2. Pre-history probability density for systems driven by white noise 2.6. Probabilities of fluctuational transitions between coexisting stable states of noise-driven systems 2.6.1. Method of optimal path in the problem of fluctuational transitions

CONTENTS 25 26 27 28 29 30 32 35

41 41 45 46 49 50 54 57 61 63 67 72 74 77 81 82 85 87 89

CONTENTS

vii

2.6.2. Transition probabilities for particular types of noise 2.6.3. Quasi-monochromatic noise 2.7. Conclusion

91 93 96

Chapter 3. Percolation Shlomo Havlin and Armin Bunde 3.1. Introduction 3.2. Critical phenomena 3.3. Fractal properties 3.3.1. The fractal dimension df 3.3.2. The graph dimension df 3.3.3. Fractal substructures 3.4. Transport properties 3.4.1. Transport on fractal substrates 3.4.2. Transport on percolation clusters 3.4.3. Fluctuations in diffusion 3.5. Fractons 3.5.1. Elasticity 3.5.2. Vibrational excitations and the phonon density of states 3.5.3. Vibrations of the infinite cluster 3.5.4. Vibrations in the percolation system 3.6. Other types of percolation 3.6.1. Directed percolation 3.6.2. Invasion percolation 3.6.3. Correlated percolation

103

Chapter 4. Aspects of Trapping in Transport Processes Frank den Hollander and George H. Weiss 4.1. Introduction 4.1.1. Reaction kinetics 4.1.2. Smoluchowski's model 4.1.3. Extensions of Smoluchowski's model 4.1.4. Rosenstock's model 4.1.5. Other reaction schemes 4.2. Trapping in one dimension: A solvable example 4.2.1. The mean trapping time (n) 4.2.2. The survival probability S(n) 4.2.3. Large-n asymptotics for S(n) 4.2.4. Small-n behavior of S(ri) 4.2.5. Preview of extensions 4.3. What do approximations, heuristics, and numerics tell us about S(n)l 4.3.1. The Rosenstock approximation

147

103 106 108 108 Ill 114 116 116 119 125 127 128 128 131 131 133 133 135 136

147 147 149 150 151 153 153 153 154 156 157 159 160 160

viii

4.4.

4.5.

4.6.

4.7. 4.8. 4.9.

CONTENTS 4.3.2. The truncated cumulant approximation 4.3.3. Systematic corrections 4.3.4. Heuristic derivation of the asymptotic form of the Donsker-Varadhan tail 4.3.5. Numerics: Exact enumeration techniques 4.3.6. The trapping problem on a fractal Some results for (n) 4.4.1. Low trap density asymptotics 4.4.2. A rigorous inequality A rigorous look at survival at long times 4.5.1. Large deviations 4.5.2. Localization 4.5.3. Drift Extensions and generalizations 4.6.1. Trap distributions other than Poisson 4.6.2. Moving traps 4.6.2.1. Segregation of reactants in confined geometries 4.6.2.2. Perturbation techniques for low trap density 4.6.2.3. Large-£ asymptotics of survival for moving traps 4.6.3. Kinetics in the presence of decaying traps 4.6.4. Further trapping models suggested by applications to chemical reactions 4.6.5. Reversible trapping 4.6.6. The variational approach Afterword Appendix A 4.8.1. Derivation of the large-n asymptotics of (Rn) in d dimensions Appendix B 4.9.1. Derivation of the small-c asymptotics for (n) in d = 3 (equations (4.88))

Chapter 5. Stochastic Resonance: From the Ice Ages to the Monkey's Ear Frank Moss 5.1. Introduction 5.2. Historical development of stochastic resonance 5.3. The adiabatic theory of McNamara and Wiesenfeld 5.4. The nonadiabatic theory of Hanggi and Jung 5.5. The perturbation theory of Marchesoni and coworkers 5.6. Stochastic resonance demonstrated in a ring laser 5.7. . . . and in electron paramagnetic resonance

162 163 164 168 170 173 173 176 177 177 178 179 181 181 183 183 185 187 188 189 191 192 193 194 194 197 197 205 205 211 214 217 220 223 227

CONTENTS 5.8. 5.9. 5.10. 5.11.

. . . and in a free-standing magnetoelastic beam Analog simulations of stochastic resonance Some speculations on applications The probability density of residence times as an alternative to the power spectrum 5.12. Stochastic resonance in the periodically modulated random walks of Weiss and coworkers 5.13. Noise-induced switching in periodically stimulated neurons 5.14. Summary and speculations on future developments

ix 228 232 239 242 244 245 248


Contributors

Armin Bunde, Institut fiir Theoretische Physik, Universitat Hamburg, D-20149 Hamburg, Germany Peter Clifford, Statistics Department, Oxford University, 1 South Park Road, Oxford OX1 3TG, United Kingdom Frank den Hollander, Mathematical Institute, University of Utrecht, P.O.Box 80.010, 3508 TA Utrecht, the Netherlands Mark Dykman, Department of Physics, Stanford University, Stanford, California 94305, and Institute for Nonlinear Science, University of California at San Diego, La Jolla, California 92093-0340 Nicholas J. B. Green, Chemistry Department, King's College, London University, Strand, London WC2R 2LS, United Kingdom Shlomo Havlin, Department of Physics, Bar Ilan University, Ramat-Gan, Israel Katja Lindenberg, Department of Chemistry and Institute for Nonlinear Science, University of California at San Diego, La Jolla, California 92093-0340 Frank Moss, Department of Physics and Astronomy, University of MissouriSt. Louis, 8001 Natural Bridge Road, St. Louis, Missouri 63121-4499 George H. Weiss, Division of Computer Research and Technology, National Institutes of Health, Bethesda, Maryland 20892

XI


Preface

The founders of physics as we know it. Kepler. Galileo, and Newton, recognized only a deterministic world in which each phenomenon had an associated cause. The philosophy of determinism is an integral part of Newton's mechanics, which consists of a very specific set of rules for the analysis of any mechanical system. One of the first successes of Newton's theory was the complete analysis of the motion of two point particles under mutual gravitational attraction. This was immediately and successfully applied to the solution of many problems in celestial dynamics. However, the extension of this analysis to systems comprised of three or more bodies poses not inconsequential mathematical problems which have not been entirely overcome at this time, although a considerable amount is known about such systems following three centuries of research. Since classical mechanics cannot furnish an exact solution to the three-body problem it is hardly imaginable that dynamical properties of mechanical systems consisting of larger numbers of particles can be studied in any exact way without introducing artificial restrictions on the nature of the system. Because of this consideration the notion of statistical methodology as being applied to the analysis of physical systems became a candidate for consideration, particularly for the analysis of gases, for which numbers of the order of 1023 are the important ones. Early work along these lines was initiated by an under-appreciated physicist named Waterliouse in the 1840s, and later and more comprehensively by the better-known Maxwell and Boltzmann. Disorder characterizes much of the physical universe. An early example of this is the phenomenon of Brownian motion, which was known to van Leeuwenhoek (and surely regarded by him as a considerable nuisance). The earliest version of statistical mechanics was considered as a supplement to the normative deterministic formulations in nineteenth century physical science, a regrettable necessity invented to overcome the mathematical difficulties in rigorously analyzing physical systems consisting of more than two particles. Nevertheless, the success of statistical methods in the physical sciences has given these methods an honored place in the armamentarium of the physical scientist. The much more profound question as to why methods based on probability theory can be used to replace a mathematical approach taking xin

xiv

PREFACE

all interactions into account has not been answered in sufficient generality to satisfy either the physicist or the mathematician. In more recent years the methods of statistical physics have been applied in a more phenomenological way to describe the world around us; i.e., in such applications there is only an indirect relation between more basic physical laws and mathematical formalism used to describe physical phenomenon. A simple example of this is the derivation of the laws of diffusion by applying scaling to the purely mathematical random walk model. A more contemporary paradigm can be found in the currently fashionable research area of fractals. While the mathematical notion of the fractal can be traced back to the beginning of the century, only in the past fifteen years has it been popularized as a tool in the physical sciences by Mandelbrot and others. (A recent useful introduction to this field is found in the collection of articles in Fractals and Disordered Systems, by Bunde and Havlin [Springer-Verlag, 1991].) The basic notion behind the use of the theory of fractals in physics requires accounting for at least two sources of randomness in describing the motion of bodies in a disordered medium. The first is that inherent in a description of the motion of the body which may be diffusive even in a completely homogeneous medium, and the second is the randomness of the medium itself. Understanding phenomena exemplified by the percolation of fluids through rock, the transport of matter in rivers, the kinetics of chemical reactions, and the deposition of particles poses problems that will challenge the applied mathematician, the physicist, and the theoretical chemist. Other areas of investigation in statistical physics relate to unintuitive mathematical and physical phenomena, such as stochastic resonance, in which the presence of an oscillating field can affect the way random noise acts on a physical system in sometimes surprising ways. A number of these topics are taken up in the present set of articles, which is meant to introduce the applied mathematician to problems of contemporary interest in the physical sciences without requiring an extensive background in statistical physics. In the main, the class of problems described here requires only a good grasp of the principles of several areas in pure and applied mathematics. Although the material discussed by the several authors of this collection is meant to be introductory, there is also a discussion of a number of contemporary unsolved problems, some requiring the application of quite deep mathematical tools; others requiring analysis by conventional mathematical tools combined with simulation and other numerical methods. Thus, a rich feast and a number of potential thesis topics suggest themselves for the graduate student, and a number of problems to chew on are provided for the practicing applied mathematician. George H. Weiss Division of Computer Research and Technology National Institutes of Health

Introduction

Probability theory originated in the analysis of some very practical problems. Since its inception as a formal mathematical theory the number and variety of applications have increased to such an extent that it is a crucial element of very many scientific disciplines. Applications of probability theory to the physical sciences first appeared in the mid-nineteenth century, but in recent years probabilistic methods have provided an especially important set of tools for analyzing a number of significant problems in both chemistry and physics. As in all of the applications of probability theory to specific subject areas, the interaction is marked by a process of mutual fertilization. On the one hand, probability theory has provided a significant number of mathematical tools to analyze physical phenomena, and on the other, problems in the physical sciences have suggested entirely new subject areas for investigation by probabilists, even those whose toolbox consists of the most abstract theoretical elements. A cursory glance at such journals as Physical Review or the Journal of Chemical Physics will provide convincing evidence of the importance of both traditional and newer fields of probability theory in the physical sciences. This collection of articles should be taken as a plate of hors d'oeuvres rather than a full meal in introducing the applied mathematician to some problems in contemporary statistical physics. An attempt has been made to keep the material pedagogically oriented, requiring only a minimal background in physics from the reader. The mathematical level is geared at the beginning graduate student level and the material covered ranges from problems that are basically well understood to problems whose solutions are at present unknown arid currently considered hot topics in the physical sciences. Five articles describing different classes of problems in statistical physics are included in this collection. These are: 1. "Diffusion Kinetics in Microscopic Nonhomogeneous Systems," by Peter Clifford and Nicholas J.B. Green. 2. "Fluctuations in Nonlinear Systems Driven by Colored Noise," by Mark Dykman and Katja Lindenberg. 3. "Percolation," by Shlorno Havlin arid Armin Bunde. xv

xvi

INTRODUCTION 4. "Aspects of Trapping in Transport Processes," by Frank den Hollander arid George H. Weiss. 5. "Stochastic Resonance: From the Ice Ages to the Monkey's Ear," by Frank Moss.

The first of these focuses on the study of simple reactions as exemplified, in the notation of chemistry, by the reaction A + B —> C. The method for writing down a rate equation to describe the kinetics of this reaction is a staple of freshman chemistry. Some 75 years ago the Polish physical chemist M. von Smoluchowski proposed bridging the gap between a macroscopic description of the kinetics of the reaction and a microscopic picture based on the rate of disappearance of the A particles. By the macroscopic description of the system kinetics we will mean the rate equation

and by the microscopic description we will mean one that takes into account the dynamics of the molecules participating in the reaction. In (0.1), [A] refers to the concentration of the A species, assumed to be constant over the entire volume, and k is a constant referred to as the rate constant. It is far from trivial to establish a rigorous correspondence between the two levels of description, the microscopic and the macroscopic. A number of questions are raised by the form of the rate equation in (0.1). Smoluchowski 's original formulation of a microscopic model for (0.1) is oversimplified in many significant ways (despite which it yields results in good agreement with a number of experiments), but nevertheless manages to produce the kinetic equation in (0.1) in an appropriate limit. An obvious difficulty with the rate equation in (0.1) is the assumption that only bulk concentrations appear in it, which totally disregards local fluctuations in particle numbers. Seventy-five years of research by both chemists and physicists have been devoted to trying to account for, and to correct, simplifications in the original analysis. There is a large body of literature on what is now referred to as the theory of diffusioncontrolled reactions. A diffusion-controlled reaction is one in which the time for two molecules to diffuse into close enough proximity with one another for a reaction to occur sets the main time scale of the reaction process. The article by Clifford and Green discusses several of the issues related to the theory of diffusion-controlled reactions. Their discussion retains the original Smoluchowski framework, which deals with a diffusive system consisting of two particles only, but allows for the existence of forces acting between the particles. The related article by den Hollander and Weiss attacks a second aspect of the general problem, in which one tries to deal more seriously with the many-body aspect of the physical problem, which is absent from the original Smoluchowski model. This article summarizes a body of mathematics phrased in terms of what is commonly referred to as the trapping model. The trapping model relates to a random walk that moves in a field of an infinite

INTRODUCTION

xvii

number of randomly placed traps. In this picture the mathematical analog of a chemical reaction between the two species is an encounter of the random walker with the trap. At such an encounter the random walker disappears and the trap remains fixed in place. The descriptor of primary interest in such a model is the survival probability of the random walker as a function of time. When a finite concentration of traps is distributed uniformly over the space the survival probability will decay to zero with increasing time. A naive guess as to the form taken by the kinetics of this decay might suggest a simple exponential decay. This, however, is not the case. If the reaction process occurs in one dimension, i.e., if the particles diffuse along a line, the survival probability can be shown to decay according to the somewhat surprising decay function exp [—(t/T)3J, where T is a constant. The basic trapping model has been applied not only to the study of the original formulation of the theory of diffusion-controlled reactions, but also to a variety of problems in solid-state physics and metallurgy. It is interesting to note that the trapping problem is related to one of the hottest mathematical research areas at the present time. This is the class of problems which attempts to account for the behavior of random walks in a random environment. Because this general subject area is suggested by models in a large number of applications, some very high-powered mathematical methods have been applied to the analysis by mathematicians. Much is also known from the work of physical scientists, based on simulation and more heuristic matheniatico-physical arguments. A large part of this literature is reviewed in the article by den Hollander and Weiss, from which the reader can learn which aspects of the theory are well understood, and which remain yet to be resolved. A common thread connects the two articles described so far, and that by Havliri and Bunde, which describes properties of percolation systems. Rather than giving a mathematically precise definition of what is meant by a percolation system, Havlin and Bunde define a simple version of such a system. Suppose that one considers a translationally-invariant lattice which is modified by randomly deleting bonds between the lattice sites, the probability of deleting any given bond being given by a constant, p. The result of this procedure is to produce a random, or disordered, lattice for which both the structural and transport properties, i.e., properties related to the motion of random walks on such a lattice, find use as models in applications to chemistry and physics. Havlin and Bunde's article discusses some of the issues, techniques, results, and open problems related to percolation systems. The remaining two articles in this collection deal with topics unrelated to the three that we have already mentioned. It is known that the Ito and Stratonovich formulations of the solution of stochastic differential equations subject to additive random white noise can by analyzed in terms of the solution of certain parabolic differential equations. Dykrnan and Lindenberg consider the situation in which the assumption that the noise is white is replaced by a

xviii

INTRODUCTION

more general model for properties of the noise as a function of time. Let the noise at time t be denoted by n(i). Stationary white noise is characterized by the properties

in which E{ } denotes an expectation. Colored noise replaces the second of these conditions by the more general

The most frequently used form for the function f ( t ] is a negative exponential. When the noise is colored it is no longer possible to convert the problem of determining properties of the dynamical system into that of solving a diffusionlike equation without introducing approximations. The class of problems suggested by the trivial change of the correlation function is huge, and there are certainly more issues to be resolved than there are answers, principally because most of what is known in the area is based on heuristic analysis. The final article in this collection, by Frank Moss, deals with the relatively new research area known as stochastic resonance. This deals with the interaction between noise and periodic forcing terms in determining the qualitative behavior of a dynamical system. The initial, somewhat surprising, discovery in this area is that deterministic periodic forcing terms can sometimes produce quite unintuitive behavior of the dynamical system. For example, when a periodic forcing term of the right frequency is applied to a class of dynamical systems, an increase of the amplitude of the noise can sometimes stabilize rather than destabilize the behavior of the system. The article by Moss concludes with a fascinating conjecture that stochastic resonance may have played an important role in the evolution of neurophysiological systems, since noise plays an important role in determining the behavior of such systems. Stochastic resonance may provide just the right sort of mechanism to turn the occurrence of noise to some advantage in the biological setting. The field of stochastic resonance is still in its infancy and much research remains to be done to tease out the many possible subtle effects. I hope that this small sampling of current problems in statistical physics will not only provide the interested reader with an idea of the definition and scope of such problems, but will also suggest many of the areas that remain open for fresh and ingenious mathematical approaches.

George H. Weiss Division of Computer Research and Technology National Institutes of Health Bethesda, MD

Chapter 1

Diffusion Kinetics in Microscopic Nonhomogeneous Systems Peter Clifford and Nicholas J. B. Green

Abstract Stochastic models of chemical reaction in microscopic systems are reviewed. Such methods are contrasted with those based on the more traditional theory of reaction-diffusion equations. Stochastic techniques are shown to be capable of dealing with some of the complexities of radiation chemistry, in particular, the initial nonhomogeneous clustered spatial distribution of reactants, the motion of reactants in solution, and the behavior of reactive species on encounter. Reaction theory with simple diffusive motion is criticized and more general models for molecular motion involving integrated diffusion processes are introduced. Emphasis is placed on the efficient design of simulation techniques and their use in validating analytical approximations. The assumption of pairwise independence in a multispecies radiation spur is shown to lead to the Smoluchowski theory of reaction. We consider modifications to this theory, which lead to substantial improvements in predictions of the time-evolution and product yields in small reactive systems. Finally, we consider models of boundary behavior for processes in which reaction is not certain on encounter. We show how such processes can be treated analytically and we describe efficient methods for their simulation. 1.1.

Introduction

A chemical reaction in solution, such as

is generally considered to take place in two stages. Since reaction cannot occur unless the A and B particles concerned are close to each other, the first stage envisaged is the encounter of the two particles during the course of their diffusive motion. The subsequent interaction between A and B while in proximity, with the possible formation of C, is the activation stage. The relative diffusion of the two reactive particles evidently depends simply on parameters governing the rate at which they move through the solution, whereas the success and speed of activation may depend on many factors, such as the details of the quantum states of the particles involved, their orientations and their energies, as well as dynamical interactions with the solvent. 1

2


If an A and a B particle in proximity typically have a high probability of separating before the product is formed, many encounters may be necessary between different pairs before a reaction finally occurs. Under these conditions the reaction (which is usually termed activation-controlled) proceeds at a rate which is limited by the activation process, and it is necessary to understand the activation mechanism in order to describe the rate theoretically. If, however, the converse is true, i.e., activation is rapid and the product is very likely to be formed before the pair can disengage, then the reaction is said to be diffusion-controlled, and it is not necessary to have a detailed understanding of the activation mechanism to study the reaction rate, since the encounter rate is simply limited by the transport of the reactive particles as they undergo their diffusive motion in the solution. There is extensive experimental evidence that the diffusion-controlled limit is approached in many chemical reactions [70] , and it is therefore important to have a reliable theory of these processes. Since the rate of a diffusion-controlled reaction depends simply on the rate at which the reactive particles encounter, it can depend strongly on the initial distributions of the particles. Under classical chemical conditions the marginal spatial density of each reactive species is uniform (although there may be pair correlations between the positions of particles). The system contains very many particles of each type, typically of the order of 1023, and the concentrations extend over the entire reaction vessel. Under such conditions, as will be found in any elementary chemistry text, the concentrations of the two species involved are observed experimentally to follow a phenomenological law

where [A] and [B] are the concentrations of the species concerned and k is a constant for the reaction known as its rate constant. Understanding the mathematical basis of this phenomenological law, and formulating a theory of the rate constant k, however, is far from elementary, as can be seen from the review by Weiss and den Hollander (in this volume). In this article we are not concerned with chemical reactions under the usual conditions of extended distributions and large numbers of particles, but rather with systems in which the reactive particles are found in more-or-less isolated clusters, each of which contains only a few particles. Although the approaches we will discuss were originally developed for these clustered systems, they are not limited to them, and they also help to illuminate the basic approximations behind the usual theories of extended systems. Systems in which the reactive particles are initially clustered in small numbers are of great intrinsic interest because of the possibility of finding exact solutions for the reaction rates, if not explicitly then by numerical methods or by simulation. In addition if three-particle clusters can be analyzed accurately, then corrections can also be found for the usual theory of diffusion kinetics in

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

3

extended systems, in which the hierarchy of correlations is truncated with pair distributions [76], [84], [63], [83]. Moreover, nonhornogeneous systems occur in many places in nature. The passage of ultraviolet light through a solution leads to the production of reactive particles in correlated pairs, which can remain effectively isolated for a long time [17]. Ionizing radiation leaves tracks which consist of small clusters of highly reactive particles (typically two to six [66]). It is also possible to study reactions in confined space such as micelles, each of which is effectively an isolated microscopic reaction vessel [80] where there is not room for large numbers of particles. One of the major features of all of these processes is that the normal chemical approach to nonhomogeneous systems, through diffusion-reaction equations, is not appropriate [23]. This failure of the usual phenornenological rate law arises for two reasons: first, the very small number of particles involved, which necessitates a statistical approach; second, the fact that the nonhomogeneity is on a microscopic scale, of the same order as the scale of the "concentration gradients" which are set up under more normal chemical conditions [63], so that "experimental" rate constants measured under homogeneous conditions cannot be applied. On the other hand, one major disadvantage of these nonhomogeneous processes is that the initial spatial distribution, and sometimes even the initial number of reactive particles in the cluster, is not known. Indeed, the major role of theory in such systems is to elucidate the initial conditions from the experimentally observed rates of reaction or yields of product. This means that it is often not possible to distinguish between different theories on the basis of experimental results: different theories may fit the experiment equally well, but from the basis of different initial distributions. In such a case it is necessary to assess theories using an alternative standard where the initial distributions can be precisely controlled. This type of study was not possible until the relatively recent development of simulation techniques on high speed computers. Since the reactions of interest in this article are limited by the transport of particles through the solution we start our discussion by describing different theoretical methods for describing the diffusive movement of a particle in solution. These methods range from the numerical solution of classical equations of motion for all the particles in the system, in the simulation technique usually known as molecular dynamics, through to the use of a stochastic differential equation in which the velocity of the particle is formally nonexistent and all the dynamical information is reduced to a single parameter, the diffusion coefficient. In §1.3 we discuss the few cases in which the diffusion formalism can be solved exactly for rates of reaction. These are generally limited to systems consisting of a single pair or to one-dimensional systems. In §1.4 we discuss the extent to which similar results can be obtained for an alternative model in which the velocities of the particles are modelled as diffusions, so that the

4


configuration is an integrated diffusion process. The results presented in this section are new. It is very difficult to extend exact results beyond the simplest systems, and so at present it is necessary to use approximations to describe nonhomogeneous systems. However, because of the additional difficulty discussed above with regard to comparisons with experiment we must find alternative means of testing the approximations. For this purpose we have developed computer simulation techniques which we describe in §1.5. In addition to the usual problems of time discretization which must be addressed in such simulations we also deal with the problems that arise when particles react with each other or reflect off each other. Simulation enables synthetic data to be created corresponding to theoretical models which are analytically intractable. It is thereby possible to make critical assessments of general approximation principles, isolating the effect of specific elements of the approximations and testing them one by one. In §1.6 we describe the usual approximation made in diffusion kinetics, that of pairwise independence, and we show how this approximation leads to the Smoluchowski theory, and how it can be applied in nonhomogeneous kinetics. In §1.7 we describe a small modification of the approximation, which allows us to take into account correlations that may exist in the initial distribution, but still makes the independence approximation for the subsequent evolution of the system. This modified approximation is generally found to be more accurate than the original approximation, and we present some new results for the rate constant that might be appropriate for the scavenging of a single pair. 1.2.

Models of molecular motion

1.2.1. Molecular dynamics. The most detailed and successful dynamical model of the motion of molecules in liquids is the method known as molecular dynamics, devised by Alder and Wainwright [1], and developed in many ways in the intervening years [2]. In this method a system is set up consisting of a number of molecules, which interact according to a predefined potential energy function. The system is repeated periodically in all directions to simulate the effect of a macroscopic liquid. The particles move according to the laws of classical dynamics, which are realized in the computer model by numerical integration of Newton's equations of motion. It is possible to use molecular dynamics simulations to model the bulk and the microscopic, static, and dynamic properties of the liquid, and with an accurate potential function, results are generally in agreement with experiment. (Of course, many bulk properties of a liquid are measurable.) One important dynamical function which can be calculated from the molecular dynamics simulation is the velocity autocorrelation function (VACF) for a single particle


5

This function is also accessible experimentally from neutron and light scattering experiments. Typical results from the computer simulation of carbon tetrachloride in [52] are shown in Fig. 1.1, which shows that the simulated VACF falls rapidly from its zero time value of 1, and has a negative excursion before the correlation decays towards zero. Molecular dynamics simulations are computationally expensive. It is possible to simulate a system containing of the order of 1000 particles for a time period of the order of a nanosecond, but there are severe problems in using molecular dynamics to simulate reaction kinetics on this tirnescale. First, each simulation can only give one single history of the cluster, i.e.. one set of reaction times. In order to obtain statistically significant kinetics it is necessary to simulate many thousands of independent systems. Unfortunately this is not practicable at present. Even if it were practicable there are technical problems with the periodic boundary condition: although the solvent molecules must be periodically repeated in order for the liquid to have the correct properties, it is not physically realistic for the reactive particles in the cluster to be repeated in the image cells unless the cell is very large. For these reasons molecular dynamics simulation is not used, except to investigate details of particle encounters and dynamics on very short timescales, or to parametrize alternative descriptions of particle motion which do not require the explicit inclusion of all the solvent molecules.

FlG. 1.1. Velocity autocorrelation function for carbon tetrafluoride simulated by molecular dynamics [52].

6


1.2.2. Langevin equations. Although the motion of a single particle in the liquid phase can be described very successfully using the deterministic molecular dynamics method when all the surrounding particles are included, in isolation the trajectory of a single particle appears to be a random process. The diffusive motion of the particle can be characterized by a random force representing the jostling of all the surrounding molecules. We can construct a simple yet detailed stochastic model of particle motion if we parametrize it using information from molecular dynamics simulations. The model is based on two hypotheses about the velocity of the particle, which is assumed to be a continuous stochastic process with a known velocity autocorrelation function: (i) orthogonal components of the velocity are statistically independent; (ii) successive velocities of the particle constitute a stationary Gaussian process. The VACF is taken from the results of a molecular dynamics simulation, and is the only parameter required by the model. Each component of velocity in this model obeys an equation of the form

where (3 is an autoregression function, <j is a scale factor, and dW represents standard Gaussian white noise, as defined in Karlin and Taylor [49]. The equation is an example of what is known in the physical literature as the Generalized Langevin Equation [58]. There have been suggestions in the literature [52] that the velocity process is non-Gaussian. To support this view, Lynden-Bell et al. estimated the probability density function of 0(£), the deviation in direction of motion suffered by a particle, as a function of time delay t, i.e.,

Simulations have also been reported of the probability density of cos0 conditioned on the initial velocity. Until recently it was believed that these simulations showed evidence of non-Gaussian behavior, and that particles could be divided into two groups, those which "rattle" in a cage of surrounding molecules giving rise to the negative excursion of the VACF and those which lose their "memory" rapidly in an essentially diffusive motion. However. Atkinson et al. [5] have shown from analysis of the simple stochastic process described above that all of these simulated densities can be accounted for perfectly adequately given the VACF. It seems likely, therefore, that the stochastic model based on the Generalized Langevin Equation is a good representation of molecular motion.


7

1.2.3. Stochastic differential equations. The simplest model of molecular motion treats the stochastically evolving trajectory of the particle as a diffusion process. This follows the formalism developed by Einstein [30] and Wiener [86], amongst others. In modern notation, the evolution of the particle position is described in terms of a stochastic differential equation (sde),

where U. expressed in units of kT, is the potential due to external effects such as electric fields or the proximity of other particles, D is the diffusion coefficient of the particle, aaT = 2D, and dW represents three-dimensional white noise, which must be defined using a suitable limiting procedure [49]. The major problem with this formalism is that the particle velocity is not denned because the sample paths of a diffusion process are nowhere differentiable. However, over sufficiently long periods the central limit theorem ensures that the diffusion equation accurately describes the evolution of the particle density. Indeed, molecular dynamics simulations are frequently tested by checking that the mean square displacement of a particle increases linearly with time after a short period during which the "memory" of the initial velocity dies out. Most kinetics, even in the transient systems of interest here, take place on timescales of tens of picoseconds and longer, which are significantly longer than the velocity correlation time, and so use of a diffusion process to model the particle motion can be justified. However, in the last few years the experimentally accessible timescale has been pushed back to the order of 100 fs, and it is almost certainly inappropriate to use a diffusion equation on such short timescales. In view of these considerations, and the fact that diffusion processes are much more easily analyzed than the alternatives described above, virtually all theoretical studies of diffusion-limited reactions have remained within the diffusion formalism. In §1.3 we discuss some of the methods that have been used, and some of the exact results that have been obtained for diffusioncontrolled reactions in nonhomogeneous systems. 1.2.4. Singular diffusions. Some analytical progress can be made in the study of more realistic models of particle motion by modelling particle velocity as a diffusion and obtaining position by integration. A limiting case of equation (1.4) in dirnensionless form gives the sde

which is known as the Ornstein Uhlenbeck process. Together with the equation

again written in sde form, the pair (x,v] form what is known as a singular diffusion, i.e., the random input dW enters into only one of the pair of sde's.

8


A great deal is known about the Ornstein-TJhlenbeck process. It is a Gaussian process and therefore isotropic when each of the velocity components is independent and identically distributed. The VACF has a simple, exponentially decreasing form. In §1.4 we will show how first-passage problems for integrated diffusions can be tackled. Singular diffusions arise in other models of particle mo.tion. Statistical studies of molecular dynamics simulations of liquids suggest that particles tend to return to the position that they occupied at some recent time in the past. In solids the random oscillations center about some fixed point in space. To model the liquid behavior an equation of the form

is suggested, or replacing the simple average velocity by an exponentially weighted one,

Substituting z for JQ exp(—a(t — s]}v(s)ds leads to the pair of sde's

This is a flexible formulation. Motion remains isotropic and both the integrated Ornstein -Uhlenbeck process and randomly driven simple harmonic motion can be recovered as particular cases. Furthermore [3], the VACF can be evaluated explicitly as

where illustrated in Fig. 1.2. 1.3.

Typical VACFs within this famiily are

First-passage times for diffusions

1.3.1. Geminate recombination. In a simple type of photochemical experiment a short flash of light causes molecules in solution to dissociate, forming pairs of correlated reactive particles, which are said to be geminate. In many experiments these pairs are uncharged [17]. but recently similar experiments have been performed in which the particles are produced with equal and opposite electrical charges [11]. These correlated pairs are sufficiently dilute in the solution that they can be considered to be isolated from one another, so that all the reaction that is observed comes from recombination of two correlated particles. In the simplest case we can assume that the reaction is certain to occur on encounter. The time-dependent yield of imreacted pairs is thus simply proportional to the time-dependent survival probability of a typical pair. The aim of the analysis is to calculate this survival probability.


FIG. 1.2. Typical VACFs calculated by using (1.12): 9 = 5,/x = 2.5

5.0.

1.3.2. Simplifications. Before analysis a number of simplifying (but reasonable) further assumptions are generally made about the particle motion: (i) the two particles diffuse independently of one another except perhaps through a long-range force which is included in the drift term of the stochastic differential equation, (ii) the liquid is isotropic, (iii) any forces acting on the particles are central, i.e., the force on a particle is parallel to the interparticle vector, (iv) the particles are spherical. These assumptions are not essential to the model, but induce considerable simplifications. There are, for example, situations in which we can make the diffusion coefficient depend on the configuration of particles [15]. There are other systems in which the space is anisotropic and yet others where external fields act on the particles or the interparticle force is not parallel to the interparticle vector. All these generalizations have also been made [62], [67]. However, in by far the most important experimental systems approximations, (i) to (iv) are appropriate. As long as any external fields are uniform, approximation (i) leads to a separation of the six-dimensional diffusion equation describing the joint density of the two particles into an equation for the relative diffusion of the pair and

10


an equation for the "diffusive center of gravity of the pair," which is of no consequence for the kinetics; an equivalent separation can be derived from the stochastic differential equation. Approximation (i) therefore allows us to reduce the six-dimensional equation to a three-dimensional diffusion equation in the relative coordinate space where the relative diffusion coefficient is simply equal to the sum of the single particle diffusion coefficients. Assumptions (ii) to (iv) guarantee that the diffusion is isotropic and that any interparticle force is directed along the radial direction in the relative coordinate space, so that the kinetics are independent of the relative orientation of the particles, depending only on r, the distance between the particles. Under these conditions, the density of the interparticle distance satisfies the equation

where D' is the relative diffusion coefficient. 1.3.3. Boundary conditions. In order to solve the diffusion equation it is necessary to impose boundary conditions. The initial condition prescribes the initial distribution of the interparticle distance. The outer boundary generally recognizes that the density should decrease to zero as r, the distance between the particles, increases without limit (although if allowance is made for a nonzero density of other pairs this boundary condition can be modified [60]). Finally, and most importantly, we must have a method to include reaction. For a fully diffusion-controlled reaction an absorbing boundary condition is employed. This condition, which is known as the Smoluchowski boundary condition in the chemical literature [70], states that the density is zero when r = a, where a is the reaction distance. The use of an absorbing boundary implies that reaction is instantaneous on encounter, which is clearly unphysical. but many reactions do seem to approach this limit. We consider alternative descriptions later in the section. 1.3.4. The backward equation. Although it may be possible to solve (1.13) for a specific initial particle density, it is much more convenient to have the set of solutions corresponding to initial distributions concentrated at each fixed distance TO for TO > a. This is because any other solution can be constructed by convolution over the relevant initial distribution, and because Pabs(rQirit)i the solution for a concentrated initial distribution, obeys not only the forward diffusion equation (1.13), but also a backward diffusion equation, which is its adjoint. In this context the backward diffusion equation describes how the density depends on TO, the initial distance between the particles. It is also straightforward to show that H(ro.t). the time-dependent survival probability for initial separation TO, obeys the same backward equation, so that a single solution for J7 gives the kinetics as a function of TO and / (H is simply the integral of £>abs(ro- r -1) with respect to r for r > a). Solving the


11

backward equation

with boundary conditions

is evidently a very convenient way of obtaining the kinetics of a geminate pair. Although the backward equation has been known in the literature of stochastic processes for many years [49], and has been applied to problems of this type in other applications, it does not seem to have found favor in chemistry until its rederivation by Sano and Tachiya [74]. 1.3.5. A general method. When p(ro,r,t), the transition density (Green's function) of the unconstrained diffusion process (i.e., without an absorbing barrier), is known, it can be used to construct solutions for £>abs> the density with absorption, and the survival probability 0. The method makes use of the continuity of the sample paths and the strong Markov property of the diffusion process [49]. The path from TQ to a can be decomposed into two parts. The first part is the diffusion up to the time when a is first hit, and the second part is the subsequent diffusion ending at a at time t. If the diffusion is at a at time t then it must first have hit a at some time u where 0 < u < t. If first passage to a occurs at u then in the subsequent period t — u the path must diffuse from a to a; the transition density for this is p(a,a,t — u). Let w(ro,a,u) be the probability density that the first passage to a occurs at time u. This of course is the density we are after. The strong Markov property ensures that the diffusion starts anew after first passage to «, which occurs at a Markov time. Thus we have

The first-passage time density w can therefore be obtained from p by deconvolution, which can be done, for example, by Laplace transforms:

The survival probability can be obtained from w since

The advantage of this method is that it is generally applicable to any one dimensional diffusion process whose transition density is known.

12


A similar method can be used to obtain the distribution of the distance between surviving particles. Paths from TQ to t can be decomposed into two classes, those that pass through a and those that do not. The probability density we require corresponds to the class that does not pass through a and so is not absorbed. Using arguments similar to those above, the required density is seen to be

1.3.6. Zero interparticle force. The simplest and by far the most useful exact solution for a single pair is the case where there are no forces between the particles. In this case the diffusing interparticle distance is a process known as the Bessel process, and obeys the stochastic differential equation

where n is the dimensionality. The transition density for an n-dimensional Bessel process is well known [48] to be

and performing the deconvolution for the case n = 3 gives

An alternative method of obtaining this result is to note that the backward equation for Q is simply the diffusion equation with spherical symmetry and no drift. The corresponding heat flow problem (initial uniform temperature, space bounded internally by a spherical surface with fixed temperature) was solved many years ago [16], and (of course) yields the same solution. Solutions are also known for the corresponding one- and two-dimensional problems (and higher). One interesting feature of the solution in any dimensionality higher than two is that there is a nonzero probability that the two particles will diffuse apart into oblivion without ever meeting. This probability is zero in one and two dimensions; it is 1—a/ro in three dimensions. The yield of escaped particles can often be measured experimentally as a plateau in survival yield between the fast geminate recombination and the much slower bulk recombination. 1.3.7. Other cases. Exact solutions for the transition density and the firstpassage time distribution are well known for many other diffusion processes. notably the Wiener process with drift [27] and the Ornstein-Uhlenbeck process [69]. Most of these solutions have not found any major application in chemical kinetics. However, one problem which has attracted a great deal of attention is the problem of geminate recombination in three dimensions with a central


13

force acting between the particles. Approximate methods have been proposed which can be applied under conditions where forces are weak [75]. [21]. [44] or strong [88], [42]. It is also straightforward to solve the stationary backward equation for the probability that the pair escapes without encounter. In three dimensions the solution is given by

The most important source of a long-range force acting between the particles is the Coulomb interaction between two ions, which is mediated by the solvent. In units of kT this interaction has a potential energy of the form

where rc is a constant characterizing the range of the interaction as mediated by the solvent, and is known as the Onsager radius. U is negative for an attractive interaction between opposite charges, and positive for a repulsive interaction between charges of the same sign. In addition to the use of approximate methods an exact solution for the Laplace transform for the survival probability of an ion pair has also been reported [47]. The solution is exceedingly complicated, and the Laplace transform must be inverted numerically if anything more detailed than the asymptotic behavior is required. In practice it is generally simpler to solve the backward equation by finite difference methods. 1.3.8. Many particles in one dimension. Another class of diffusionreaction problem for which exact analytical solutions for the kinetics can be obtained arises where the motion of the particles is limited to one dimension. In these systems particles move independently of one another: their motion can be modelled either as continuous-time random walks with exponentially distributed waiting times between jumps, or as standard Browniaii motions in one dimension. Their chemical reaction, which occurs instantly on contact, either leads to annihilation of both particles (annihilating random walk or annihilating Browniaii motion) or. alternatively, to only one of the two particles being removed and the other left undisturbed (coalescing random walk or Browniaii motion). Many mathematical techniques can be applied to the solution of this onedimensional many-body problem, and the problem has received a great deal of attention in the literature [10]. 81]. [51]. [7]. [12]. One useful technique proposed by Torney and McConnell [81] and generalized by Balding [6] is to consider periodic Browniaii motion on a ring containing a finite number of particles and to take the limit as the size of the ring and the number of particles increase. Another promising approach suggested by Balding et al. [7] is to construct a random graph known as the percolation substructure. In this graph a time axis

14


is associated with each point on a one-dimensional lattice. On each of these axes at exponentially distributed intervals a lateral arrow is put down (see Fig. 1.3) joining the axis with one of its two neighbors. With appropriate rules for the interpretation of this graph it can be used to construct realizations of the annihilating random walk, the coalescing random walk, and a third random process known as the Invasion Process [25]. The resulting dualities between the annihilating (or coalescing) random walk and the Invasion Process enable explicit solutions for time-dependent single-site occupancy probabilities and for spatial correlations to be obtained. Equivalent results for annihilating and coalescing Brownian motion can also be found after a suitable rescaling of time and space. Thus, for example, if the initial distribution of particles is spatially stationary, then the concentration of surviving particles is given by

for coalescing Brownian motion and

for annihilating Brownian motion, where

FIG. 1.3. Percolation substructure for the dual processes of invasion and reaction. Lateral arrows a,re thrown down between neighboring pairs of sites independently with exponentially distributed interarrival times. Particles undergo random walks with steps dictated by the arrows. Particles are annihilated when their trajectories intersect.


15

N(x] is the probability that in the initial configuration there is at least one particle in the interval [0,x), and O(x) is the probability that there is an odd number of particles in the interval. These results apply when the particles move by standard Brownian motion (which has a diffusion coefficient of ^). A different value of the diffusion coefficient D can be accommodated simply by replacing t in the solutions by 2Dt. The long-time asymptotic concentration is I/A/TT! for coalescing Brownian motion (CBM) and 77'\pK~t for annihilating Brownian motion (ABM), where 7 = lirnx" 1 f0J O(y)dy. In the case of most interest the initial distribution of particles is a Poisson point process with density 6, and 7 = • In this case we obtain

for CBM and

for ABM, which differs from the result for CBM simply by a factor of 4 in the timescale. Results can also be found for other spatially stationary initial distributions such as equally spaced particles and particles with alternating spacings. The latter case shows clearly that the parameter 7 is a measure of clustering in the initial distribution and the dependence of the asymptotic result for ABM on 7 demonstrates that some memory of the initial clustering persists even in the limit of long times. This conclusion is also borne out by a consideration of the distribution of nearest neighbor distances [10]. Although these techniques do permit simple and exact solution of many one-dimensional problems, unfortunately it does not seem that any of them can be readily generalized to (or used for) problems with higher dimensionality. 1.3.9. Other models of reaction. In all the preceding discussion the assumption has been made that reaction occurs instantaneously when two particles come into contact. This approximation can be relaxed in either of two directions. It was realized many years ago that some reactions are not fast enough to be diffusion-controlled, but yet are sufficiently fast for the rate to be influenced by transport. In the foregoing discussion chemical reaction was described using a perfectly absorbing boundary. The type of reactive boundary most frequently used to describe partially diffusion-controlled reactions is generally known in the mathematical literature as an elastic boundary [87]; in the physical and chemical literature, by analogy with the classical theory of heat conduction, it is termed a radiation boundary condition [26], [70]. " In the case of geminate recombination the radiation boundary condition assumes that the reaction rate, which is the diffusive flux across the boundary

16


(in relative coordinate space), is proportional to the spatial density at the boundary,

The adjoint of this condition, which must be used in the backward equation, is

There are many ways in which an elastic boundary can be constructed mathematically [49], [87]. One useful way to think of it is that reaction takes place with a finite rate v in local time at the boundary [38]. There is a general relationship between a solution of a diffusion equation for an absorbing boundary and that for an elastic boundary [65]. Thus. all of the cases where exact or approximate solutions for the diffusioncontrolled geminate survival probability are known can also be generalized to the radiation boundary condition. For example the geminate recombination solution in the absence of forces (cf. (1.22)) becomes

The where an and approximate and exact solutions for ion recombination have also been obtained for a radiation boundary condition [37], [44]. In many ways the radiation boundary condition is merely an expedient device. It is a recognition that the reaction rate depends on more than transport through the solution, but it makes no real effort to understand or to model the activation process which gives rise to the deviation from full diffusion control. Rather, it covers all the processes of real interest with a "grey sphere" and a single reactivity parameter v. The use of a boundary condition is not the only way in which reaction can be modelled in geminate recombination. Some reactions occur at nonzero rates even under conditions where the particles are static, for example, frozen into a glass, or at rates faster than expected for full diffusion control. Such reactions can take place even when the particles are not in contact, and must be described using a rate constant which depends on the distance between the reacting particles. If this distance-dependent rate constant is denoted k(r] and the diffusive trajectory of the interparticle distance on a particular sample path is JR(t). then the time-dependent survival probability on the path is simply


17

which is easily recognized to be a Kac functional [49] and obeys the backward equation

i.e., the same backward equation as the unreactive diffusion but with an additional sink term to describe the reaction rate. A similar sink term can be included in the (more familiar) forward equation. Two models of distance-dependent rate constant have been used in the chemical literature, deriving from different physical origins. The first type is derived from the so-called rnultipolar mechanism of energy transfer [33], and is generally used to describe fluorescence-quenching experiments. In this case k(r] is of the form a/r n , and n is usually 6 [36]. This might be an appropriate description for an experiment in which one of the geminate partners is produced in an excited state. The second type of dependence is derived from the exchange mechanism of energy or electron transfer, and can be used for electron transfer reactions in solution. This mechanism gives k(r] — a exp(—/3r) [29]. Both of these forms of k(r] have been widely used in conjunction with the Smoluchowski theory for homogeneous reactions (see, for example, the review by Gosele [36]); however, we are not aware of any application of either form to a rionhomogeneous geminate problem. It could be pointed out at this stage, as has been done by Szabo et al. [78], that the radiation boundary condition is itself a rate constant of this form. k(r} — k 6(r — a), where fi denotes the Dirac delta function. In addition, the absorbing boundary condition is of the same form in the limit k —» oc. 1.4.

First-passage times for integrated diffusions

The state space of an integrated diffusion is the pair y = (x, v) where x(/) is the position and v(t) is the velocity vector of the particle. The pair (x. v) evolves according to the pair of stochastic differential equations

o that y(t) is a Markov process. First-passage problems for the position component can be tackled by using a more genernl version of the renewal equation used in §1.3.5. For first passage to the surface of the sphere S from an initial state yo = (XQ.VQ) outside the sphere, the equation becomes

where p is the transition density of the unconstrained singular diffusion y(^). a is any point within the sphere, and w(yo.y 0 starting from XQ = 0, VQ = 0 has asymptotic form

Starting from XQ — 0, VQ = — I the asymptotic behavior becomes

where

and where K = 3F(5/4)/(2 3 / 4 7r 3 / 2 ). The behavior of i/(x) is graphed in Fig. 1.5. As can be seen in Fig. 1.6, excellent approximations to first-passage time distributions for integrated diffusions can be obtained by matching the distribution of a transformed Gamma variable to the asymptotic form of the density [4]. 1.4.3. Escape probabilities. Integrated Brownian motion is certain to return to its initial position. A basic quantity in chemical applications is the escape probability. To generate some feel for the problems involved in evaluating escape probabilities for integrated diffusions we will consider briefly the case in which the velocity process is Brownian motion with drift d. The Radon-Nikodym derivative of Brownian motion with drift relative to the process without drift is exp(—d 2 t/2 — d — dv(t]}. It follows that the joint density of the hitting time and hitting velocity is

Integrating f ( t . v ) over both t and v gives the probability of return. Atkinson and Clifford [4] have shown that for small values of d the escape probability behaves like

when the process has been scaled so that v — 1. It is conjectured that the escape probability is exactly erf (3d)1'2. There is some suggestion from numerical studies of other integrated diffusions that the square root behavior may be a common feature of systems with small escape probabilities.


21

FIG. 1.5. Numerical evaluation of the factor v(x] which determines the asymptotic behavior of the first-passage density to level x for integrated Brownian motion. The horizontal axis is x 1 / 6 sign(x). See equation (1.44).

FIG. 1.6. Comparison of the distribution function of the first-passage time density for integrated Brownian motion: — determined numerically and • • • analytic approximation given in [4].

22 1.5.

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS Computer simulation methods

Unfortunately, exact solutions for diffusion-controlled kinetics are limited for the present to one-dimensional systems and to two-particle systems, even within the diffusion approximation. This being the case, the only way at present to obtain "exact" solutions outside these limits is by computer simulation. In such a simulation the trajectories of the diffusing particles are simulated using a suitably discretized random walk, and reactions are modelled when particles encounter one another. The principle is relatively simple, but there are one or two slightly tricky points which need to be considered. The most important point is that any method chosen must be tested by comparison with exact solutions wherever appropriate to ensure that there is no systematic undercounting (or overcounting) of reactions. 1.5.1. Particle motion. As discussed above, if the diffusive motion of the reactive particles is described by a diffusion equation then the path of the particle obeys an sde

The sde contains two terms, a drift term, which contains any external forces on the particle and long-range forces acting between particles (generally assumed to be pairwise additive) through the gradient of the potential energy function [7, and a random dispersion term representing the jostling of the surrounding solvent molecules. The trajectory of the particle is obtained by numerical integration of the sde. As with ordinary differential equations (ode's), this can be done by several methods [55], [56], [71], [64], for example, generalizations of the Euler. Heun. or Runge Kutta methods for ode's; each involves discretization of time. The simplest and most direct method (although not the most accurate) is the Euler method in which the discretized sde becomes

where r = 6t is the time increment and N is a vector of three independent normal random variables of mean 0 and variance 1. This method has the virtue that if there are no external or interparticle forces the discretization is an exact sample of the correct particle trajectory. In the presence of interparticle forces the accuracy of the method depends on the time-step being short enough that the forces remain effectively constant over the likely distances travelled by the particle during the time-step. 1.5.2. Absorbing boundary. The most obvious way to simulate a diffusion-controlled reaction is to evaluate the particle trajectories until particles are found in a reactive configuration (i.e., overlapping). This method is accurate if the time-step is infinitesimally small. However, with discrete steps. a systematic undercounting of reaction occurs, since even when particles do


23

not overlap either at the start or at the end of a time-step there is a nonzero probability that encounter has taken place during the time-step [18]. This type of error arises even when the trajectories are sampled exactly, and can be tackled in two different ways. Either the time-step can be reduced when two particles approach one another, which reduces the magnitude of the problem at the expense of increased computational time, but does not remove it altogether. The alternative is to estimate the probability that two particles have encountered during a time-step conditional on their positions at the start and at the end of the step, and to generate a uniform random variable to decide whether encounter has taken place. It is particularly easy to calculate this probability if we characterize the relative particle positions simply using the interparticle distance. The probability of surviving the time-step is the probability that the innmum of the interparticle distance over the time-step r is greater than the encounter distance a, conditional on an interparticle separation of x at the start of the time-step and an interparticle distance of y at the end of the time-step. This quantity is the transition density from x to y for the diffusion absorbed at a (i.e., the transition density limited to those trajectories that do not pass through a) divided by the transition density from x to y for the unbounded diffusion

If there are no interparticle forces then the interparticle distance diffuses as a three-dimensional Bessel process, for which both p and pabs are well known (see §1.3.5) and £)a is given by [39]

In the presence of interparticle forces the sampling of the particle trajectories is no longer exact and the time-step must be sufficiently short that the interparticle drift remains effectively constant throughout the time-step. Under these conditions the interparticle distance is approximately a Wiener process with drift. Once again, pa\^ and p have a simple form for this process and the resulting equation for fia is

Simulation is now very simple. If particles are found in a reactive configuration then they must have encountered during the time-step, and so they are removed. Otherwise the encounter probability is calculated according to one of the equations above, and a random number is generated to decide whether or not encounter has occurred. The effect of failing to include this conditional reaction probability is illustrated in Fig. 1.7, which compares simulations and the exact solution for the recombination of a single pair without interparticle forces (equation (1.50)). Even for the rather small time-step (1 ps) and

24


FIG. 1.7. Simulated geminate recombination probability. Absorbing boundary condition; TQ = 15, a = 10, D' — 1. — analytic solution (complement of (1.22)); o o o o simulation with bridging process (see (1.50)); A A A A simulation without bridging process. In both cases 105 realizations were performed with a constant time-step of 1 ps [39]. large encounter distance (10 A) it can be seen that there is a significant underestimate of the amount of recombination if the bridge reactions are not included. 1.5.3. Reflecting boundary. In simulations with an absorbing boundary, reaction rates are unchanged if particles are permitted to overlap one another since it is only necessary to discover whether the particles have encountered. If reactions are not completely diffusion-controlled, however, it must be recognized that particles may not occupy the same space and so when the trajectories pass through a forbidden region they must be modified to allow for reflection. Reflection occurs only if the infimum of an interparticle distance in the sampled trajectories over the time-step is less than the interparticle separation a. The infimum conditional on the sampled separations x and y at the start and end of the step, respectively, has probability distribution given by equation (1.51), and a suitable random variable with this distribution can be generated from a uniform random number by the inversion method [28]

where U is a uniform [0.1] random number. If M < a then reflection occurs and the trajectories must be modified. Consider two particles whose three-dimensional trajectories are given by Xj(£) and X2(£). The interparticle vector r = xi — X2 is stochastically in-


25

dependent of the diffusive center of gravity s = [Dî + Dix 2 ]/(Di + ^2)- Thus reflection is only going to affect r and not s over a time-step. Furthermore, if the step is small enough that the interparticle drift is effectively constant, then diffusion can be separated into three orthogonal independent componentsone component along the interparticle direction and the others orthogonal to it. Again, to first order, only the component along the interparticle direction (which is to the same order as the interparticle distance) need be modified. Once again, and for a similar reason as in the case of the absorbing boundary, the simplest method of modifying the interparticle distance for reflection is not correct. In this method, if the particles are found with a separation y < a at the end of a step, then the separation is modified to 2a — y. This method is correct if the separation is a simple Wiener process without drift, when it corresponds to the well-known reflection principle [32] . If there is nonzero interparticle drift this description is not exact, but improves as the step size is reduced. However, this computational expense is not necessary. Another construction of reflected Brownian motion replaces y by y + a — M if M < a, i.e., if encounter has taken place [46]. This construction is exact not only for the usual Wiener process, but also for the Wiener process with drift [39]. Since M has already been generated to ascertain whether the pair has encountered, it is almost no extra computational effort to use this prescription for the reflected interparticle distance. Once the new distance is known it can be combined with the orthogonal components to reconstruct the interparticle vector r, which can then be combined with s to regenerate the modified interparticle positions xi and X2. 1.5.4. Radiation boundary. The purpose of developing a method for implementing a reflecting boundary in the Monte Carlo simulation is to allow simulation of kinetics which are not fully diffusion-controlled. We will only discuss the radiation boundary condition here. The principle is the same as that for the absorbing boundary. It is necessary to estimate the probability of survival (or reaction) conditional on the pair separation at the start and at the end of the step. The simplest method is to compute prad/Prefl5 i- e - 5 the ratio of the surviving density from x to y at time T with the radiation boundary condition, to the transition density with reflection only. Even though the function can be evaluated explicitly, it is very expensive to evaluate at every step. The computational efficiency can be greatly improved by dividing trajectories into two classes, those that hit a and those that do not. The survival probability restricted to those trajectories that hit a is

26


which can be evaluated explicitly, when the interparticle distance is approximately a Wiener process with constant drift [39]. The effect of failing to deal with reflection correctly is shown in Fig. 1.8. It is seen to be of a similar order in this case to the error incurred by failing to include the bridging reaction probability in the simulations with an absorbing boundary.

FIG. 1.8. Simulated geminate recombination probability. Radiation boundary condition; ro — 5, a = 5, D' = 1, v = 1. — analytic solution based on (1.32); o o o o simulation based on (1.52); A A A A simulation based on approximate reflection principle (see text). In both cases 105 realizations were performed with a constant time-step of 1 ps [39].

1.6.

Independent pairs

Since it is too hard at present to find exact solutions for diffusion-controlled kinetics in multibody systems, it is necessary to resort to approximate methods. Once the essential nonhomogeneity of the problem was recognized, a series of theories based on macroscopic diffusion-reaction equations were proposed [73], [57], [61], [50], [82], [14]. However, although such descriptions are perfectly valid for problems where there are many reactive particles and the inhomogeneities are macroscopic, it has been demonstrated conclusively that they make serious errors of principle in microscopic systems, leading both to incorrect kinetics and to the wrong product yields [20], [23].

The most successful approximation to have been proposed to date for microscopic nonhomogeneous systems is the independent pairs approximation [19], [23]. Rather than taking a macroscopic theory and attempting to extend it to microscopic systems, this approximation starts with the microscopic description of the geminate pair and uses an independence approximation to extend the treatment to systems of more than two particles.


27

1.6.1. Independently and identically distributed distances. We start, therefore, with a pair of particles whose distance apart is a random variable with density f ( r } . The survival probability of this pair is evidently given by

The probability of the pair encountering in the time interval [t,t + d t ] , given that it survives to time t, is therefore —(d\nTL/dt)dt, which we denote \(t}dt. In chemical terms this is effectively a first-order time-dependent rate constant for the encounter in the sense that the survival probability trivially obeys the rate equation

Now suppose that at time zero, there are A^ particles whose locations are independently and identically distributed (iid) in some region of space. In this case the marginal distributions of the interparticle distances are also identical, but they are clearly not independent, since they are constrained to obey triangle inequalities (for example). The simplest form of the independent pairs approximation consists in assuming that the interparticle distances are also independent, except that when reaction occurs all the distances involving the reacting particles are removed along with the particles. Thus, given that there are N unreacted particles at time t, there are N(N — l)/2 interparticle distances, each of which is equally likely to be the next pair to encounter, and each of which will make a reactive encounter in the interval [t,t + dt} with probability X ( t } d t . Since the distances are assumed to be independent, the probability of an encounter in this interval is ±X(t)N(N-l)dt. The state of the system is characterized by the number of remaining particles AT, and the theory gives an equation for the probability distribution

In this particular case the master equation can be solved exactly [19] following a change of timescale T = J0 X(t}dt and using the generating function method [54], and gives for the expectation of number of particles remaining

where

28

and the summation in (1.57) is for all N < NQ such that NO — N is even. The method can easily be generalized to clusters containing more than one species of particle with different initial distributions [23], and to partially diffusioncontrolled reactions [45] although, except in special cases, the set of coupled differential equations must be solved numerically. One of these special cases which is worthy of note is that of a cluster containing equal numbers of two types of particles, where only reactions between unlike types can occur. Such systems are realized in nature in radiation tracks in low permittivity solvents where forces between ions are so strong that to all intents encounters between similarly charged particles are impossible, whereas oppositely charged particles are strongly attractive. The solution to the independent pairs approximation in this case is [45]

where N represents the number of ion pairs present and

1.6.2. The Smoluchowski theory. Many workers in the past have used the diffusion equation, and sometimes its adjoint, to demonstrate a link between the two-particle geminate survival probability and the many-body Smoluchowski rate constant for reaction of a particle with one of a sea of scavenger particles surrounding it [77], [35], [79]. Most of these studies treat the relationship as exact because they fail to notice the fundamental approximation of the Smoluchowski theory discussed by Weiss and den Hollander (this volume) and Noyes [63], which can be stated as follows. The central particle is a stationary sink about which the scavengers diffuse independently, and which swallows up any scavengers which hit it. The theory is frequently applied to systems where the central particle is not stationary and where the central sink is also destroyed by reaction. It is not immediately obvious how the formalism can describe either of these effects. The independent pairs approximation can be used to throw some light on both of these questions [37]. First, we consider the implication of using Smoluchowski's relative diffusion equation. This is a diffusion equation for the scavenger particles in the frame of reference of the central particle. The use of such an equation implies that the scavengers diffuse independently, not in real space, but in the frame of reference of the central particle. Brief consideration of the case where the central particle is mobile and the scavengers are static, discussed by Weiss and den Hollander, suffices to show that this is an approximation. But in fact it is the same approximation that we have just applied to microscopic cluster kinetics. Thus, in a sense, the theory outlined in §1.6.1 is a generalization of the Smoluchowski theory to microscopic


29

systems. If this is so then it should be possible to derive Smoluchowski's results probabilistically. The derivation here follows [37]. Consider a single particle initially at the origin in a volume V , containing scavengers that are distributed according to a Poisson point process with density c(x), which is such that limc(x) = Co as x —>• oo. The probability that the volume V contains exactly TV particles is given by the Poisson distribution (C0M)N exp(-C 0 M)/JV! where C0M = Jv, c(x)dx. Using fj(x, t] to denote the probability that a particle with initial vector x has not come within a distance a, of the central particle by time t, and taking account of the spatial distribution of the particles c(x)/(CoM) we obtain the marginal probability of survival

In the independent pairs approximation the central ion survives in the presence of A; particles if none of the independent pair distances attains a; the probabilit of this is IIfc. Summing over all possible values of k we find for the survival probability of the central particle (assuming that it is destroyed on encounter)

so that the pseudo first-order time-dependent rate constant for the scavenging is iven by

Application of the most important case in which the initial concentration is uniform, c(x) = Co. and H is given by (1.22) gives Smoluchowski's result:

This formula has been derived in other ways [70]. but this derivation shows very clearly that the first objection to the Smoluchowski theory (that of the stationary central particle), is equivalent to the independent pairs approximation, and that the second objection (that of the indestructible sink) is not necessary (as also pointed out by Steinberg and Katchalski [77] and Tachiya [79]). since the derivation given above made no such assumption, either implicitly or explicitly. 1.7.

The independent reaction times approximation

The approximation discussed in §1.6 has a number of operational disadvantages. The introduction of each new chemical species into the cluster (or particles of the same species but with different initial distributions) necessitates

30


an expansion of the dimensionality of the state space. Thus the computational size of the problem can quickly get out of hand, and for systems containing several types of particles with different initial distributions it is desirable to have an alternative way of applying the independent pairs approximation. This alternative method, the Independent Reaction Times (IRT) method, can be formulated either as a stochastic computer simulation technique, or more formally and analytically. It was first suggested in this context by Clifford et al. [20], and has been used by several groups in the intervening years [12], [13], [22], [24], [42], [43]. 1.7.1. The IRT simulation method. The idea behind the IRT method is very simple. Reaction times are generated at random according to a suitable prescription for every possible pair of reactive particles in the system. This gives a set of N(N — l)/2 reaction times. The array of times is searched for the minimum time, and the corresponding pair of reactants is deemed to have reacted at this time and is replaced by products. If the products are unreactive, then all the other times generated for the two particles which have now reacted are removed from the array. The array is then searched again for the minimum reaction time for the remaining particles and the process is continued until no further reaction is possible, which can occur either because there are no reactive pairs left or because all the remaining reaction times in the array are infinite. Such a simulation gives a chemical realization of the system as a set of reactions and the times at which they occur, just as the more detailed Monte Carlo simulation method does. And just as for the Monte Carlo method, the IRT simulation must be realized many thousands of times before the simulated kinetics attain an acceptable level of significance. However, whereas the Monte Carlo method follows the trajectories of the diffusing particles in detail, and is computationally intensive, one realization of the IRT method simply requires the generation of random reaction times for each pair from a suitable distribution and a few searches through the array of times. It is generally orders of magnitude faster than the full Monte Carlo simulation. At the heart of the IRT method is the generation of the random reaction times, and this is where the independent pairs approximation comes in. In the simplest case the IRT method is identical to the master equation described in §1.6.1. In this case all particles of a given type are iid from a model distribution. Given these distributions it is simple in principle (but often inconvenient in practice) to calculate the marginal density of the separation distance for each type of pair present. Let us denote this density for pairs containing particles of types i and j by gij(r). In the independent pairs approximation each pair distance is assumed to be independent of all the others. The marginal survival probability for a pair of type ij is obtained by integrating the geminate pair survival probability fl,j(r, £) over the density


31

of the initial separation gtj ,

The random reaction time is then sampled from this distribution function by standard methods, described, for example, by Devroye [28]. The function II, however, is often rather complicated and inconvenient to sample from, although it can always be done by sampling a random distance from gij and then generating a time from JT^j conditional on this distance. However, if we are going to go to the extent of generating all the distances at random, why not start the simulation from the required initial particle distribution and calculate the distances from the initial configuration? The great advantage of the IRT simulation method is that it can be started from a real initial distribution of particles. If this is done then the underlying approximation is more subtle than the simplest independent pairs approximation. The random reaction times are generated independently for each pair from the correct marginal distribution function f^j conditional on the true interparticle distance in the initial configuration. However, whereas in the straight independent pairs approximation the initial distances are also assumed to be independent, in this modified approximation the initial distances obey all the correct constraints (triangle inequalities, etc.) automatically because they have been calculated from a real particle configuration. They are therefore not independent. Because of the difficulty of solving these many-body diffusion problems exactly the approximation behind the independent pairs approximation is generally tested by comparison with Monte Carlo simulations for the same initial configuration. Tests have been performed for simple clusters with fixed and random initial configurations, for charged and uncharged particles [42], [23], [24], for partially diffusion-controlled reactions (radiation boundary condition) [45] , for overlapping spurs [68] , and for segments of a radiation track [43]. Some typical results are shown in Fig. 1.9. It is possible to find initial configurations where the approximation breaks down, for example, where there is a high concentration of positive ions at the center of the configuration and the negative ions are more thinly spread out; in this case the independent pairs approximation takes no account of the ways in which the interparticle forces reinforce each other, and underestimates the rate of reaction [42]. However, examination of Fig. 1.9 shows that for most situations of interest the approximation is extremely accurate. It is also possible to generalize the method to permit the products of reaction to remain reactive. In this case, following a reaction, new reaction times must be generated for the possible reactions of the product molecule. The method loses much of its elegance at this point, because it is sometimes necessary to give the particles new positions at this time, but three methods have been suggested within the spirit of the basic approximation for extending the

32


FIG. 1.9(a). Comparison of the Monte Carlo (MC) and IRT simulation methods. Spur of two A particles and two B particles; initial positions iid spherical normal with standard deviation 10; encounter distances a = 10; diffusion coefficients D — 0.5, all particles uncharged, o o o o MC simulation (104 realizations, variable time-step); — IRT simulation (105 realizations) [23].

scheme to include reactive products [18]. These will not be discussed further here. Comparison of the different types of simulation show that the IRT method with real initial configurations of particles generally gives a more accurate description of the kinetics than the master equation (or equivalently the IRT method with independent initial distances) [18]. This observation is presumably explained by the fact that the real initial configuration recognizes all the correlations between the initial distances. 1.7.2. Analytical formulation of the method. In the preceding subsection the IRT approximation was described as a method of stochastic simulation. It is also possible to formulate the method (without reactive products) as a death process and to find its infinitesimal generator [41]. The state of the system is characterized by the identities of the particles remaining. Suppose that the system starts with particles at locations {xi} for i = 1,2,.... Consider a subset A of the particles whose cardinality is n (and let the complement of A be A'}. If we take particles in pairs, as we must in order to model reaction between them, then there are n(n — l)/2 such pairs of particles. Let the set of all possible pairs of particles in A be denoted by A^.


33

FIG. 1.9(b). Comparison of the MC and IRT simulation methods. Spurs of three ion pairs. rc = 29 nm, encounter distance a = 1 nm, diffusion coefficients = 0.5 nrn2 ps l; initial positions sampled from centered spherical normal distributions. MC simulation (104 realizations, variable time-step]; — IRT simulation (105 realizations] [42]. (top] cr+ = a^ = 1 nm; (bottom] cr+ = 1 nm; a_ = 8 nm.

If a specific pair a = {i, j} were in isolation then the probability density of its reaction time would depend on the initial locations x; and Xj as well as the reactive distance and time t. For convenience, we denote the density by uja(t). According to the IRT approximation, if the cluster is in state A at time t, the probability that it will still be in state A at time t + h is the probability that none of the pairs in A% have reacted in the interim, which is given by

34


FIG. 1.9(c). Comparison of the MC and IRT simulation methods. Nonrandom initial configurations chosen to accentuate correlations between ionic forces. rc — 29 nm, encounter distance a = I nm, diffusion coefficients = 0.5 nm2 ps~l. o o o o MC simulation (104 realizations, variable time-step); — IRT simulation (10° realizations) [42].


35

State A can be formed from state A U a if a and A are disjoint and the two particles of a react. The forward equation for the Markov process thus becomes

In the special case where all the initial distances between particles of a given type are iid, this gives the master equations discussed in §1.6, however, this formulation is evidently more general and like the IRT simulation can be applied to any initial configuration of particles. 1.7.3. Competition between scavenging and geminate recombination. As an illustration of the type of improvement that can be achieved by recognizing the initial correlations we consider a simple system in which an identical pair of particles is produced with some initial distribution, for example, by photolysis. The two particles can recombine or be scavenged by a reactive solute, which is uniformly distributed throughout the solution. Samson and Deutch [72] have formulated and solved the relevant diffusion equation for the case where the two radicals are stationary. This method, which takes advantage of the separability of the Laplacian in bispherical coordinates [59], cannot easily be generalized to the case where the two radicals are mobile; however, Monte Carlo simulation is straightforward, and has been reported [22], [9]. One early paper noted that the normal Srnoluchowski rate constant appeared to overestimate the amount and rate of scavenging, and ascribed this to the first scavenging reaction depleting the scavenger distribution in the neighborhood of the pair, thus reducing the rate of the second reaction. We decided to investigate whether this local depletion was indeed the source of the overestimate. To do this we performed IRT simulations for systems in which the two radicals are permitted to diffuse without interaction (i.e., to penetrate within each other) but react by diffusion control with a scavenger. The first and second reactions to occur were considered separately. The results of the IRT simulation are that the first reaction as well as the second reaction are slower than expected on the basis of the simple Srnoluchowski theory, with the largest discrepancy occurring when the two particles are initially coincident. Similar results are found in Monte Carlo simulations (although the statistics are not good enough to extract a timedependent rate constant). Some light can be shed on this phenomenon by the following mathematical analysis. Consider a typical scavenger particle in a system with uniform scavenger density Co. In the Smoluchowski theory the reaction times of this scavenger with each radical are assumed to be independent. If the marginal density of one of these times is ,/(t), then the joint density of the two times is f ( t \ ) f ( t 2 } because the theory assumes that the two radical-

36


scavenger distances are independent. However, this is clearly not the case if the two radicals are close together; the two distances together with the radical-radical distance are constrained to obey triangle inequalities, so that if one radical-scavenger distance is long (short), then the other one will be long (short) too, and this leads to a correlation between the reaction times generated for the two radicals with each scavenger in the IRT simulation. The modification of the Smoluchowski theory implicit in the IRT simulation is thus as follows: reaction times for each radical with the scavenger particle are generated independently conditional on the initial interparticle distances. Thus if the scavenger is at x, and the two radicals are at xj and X2, then using the notation of §1.6, we have for the joint density of t\ and £2,

and the probability that the reaction times exceed t\ and £2, respectively, is then

If there are N scavengers in the volume V, which has the Poisson probability (Co\V\)N /Nl, then the joint density of the ensemble of IN reaction times is YliLi f(tii,t2i), in which the scavengers are assumed to be independently distributed, but the correlation between the two reaction times for each scavenger is retained. The probability that no reaction has occurred by time t is then simply the probability that all 2N reaction times are greater than £, i.e., F(t,t)N . After Poisson weighting over the distribution of TV this gives for the probability of the pair surviving

which gives the time-dependent rate constant for the first reaction

In the limit where the two radicals are initially coincident the correlations are greatest, and this gives the explicit result

which is very similar to the Smoluchowski rate constant, except that the transient term is reduced by a factor of v/2- In the case where the two radicals are not coincident but have separation e/, the Laplace transform of F can be obtained, and by consideration of the asymptotics of the Laplace transform it can be shown that at short times the normal Smoluchowski rate constant is obtained (for d > a), and at long times the formula above is approached.

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[66] S. M. Pimblott, J. A. LaVerne, A. Mozumder, and N. J. B. Green, Structure of electron tracks in water. 1. Distribution of energy deposition events, J. Phys. Chem., 94 (1990), pp. 488-495. [67] S. M. Pimblott, A. Mozumder, and N. J. B. Green, Geminate ion recombination in anisotropic media. Effects of initial distribution and external field, J. Chem. Phys., 90 (1989), pp. 6595-6602. [68] S. M. Pimblott, C. Alexander, N. J. B. Green, and W. G. Burns, Effects of spur overlap in radiation chemistry: Reaction in two nearby spurs, J. Chem. Soc. Faraday Trans., 88 (1992), pp. 925-934. [69] N. U. Prabhu, Stochastic Processes: Basic Theory and Its Applications, Macmillan, New York, 1965. [70] S. A. Rice, Diffusion-Limited Reactions, Elsevier, Amsterdam, 1985. [71] W. Riimelin, Numerical treatment of stochastic differential equations, SIAM J. Numer. Anal., 19 (1982), pp. 604-613. [72] R. Samson and J. M. Deutch, Exact solution for the diffusion-controlled rate into a pair of reacting sinks, J. Chem. Phys., 67 (1977), pp. 847-853. [73] A. H. Samuel and J. L. Magee, Theory of Radiation Chemistry. I. Track effects in radiolysis of water, J. Chem. Phys., 21 (1953), pp. 1080-1087. [74] H. Sano and M. Tachiya, Partially diffusion-controlled recombination, J. Chem. Phys., 71 (1979), pp. 1276-1282. [75] P. Sibani and J. B. Pedersen, Alternative treatment of diffusion-controlled bulk recombination, Phys. Rev. Lett., 51 (1983), pp. 148-151. [76] M. von Smoluchowski, Mathematical theory of the kinetics of the coagulation of colloidal solutions, Z. Phys. Chem., 92 (1917), pp. 129-168. [77] I. Z. Steinberg and E. Katchalski, Theoretical aspects of the role of diffusion in checmical reactions, fluorescence quenching, and nonradiative energy transfer, J. Chem. Phys., 48 (1968), pp. 2404-2410. [78] A. Szabo, G. Lamm, and G. H. Weiss, Localized partial traps in diffusion processes and random walks, J. Statist. Phys., 34 (1984), pp. 225-238. [79] M. Tachiya, Theory of diffusion-controlled reactions: Formulation of the bulk reaction rate in terms of the pair probability, Radiat. Phys. Chem., 21 (1983), pp. 167-175. [80] , Reaction kinetics in micellar solutions, Canad. J. Phys., 68 (1990), pp. 979991. [81] D. C. Torney and H. M. McConnell, Diffusion-limited reactions in one dimension, J. Phys. Chem., 87 (1983), pp. 1941-1951. [82] C. N. Trumbore, D. R. Short, J. E. Fanning, and J. H. Olson, Effects of pulse dose on hydrated electron decay kinetics in the pulse radiolysis of water. A computer modeling study, J. Phys. Chem., 82 (1978), pp. 2762-2767. [83] N. G. van Kampen, Cluster expansions for diffusion-controlled reactions, Int. J. Quantum Chem.: Quantum Chemistry Symposium, 16 (1982), pp. 101-115. [84] T. R. Waite, Bimolecular reaction rates in solids and liquids, J. Chem. Phys., 32 (1960), pp. 21-23. [85] G. H. Weiss and F. den Hollander, Aspects of trapping in transport processes, Chap. 5, this volume. [86] N. Wiener, Differential Space, J. Math. Phys., 2 (1923), pp. 131-145. [87] D. Williams, Diffusions, Markov Processes, and Martingales, Vol. 1, Wiley, Chichester, 1979. [88] F. Williams, Kinetics of ionic processes in the radiolysis of liquid cyclohexane, J. Amer. Chem. Soc., 86 (1964), pp. 3954-3959.

Chapter 2

Fluctuations in Nonlinear Systems Driven by Colored Noise Mark Dykmari and Katja Lindenberg

Abstract We consider the spectral density of the fluctuations as well as rare large fluctuations in nonlinear systems driven by colored Gaussian noise. Special emphasis is placed on the review of recent results on the application of the method of optimal paths to the analysis of large fluctuations. We formulate the variational problems for the optimal paths along which the system moves with an overwhelming probability in the course of a fluctuation that brings the system to a given point in the phase space, and also in the course of the escape from a metastable state. The formulation relies on knowledge of the shape of the power spectrum of the noise, which can usually be determined experimentally. The solutions of the variational equations are considered for various shapes of the power spectrum, including the case1 of a spectrum with a sharp peak at finite frequency (quasi-monochromatic noise) where qualitative features of large fluctuations related to the noise color are distinctly pronounced (e.g.. the occurrence of multiple crossings of a saddle point by the optimal path without a transition to another stable state). We analyze the problem of the probability density for the system to pass a given point at a given time before arrival at another point in the course of a large fluctuation (the prc-history problem). The maximum of this probability density lies on the optimal path. The results of recent analog simulations of large fluctuations in systems driven by Gaussian noise, including ones where the optimal paths have been visualized via the analysis of the pre-history, are discussed. 2.1.

Introduction

The understanding of the pattern of fluctuations in dynamical systems driven by noise poses one of the fundamental problems of physical kinetics. The problem was formulated originally by Einstein [I] and Sinoluchowski [2] in the description of the Brownian motion of a macroparticle. It is of immediate current interest in the context of a vast array of physical phenomena, starting with transport phenomena in solids (for instance, the kinetics of electrons interacting with phonons and/or impurities) [3], [4], and including kinetics of laser modes [5], [6] and kinetics of Josephson junctions [7]. The problem of noise-induced fluctuations is also immediately related to

41

42


physical measurements: a physical instrument is a dynamical system driven by fluctuations from various sources, including the very quantity being measured. The features of the fluctuations in a system depend on the character and intensity of the driving noise f ( i ) and the way in which it couples to the system. In general, f ( t ) may depend on the state of the system, i.e., the properties of the noise may be different in different states. Most noise of physical relevance can be described fully by its correlation functions {/(£)}, (f (t\) f (ti}} •> • • • ? {/(î) ' ' ' f ( t n ) ) i • • • 5 where the brackets {• • •) represent an average over an ensemble of statistically equivalent realizations of the noise. In many cases the correlation functions are mutually interrelated. A common situation arises when the noise driving a system originates from its coupling to a macroscopic system of N dynamical degrees of freedom, with TV ^> 1 (e.g., a thermal bath). Such noise is typically a superposition of a large number of "elementary" fields or forces,

In the simplest cases the fi(t) with different i refer to different elementary excitations in the bath and are mutually uncorrelated. If this is the case, then under some conditions (e.g., if the fi(t) are each of order A/"""1/2), in the limit N —> oo, f ( t } is Gaussian according to the central limit theorem of probability theory [8], [9]. In this case all the correlation functions can be expressed in terms of the two lowest-order ones. We shall deal only with stationary noise, i.e., noise whose statistical distribution does not change with time. The only characteristic of a stationary zero-mean ((/(£)) = 0) Gaussian noise is the time correlation function 0(£) or, equivalently, its Fourier transform $(u;),

All odd-order correlators vanish, while all the even-order ones are expressed in terms of (f)(t). For example, the fourth-order correlator is

(It is easy to understand (2.3) if f(t] is equal to the sum of a large number N of weak uncorrelated forces fi(t) oc Af" 1 / 2 by noting that the omission of N terms in the sums with more than two coinciding i (or, for that matter. of any A^ terms) introduces only small errors of O(N~l) in the fourth-order correlator.) The function $(u;) in (2.2) is called the power spectrum of the noise f ( t ) — it is precisely 4>(o;) that is often measured to characterize the noise. The measurements usually assume that the noise is ergodic. i.e.. that an ensemble average is equivalent to a time average. If this assumption is valid (as is

FLUCTUATIONS IN NONLINEAR SYSTEMS

43

usually the case in physical systems), then according to the Wiener-Khintchine theorem [5], [6]

and measurement of $(0;) simply involves recording f ( t ) over a sufficiently long time 2t0 and calculating a Fourier transform. Since $(cj) is a physical observable it is advantageous to express the characteristics of a noise-driven system in terms of it. The shape of the power spectrum depends on the source of the noise. For example, if the noise results from coupling to a thermal bath then 4>(o;) is determined by the density of states of the elementary excitations of the bath, the coupling constants between the bath and the system of interest, and the temperature. The shape of $(u) is used to differentiate, very roughly, between two types of noise: white and colored. White noise is characterized by a totally flat spectrum (o;) must necessarily vanish as uj —-» oo, since otherwise the time correlation function 0(t) would diverge as t —> 0, i.e., the mean-square value of the noise would be infinite. However, if all the characteristic eigenfrequencies and reciprocal relaxation times of a noise-driven system are small compared to the frequencies over which oo). The shape of this distribution depends substantially on whether dissipation and fluctuations in the system are both due to its coupling to a thermal bath (i.e., whether the noise is of thermal origin), or whether the driving noise is of nonthermal origin. In the first case there is a relation between the noise driving the system and the dissipative forces that extract energy from the system (cf. §2.2). As a consequence, the shape of the stationary distribution of the system (which in this case is an equilibrium distribution in the thermodynamic sense) for sufficiently weak coupling is Gibbsian regardless of the shape of the power spectrum $(0;). If the power spectrum is not flat, i.e., if the noise is colored, the relaxation toward the equilibrium state occurs via an equation of motion that is not time-local (contains explicit memory terms), that is, the relaxation is "retarded." The fluctuation-dissipation relation can sometimes be described phenomenologically, as in the case of Brownian motion [1] (cf. [18] and [19] and references therein for a discussion of fluctuation-dissipation relations in systems driven by colored noise). In some cases the noise and dissipation can be calculated from a microscopic model of the dynamical system of interest, a heat bath, and their coupling. The latter approach was first applied by Bogoliubov [20] to the problem of a linear oscillator coupled to a phonon bath; the corresponding quantum analysis of the dynamics of a linear oscillator coupled to a heat bath was first given by Schwinger [21] (see also Senitzky [22]). If the noise driving a system is of nonthermal origin, the statistical distribution of the system in the stationary state (if indeed one exists) is nonGibbsian. However, for sufficiently weak noise and nonbifurcational parameter values, the distribution still has a maximum (maxima) at the attractor(s) and is Gaussian near the maximum (maxima), just as in the case of thermal equilibrium. The dependence of the parameters of the Gaussian distribution(s) on the shape of the power spectrum of the driving noise is discussed briefly in §2.2. The spectral densities of the fluctuations of systems driven by thermal noise and bv nonthermal colored noise are also considered there.


45

It is not only important to understand the behavior of a fluctuating system near its most probable states; the tails of the distribution are also of great interest for various experimental measures. The tails describe the distribution for states of the system that are reached only rarely and come about mainly from occasional large "outbursts" of noise that "push" the system far away from the small-fluctuation region in phase space. In §2.3 we present an idea for a systematic approach to the problem of occasional large fluctuations in systems driven by Gaussian noise with a power spectrum of arbitrary shape. The approach is based on the concept of the optimal path, i.e., on the physical assumption (proved a posteriori) that the paths along which the system moves to a given point in the course of fluctuational "outbursts" are concentrated around a particular most probable path, called the optimal path. In §2.4 the application of the method of the optimal path is illustrated via an example of a system driven by "strongly colored" (quasi-monochromatic) noise, i.e., noise with a power spectrum that exhibits a narrow peak at a finite frequency. The pattern of fluctuations caused by such a noise differs drastically from that of a white-noise-driveri system. In §2.5 we further explore the idea of optimal paths by investigating the statistical distribution of the paths along which a system arrives at a given point. The corresponding pre-history problem is formulated, and recent experimental data on the first direct observation of optimal paths is discussed. A qualitative feature of the kinetics of bistable (and multistable) systems is the onset of noise-induced transitions (noise-induced switching) from one (meta)stable state to another. Such transitions occur as a result of large fluctuations, and they can be analyzed within the scope of the method of the optimal path. The analysis and its application to systems driven by noise with power spectra of various shapes, including exponentially correlated and quasi-monochromatic noise, are considered in §2.6. Some further perspectives for future research in the area of large fluctuations in systems driven by weak noise are outlined in §2.7. 2.2.

Spectral density of fluctuations and statistical distribution near a stable state

A ubiquitous (but by no means all-inclusive) description of the dynamics of a system driven by nonthermal noise is a set of coupled stochastic differential equations (frequently called Langevin equations) of the form

Here the gn(x) are functions of the state x of the system at time t. The fn(t) are the components of a zero-mean ((fn(t)) — 0) Gaussian noise, with correlation functions and spectral densities

46


Thus, the components fn(t] for different n may be interrelated, as embodied in those correlators ^nm(t] that do not vanish. The fn(t] in (2.7) are assumed to be independent of the state x of the system, i.e., the noise is "additive." In general the noise is not of thermal origin, and in that case the dissipative contributions to gn(x) are not related to the noise f n ( t ) . Systems driven by noise of thermal origin are discussed in §2.2.3. If the noise is sufficiently weak, the system experiences mostly small fluctuations about its stable state(s) (attractor(s)). We shall only consider systems with attractors of the simplest form, namely, fixed points in phase space (foci or nodes). The generalization to limit cycle attractors (tori) is straightforward, but the case of more complicated attractors associated with dynamical chaos (see Thompson and Stewart [23] and references therein) and the interplay of noise-induced fluctuations and dynamical chaos lie outside the scope of the present chapter. 2.2.1. Spectral densities of fluctuations in nonthermal systems. Let us begin by considering the spectral densities of the fluctuations of the variables

with

These experimentally accessible spectral densities constitute a useful way to characterize the dynamical behavior of the fluctuating system in the stationary state. Here xst is the stable state of the noiseless system, that is, the stationary solution of (2.7) in the absence of the noise:

Simple expressions for the Qnm(u) can be obtained by noting that, according to linear response theory, the response of the system to weak forces, including random ones (i.e., noise of low intensity), is given in terms of linear susceptibilities Xnm(t) [24]. Specifically, the noise-induced change 6xn(t) in xn is of the form

If one identifies 6xn(t] with the difference xn(t) — x^ that appears in (2.10), one finds from (2.9), (2.10), and (2.12) that the spectral densities Qnm((jj} of the fluctuations can be expressed in terms of the power spectra $nm(u;) of the components of the noise as


47

where

and Xnm(^) is the one-sided Fourier transform of the linear susceptibility,

Care should be taken when evaluating the spectral densities of the fluctuations from the formulas (2.9) and (2.10) since (2.7) is time-irreversible and therefore fluctuations and initial values of the xn are eventually "forgotten" when time goes forward but not when time goes backward. In (2.9) and (2.10) it has been assumed that the system was "prepared" at t = — oo. The susceptibilities Xnm(^) can be calculated by diagonalizing the matrix g whose elements are gnm = (dgn/dxm}x=xst- In terms of the unitary matrix A that performs the diagonalizing transformation,

we can write

Because of the assumed stability of the state xst, the real parts of the eigenvalues o>n of the matrix g are positive. Equations (2.13)- (2.17) describe in explicit form the dependence of the spectral densities of the fluctuations of a noise-driven system on the shape of the power spectrum of the noise: Qnm(^} is simply the sum of the products of the spectral components $ n /. m /(u;) of the noise and the Green functions G";',;"'(u;). The Green function G^™'(u] expresses the spectral density of fluctuations of the variables xn, xin if the driving noise is white with correlators (/A'(0//(O) = Dfi(t—t')fikji'bi?n'- For a one-dimensional system (L = I in (2.7)). the relations (2.13) and (2.14) reduce to

where

and

The simple and instructive expression (2.18) also holds for the spectral density of the fluctuations of a particle with coordinate q and momentum p — q driven by colored noise and described by the equation of motion

48


This equation of motion can be written in the form (2.7) with x\ — g, g\ — p, X2 = p, and g2 = —2Tp — U'(q). Note that only #2 is directly driven by the colored noise; the equation for x\ contains no explicit noise term. The spectral density of the fluctuations of the coordinate,

is given by (2.18) with

Here qst is the position of the particle at the minimum of the potential U(q), i.e., the solution of The relations (2.13) between the spectral densities of the fluctuations of the system and the noise, and, in particular, the special case (2.18), show that the system "filters" the noise, and that the spectrum of a system driven by colored noise reflects features of the spectrum of the noise as well as characteristics of the system itself. This makes it possible to use such systems in a judicious way to investigate the power spectrum of an unknown noise. A particularly useful system for this purpose is the underdamped oscillator described by (2.21) with a tunable frequency LJO and with a damping F (the "bandwidth" of the function G(u)) that is much smaller than u0 and than the characteristic bandwidth of the power spectrum 4>(o;) under investigation. In this case the spectral density Q(u] of the fluctuations of the oscillator contains a narrow peak near u;0 of halfwidth F and height $(a;0)/4F2a;Q. We note, however, that a delicate problem may arise if the noise is not very weak. As pointed out by Ivanov et al. [25] in the context of the problem of the absorption spectra of localized vibrations of impurities in crystals and analyzed in detail by Dykman and Krivoglaz [26] (see also [27]), an anharmonicity in the potential U(q) of a noise-driven underdamped oscillator may result in a nonlinear response to noise as evidenced by a strong distortion of the peak of the spectral density of the fluctuations. The distortion is caused by the fact that the anharmonicity leads to an amplitude-dependence of the vibration frequency. As a consequence, noise-induced fluctuations of the amplitude cause fluctuational frequency straggling. If this straggling exceeds the small frequency "uncertainty" F associated with the damping, the shape of the spectral peak is substantially modified from the Lorentzian shape obtained from the linear approximation, and its height differs from ^(uJoJ/^Tu!^. The noise-induced distortion of the peak of Q(u) as described by Dykman and Krivoglaz [26] has been observed and investigated in detail quantitatively in analog electronic experiments [28]. The intensity of the noise that leads to a distortion of the spectrum is much smaller than that resulting in the distortion of the probability distribution caused by the nonlinearity of the potential. One general consequence of (2.18)-(2.20) is that the time correlation function of a one-degree-of-freedom system driven by colored noise is a smooth

FLU C T U ATI ONS IN NON LI NEAR SYSTEMS

49

function near t = 0 (cf. Sancho [29], who considers the particular case of exponentially correlated noise). As can be seen from (2.10) and (2.18)-(2.20), this function is given by the expression

If $(u;) were constant (white noise), the derivative Q'(t) would be singular as t —> 0 since the integral over a; of a;/(a2 + a;2) diverges. If, however, (u) and hence /3 scale as D. As a result, it is evident in (2.29) and (2.30) that the width of the Gaussian distribution is proportional to the noise intensity D, i.e., the distribution becomes increasingly narrower with decreasing noise. The distribution also becomes narrower as the maximum of the power spectrum of the noise moves away from the range of large susceptibility of the system, i.e., the range where the Green functions le-

2.2.3. Spectral density of fluctuations in thermal equilibrium. Thermal equilibrium fluctuations in a dynamical system arise through its coupling to a thermal bath which is itself a dynamical system of many degrees of freedom [24]. In some cases, the description of the entire coupled system can be reduced to a set of stochastic equations of motion for the dynamical variables of interest in which the effects of the bath appear as potential shifts (and/or mass renormalizations), dissipative contributions, and random forces (noise). However, if the random forces resulting from this reduction have finite correlation times (colored noise), the form of these equations differs from (2.7) even in the simplest cases. Correspondingly, the shape of the spectral density of the fluctuations of the system differs from that given in §2.2.1. We briefly illustrate these points through a simple specific "generic" model [5], [6], [20]-[22], [30], in which the coupling between the system and the bath is assumed to be linear in the coordinate of the system. The coupled system-bath Hamiltonian function H for this model is of the form

where


51

Here q and p are the coordinate and momentum of the system, HQ and H^ are, respectively, the Hamiltonian functions of the system and the bath in the absence of coupling, U(q) is the potential energy of the isolated system, and H is only a function of the dynamical variables of the bath, so that the dependence of the coupling energy on the system of interest is linear in the system coordinate q (an analysis similar to that outlined below can be carried out for the case of coupling proportional to the momentum p). Not only the evolution of the system but also that of the bath is affected by the coupling between the two. If the coupling is sufficiently weak, the response of the bath to the perturbation Hi is given by linear response theory and hence only requires consideration of the evolution of the bath in the absence of the system. This linear response can be described with the help of a generalized susceptibility K(t) as

where it has been assumed that the coupling was switched on at t = — oo. Here f ( t ] is the instantaneous value of the bath-dependent quantity E! in the absence of the bath-system coupling. This instantaneous value fluctuates in time. If H is itself a sum of many "elementary" uncorrelated contributions arising from different degrees of freedom of the bath (e.g., a bath of oscillators or particles each interacting individually with the system), then the discussion in the Introduction leads to the conclusion that /(t) is a Gaussian random process which we take to have a zero-mean (a nonzero-rnean can be removed by a proper redefinition of variables). Since the bath has been evolving since t = —oo, the random process is stationary. The second term in (2.33) arises from the system-bath interaction and represents the way in which the bath dissipates excess energy of the system that is introduced by the interaction. It follows from the fluctuation-dissipation theorem [24] that the Fourier transform of the susceptibility K(t] is related to the power spectrum (oj) of the noise f ( t ] and the temperature T of the bath (in energy units) as

where

and P denotes the principal value integral. The expressions (2.32) and (2.33) lead to a stochastic equation for the dynamical variable g,

The integral in (2.36) describes a dissipative delayed "self-action" of the system mediated by the bath. According to (2.34) and (2.35), if the noise f ( t ) were

52


white the susceptibility K(t] would be proportional to the derivative of the ^-function (cf. Ford et al. [30]), K(t) oc dd^/dt, and the integral in (2.36) would be proportional to q(t), leading to the usual dissipation linear in the instantaneous value of the momentum, as in Brownian motion (cf. (2.21)). We note, however, that this limit leads to a self-energy divergence problem similar to the well-known divergence encountered in quantum electrodynamics: the real part of the susceptibility K(UJ) diverges and as a result the renormalized (because of the coupling to the bath) frequency of the small-amplitude vibrations of the system is infinite. There is no physical significance underlying this divergence: it can be removed in a way that is standard in quantum electrodynamics; furthermore, it does not arise if the power spectrum of the noise decays to zero as u —> oo, i.e., for colored noise. An important case where the dynamics of the system can be described within a quasi-white-noise approximation and where divergences do not arise occurs when the coupling is weak compared not only to the characteristic frequencies of the bath but also to those of the system: if |A"(u;)| is small compared to the squared frequencies u2(E) of the eigenvibrations with energies E < T and if the dependence of K(UJ] on u is smooth for uj ~ u(E], the dynamics of the energy and of the slowly varying portion of the phase of the system on a time scale coarse-grained over t ~ u)~l(E} are the same as those of white-noise-driven Brownian vibrations. The noise intensity for such motion is $[u(E}} « const, and the friction coefficient is given by lm.K(uj(E}]/u(E} ^ const, (cf. Bogoliubov [20], where a corresponding microscopic derivation was given for the first time; see also [30]). The perturbative corrections to these results due to the color of the noise have been considered by Carmely and Nitzan [32] (see also references therein). The explicit solution of (2.38) for colored noise f ( t ) with an arbitrary spectrum $(u;) can be obtained when the force U'(q) is linear, or when the noise is sufficiently weak that the force can be linearized about the equilibrium position qst, U'(q) ^ u^(q — qsi}. The spectral density of the fluctuations within this approximation can be seen from (2.36) to be given by

It is instructive to compare (2.37) with the undelayed relaxation result (2.18) with (2.23). In both cases the spectral density Q(UJ] is proportional to the power spectrum $(0;) of the driving noise, but the coefficients are different. In particular, in (2.37) the form of the coefficient depends, through K(UJ), on the shape of (I>(u;) itself. Therefore, if the coupling is sufficiently strong that \K(u)\ > CO>Q, the structure of $(u;) near the maximum is not reflected directly in Q((JO]. However, for weak coupling and low-frequency noise, where both \K(u)\1'2 and the characteristic width of $(u;) are small compared to CJD, the features of $(cj) are clearly reproduced in Q(UJ).


53

We note that (2.36) and (2.37) hold regardless of the inter-relation between the characteristic frequencies of the system and \K(u)\1/2 provided that the system-bath coupling is weak compared to the characteristic frequencies of the bath. At the same time, (2.36) is exact for a particular model of a harmonic oscillator and a bath composed of a set of harmonic oscillators with a couplingfunction E in (2.32) that is linear in the bath oscillator coordinates [30] (cf. [33], where Q(UJ] was given for the corresponding quantum problem). Thermal equilibrium fluctuations in systems described by (2.36) have been investigated for various forms of the potential U(q) and of the spectrum $(cj) (sometimes the "retarded" term in (2.36) is written in a form where q(r] is replaced by p(r) and the kernel is transformed accordingly: cf. [34] and references therein). Among the most recent results related to the color of the noise we mention the observation by means of analog electronic simulation [35] of the onset of an additional peak in the spectral density of the velocity fluctuations (i.e., of the fluctuations in q) in a system with a periodic cosine potential. The simulated system was described by (2.36) with a cosine potential U(q] and with exponentially correlated noise (cf. (2.6)), but the effects of the renorrnalization of the potential arid the mass related to the real part of K(UJ] were omitted. A double-peaked spectrum was observed (see Fig. 2.1), and the peaks were attributed to intrawell vibrations and to the interplay of motion over the barrier and the color of the noise. To understand this behavior, consider the special case of a harmonic oscillator described by (2.37). The spectral density of the velocity fluctuations is given by u}'2Q(u). If the spectrum of the noise is of the form $(u;) = AT I(I + u 2 t 2 ) and we neglect ReK(u) in (2.37), then u2Q(uj) exhibits a sharp peak at the oscillator frequency UJQ provided that UJQ 3> A/4(1 -fo; 2 t 2 ). On the other hand, for small u0 such that u0 2, the spectrum u}2Q((jj} shows a peak at the frequency u; ^ (yl/t 2 ) 1 / 3 . This peak only appears when the noise is colored and does not arise for white noise. The motion in a. cosine potential is characterized by a broad spectrum of eigenfrequencies. all the way down to zero, and therefore both of these peaks may coexist. The position of the color-induced peak in Fig. 2.1 is satisfactorily reproduced by a calculation using a flat potential U(q) — const, (equivalent to setting LJO = 0 in iJ2Q(uj}} with the appropriate noise parameter values [35]. The results of this section indicate that the statistical distribution of the fluctuations of a noise-driven system near its maxima and the spectral density of the fluctuations can be found explicitly for sufficiently weak colored noise. Explicit solutions can be found for systems driven ly noiithermal noise (with iinretarded relaxation), and also for equilibrium systems driven by thermal noise (whose relaxation is retarded). The shape of the distribution near a maximum is Gaussian in all cases, while the shape of the spectral density of the fluctuations is given by relatively simple expressions and is in general proportional to the power spectrum of the driving noise, with the coefficient

54


FIG. 2.1. Spectral density of fluctuations of the velocity of a system in "thermal equilibrium" at "temperature" T fluctuating in a cosine potential U(x) = —Acosx as measured in an analog experiment (the circles show the results of a digital simulation) [35]. The low-frequency peak is due to the color of the noise. The data are for exponentially correlated noise, with tc = 10/V^4, D = 2A\fA, T = A; the frequency is measured in units of the eigenfrequency of intrawell vibrations.

depending on the frequency and, in the case of thermal fluctuations, on the shape of the power spectrum of the noise. 2.3.

Large fluctuations: Method of the optimal path

The small fluctuations analyzed in §2.2 are the most probable fluctuations, and as such they determine the noise-induced "smearing" of the system about its stable positions. Another important problem for noise-driven systems is that of determining the probabilities of occasional large fluctuations. Large fluctuations result from "outbursts" of the driving noise which cause the system to move far from the stable states in phase space. These large fluctuations determine the shape of the tails of the statistical distribution of the system where the distribution is small. In this section we study the probabilities of large fluctuations for monostable systems driven by colored noise. To demonstrate the ideas and to focus sharply on those results that are specifically related to the color of the noise we concentrate on the simplest type of systems,


55

namely, those described by the equation of motion (2.7) with a single dynamical variable,

We shall call x a "spatial" variable. If the system is monostable, i.e., if the potential U(x] has only one minimum at x = xst [Uf(xst] = 0], its intrinsic motion is characterized by the relaxation time tr = l/U"(xst). The spatial scale of the most probable fluctuations of the system is given by the root-mean-square displacement Ax — (i1/2 about the stable position xst. According to (2.25), Ax is proportional to the square root of the noise intensity D (cf. the last paragraph in §2.2.2). On the other hand, the decay of correlations of the noise is characterized by the correlation time tc given by the reciprocal width of the narrowest peak (or dip) of the power spectrum t0) the system fluctuates mostly near the stable state. Then the system makes an "excursion" to x of characteristic duration t0. The successive excursions to x are thus statistically independent of one another, since the previous excursion has been forgotten by the time the next excursion occurs. Of course, in addition to the excursions to a given x, other excursions to remote points may be taking place. The duration of each of them is of order t 0 , since this is the only time that characterizes the correlation of fluctuations or the deterministic evolution of the system (whichever takes more time). The intervals between excursions to any extreme value that is sufficiently far from the stable state are also statistically independent of one another. It is evident from the "physical" notion of the probability p(x}dx as the relative length of time spent in a small vicinity dx around the point x (ergodic hypothesis) that p ( x ) / p ( x s t ) ~ t0/T(x). It is also clear from the above picture that T(x) might be called a mean first-passage

56

CONTEMPORAR PROBLEMS IN STATISTICAL PHYSICS

time (MFPT) to the point x from the regime of small fluctuations about the stable state. A typical trajectory x(t) of a noise-driven system that illustrates the arguments presented above is sketched in Fig. 2.2. One can single out a path (a portion of this trajectory) that arrives at a given x having started at some point within Ax ~ D1'2 of xst (it is not useful to specify the starting point of the motion to an accuracy sharper than the fluctuational smearing of xst). Each such path is noise-driven and therefore random. Furthermore, different paths that arrive at x are mutually independent. Therefore, these paths can themselves be described as random processes, and one can associate a probability density with the realization of a particular path x(t). This probability density is a functional. p[x(t}}, since the random quantity is itself a function and not a variable [36] (see also [37] and references therein). Since the point x lies far from the attractor. the probability density for the realization of any particular path that reaches x at time t is small. Furthermore, the probability densities for paths that on their way to x pass different points at a given time prior to reaching x differ considerably (exponentially for systems driven by Gaussian noise) if these points differ by an amount that substantially exceeds Ax. Therefore, one might expect that there is a group of paths that are close to one another (lying within a range Ax of one another) along which the system is most likely to move toward a given x at time t. One can further imagine that these paths surround an "optimal path" which represents the most probable path for arrival at .r at

FIG. 2.2. A sketch of the trajectory ,r(t) of a noise-driven system exhibiting a large occasional fluctuation.


57

time t. Recent experimental data on the visualization of an optimal path in a noise-driven system, and the related problem of the motion on the way to a given point (the pre-history problem) are considered in §2.5. As shown below, the probability density that the system is found in a state x (the statistical distribution p(x)} in a system driven by weak Gaussian noise decreases exponentially sharply with increasing separation x — xst . It is therefore important to be able to calculate p(x] at least to logarithmic accuracy, i.e., to find the leading terms in lnp(x). To do this it suffices to calculate only the probability density for the realization of the optimal path. The ideas underlying this calculation are outlined in the next subsection. 2.3.1. Variational problem for the optimal path. A convenient approach to the analysis of large fluctuations in noise-driven systems is based [33], [38] on Feynman's idea [36] of the direct inter-relation between the probability densities of the paths of the system and those of the noise (see also [39]- [45]). This inter-relation arises from the fact that each path of the noise results in an associated path of the dynamical variables; in particular, each path f ( i ) in (2.38) results in an associated path x(t}. As a consequence, the probability density for reaching a given point in the phase space of the system at a given instant is determined by the integral of the probability density functional for the noise over those noise trajectories that bring the system to that point at that instant. For a point that is remote from the stable state, the probability densities for all appropriate trajectories of the noise (and of the system) are very small and, as noted earlier (see also below), for Gaussian noise the probability densities differ exponentially for different trajectories. Therefore, the integral over the paths can be calculated by the method of steepest descent, and it is precisely the optimal path that corresponds to the extremal solution. The path-integral approach was applied recently to large fluctuations in colored-noise-driven systems in several papers in addition to those cited above [46], [47]. To find the stationary distribution p(x) of a monostable system it is convenient [38] to express it in terms of the transition probability density u>(z,0; xa, ta] for the transition from a point xa occupied at some instant ta < 0 to the point x at the instant t — 0. Since the initial state of the system and of the noise are forgotten over the time interval t0 given in (2.39), one has

The transition probability density w can in turn be written in terms of a path integral over the trajectories of the driving noise. For the system (2.38),

Here p[/(£)] is the probability density functional for the noise trajectories f ( t ) . Equation (2.41) expresses the fact that iy(x,0; x a , ta) is the integral over all

58


realizations of the noise f(i] which move the coordinate x(i) from the value xa at t — ta to x at t = 0. The weighting factor p [/(£)] gives the probability of a realization, and the denominator in (2.41) is simply a normalization factor. For Gaussian noise the probability density functional p[/(£)] is Gaussian [36], i.e., it is exponential in a bilinear functional of /(£). For white noise with $(0;) — D the form of this functional can be argued as follows. We begin by discretizing the noise f(t] into a sequence of values f ( t i ) where A = ti+\ — ti is very small (A —> 0). The quantities f ( t i ) are random numbers rather than random functions. If we carry out this discretization process for white zeromean Gaussian noise of intensity D such that (f(t)f(tf)) = D6(t — t'), we obtain (f(ti)f(tj)) = (D/A)fiij. The distribution function of the f ( t i ) (which is not a functional) to within a normalization factor is of the form

In the limit A —>• 0 the multivariable distribution function (2.42) goes over into the probability density functional

The path-integral formalism was used for white noise by Wiener [48]. When the noise driving the system is colored, the form of the exponent in the expression for fp[f(t)] is more complicated. It is important to note, however, that it can be expressed entirely in terms of the power spectrum $(u;) of the noise. To do this we observe that due to the stationarity of the noise, the Fourier components /w of the noise are ^-correlated in frequency,

The form of the probability density functional p[fu] for ^-correlated noise is similar to (2.43):

(cf. [36], [39], and [42]). In writing (2.45) we have taken into account that $(LJ) is an even function of a;, which in turn is a consequence of the fact that the time correlation function (£) of the noise is even in t. It follows from (2.44) can be written as and (2.45) that the probability density functional

In going from (2.45) to (2.46) we assumed that $~l(ui) can be expanded as a series in u; that converges for finite u. The operator F(—id/dt) is then selfadjoint within the class of functions f ( t ) that are sufficiently smooth and that


59

vanish as t —> ±00, and for which the integral j dt f(t)F(—id/dt)f(t] exists. The smoothness condition on f(t) imposes no physical restriction in practice since noise generated by real physical sources does not have singularities. The results considered below refer mostly to the case where $~1(o;) is a finite series in cj; in particular, for exponentially correlated noise as in (2.6), — oo express this fact to within an error proportional to Z)1/2 (reflecting the uncertainty in the position of the system near xst and of the noise near zero). This error can be neglected since all contributions to R(x] that vanish when D —->• 0 are ignored. The conditions (2.49) for t > 0 follow from the fact that the calculation of the statistical distribution p(x) to logarithmic accuracy does not require us to follow the further evolution of the system once it has reached the -given point x; therefore, the driving noise f ( t ) can be allowed to decay back toward zero for t > 0 "on its own" independent of x(t), and therefore one can set X(t) = 0 (this boundary condition has also been taken into account in (2.48) by setting the upper limit in the second integral in 3ft equal to zero). We note that X(t) can be discontinuous for t = 0. At the same time, f ( t ) itself and also several derivatives of f ( t ) are continuous (except for the case of white noise; see below). Because of this continuity, there arises a "postaction" : the decay of f ( i ] for


61

f > 0 does influence the behavior of f ( t ) for t < 0 even though it does not contribute explicitly to R(x}. An alternative procedure that accounts for the inter-relationship between f ( t ) and x ( t ) in the functional integral (2.41) is to multiply the integrand in the numerator of (2.41) by the functional (whose effect must be accommodated in the prefactor C(x) in (2.47))

Note that iT>z(i] is closely related to \(t) in (2.48). The Gaussian integral over T)f(t) can then easily be calculated using standard methods [36], and the statistical distribution p(x] to logarithmic accuracy is obtained by minimizing the remaining functional of ,r(f) and of the auxiliary variable z ( t ) . This approach was used by Liiciani and Verga [46]. In contrast to the differential variational equation for the functional 9R in (2.48) found by the first procedure and considered in the next subsection, the variational equations obtained from the approach of Luciani and Verga are integral equations. An advantage of the present formulation is that it deals with the functions /(/) and x ( t ] that correspond to physical observables. so that intuitive arguments can be used when seeking the solution for the optimal paths. The approach also allows one to formulate the boundary conditions needed to obtain the statistical distribution and the transition probabilities (see also §2.6) so that the substantial difference between the twro problems becomes obvious. Another advantageous feature of the approach is that the solutions can be immediately tested experimentally. Since $(u;) is proportional to D. the operator F ( — i d / d t ) = D/3>( — i d / d t } in (2.48) does not change with a rescaling of the noise intensity, and therefore the function R(x] is independent of D. The dependence of p ( x ) on D as given in (2.47) is thus of the activation type, and R ( x ] can be called an '"activation energy" for reaching a point .r. The concept of the activation energy is meaningful and the approximation (2.47) holds provided that

In this case, the distribution p ( x ) is exponentially small for a given .r and. as noted earlier, the average interval between successive outbursts of the noise that bring the system to a given x (the MFPT). T(x) ~ f ( ) exp[/?(.r)/D]. greatly exceeds both the relaxation time tr of the system and the correlation time t(. of the noise. 2.3.2. Variational equations and their analysis in limiting cases. The (deterministic) set of variational equations describing the optimal paths fopt(t] and xopt(t) follows from (2.48):

62


with

The set (2.52) and (2.53) with the boundary conditions (2.49) constitute a boundary-value problem. This problem can be solved numerically for an arbitrary system potential U(x] and for an arbitrary shape of the power spectrum $(u;) of the noise. A simple procedure can be followed when F(UJ) oc $~l(u) is a polynomial in a;2 of finite degree M. The procedure uses the fact that the function U'(x) is linear in x — xsi near the stable state xst . As a consequence, (2.52) and (2.53) are linear for t —> — oo where x is close to xst , and the solution for f ( t ) , A(£), and x(t) — xst for t —> — oo is a linear combination of the exponentials exp(/it) with Re/i > 0. The values of // are obtained from the secular equations

The solution contains M + I coefficients. They can be found from the condition that x(0) = x and from M relationships between /(t), d f / d t , . . . , d2M~lf/dt2M-1 for t = 0. To arrive at these relationships we first note that the function f ( t ) and its derivatives d f / d t , . . . , d2M~1f/dt2M~~1 are continuous at t = 0. This follows from the fact that (2.52) is a 2Mth-order differential equation for /(t), and the function X(t) on the right-hand side of this equation is seen from (2.53) to be continuous for t < 0 and for t > 0 (where A = 0). Thus, a discontinuity of A can only occur at t = 0. Hence, f ( t ) and its derivatives should be continuous for all t. On the other hand, the solution of (2.52) for f ( t > 0) where A = 0 is of the form

Because of the continuity of f ( t ) and its derivatives, the An, Bn in (2.56) are functions of the coefficients of the solution for t —> — oo, and it follows from the condition f ( t ) —> 0 for t —> oo that the M functions that would lead to divergence of f(t] vanish, i.e., that A\ = 0 , . . . , AM = 0. The solutions of the variational equations (2.52) and (2.53) can be obtained in explicit form for several limiting cases. The simplest case is that of fluctuations in a quadratic potential,

A Fourier transform of (2.52) and (2.53) over time yields


63

The statistical distribution p(x) obtained from (2.47) and (2.58)-(2.60) coincides with the result (2.30), (2.24) which was obtained in an entirely different way. 2.3.3. Activation energy for noise of small correlation time. Comparison to other approaches. Another limiting case for which R(x] can be obtained explicitly is that of "weakly colored" noise, i.e., noise whose characteristic correlation time tc is small compared to the relaxation time of the system, tr — \U"(xst}\~1. This is the limit in which t~l is much smaller than all characteristic frequencies of the power spectrum $(u;) of the noise, as illustrated in Fig. 2.3(a). One expects the system to be influenced primarily by low-frequency noise fluctuations, u ~ t"1, while high-frequency fluctuation are mostly filtered out because the system can not follow them (see, however, §2.4). Therefore, it is the low-frequency part of 3>(u;) that determines the main features of the fluctuations of the system. The limit where the finiteness of the correlation time of the noise can be neglected altogether corresponds to a white noise driver. In this limit, i.e., to zeroth order in t c /t r , the optimal path is described by the equations

We note that the optimal path xopt(t) as given by (2.62) is a "time-inverted" path of the system in free motion, i.e., in the absence of external noise, as described by (2.38) with f ( t ) — 0. We also note that for a white-noisedriven system the optimal path of the noise f0pt(t) is discontinuous, which is reasonable: since the noise is temporally uncorrelated, it can be assumed to vanish (i.e., it can be forced to achieve a root-mean-square value equal to zero within the optimal path method) immediately upon having "brought" the system to a given point. At first glance, one might expect the corrections to (2.62) and to the corresponding expression for R(x] due to a finite correlation time of the noise to be of order t^/t2, because F ( — i d / d t ] in (2.52) is a series in powers oft2d2/dt2. However, because of the discontinuity of the optimal path f0pt(t] at t = 0 there

64


FIG. 2.3. A sketch of the power spectrum <J?(u;) of colored noise with all correlation times being of the same order of magnitude. The spectral components of the noise that influence a noise-driven system most substantially are those with frequencies smaller than or of the order of the characteristic reciprocal relaxation time t~l of the system. The corresponding areas of3>(ijj} are shaded. It is evident that if the correlation time of the noise is small compared to tr (a) it is the shape ofQ(uj} for small cu that determines the fluctuations in the system, while for the large correlation times of the noise (b) nearly all Fourier components of the noise are involved in determining the fluctuations of the system.

appears a quickly varying contribution to fopt(t} for \t\ ~ tc that gives rise to a correction of order tc/tr to the activation energy R(x). One obtains from (2.52) and (2.53) [45]

where

and (fr(t) is the time correlation function of the noise as given in (2.2). Note that the ratio (f)(t)/D is independent of the noise intensity. The first term in


65

R(x] in (2.63) is the well-known result for systems driven by white noise and can be obtained directly from (2.62). It leads to a Boltzmann distribution p(x) in (2.47) [24] with an "effective temperature" of D/2F(0) = (1/2)$(0). This is in agreement with the observation (cf. Fig. 2.3(a)) that the effective intensity of the "white" noise is proportional to $(0) for \tc 0 since R(x] then exceeds its white-noise value. The latter occurs if tc < 0. An example of a power spectrum (o;) that leads to a negative finite-correlation-time-induced correction to R(x] is considered in §2.4. If the noise is exponentially correlated (cf. (2.6)), the correlation time tc in (2.64) is precisely the time tc that parametrizes the correlation function of the noise, and the correction to R(x] is then positive. Equation (2.47) with (2.63) in this case coincides with the result of others (see [49] and [50] and also [17], [51]-[53], and references therein). These results have been obtained by several methods. In particular, the approaches based on the derivation of "effective Fokker Planck equations" for the statistical distribution by various approximation procedures have been reviewed in detail by Lindenberg et al. [17]. The approach of [49] and [50] exploits the fact that the exponentially correlated noise /(t) itself, and the composite system [x(t), /(£)] consisting of the dynamical system of interest and the noise, can be viewed as Markov processes, with the equation for x(t) being of the form (2.38) and that for f ( t ) being of the form

where £(£) is Gaussian white noise. The evolution of the joint probability density p = p(x, /; t) of the variables x, / is described by the Fokker-Planck equation

For small noise intensities the eikonal approximation can be used to solve this equation. In particular, one can seek a stationary solution of the form [54]

(see also [49] and [50] and references therein). To lowest order in D the equation for S(x, /) that follows from (2.66) is a first-order nonlinear differential

66


equation of the form of a Hamilton-Jacob! equation,

so that S(x, /) can be associated with the mechanical action of an auxiliary dynamical system described by the variables x and /, the associated momenta dS/dx and dS/df ', and the Hamiltonian function H. In the general case of finite tc this equation cannot be solved analytically because of the lack of detailed balance [15] in the system. (We note that the eikonal approximation was applied in [55] and [56] when considering Markovian physical systems without detailed balance, in particular, the problem of the transitions between stable states of such systems [56]; in [38] a path-integral formulation was applied to this problem.) For small tc, however, the solution (2.68) can be obtained easily. To lowest order in tc it is of the form

By substituting (2.69) into (2.67) and integrating over / one arrives at the "Boltzmann" distribution p(x] oc exp[— 2U(x)/D]. The corrections to 5(x, /) that lead to the distribution (2.47) with R(x] given by (2.63) can be obtained by perturbation theory in tc [49], [50]. The method also makes it possible to find, again for exponentially correlated noise, the color-induced corrections not only to the exponent of the distribution (as in (2.63)) but also the terms of order tc in the prefactor. There is an immediate parallel between the path-integral formulation presented above as applied to the particular case of exponentially correlated noise, and the eikonal equation (2.68). To show this we first note that, for the extreme path, the integral /f^ dt f ( t ) F ( — i d / d t ) f ( t ) in the functional 3ft in (2.48) can be replaced by /^ dt f(t)F(-id/dt)f(t) since X(t) = F(-id/dt)f(t) vanishes for t > 0. Furthermore, the functional 3ft can be considered as a function of time £', 3ft[/(t), x(t); tf], if the upper limits in both integrals in 9ft are replaced by the running time t' . For exponentially correlated noise F(—id/dt) = 1 — t^d2/dt2 and the first term in (2.48) can be integrated by parts, so that the resulting functional can be written as

(Here in the integration by parts we have taken into account that for the optimal path f(tf) — — t ~ l f ( t ' } at the upper limit t' of the integral: cf. (2.52). (2.53), and (2.56).) The functional 3? in (2.70) can be viewed as a mechanical


67

action of a system with dynamical variables x, / and with Lagrangian L. The generalized moments px,p/ are of the standard form [57]

and the Hamiltonian function H(x,f,px,pf] is given by the Legendre trans-

form

The expression (2.73) for the Hamiltonian function obviously coincides with (2.68) provided that account is taken of the fact that px — dS/dx and Pf — dS/df [57]. The mechanical action S in (2.68) is just the minimum of the functional 5R in (2.70), and the equation H — 0 corresponds to the fact that this minimal value is independent of t' for given x. Thus, the path integral formulation and the eikonal approximation in the Fokker-Planck equation give identical results for the case of exponentially correlated noise. We note that there is yet another alternative way of reducing to a mechanical problem the calculation of the tails of the statistical distribution for systems driven by noise that is itself a Markov process and/or a component of one. The method is based [38] [43] on expressing the white noise that drives the noise that in turn drives the system only in terms of the dynamical variable of the system and thereby excluding the colored noise at this point, substituting this expression into the probability density functional (2.43) for the white noise [36] , and minimizing the resulting functional of the dynamical variable in the argument of the exponent. This program is very similar to that presented in §2.3.1. The advantage of the formulation in §2.3.1 (see also [45 ) is that it is based directly on the power spectrum of the noise that drives the system and is not limited to Markov processes. A further advantage is that it is straightforward to write the boundary conditions (see (2.49)). Yet another advantage, as mentioned earlier, is that the formulation presented here appeals to physical intuition and that therefore many peculiar features inherent to systems driven by colored noise can be understood in physical terms (in this context, see §2.4 below). Finally, we note that the analysis of the tails of the stationary distribution for certain types of Markov systems without detailed balance was performed in a different way in the mathematical paper of Ventzel and Freidliri [58] (see also [59]). 2.3.4. Statistical distribution for noise with large correlation time. The variational equations (2.52) and (2.53) can also be solved analytically when all correlation times of the noise greatly exceed the relaxation time of the system, i.e., tc 3> tr. Thus t~l exceeds all characteristic frequencies of the power spectrum 3>(u) of the noise, as shown in Fig. 2.3(b), so that

68


all Fourier components of the noise are "incorporated" in and affect the dynamics of the system. In this case one can physically picture the optimal fluctuation path as follows [60] (see also [61]): the noise f ( t ] varies with a slow characteristic increment/decrement time tc, and the system follows this variation adiabatically, i.e., the value of the coordinate x(t) is given by the expression

In other words, when f ( t ) varies slowly the system occupies the time-dependent minimum of the "adiabatic" potential U(x) — x f ( t ) , To calculate the activation energy R(x] to reach a given point x in the approximation (2.74) it is convenient to change from the differential equation (2.52) for f ( t ) to an integral equation that relates f(t] to \(t). Allowing for the inter-relations (2.2) and (2.46) between F(LJ) and the time correlation function (f>(t) of the noise, and also for the boundary conditions (2.49), one obtains from (2.52)

The activation energy R(x) can in turn be written as

The subsequent analysis depends on whether the function U"(x) is positive in the interval (xst,x), i.e., whether |t/'(:r)| increases monotonically as the coordinate moves away from the stable-state value xsi to a given x, or whether in this interval the potential U(x) has an inflection point xmfl where U"(x] changes sign. The adiabatic approximation (2.74) only holds in the former case, since U"(x] is a measure of the local reciprocal relaxation time around the minimum of the adiabatic potential U(x) — x f ( t ) , and the criterion for the slowness of the noise in (2.74) can therefore in general be expressed as

The evolution of the adiabatic potential with increasing f ( t ) for these two situations, namely, one in which U"(x] is positive throughout and one in which U"(x] changes sign, is shown, respectively, in Figs. 2.4(a) and (b). If (2.77) is in fact satisfied so that the adiabatic approximation holds, then the function X(t) is seen from (2.53) to vary (increase in absolute value) much more rapidly than f ( t ) for t < 0,

Bearing in mind that x(t — 0) = x one obtains from (2.74). (2.75), and (2.76)


69

FIG. 2.4. Evolution of the adiabatic potential U(x) — xf with increasing force f for "bare" potentials U(x) that (a) do not have and (b) do have an inflection point m l m l = 0. In the case (b) the initial minimum of the adiabatic potential x J . U"(x' f ) becomes more shallow with increasing f /U'(xin^1} and eventually disappears when f / U ' ( x " l f l ) > 1.

and

so that finally [45]

For exponentially correlated noise D/(0) = 2tc, and (2.81) then yields precisely the result obtained in [43], [46], and [60]. When the correlation time of the noise is large, it is possible not only to determine the argument of the exponential in the expression for the statistical distribution (as done above) but also the prefactor of the distribution. Because

70


of the adiabatic character of the response of the system, the probability p(x)dx for the dynamical variable of the system to lie in the interval dx around a given x is equal to the probability p(f)df for the noise to lie in the interval df about / = U'(x). Therefore,

For values of x relatively close to xst, where U'(x) ~ U"(xst}(x — xst), equation (2.82) agrees with equation (2.30) obtained for this range of x with a noise power spectrum of arbitrary shape, if one allows in (2.24) for the fact that U"(xst] substantially exceeds all characteristic frequencies of the noise. If the potential U(x) has an inflection point xtnfl so that U"(xm^1) = 0, then for points x lying close to xm-^, and for points x that lie on the opposite side of xmfl than does xst , the adiabatic approximation (2.74) does not hold since the local relaxation time of the system becomes very large where U"(x] is small. It is obvious from Fig. 2.4(b), however, that the probability of reaching the point xmfl and also the region beyond xin^ is simply determined by the probability of the force f ( t ) reaching the "critical" value U'(xinfl). Having been brought to the inflection point by a large outburst of noise, the system does not need strong additional forcing to move further; in effect, it moves further "on its own." The dominant term in R(x] can therefore be written as

(see [43], [46], [60], and [61] for exponentially correlated noise and [45] for the general case). It is obvious from (2.83) that the distribution beyond xmfl is flat, i.e., that R(x] in this region is independent of x. This flatness is apparent in the results of the numerical calculations of R(x) for exponentially correlated noise carried out by Bray et al. [43] for the case of a quartic bistable potential of the form

The function R(x] is plotted in Fig. 2.5 (we have used only the data for the region x < 0). The numerical data clearly demonstrate the evolution of the shape of R(x) with varying noise correlation time, from R(x) oc U(x] in the white-noise limit tc —> 0 to a function with a nearly flat section between xmfi _ _]_^y/3 anci x — o for large tc. The data were obtained by solving equations of the type (2.52) and (2.53). More precisely, instead of a system of equations consisting of a second-order equation for f ( t ) and first-order equations for X(t) and x(t), a fourth-order equation for x(t) was constructed (the force f ( t ) was excluded at the initial stage of the path integral formulation, as mentioned above) [43]. A self-similar solution of the form ofy(x) — x(i] was


71

FIG. 2.5. Change of the activation energy R(.i') to reach point x with changing correlation time t(. of the exponentially correlated driving noise [43]. The system is described by (2.38) with the quartic bistable potential (2.84). The data for the stable state x = —I are plotted. The curves are labeled by the value o f t c . The tc = 0 (white noise) curve corresponds to the Boltzmann distribution.

sought, so that the problem was reduced to a second-order equation for y(x}. We note that this approach, although very effective in the problem considered in [43], is not applicable in the general case of colored noise (in particular in the case of quasi-monochromatic noise considered in the next section), since x ( t ) takes on different values for given x for different t along the optimal path. The slowing-down of the motion of the system gives rise to corrections to the activation energy that are norianalytic in tc [43]. [45]. The order of magnitude of the corrections can be estimated by noting that in the vicinity of the point of inflection the equation of motion of the system is of the form

If the motion in this region lasts for a time ot Xf — xs , i.e.. for x between xst and xj the dispersionCT(X,XJ)does become large. It is evident from Fig. 2.9 that the expression (2.121) for a(x;x/) explains the experimental data, at least qualitatively. The discrepancy between theory and experiment is most likely related to the fact that the noise intensities investigated experimentally were not sufficiently small for the theory to be applicable; in particular, the width of the peak of ph is seen from Fig. 2.8 to be comparable to the distance between the singular points of the potential U(x] of the system, while in the theory the width of the peak has been assumed to be much smaller. Another important point to be addressed in the future is the interpretation of the data when Xf approaches a saddle point where a(x]Xf] diverges arid the present theory becomes inapplicable. It follows from the results of the present section that the formulation of the pre-history problem has made it possible to visualize optimal paths in noisedriven systems and to investigate their statistical distribution. Through this approach it has been possible to provide direct experimental verification of the fundamental concept of the optimal path. 2.6.

Probabilities of fluctuational transitions between coexisting stable states of noise-driven systems

It is a feature of many physical systems that they have two or more coexisting stable states. Among the many examples of such systems we mention interstitial atoms or molecules in solids that can occupy any elementary cell with equal probability [3], [4], active (lasers) and passive optically bistable and multistable devices (see [68] -[70] and references therein), a relativistic electron in a Penning trap that displays Instability when excited by cyclotron resonant radiation [71], and biased Josephson junctions with coexisting oscillatory and steady states [7], [12]. A feature common to systems with coexisting stable states is the possible occurrence of nuctuational transitions (switchings) between these states. Because of its broad importance and interest, the problem of the transition probabilities between stable states has been considered in a large number of papers (see the reviews [12]- [17]. [33]. and [59] and references therein): it was probably Kramers' paper [72] that most influenced the modern developments in this field. (We note that in spite of the apparent simplicity of the formulation, the complete solution of the Kramers problem of the escape of a white-noise-driven particle with one degree of freedom from a potential well has only been obtained recently (cf. [73] and references therein).) The effects of color of the driving noise are still a matter of vivid discussion, although some results and some concepts have already been well established. The physical concept of the probability \Y of a transition between stable states or. equivalent ly. of the probability of escape from a stable state, is based

88


on the very fact that this probability is much smaller than both the relaxation rate t~l of the system and the inverse correlation time t~l of the noise,

If the condition (2.123) is fulfilled, there are two clearly distinct time scales. Within a time ~ t0l the system placed initially somewhere in the range of attraction of a given stable state (attractor) in the phase space approaches this attractor with an overwhelming probability for sufficiently weak noise, and forgets the initial position. Simultaneously, the noise correlations decay, i.e., the initial state of both the system and the noise are forgotten. For a while the system fluctuates about the attractor. The escape from the attractor occurs as a result of a large occasional outburst of the noise which drives the system away from the region in phase space associated with the initially occupied attractor (drives the system to another attractor around which the system now fluctuates). The average frequency of such outbursts is given by W, and therefore the probability that such an outburst occurs over the time t0 is very small according to (2.123), thus leading to a self-consistent picture. Only if this description is valid can one appeal in a meaningful way to the notion of the probability of the transition (escape) from a given stable state; otherwise the transition probability would depend on the initial position of the system and/or the initial state of the noise, and one would arrive at a continuum of "transition probabilities" when considering a distribution of initial states. Such a continuum does not have much in common with an intuitively clear rate of the transition under consideration. Since a statistical distribution in the vicinity of a stable state is generated over a time of order t0 regardless of the initial state of the system, it is obvious from the above picture that the results for the distribution of noise-driven systems considered in §§2.2-2.5 hold not only for monostable, but also for bistable and multistable systems. However, in these latter cases they yield not a stationary but a quasi-stationary statistical distribution for the population around an attractor; its integral over the corresponding region of the phase space (the total population around the attractor) slowly evolves in time because of transitions away from the attractor. The criterion (2.123) places a restriction on the intensity of the noise: only for sufficiently weak noise is the concept of a transition probability sensible. For example, if a system is fluctuating in a double-well potential (cf. Fig. 2.10) and the noise is so strong that motion over the barrier is strongly excited, the concept of transitions between potential wells is obviously meaningless because the system is in fact not located in any single well. It was demonstrated earlier that, for Gaussian noise, the probabilities of the large outbursts, including those necessary for a transition to occur when the noise is weak, are exponentially small. As before, it is therefore most interesting to calculate these probabilities to logarithmic accuracy in the noise intensity. The results are presented below.


FIG. 2.10. Sketch of a bistable potential U(x}; positions, xs is the saddle point.

89

are. the stable equilibrium

2.6.1. Method of optimal path in the problem of fluctuational transitions. As in the case of the rare fluctuations that determine the tails of the statistical distribution, the probabilities of different fluctuations (realizations of the paths of noise and of the corresponding paths of the system) that result in transitions between stable states differ exponentially strongly from each other when the noise is weak. Therefore, to logarithmic accuracy, the value of the probability Wij of a transition from the iih to the jth stable state is given by the probability of realization of the most probable (optimal) appropriate path of the noise and, correspondingly, of the optimal path of the system. We shall consider Wlj for the simplest case when a system is described by one dynamical variable x(t) and the equation of motion is of the form (2.38). As always, f ( i ] is zero-mean Gaussian noise with the power spectrum (2.2) and the probability density functional (2.46). We assume the potential U(x) of the system to be a double well (see Fig. 2.10); the stable states of the system are positioned at the minima of the potential, x\ and £2, and the local maximum of U ( x ] , x s , is the saddle point. The difference between the problem of the tails of the statistical distribution and the problem of the transition probabilities lies in the following. In the former problem the further destiny of the system after its arrival at a given point in the phase space as a result of the large fluctuation was not of interest. The force f ( t ] did not vanish at the moment t of arrival (t =• 0 in the variational functional (2.48)), and in the course of its decay for t > t this force drove the system back toward the stable state occupied initially. In the case of

90


the optimal path resulting in a transition, on the other hand, the system that was initially in one stable state (the state z) is to be in another stable state (the state j ] when the optimal fluctuation of the noise has died out and f ( i ) and its derivatives have become zero (to within their root-mean-square values proportional to D1'2). Therefore, to logarithmic accuracy we write

where, for the system described by (2.38), R{ is the solution of the variational problem

Equations (2.124) and (2.125) are similar to (2.47) and (2.48), and just as in (2.48) the minimum here should be taken with respect to f ( t ) and x ( t ) independently; X(t) is an undetermined Lagrange coefficient. However, in contrast to (2.48), the upper limit in the second integral is not the instant of arrival at a given point but +00: as explained above, it is necessary to know in the present problem where the system is when f(t] has decayed to zero, i.e., for t —> -j-oc. Obviously, by that time the system can be not only in the stable state j but at any point in the range of attraction of the state j, including the boundary point xs (the probability of a transition from xs to Xj caused by the small fluctuations is ~ 1/2). The point xs is precisely the point where the optimal path of the system should end in the problem of the transition probability. Indeed, this is a stationary point, U'(xs) = 0, and thus there occurs a slowing down of the motion of the system for f(i] = 0, i.e., the conditions of approachig xs and of vanishing of f ( t ) and its derivatives are fulfilled self-consistently for t —> +00, x —> xs. Self-consistency requires that A(t) also vanish for t —> oo, x —>• xs; this follows from the variational equation \(t) — U"(x)X(t) (cf. (2.52) and (2.53)) and from the fact that U"(xs) < 0. Therefore, the boundary conditions for the variational problem (2.125) for the transition from the /th stable state are of the form

The boundary conditions are obviously of great importance in the problem of the transition probability. It is an advantage of the present pathintegral formulation based on physically clear concepts that it facilitates the formulation of boundary conditions. We note that in the important case of noise f ( t ) that is a component of an TV-component Markov process (this occurs when F(LU) oc &~I(LJ) is a polynomial of degree N in a;2 and includes the case


91

of exponentially correlated noise), the conditions (2.126) can be obtained in a different way [45]. In this case one can consider fluctuational transitions between stable states of an (N + l)-component Markov system (the (N + l)st component is the dynamical variable x(t) itself) driven by white noise. To cause a transition, this noise must bring the system to a hypersurface that separates the ranges of attraction of two different stable states; the system will then go on to the other stable state from that originally occupied with a probability ~ 1/2. If we are interested in calculating the transition probability to logarithmic accuracy we must optimize not only over the paths of the noise but also over the final point of the system on the separating hypersurface [38]. It is precisely the point / = / = • • • = f ( N ~ 1 ^ = 0, x — xs (the saddle point of the multidimensional process) that gives the maximum probability, in complete agreement with (2.126). The variational equations for the optimal paths fopt(t} of the noise and %opt(t) of the system that follow from (2.125) are of the form (2.52) and (2.53). It is a general feature of systems driven by colored noise, however, that the solutions for the problem of the tails of the statistical distribution and for that of the transition probability are quite different: it does not follow from the fact that the system has reached a saddle point that it will then go to a different stable state with probability ~ 1/2. In fact, in the general case (an example is given in the next subsection) it will come back to the initially occupied state with overwhelming probability, i.e., Here R(xK]X^ is the activation energy (2.48) for reaching the point xs if the stable state i was occupied initially. The inequality (2.127) shows that the mean first-passage time to the point xs does not give the reciprocal transition probability - the latter is in general exponentially larger than the former. 2.6.2. Transition probabilities for particular types of noise.

The

activation energy 7?,7 for the transition from the iih stable state can be evaluated in explicit form in some limiting cases. The simplest case is that of a short correlation time tc of the noise, such that the bandwidth of the noise substantially exceeds the reciprocal relaxation time of the system (see Fig. 2.3(a)). To zeroth order in t c , i.e., in the white-noise limit, the solution of the variational equations for /(t), x ( t ] , A(t) is of the form (2.62). The colorinduced correction can be obtained from (2.125) by noting that the operator F is a series in t^cP/dt2, to find the lowest-order correction in tc it suffices to allow for the linear term in this series in (2.125), while keeping for f ( t ) the corresponding zero-tc approximation (2.62). (This is a standard trick in the perturbation theory for variational problems [57] which does not work, however, for corrections to the statistical distribution that are nonanalytic in t2r\ see §2.3.) The result is of the form [45]

92


where

The sign of the color-induced correction coincides with that of F"(0). This means that if the power spectrum 4>(o;) of the noise has a maximum at u = 0, then the correction is positive, i.e., the color causes the transition probability to decrease, while in the opposite case (as, for example, when noise is produced by a harmonic oscillator described by (2.88) with u% > 2F2 that filters white noise) the transition probability exceeds its white-noise-limit value. We note that the change of the transition probability due to the noise color is exponentially strong when the correction to Ri, although small compared to the main term, is nevertheless large compared to D. In the particular case of exponentially correlated noise (2.6), F"(0)/F(0) = 2t2, and (2.128) goes over into the result of Klosek-Dygas et al. [49] obtained by seeking the solution of the appropriate Fokker-Planck equation (2.66) (or its adjoint equation for the mean firstpassage time) in the eikonal approximation; not only the argument of the exponential but also a prefactor in the expression for the transition probability were obtained in [49]. A systematic analysis of the color-induced corrections to transition probabilities for exponentially correlated noise was given by Bray et al. [39], [43] using a path-integral formulation somewhat different fromhe present one (see §2.3); by making use of an instanton technique it was also possible to obtain a prefactor within this formulation for tc (o;) of the noise is small compared to t~l (see Fig. 2.3(b)), as explained in §2.3 the system follows the noise adiabatically and occupis a minimum of the adiabatic potential U(x) — x f ( t ) for f(t}/U'(xmf1} < I (xinfl is the inflection point of U(x)); see Fig. 2.4. When f ( t ) = U'(xinfl) this minimum transforms into the inflection point. For even larger f(t)/U'(xmfl) the special character of this point disappears, and the system "rolls down" to another stable state. Therefore, the transition probability to logarithmic accuracy is equal to the probability for f(t] to take on the value U'(xmfl) (these arguments were given by Tsironis and Grigolini [60] specifically for exponentially correlated noise, but they certainly hold for other types of noise as well), i.e.,

(see Dykman [45]; the corresponding expression for exponentially correlated noise was first obtained by Luciani and Verga [46]). As explained in §2.3, the correction to (2.130) is nonanalytic in tr/tc; it was obtained for exponentially correlated noise by Bray et al. [43] and is of the form


93

We note that the onset of substantial corrections to (2.130) related to the slowing down of the motion of the system near the minimum of the adiabatic potential when /(t) approaches U'(xin^1} was noticed in [61]. The numerical results obtained by Bray et al. [43] for the activation energy of a transition of a system with a symmetric double-well potential of the form (2.84) driven by exponentially correlated noise are shown in Fig. 2.11. As might be expected upon inspection of (2.128)-(2.131), although the activation energy as a function of the correlation time of the noise, Ri(tc~), increases monotonically with increasing tc, the first derivative R^tc] is nonmonotonic. To make the features of Rz(tc) more evident, the renormalized quantity [43] proportional to [Ri(tc) — Ri(0)]/tc has been plotted in Fig. 2.11. It has a maximum at log(t c /t r ) ~ 1.1. These results and the nonmonotonicity of [Ri(t(.) — Ri(0}]/tc in particular have been confirmed quantitatively in detailed Monte Carlo simulations in [751.

FIG. 2.11. The activation energy R = RI = R^ for the transitions between stable states of a symmetrical system with the potential U(x) = — (l/2)x 2 + (l/4),x 4 as a function of the correlation time t(, of the exponentially correlated driving noise [43]. The asymptotes given by the expressions (2.128) (the small tc limit), and (2.130), (2.131) (the large tc limit) are shown dashed. The value of R° is that of R for tc = 0. while R°° is given by (2.130), Rx — 4£ c /27, for the particular type of noise and the potential U(x) considered in [43]. 2.6.3. Quasi-monochromatic noise. The features of the escape from a stable state related to the color of the driving noise are even more distinct when the noise has "true color," i.e., when its power spectrum 3>(u;) contains a narrow peak at a finite frequency. We illustrate these features by considering as an example the QMN considered in §2.6. The shape of the power spectrum of this noise is given by (2.89), and we assume that the position LJO of the

94

CO NTE MPORARY [PROB LEMS IN S TA TISTICAL PHYSIC S

maximum of the spectral peak, the halfwidth F of the peak, and the relaxation time tr of the system satisfy the inequality (2.92), T £, the infinite cluster can be regarded as homogeneous. Since the correlation length is the only relevant length scale we expect that, similar to

120


the static properties, the transport properties can also be described by simple scaling laws. We begin by discussing diffusion (a) only on the infinite cluster and (b) in the whole percolation system, where finite clusters are also present, and relate the diffusion exponents to the conductivity exponents. (a) Diffusion on the infinite cluster. The long-time behavior of the mean square displacement of a random walker on the infinite percolation cluster is characterized by the diffusion constant D. Let us denote by D' the diffusion constant of the whole percolation system. It can be easily related to D. The dc conductivity of the percolation system increases above pc as u ~ (p — pc}^ (see (3.16)). Thus, due to the Einstein equation (3.24), the diffusion constant D' must also increase this way. The mean square displacement, and therefore D', is obtained by averaging over all possible starting points of a particle in the percolation system. It is clear that only those particles which start on the infinite cluster can travel from one side of the system to the other and thus contribute to D'. Particles that start on a finite cluster cannot leave this cluster, and thus do not contribute to D''. Hence D' is related to D by D' = DPoc, implying

Combining (3.21) and (3.27), the mean square displacement on the infinite cluster can be written as [84]-[86]

where

describes the time scale the random walker needs, on the average, to explore the fractal regime in the cluster. Since £ ~ (p — pc}~" is the only length scale here, it follows that t^ is the only time scale, and we can combine the short-time regime and the long-time regime by a scaling function /(£/^),

To satisfy (3.28), we require f ( x ) ~ x° for x 1. The first relation trivially satisfies (3.28). The second relation gives D = limtôc (r2(t))/2dt - t\/dw~l, which together with (3.27) and (3.29) yields a relation between dw and JJL [84]-[86]:

Comparing (3.25) and (3.31) we identify the relation

PERCOLATION

121

(b) Diffusion in the percolation system. To calculate (r 2 (t)) for random walks in the percolation system [84], [85] (which consists of all clusters), one must average over all starting points of the walkers that are uniformly distributed over all occupied sites. To this end, we average first over all random walks that start on clusters of fixed size s, and thus obtain the mean square displacement (r](t}} of a random walker on an s-site cluster. Then we average (r|(t)} over all cluster sizes using the cluster size distribution n s (p), which at p = PC-, is described by the power law ns(p) ~ s~T (see, e.g, [2]). The mean radius Rs of all clusters of s sites is related to s by s ~ Rsf. For short times the random walkers travel a distance smaller than Rs, diffusion is anomalous, and (ri(t)} ~ t2/dw. por verv long times, however, since the random walker cannot escape the s-cluster, (ri(t)} is bounded by R2. Hence we can write

From (rf (t)} we obtain the total mean square displacement (r 2 (i)) by averaging over all clusters,

According to (3.33), there exists, for every fixed time t, a crossover cluster size S*(t) ~ Rd,f ~ tdf/d«>: For s < 5 x (t), (ri(t)> - R2S, while for s > S'x(t), (r1(t)) ~ i 2 / d «'. Accordingly, (3.34) can be written as

The first term in (3.35) is proportional to [Sx (t)] 2 "" r+2 / d / and the second term is proportional to [Sx(t}}2~Tt2/dw . Since Sx(t) ~ t d // d u i , both terms scale the same and we obtain

with the effective exponent d'w = 2/[(df/dw)(2 — T -\- 2/d/)]. Using the scaling relations r = 1 + d/d/ and df = d - /3/zx we find [84]- [86]

Note that d'u, > dw since the finite clusters slow down the motion of the random walkers compared to those on the infinite cluster. The probability of a random walker to be at the origin at time t can be calculated in the same way, starting from the expression for finite s-clusters and performing the

122


average over all clusters. This procedure is described also in §3.5 where the fracton density of states, which is closely related to (P(0, £)}, is discussed. In the percolation system where particles diffuse on the finite clusters, (-P(0, £)) tends to a constant at large times. The leading time-dependent correction is

where d's — 2d/dw. Equation (3.38) is derived using similar arguments leading to (3.36). The constant (P(0, oo)} is due to the lower limit in the analog of (3.35). The above result, (3.36), obtained at pc can be generalized to p > pc. As in (3.30), we assume that (r2(t)) can be written as

where g(x) ~ x° for x ™ for x ^> 1. The first relation satisfies (3.36); the second relation yields D' ~ Iimt-ôo(r2(t))/t ~ t^ ~ (P ~~ Pc)î in agreement with the result for the dc conductivity, (3.16) and (3.24). It is important to note, however, that according to the above derivation, d'w represents the exponent characterizing the second moment of the distribution function. Other moments (rk(t)) ~ £ fc / d w( fc ) can be calculated in a similar way. It is easy to show that

Hence different moments are characterized by different exponents d'w . This is in contrast to diffusion in the infinite percolation cluster where dw(k) = dw and does not depend on the moment k. (c) Conductivity in the percolation system. Next we consider the total conductance, S(p, L) = /9"1, of a random insulator-conductor mixture of size Ld and concentration p of conductors and (1 — p) of insulators, above the critical concentration pc. On length scales larger than the correlation length £ the system is homogeneous and (3.16) and (3.17) hold, while on length scales smaller than £ the clusters are fractals and (3.18) holds. Again, since £ is the only length scale here, we can satisfy both regimes by the scaling ansatz (see, e.g, [2])

Here F(x] ~ x° for x 1 we require F(x) ~ X& and the relation

PERCOLATION

123

to satisfy (3.16) and (3.18). Equations (3.41) and (3.42) can be used to determine n from measurements of the total conductivity as a function of L close to pc. (d) Rigorous bounds. In percolation dw and £ cannot be calculated exactly, but upper and lower bounds can be derived, which are very close to each other in d > 3 dimensions. The resistance between any two sites on the infinite percolation cluster at criticality is given by the resistance of the backbone connecting these sites. This is because the backbone is the only part of the cluster on which the current flows. The backbone can be viewed as a chain consisting of blobs connected by red bonds [70] (see Fig. 3.9). An upper bound for the resistance can be obtained by assuming that the effect of loops can be neglected and each blob is replaced by a single shortest path. A lower bound can be obtained by assuming that the resistance of the blobs in the backbone can be neglected. The reason for this is that cutting the loops increases the resistance, and taking the blob resistance as zero decreases the resistance. Thus the values derived for £ and dw when neglecting loops or blobs can serve as upper or lower bounds, respectively [87]. In a loopless cluster, there exists only one path of length i between two sites, and the resistance p between these sites is proportional to the chemical distance i between them. Since the chemical distance t scales with the Euclidean distance r as r ~ 1D we obtain

where dm[n is the fractal dimension of the minimum path. Hence the static exponent v characterizes the dynamical properties when loops can be neglected. Assuming that the blobs have zero resistance, the total resistance is simply proportional to the number of red bonds, nreci, between both sites. Since r 1 /^ [63], [64], one has

Thus, the bounds for £ are

From (3.23) follow bounds for dw and d s ,

The Alexander and Orbach [78] conjecture that ds = 4/3 or dw — 3d//2 for 2 < d < 6 is a good estimate for percolation.

124


(e) Probability density. On fractals, the distribution function (P(r, £)), averaged over all starting points on the fractal, is no longer Gaussian [41]. Following arguments similar to those used by Fisher [88], [89] and Domb [90] (see also de Gennes [91]) to describe the distribution of self- avoiding random walks, one obtains that (P(r,t)) for r/(r2}1/2 >> 1 is described by a stretched Gaussian [41], [88]-[90]

where the exponent u is related to dw by

Numerical data supporting (3.47) are shown in Fig. 3.12.

FIG. 3.12. Plot of rP(r, t) as a function ofr/tl'dw for different values of r and t. The solid line represents the theory of equation (3.47). For r/{r2)1/2

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