Chaotic dynamics and transport in classical and quantum systems

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Chaotic Dynamics and Transport in Classical and Quantum Systems

NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I. II. III. IV. V.

Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences Science and Technology Policy

IOS Press Kluwer Academic Publishers IOS Press Kluwer Academic Publishers IOS Press

The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of meeting are supported from time to time. The NATO Science Series collects together the results of these meetings. The meetings are co-organized bij scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above. Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series. http://www.nato.int/science http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm

Series II: Mathematics, Physics and Chemistry – Vol. 182

Chaotic Dynamics and Transport in Classical and Quantum Systems edited by

P. Collet Ecole Polytechnique, Paris, France

M. Courbage Université Paris 7-Denis Diderot, France

S. Métens Université Paris 7-Denis Diderot, France

A. Neishtadt Space Research Institute, Moscow, Russia and

G. Zaslavsky New-York University, New York, NY, U.S.A.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW

eBook ISBN: Print ISBN:

1-4020-2947-0 1-4020-2946-2

©2005 Springer Science + Business Media, Inc. Print ©2005 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at:

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Contents Preface

vii

Participants

ix

Part I : Theory P. Collet : A SHORT ERGODIC THEORY REFRESHER

1

M. Courbage: NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT 15 Valentin Affraimovich, Lev Glebsky: COMPLEXITY, FRACTAL DIMENSIONS AND TOPOLOGICAL ENTROPY IN DYNAMICAL SYSTEMS

35

G.M. Zaslavsky, V. Afraimovich: WORKING WITH COMPLEXITY FUNCTIONS

73

Giovanni Gallavotti: SRB DISTRIBUTION FOR ANOSOV MAPS

87

P. Gaspard : DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY

107

Walter T. Strunz: ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS

159

Eli Shlizerman, Vered Rom Kedar: ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. 189 Monique Combescure: PHASE-SPACE SEMICLASSICAL ANALYSIS. AROUND SEMICLASSICAL TRACE FORMULAE 225

Part II : Applications

Ariel Kaplan, Mikkel Andersen, Nir Friedman and Nir Davidson: ATOM-OPTICS BILLIARDS

239

Fereydoon Family, C. Miguel Arizmendi, Hilda A. Larrondo: CONTROL OF CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS

269

v

vi F. Bardou: FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE MEAN WAITING TIMES

281

Xavier Leoncini, Olivier Agullo, Sadruddin Benkadda, George M. Zaslavsky: ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA TURBULENCE

303

Edward Ott,Paul So,Ernest Barreto,Thomas Antonsen: THE ONSET OF SYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC AND PERIODIC DYNAMICAL UNITS

321

A. Iomin, G.M. Zaslavsky: QUANTUM BREAKING TIME FOR CHAOTIC SYSTEMS WITH PHASE SPACE STRUCTURES

333

S.V.Prants: HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM ELECTRODYNAMICS 349 M. Cencini,D. Vergni, A. Vulpiani: INERT AND REACTING TRANSPORT

365

Michael A. Zaks: ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF VISCOUS FLUIDS 401 J. Le Sommer, V. Zeitlin: TRACER TRANSPORT DURING THE GEOSTROPHIC ADJUSTMENT IN THE EQUATORIAL OCEAN 413 Antonio Ponno: THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC LIMIT 431

Lectures

441

Preface

From the 18th to the 30th August 2003 , a NATO Advanced Study Institute (ASI) was held in Cargèse, Corsica, France. Cargèse is a nice small village situated by the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese provides ∗ a traditional place to organize Theoretical Physics Summer Schools and Workshops in a closed and well equiped place.The ASI was an International Summer School* on "Chaotic Dynamics and Transport in Classical and Quantum Systems". The main goal of the school was to develop the mutual interaction between Physics and Mathematics concerning statistical properties of classical and quantum dynamical systems. Various experimental and numerical observations have shown new phenomena of chaotic and anomalous transport, fractal structures, chaos in physics accelerators and in cooled atoms inside atom-optics billiards, space-time chaos, fluctuations far from equilibrium, quantum decoherence etc. New theoretical methods have been developed in order to modelize and to understand these phenomena (volume preserving and ergodic dynamical systems, non-equilibrium statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum dissipative processes etc). The school gathered a team of specialists from several horizons lecturing and discussing on the achievements, perspectives and open problems (both fundamental and applied). The school, aimed at the postdoctoral level scientists, non excluding PhD students and senior scientists, provided lectures devoted to the following topics : Statistical properties of Dynamics and Ergodic Theory Chaos in Smooth and Hamiltonian Dynamical Systems Anomalous transport, fluctuations and strange kinetics Quantum Chaos and Quantum decoherence Lagrangian turbulence and fluid flows Particle accelerators and solar systems More than 70 lecturers and students from 17 countries have participated to the ASl. The school has provided optimal conditions to stimulate contacts between young and senior scientists. All of the young scientists have also received the opportunity to present their works and to discuss them with the lecturers during two posters sessions that were organized during the School. The proceedings are divided into two parts as follows: I. Theory This part contains the lectures given on basic concepts and tools of modern dynamical systems theory and their physical implications. Concepts of ergodicity and mixing, complexity and entropy functions, SRB measures, fractal dimensions and bifurcations in hamiltonian systems have been thoroughly developed. Then,

∗

http://www.ccr.jussieu.fr/lptmc/Cargese/CargeseMainPage.htm

vii

viii models of dynamical evolutions of transport processes in classical and quantum systems have been largly explained. II. Applications In this part, many specific applications in physical systems have been presented. It concerns transport in fluids, plasmas and reacting media. On the other hand, new experiments of cold optically trapped atoms and electrodynamics cavity have been thoroughly presented. Finally, several papers bears on synchronism and control of chaos. We also provide most recent references of the other given lectures at the school. These lecture notes represent, in our views, the vitality and the diversity of the research on Chaos and Physics, both on fundamental f and applied levels, and we hope that this summer-school will be followed by similar meetings. The summer-school was mainly supported by NATO and the staff of the University Paris 7. M. Courbage was a coordinator off the Summer-School, G. Zaslavsky was a director of the NATO-ASI and A. Neishtadt was a co-director. We would like to thank all the institutions who provided support and encouragements, namely, NATO ASI programm, the European Science Foundation through Prodyn programm, the Collectivités Territoriale Corse, the Laboratoire de Physique Théorique de La Matière Condensée (LPTMC) and the Présidence of the University Paris 7. Thanks also to the Centre de Physique Théorique de l'Ecole Polytechnique de Paris. The meeting was an occasion for a warm interactive atmosphere beside the scientific exchanges. We want to thank those who contributed to its success: the director and the staff of the Institut d'Etudes Scientifiques de Cargèse and, the team of the Université Paris 7, especially Evelyne Authier, Secretary of the LPTMC, who provided essential help to organize this ASI. P . Collet, M. Courbage, S. Metens, A. Neishtadt, G.W. Zaslavsky

PARTICIPANTS Valentin Afraimovich IICO UASLP. Communication Optica. Universitet Autonomny de San Luis. Karakorum 1470, Iomas, 4eme section. CP78200 San Luis Potosi. SLP Mexico [email protected] Omar Al Hammal Institute “Carlos I” for Theoretical and Computational Physics. Universidad de Granada Avda. Fuente Nueva, s/n, 1 18071 Granada Spain [email protected] Athanasios Arvanitidis Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Geece [email protected] Francois Bardou IPCMS-CNRS, 23, rue du Loess, BP 43 67037 Strasbourg Cedex 2 France [email protected] Maxim Barkov Space Plasma Physics-Space Research InstituteProfsoyuznaya str. 84/32 117997 Moscow Russie [email protected] Jacopo Bellazzini Universita’ di Pisa - Dipartimento Ingegneria AerospazialeVia Caruso 120 56100 Pisa Italie [email protected] Viatcheslav Belyi Theoretical department-Russian Academy of Sciences IZMIRAN 142190 Troitsk, Russia [email protected] Cristel Chandre CPT -- CNRS-Centre de Physique Théorique Campus de Luminy - case 907 13288 Marseille, France [email protected] Hugues Chaté SPEC - CEA - Saclay, 91191 Gif-sur-Yvette, France [email protected] Guido Ciraolo Centre de Physique Theorique,CNRS case 907 13288 Marseille cedex 9, France [email protected] Steliana Codreanu Department of Theoretical Physics-Babes-Bolyai University Kogalniceanu str. 1 2400 CLUJ-NAPOCA Roumanie [email protected] Pierre Collet Centre de physique theorique-CNRS Ecole Polytechnique, route de Saclay 91128 Palaiseau cedex France [email protected] Pieter Collins Control and computation-CWIKruislaan 413 1098 SJ Amsterdam Nederland [email protected]

ix

x Monique Combescure Institut de Physique Nucléaire de Lyon, CNRS-Bât Paul Dirac, 4 rue Enrico Fermi 69622 VILLEURBANNE France [email protected] Maurice Courbage LPTMC case 7020-Université de Paris 7, 2 place Jussieu 75231 Paris Cedex 05 France [email protected] Giampaolo Cristadoro Center for Nonlinear and Complex Systems and Dipartimento di Scienze Chimiche, Fisiche e MatematicheUniversita’ dell’Insubria (sede di Como)-via Valleggio,11 22100 Como Italie [email protected] Nir Davidson Dept. of Physics of Complex Systems-Weizmann Institute of Science Rehovot 76100 Israel [email protected] Filippo De Lillo Dipartimento di Fisica Generale-University of TorinoVia Giuria,1 10125 Torino Italie [email protected] Jean-Pierre Eckmann Departement de Physique Theorique and Section de Mathematiques Universite de Geneve-32, Bld D’Yvoy 1211 Geneva 4 Suisse [email protected] Massimiliano Esposito Service de Chimie Physique CP 231-Universite Libre de Bruxelles Boulevard du Triomphe B-1050 Bruxelles Belgique [email protected] Fereydoon Family Physics Department-Emory UniversityPhysics Department, Emory University GA Atlanta USA [email protected] Stefano Galatolo Dipartimento di matematica applicata-Universita di PisaVia Bonanno Pisano 25b 56126 Pisa Italie [email protected] Govanni Gallavotti Fisica -Univ. Roma 1P.le Moro 2 00185 Roma Italie [email protected] Pierre Gaspard Center for Nonlinear Phenomena and Complex SystemsCampus Plaine, CP 231 B-1050 Brussels Belgique [email protected] Alessandro Giuliani Physics department-Universita’ di Roma “La Sapienza”- Via Ivanoe Bonomi, 92, 00139 Roma, Italy [email protected] Vasiliy Govorukhin Rostov State University - Computational mathematics Zorge str. 5 344090 Rostov-on-Don Russie [email protected]

xi Seiichiro Honjo University of Tokyo, Graduate School of Arts and Sciences-Department of Basic Science, Kaneko LaboratoryKomaba 3-8-1, Meguro-ku 153-8902 Tokyo Japon [email protected] Alexander Iomin Department of Physics, Technion 32000 Haifa Israel [email protected] Alexander Itin Lab. 627 Space Research Institute-Profsoyuznaya str. 84/32 117997 Moscow Russia [email protected] Brunon Kaminski Faculty of Mathematics and Informatics Nicholas Copernicus University, ul Chopina 12/18, 87-100 Torun, POLAND [email protected] Janina Kotus Warsaw Univ.of Technology-Department of MathematicsPlac Politechniki 1 00-661 Warsaw Pologne [email protected] Alexandra Landsman Princeton University, 235 Thunder Circle PA 19020 Ben Salem USA [email protected] Jacques Laskar Astronomie et Systemes Dynamiques-IMC77, Av. Denfert-Rochereau F-75014 PARIS France [email protected] Xavier Leoncini PIIM-Université de Provence, Centre de St Jerome, case 321 13396 Marseille Cedex 20 France [email protected] Fabio Lepreti Section of Astrophysics, Astronomy, and Mechanics-Department of Physics, Aristotle University of ThessalonikiDepartment of Physics, Aristotle University of Thessaloniki 54124 Thessaloniki Grèce [email protected] Emanuel Lima Instituto de Física de São Carlos-Universidade de São PauloMajor Júlio Salles 870, Vila Pureza, 13561-010 16 São Carlos Brazil [email protected] Helen Makarenko Department of Physics-V.Karazin National University4 Svobody square 61077 Kharkiv Ukraine [email protected] Amar Makhlouf Université de Annaba-Labo de Mathématiques, 14 Rue Zighoud Youcef, DREAN 36 ELTARF, ALGERIE [email protected] Miguel Manna Physique Mathematique et Theorique CNRS-UMR5825-Universite Montpellier 2Place Eugene Bataillon 34095 Montpellier, France [email protected]

xii Paul Manneville Laboratoire d’Hydrodynamique-CNRS-Ecole PolytechniqueEcole polytechnique 91128 Palaiseau, France [email protected] Stéphane Métens LPTMC, case 7020-Université Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05, France [email protected] Stavros Muronidis Dept. of MathematicsAristotle University of Thessaloniki 54006 Thessaloniki, Grèce [email protected] Stefano Musacchio Dipartimento di Fisica Generalevia Pietro Giuria 1 10125 Torino, Italie [email protected] Anatoly Neishtadt Space Research Institute -Russian Academy of Sciences, Profsoyuznaya 84/32 Moscow 117997, Russia [email protected] Tali Oliker Technionyuvalim 56 20142 d.n. misgav, Israel [email protected] Edward Ott I.R.E.A.P.-University of Maryland, City COLLEGE PARK 20742 MARYLAND, USA [email protected] Séverine Pache ITP-EPFL, Ecublens 1015 Lausanne, Suisse [email protected] Antonio Politi ISTITUTO NAZIONALE DI OTTICA APPLICATA, LARGO E. FERMI 6 50125 FIRENZE, Italy [email protected] Antonio Ponno Dipartimento di Matematica-Universita’ di Milano, Via Saldini 50 20133 Milano, Italie [email protected] Serguei Prants Institution Pacific Institute of the Russian Academy of Sciences 43 BALTIISKAYA St. 690041 VLADIVOSTOK RUSSIA [email protected] Saar Rahav Technion-Department of Physics 32000 Haifa Israel [email protected] Vered Rom-Kedar Weizmann InstituteP.O. Box 26-Department of Computer science and applied mathematics 76100 Rehovot, Israel [email protected] Majid Saberi

LPTMC, case 7020-Université Paris 7-Denis Diderot, 2 place Jussieu 75231 Paris Cedex 05, France Marc Senneret LPTMC, case 7020-Université de Paris 72 place Jussieu 75251 PARIS CEDEX 05 France [email protected]

xiii Michael Shlesinger Office of Naval Research, 800 n°Quincy Str., Arlington, VA22217-5660, USA [email protected] Eli Shlizerman Computer Science and Applied Mathematics - Dynamical Systems- Weizmann Institute of Science 76100 Rehovot Israel [email protected] Dominique Simpelaere Université Paris 6-Pierre et Marie Curie, Lab. De Probabilié, 4, Place Jussieu 75252 Paris cedex 05, France [email protected] Tom Solomon Bucknell University Lewisburg, PA 17837, USA [email protected] George Stilogiannis Dept. of Mathematics Aristotle University of Thessaloniki 54006 Thessaloniki, Grèce [email protected] Ion Stroe Mechanics-POLITEHNICA University of Bucharest Splaiul Independentei 313 RO-77206 Bucharest ROMANIA [email protected] Walter Strunz Physikalisches Institut, Universitat Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany [email protected] Vladimir Ten Dept of Mathematics and Mechanics Moscow State University-MSU, Vorobevy gory 119899 Moscow Russia [email protected] Sandro Vaienti University of Toulon and Centre de Physique TheoriqueCase 907 3288 Marseille Cedex 09 France [email protected] Jiri Vanicek Jefferson Physical LaboratoryMSRI, 1000 Centennial Drive CA 94720 Berkeley USA [email protected] Alexei Vasiliev Laboratory of Chaotic Dynamics Space Research Institute-Profsoyuznaya 84/32 117997 Moscow Russie [email protected] Sebastien Viscardy Universite Libre de Bruxelles-Service de Chimie-PhysiqueCampus Plaine, CP 231 1050 Brussels Belgium [email protected] Thérèse Vivier UMR5584 (CNRS) - Institut Mathématiques de Bourgogne BP47970 21078 Dijon Cedex France [email protected] Angelo Vulpiani Dipartimento di Fisica, Universita di Roma La Sapienza P.le A. Moro 2 I-00185 Roma Italie [email protected] Piotr Waz Center for Astronomy-Nicholaus Copernicus University, Gagarina 11 87-100 Torun Pologne [email protected]

xiv Tatiana Yelenina Numerical Simulation of Electrodynamics and Magnetohydrodynamics-Keldysh Institute of Applied Mathematics of RAS4, Miusskaya sq. 125047 Moscow Russie [email protected] L.S. Young Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA [email protected] Michael Zaks Humboldt University of Berlin-Dept of Stochastic Processes, Newtonstr. 15 D-12489 Berlin Allemagne [email protected] Georges Zaslavsky Courant Institute, New-York University, 251, Mercer Street, NY-10012-1182 New-York, USA [email protected] Vladimir Zeitlin Laboratoire de Meteorologie Dynamique, E N S, 24 Rue Lhomond 75231 PARIS CEDEX 05 France [email protected]

A SHORT ERGODIC THEORY REFRESHER. P. Collet Centre de Physique Théorique CNRS UMR 7644 Ecole Polytechnique F-91128 Palaiseau Cedex (France) [email protected]

Abstract

1.

We give a short refresher on some of the main definitions and results in ergodic theory. This is not intended to be an introduction nor a review of the subject. There are many very good texts about ergodic theory some of them are given in the references.

Introduction.

A system is characterized by the set Ω of all its possible states. At a given time, all the properties of the system can be recovered from the knowledge of the instantaneous state x ∈ Ω. The system is observed using the so called observables which are real valued functions on Ω. Most often the space of states Ω is a metric space (so we can speak of nearby states) and we will only consider below Borel measurable observables. As time goes on, the instantaneous state changes (unless the system is in a situation of rest). The time evolution is a rule giving the change of the state with time. Time evolutions come in different flavors. Discrete time evolution. This is a map from the state Ω into itself producing the new state after one unit of time given the initial state. If x0 is the state of the system at time zero, the state at time one is x1 = T (x0 ) and more generally the state at time n is given by xn = T (xn−1 ). This is often written xn = T n (x0 ) with T n = T ◦ T ◦ · · · ◦ T n-times. Continuous time semi flow. This is a family (ϕt )t∈R+ of maps of Ω satisfying ϕ0 = Id , ϕs ◦ ϕt = ϕs+t .

1 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 1–14. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

2 The dynamics can be given by a differential equation on a manifold associated to a vector field F dx = F (x) . dt This is for example the case of a mechanical system in the Hamiltonian formalism. Under regularity conditions on F , the integration of this equation leads to a semi-flow (and even a flow). There are other more complicated situations like non-autonomous systems (in particular stochastically forced systems), systems with memory, etc. A dynamical system is a set of states Ω equipped with a time evolution. If a sigma-algebra is given on Ω, we will always assume that the time evolution is measurable. This will often be a Borel sigma-algebra. One would like to understand the effect of the time evolution and in particular the behaviour at large time. For example, if A is a subset of the phase space Ω (describing the states with a given property), one would like to know in a long time interval [0, N ] (N large) how much time the system has spent in A, namely how often the state has the property described by A. Assume for simplicity we have a discrete time evolution. If χA denotes the characteristic function of the set A, the average time the system has spent in A over a time interval [0, N ] starting in the initial state x0 is given by N 1 AN (x0 , A) = χ (T j (x0 )) . (1) N + 1 j=0 A It is natural to ask if this quantity has a limit when N tends to infinity. The answer may of course depend on A and x0 , but we can already make two important remarks. Assume the limit exists and denote it by µx0 (A). First, it is easy to check that the limit also exists for T (x0 ) and for any y ∈ T −1 (x0 ) = {z | T (z) = x0 }, and moreover µT (x0 ) (A) = µy (A) = µx0 (A) .

(2)

Second, the limit also exists if A is replaced by T −1 (A) and is equal, namely (3) µx0 (T −1 (A)) = µx0 (A) .

3

A short ergodic theory refresher.

If one assumes that µx0 does not depend on x0 at least for measurable sets, one is lead to the notion of invariant measure. By definition a measure µ is invariant if for any measurable set A µ(T −1 (A)) = µ(A) .

(4)

Similar considerations and definitions hold for continuous time. Unless otherwise stated, when speaking below of an invariant measure we will assume it is a probability measure. We will denote by (Ω, T, B, µ) the dynamical system with state space Ω, discrete time evolution T , B is a sigma-algebra on Ω such that T is measurable with respect to B and µ is a measure on B invariant by T . We can now come back to the question of the asymptotic limit of the ergodic average (1). This is settled by the ergodic theorems. The ergodic theorem of Von-Neumann [22] applies in an L2 context, while the Birkhoff ergodic theorem [2] applies almost surely (we refer to [14], [20] for proofs and extensions). We now state the Birkhoff ergodic theorem. Theorem 1 Let (Ω, T, B, µ) be a dynamical system (recall that T is measurable for the sigma algebra B and the measure µ on B is T invariant). Then for any f ∈ L1 (dµ) AN (x, f ) =

N 1 f (T j (x)) . N + 1 j=0

converges when N tends to infinity for µ almost every x. We now make several remarks about this fundamental result. By (2), the set of points where the limit exists is invariant (and of full measure by Birkhoff’s Theorem). Moreover, if we denote by g(x) the limiting function (which exists for µ almost every x), it is invariant, namely g(T (x)) = g(x). The set of µ measure zero where nothing is claimed depends on f and µ. One can often use a set independent of f (for example if L1 (dµ) is separable). We will comment below on the dependence on µ. The theorem is often remembered as saying that the time average is equal to the space average. This has to be taken with a grain of salt. As we will see below changing the measure may change drastically the exceptional set of measure zero and this can lead to completely different results.

4 The set of initial conditions where the limit does not exist, although small from the point of view of the measure µ may be big from other points of views (see [1]). The most interesting case is of course when the limit in the ergodic theorem is independent of the initial condition (except for a set of µ measure zero). This leads to the definition of ergodicity. A measure µ invariant for a dynamical system (Ω, A, T ) is ergodic if any invariant function (i.e. any measurable function f such that f ◦ T = f , µ almost surely) is µ almost surely constant. There are two often used equivalent conditions. The first one is in terms of invariant sets. An invariant (probability) measure is ergodic if and only if µ(A∆T −1 (A)) = 0 ⇐⇒ µ(A) = 0

or µ(A) = 1 .

(5)

The second equivalent condition is in terms of the Birkhoff average. An invariant (probability) measure is ergodic if and only if for any f ∈ L1 (dµ) N 1 j f (T (x)) = f dµ (6) lim N →∞ N + 1 j=0 µ almost surely. We now make some remarks about the definition of ergodicity and its equivalent formulations. In the condition (6), the limit exists µ almost surely by the Birkhoff ergodic Theorem (1). It is enough to require that the limit is µ almost surely constant since the constant has to be equal to the integral. In formula (6), the state x does not appear on the right hand side, but it is hidden in the fact that the formula is only true outside a set of measure zero. It often happens that a dynamical system (Ω, A, T ) has several ergodic invariant (probability) measures. Let µ and ν be two different ones. It is easy to verify that they are disjoint. One can find a set of measure one which is of measure zero for the other and vice versa. This explains why the ergodic theorem applies to both measures leading in general to different time averages. For non ergodic measures, one can use an ergodic decomposition (disintegration). We refer to [14] for more information. However in concrete cases this may lead to rather complicated sets.


5

In probability theory, the ergodic theorem is usually called the law of large numbers for stationary sequences. Birkhoff’s ergodic theorem holds for semi flows (continuous time average). It also holds of course for the map obtained by sampling the semi flow uniformly in time. However non uniform sampling may spoil the result (see [23] and references therein). Simple cases of non ergodicity come from Hamiltonian systems with the invariant Liouville measure. First of all since the energy is conserved the system is not ergodic if the number of degrees of freedom is larger than one. One has to restrict the consideration to each energy surface. More generally if there are other independent constants of the motion one should restrict oneself to lower dimensional manifolds. For completely integrable systems, one is reduced to a constant flow on a torus which is ergodic if the frequencies are incommensurable. It is also known that generic Hamiltonian systems are neither integrable nor ergodic (see [16]).

2.

Rate of convergence in the ergodic theorem.

It is natural to ask how fast is the convergence of the ergodic average to its limit in Theorem 1. At this level of generality any kind of velocity above 1/n can occur. Indeed Halasz and Krengel have proven the following result (see [10] for a review). Theorem 2 Consider a (measurable) automorphism T of the unit interval Ω = [0, 1], leaving the Lebesgue measure dµ = dx invariant. 1 ) For any increasing diverging sequence a1 , a2 , · · ·, with a1 ≥ 2, and for any number α ∈]0, 1[, there is a measurable subset A ∈ Ω such that µ(A) = α, and n−1 1 a χA ◦ T j − µ(A) ≤ n n n j=0

µ almost surely, for all n. 2 ) For any sequence b1 , b2 , · · ·, of positive numbers converging to zero, there is a measurable subset B ∈ Ω with µ(B) ∈]0, 1[ such that almost surely 1 n−1 χB ◦ T j − µ(B) = ∞ . lim n→∞ nbn j=0

6 In spite of this negative result, there is however an interesting and somewhat surprising theorem by Ivanov dealing with a slightly different question. To formulate the result we first define the sequence of down-crossings for a non-negative sequence (un )n∈N . Let a and b be two numbers such that 0 < a < b. For an integer k ≥ 0 such that uk ≤ a, we define the first down crossing from b to a after k as the smallest integer nd > k (if it exists) such that 1 ) und ≤ a, 2 ) There exists at least one integer k < j < nd such that uj ≥ b. Let now (nl ) be the sequence of successive down-crossings from a to b (this sequence may be finite and even empty). We denote by N (a, b, p, (un )) the number of successive down-crossings from b to a occurring before

time p for the sequence (un ), namely N (a, b, p, (un )) = sup {l | nl ≤ p} . Theorem 3 Let (Ω, A, T, µ) be a dynamical system. Let f be a non negative observable with µ(f ) > 0. Let a and b be two positive real numbers such that 0 < a < µ(f ) < b, then for any integer r r a n µ x N a, b, ∞, f (T (x)) >r ≤ . b

We refer to [9], [5], [12] for proofs and extensions. In order to get some information on the rate of convergence in the ergodic theorem, one has to make some hypothesis on the dynamical system and on the observable. If one considers the numerator of the ergodic average, namely the ergodic sum Sn (f )(x) =

n−1

f (T j (x))

(7)

j=0

this can be considered as a sum of random variables, although in general not independent. It is however natural to ask if there is something similar to the central limit theorem in probability theory. To have such a theorem, one has first to obtain the limiting variance. Assuming for simplicity that the average of f is zero, we are faced with the question of convergence of the sequence 1 n

2

Sn (f )(x)

dµ(x)

7

A short ergodic theory refresher. 2

=

f (x) dµ(x) + 2

n−1 j=1

n−j n

f (x) f (T j (x)) dµ(x) .

Here we restrict of course the discussion to observables which are square integrable. This sequence may diverge when n tends to infinity. It may also tend to zero. This is for example the case if f = u − u ◦ T with n 2 u ∈ L2 (dµ). Indeed, √ in that case Sn = u − u ◦ T is of order one in L and not of order n (see [10] for more details and references). A quantity which occurs naturally from the above formula is the autocorrelation function Cf,f which is the sequence given by

Cf,f (j) =

f (x) f (T j (x)) dµ(x) .

(8)

If this sequence belongs to l1 , the limiting variance exists and is given by σf2 = Cf,f (0) + 2

∞

Cf,f (j) .

j=1

If moreover σf > 0, we say that the central limit theorem holds if

t Sn (f )(x) 1 2 √ lim µ x =√ e−u /2 du . ≤t n→∞ σf n 2π −∞

We emphasize that this kind of result has been only established for certain classes of dynamical systems and observables. We refer to [6] for a review. There are also results like Berry-Essen inequalities, law of iterated logarithms, invariance principles, large deviations etc. A natural generalization of the auto-correlation is the cross correlation between two square integrable observables f and g. This function (sequence) is given by

Cf,g (j) =

f (x) g(T j (x)) dµ(x) −

f dµ

g dµ .

The second term appears in the general case when neither f nor g has zero average. The case where f and g are characteristic functions is particularly interesting. If f = χA and g = χB , we have

Cχ

A

,

χB (n) =

A

χB ◦ T n dµ = µ(A) µ(T n (x) ∈ B|x ∈ A) .

If for large time the system looses memory of its initial condition, it is natural to expect that µ(T n (x) ∈ B|x ∈ A) converges to µ(B). This

8 leads to the definition of mixing. We say that for a dynamical system (Ω, A, T ) the T invariant measure µ is mixing if for any measurable subsets A and B of Ω, we have lim µ(A ∩ T −n (B)) = µ(A)µ(B) .

n→∞

There are many other mixing conditions including various aspects and velocity of convergence. We refer to [8] for details.

3.

Entropies.

Often one does not have access to the points of phase space but only to some fuzzy approximation. For example if one uses a real apparatus which always has a finite precision. There are several ways to formalize this idea. The phase space Ω is a metric space with metric d. For a given precision > 0, two points at distance less than are not distinguishable. One gives a (measurable) partition P of the phase space, P = {A1 , · · · , Ak } (k finite or not) , Aj ∩ Al = ∅ for j = l and Ω=

k

Aj .

j=1

If there is a given measure µ on the phase space it is often useful to use partitions modulo sets of µ measure zero. The notion of partition leads naturally to a coding of the dynamical system. This is a map Φ from Ω to {1, · · · , k}N given by Φn (x) = l

if

T n (x) ∈ Al .

If the map is invertible, one can also use a bilateral coding. If S denotes the shift on sequences, it is easy to verify that Φ ◦ T = S ◦ Φ. In general Φ(Ω) is a complicated subset of {1, · · · , k}N , i.e. it is difficult to say which codes are admissible. There are however some examples of very nice codings like for Axiom A attractors (see [4] [17] and [21]). Let P and P be two partitions, the partition P ∨ P is defined by P ∨ P = {A ∩ B , A ∈ P , B ∈ P } . If P is a partition,

T −1 P = {T −1 (A)}

9


is also a partition. Recall that (even in the non invertible case) T −1 (A) = {x , T (x) ∈ A}. A partition P is said to be generating if (modulo sets of measure zero) ∞

T −n P =

n=0

with the partition into points. In this case the coding is injective (modulo sets of measure zero). We now come to the definition of entropies. There are two main entropies, the topological entropy and the so called metric or KolmogorovSinai entropy. Both measure how many different orbits one can observe through a fuzzy observation. The topological entropy is defined independently of a measure. We will only consider here the case of a metric phase space. The topological entropy counts all the orbits modulo fuzziness. We say that two orbits of initial condition x and y respectively are (the precision) different before time n (with respect to the metric d) if sup d(T k (x), T k (y)) > . 0≤k≤n

Let Nn () be the maximum number of pairwise different orbits up to time n. In other words this is the maximum number of pairwise different films the dynamics can generate up to time n if two images differing at most by are considered identical. The topological entropy is defined by htop = lim lim sup 0 n→∞

1 log Nn () . n

When an ergodic invariant measure µ is considered, the disadvantage of the topological entropy is that it measures the total number of (distinguishable) trajectories, including trajectories which have an anomalously small probability to be chosen by µ. It even often happens that these trajectories are much more numerous than the ones favored by µ. The metric or Kolmogorov-Sinai entropy is then more adequate. If P is a (measurable) partition, its entropy Hµ (P) with respect to the measure µ is defined by Hµ (P) = −

µ(A) log µ(A) .

A∈P

In communication theory one often uses the logarithm base 2.

10 The entropy of the dynamical system with respect to the partition P and the (invariant, ergodic, probability) measure µ is defined by 1 n−1 −j Hµ (∨j=0 T P) . n→∞ n

Hµ (T, P) = lim

Finally the metric or Kolmogorov-Sinai entropy is defined by hµ (T ) =

sup

P finite or countable

Hµ (T, P) .

If P is generating, and hµ (T ) < ∞, then hµ (T ) = Hµ (T, P) . One has also the general inequality hµ (T ) ≤ htop (T ) , and moreover htop (T ) =

sup

hµ (T ) .

µ, T ergodic

However the maximum may not be reached. We refer to [24] for proofs and more results. An important application of the entropy is Ornstein’s isomorphism theorem for Bernoulli shifts, we refer to [18] for more information. Another important application of the entropy is the Shannon-McMillan-Breiman theorem which in a sense counts the number of typical orbits for the measure µ. Theorem 4 Let P be a finite generating partition. For any > 0 there is an integer N () such that for any n > N () one can separate the −j P into two disjoint subsets atoms of the partition Pn = ∨n−1 j=0 T Pn = Bn ∪ Gn 

such that

µ



A ≤

A∈Bn

and for any A ∈ Gn e−n(hµ (T )+) ≤ µ(A) ≤ e−n(hµ (T )−) In other words, the atoms in Bn are in some sense the “bad” atoms, but their total mass is small. On the other hand, the good atoms (those

11


in Gn ) have almost the same measure, and of course their union gives almost the total weight. An immediate consequence is that # Gn en hµ (T ) , where # Gn denotes the cardinality of the set Gn . This is similar to the well known formula of Boltzmann in statistical mechanics relating the entropy to the logarithm of the (relevant) volume in phase space. There is also an obvious connection with the equivalence of ensembles. We refer to [14] and [15] for more information. As mentioned before, it often happens that # Gn # Bn . A simple example is given by the Bernoulli shift on two symbols. Let p ∈]0, 1[ with p = 1/2. Consider the probability measure ν on {0, 1} with ν({0}) = p, and the measure µ on Ω = {0, 1}N which is the infinite product of ν. The distance between two elements x and y of Ω is defined by d(x, y) = e− inf {i , xi =yi } . It is easy to prove that the measure µ is invariant and ergodic for the shift S on sequences. Moreover, htop (S) = log 2. The partition

P = {x0 = 0 }, {x0 = 1 } is generating. For the entropy one has hµ (S) = −p log p − q log q with q = 1 − p. However for n large we get (since p = 1/2) # Gn en hµ (S) 2n # Bn . Another way to formulate the Shannon-McMillan-Breiman theorem is to look at the measure of cylinder sets. For a point x ∈ Ω, let Cn (x) −j be the atom of ∨n−1 P which contains x. In other words, Cn (x) is j=0 T the set of y ∈ Ω such that for 0 ≤ j ≤ n − 1, T j (x) and T j (y) belong to the same atom of P (the trajectories are indistinguishable up to time n − 1 from the fuzzy observation defined by P). Then for µ almost every x we have 1 hµ (T ) = − lim log µ(Cn (x)) . n→∞ n A similar result holds using a metric instead of a partition. It is due to Brin and Katok, and uses the so called Bowen balls defined for x ∈ Ω, the transformation T , δ > 0 and an integer n by

B(x, T, δ, n) = y , d(T (x), T (y)) < δ for j = 0, · · · , n − 1 . j

j

12 These are again the initial conditions leading to trajectories indistinguishable (at precision δ) from that of x up to time n − 1 . Theorem 5 If µ is T ergodic, we have for µ almost any initial condition 1 hµ (T ) = lim lim inf − log µ(B(x, T, δ, n)) . δ0 n→∞ n We refer to [3] for proofs and related results. A way to measure the entropy in coded systems was discovered by Ornstein and Weiss using return times to the first cylinder. The result was motivated by the investigation of the asymptotic optimality of Ziv’s compression algorithms. Let q be a finite integer, and assume the phase space of a dynamical system is a shift invariant subset of {1, · · · , q}N . As before, we denote the shift by S. Let µ be an ergodic invariant measure. Let n be an integer and for x ∈ Ω, define Rn (x) as the smallest integer such that the first n symbols of x and S Rn (x) (x) are identical. Theorem 6 For µ almost every x we have lim

n→∞

1 log Rn (x) = hµ (T ) . n

We refer to [19] for the proof. A metric version was recently obtained by Donarowicz and Weiss, using again Bowen balls, in the context of dynamical systems with a metric phase space. Let for δ > 0 and the integer n

R(x, T, δ, n) = inf l > 0 , T (x) ∈ B(x, T, δ, n) . l

In other words

R(x, T, δ, n) = inf l > 0 , d(T l+j (x), T j (x)) < δ for j = 0, · · · , n − 1 . Theorem 7 For µ almost every x, we have lim lim sup

δ0 n→∞

1 log R(x, T, δ, n) = hµ (T ) . n

We refer to [7] for the proof and related results.

13

References [1] L.Barrera, J.Schmeling. Sets of “non typical” points have full topological entropy and full Hausdorff dimension. Israel J. Math. 116, 29-70 (2000). [2] G.D.Birkhoff. Proof of the ergodic theorem. Proc. Nat. Acad. Sci. USA, 17, 656-660 (1931). [3] M.Brin, A.Katok. On local entropy. In Geometric dynamics. Lecture Notes in Math., 1007, 30-38 Springer, Berlin, 1983. [4] R.Bowen. Equilibrium States and Ergodic Theory of Anosov Diffeomorphisms. Lecture Notes in Mathematics 470. Springer-Verlag, Berlin Heidelberg New York 1975. [5] P.Collet, J.-P.Eckmann. Oscillations of Observables in 1-Dimensional Lattice Systems. Math. Phys. Elec. Journ. 3, 1-19 (1997). [6] M.Denker.The central limit theorem for dynamical systems. In Dynamical systems and ergodic theory. Banach Center Publ., 23, 33-62 PWN, Warsaw, 1989. [7] T.Donarowicz, B.Weiss. Entropy theorems along the time when x visits a set. Illinois Journ. Math. To appear. [8] P.Doukhan. Mixing. Properties and examples. Lecture Notes in Statistics, 85. Springer-Verlag, New York, 1994. [9] V.Ivanov. Geometric properties of monotone fluctuations and probabilities of random fluctuations. Siberian Math. Journal 37, 102-129 (1996). Oscillation of means in the ergodic theorem. Doklady Mathematics 53, 263-265 (1996). [10] A.Kachurovskii. The rate of convergence in the ergodic theorems. Russian Math. Surveys 51, 73-124 (1996). [11] S.Kalikow. Outline of Ergodic troy.msci.memphis.edu/ quasa/kalikow/kalikow.pdf.

Theory.

[12] S.Kalikov, B.Weiss. Fluctuations of the ergodic averages. Illinois J. Math. 43, 480-488 (1999). ´ Y.Sinai, S.Fomin. Ergodic Theory. Springer-Verlag New-York, 1980. [13] I.Kornfeld, [14] U.Krengel. Ergodic Theorems. Walter de Gruyter, Berlin, New York 1985. [15] O.Lanford. Entropy and equilibrium states in classical statistical mechanics. In Statistical mechanics and Mathematical Problems. Lecture Notes in Physics 20, 1-113, Springer-Verlag, Berlin 1973. [16] L.Markus, K.Meyer. Generic Hamiltonian dynamical systems are neither integrable nor ergodic. Memoirs of the American Mathematical Society, 144. American Mathematical Society, Providence, R.I., 1974.

14 [17] M.Keane. Ergodic theory and subshifts of finite type. In Ergodic theory, symbolic dynamics, and hyperbolic spaces, 35-70. Oxford Sci. Publ., Oxford Univ. Press, New York, 1991. [18] D.Ornstein. Ergodic Theory, Randomness, and Dynamical Systems. Yale University Press, New Haven London 1974. [19] D.Ornstein, B.Weiss. Entropy and data compression schemes. stationary random fields. IEEE Trans. Information Theory 39, 78-83 (1993). [20] K.Petersen. Ergodic Theory. Cambridge University Press, Cambridge 1983. See also http://www.math.unc.edu/Faculty/petersen. [21] D.Ruelle. Thermodynamic formalism. Addison-Wesley, Reading, 1978. [22] J.von-Neumann Proof of the quasi ergodic hypothesis. Proc. Nat. Acad. Sci. USA, 18, 70-82 (1932). [23] K.Reinhold. A smoother ergodic average. Illinois Journ. Math. 44, 843-859 (2000). [24] Y.Sinai. Introduction to Ergodic Theory. Princeton University Press, Princeton 1976.

NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT. Maurice Courbage Université Paris 7 - Denis Diderot. L.P.T.M.C. Tour 24-14.5ème étage 2, Place Jussieu 75251 Paris Cedex 05 - FRANCE [email protected]

Abstract

1.

The purpose of this paper is to survey shortly some notions in the spectral theory of ergodic dynamical systems and their relevance to mixing and weak mixing. In addition, we present some dynamical systems of particles submitted to collisions with nondispersive obstacles and their ergodic and spectral properties. Transport is formulated in terms of random walk generated by deterministic dynamical systems and their moments.

Introduction

Transport in physical systems is mainly modelled by diffusion processes. For systems of noninteracting particles having ”chaotic” motion, diffusion appears in the long-time limit as a result of a random walk generated by the deterministic dynamics with invariant probability measure. The machinery is an extension of the central limit theorem to deterministic dynamical systems. Such extensions goes back to works by Sinai [26] in 1960. It is generally beleived that diffusion is related to mixing properties of the motion of the particles. However, this question is not yet quite clear. We shall discuss in the last section some recent results on central limit theorem and mixing properties. Spectral theory of Dynamical Systems provided powerful tools to study mixing and its rate. In 1931, B.O. Koopman pubished his paper entitled ” Hamiltonian systems and transformations in Hilbert space” in which he showed how a measure-preserving transformation T induces a unitary operator U on the Hilbert space of the measurable functions 15 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 15–33. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

16 of L2 type [18]. This led von Neumann to publish in 1932 a proof of the ergodic theorem in this Hilbert space (see reference in the paper of P. Collet in this volume [9]). Hopf, Koopman and von Neumann studied the relationship between mixing properties and the spectrum of U showing that for T , being weakly mixing is equivalent, for U , to having continuous spectrum [16, 19]. Later on, von Neumann and Halmos classified completely the dynamical systems having purely discrete spectrum. All these results are presented in the Halmos book in 1956 [14]. In the sixties, many progresses were made in studying the stucture of the continuous spectrum of K-systems with works of Kolmogorov, Sinai and Rokhlin [10]. In the same period, Anzai introduced a new class of dynamical systems, the class of skew products which have zero entrpoy and continuous spectrum. Continuous spectrum has been found also in other examples of dynamical systems with zero entropy, namely interval exchanges. During the last 30 years, there were many spectral studies in this kind of dynamical systems ( for a review, see [13]). On the other hand, some results on the central limit theorem in weakly dependent processes and its relation to mixing have been presented, namely by I.A. Ibragimov [17] and P. Billingsley [3], see also the review by A. Liverani [21].

2.

Preliminaries on Spectral Theory of Dynamical Systems

We shall only consider DS with an invariant probability measure. For an ergodic system, as a result of the Birkhoff theorm (see [9]) applied to the characteristic function 1A of a subset A, the invariant measure describes the frequency of visit of a typical trajectory to any given measurable set. Definition 2.1 Let X be a measurable space with a probability measure µ and let A be the family of all measurable subsets. A dynamical system (DS) is an invertible measurable transformation T on a X , which is measure-preserving, that is, µ(T −1 A)) = µ(A) for any A ∈ A. In what follows we shall denote a DS simply by (X, T, µ). As T may have many invariant measures, the measure µ will be specified in each example. The Koopman unitary operator associated to T is defined on the Hilbert space H = L2 (X, µ) by : U f (x) = f (T x)

(2.1)

Notes on Spectral Theory, Mixing and Transport.

17

In order to stress on the dependence of U with respect to T, we should use the notation UT , but we shall omit it and use it only when necessary. The spectral properties of this operator are called the spectral properties of the DS. The first important spectral property of a DS is a characterization of its ergodicity. Recall that (X, T, µ) is ergodic if any invariant measurable subset A (i.e. T A = A, eventually modulo a subset of zero measure) has either zero measure or full measure. Spectral characterization of ergodicity is given by: Theorem 2.2 (X, T, µ) is ergodic if and only if the only eigenfunction of U for the eigenvalue 1 is the constant function. The famous elementary example in this respect is the rotation of the circle. Let X = S 1 be the circle of length 1, identified with the interval [0, 1[, let α ∈ [0, 1[, T (x) = x + α and µ = µL the Lebesgue measure, then the orthonormal basis of L2 (X, µ) given by: ϕn (x) = exp(2inπx), n ∈ Z, is a family of eigenfunctions of U for the eigenvalues λn = exp(2iπnα). The above theorem implies that T is ergodic if and only if α is an irrational number. Definition 2.3 T is said to have a discrete spectrum if U has a complete orthonormal family of eigenfunctions. The spectrum of U was very important since the early stage of the Ergodic Theory when Halmos and von Neumann proved that two ergodic systems, (X, T, µ) and (Y, S, ν), both having discrete spectrum, are isomorphic if and only if UT and US have the same set of eigenvalues. This is no more true when one of these systems has not a purely discrete spectrum. Let us call, in this case, the discrete spectrum subspace of U , Hd , the subspace generated by all the eigenfunctions of U , this means that the orthogonal complement of this subspace, H⊥ d , called the continuous spectrum subspace, Hc , is not reduced to zero. A useful quantity in order to characterize the continuous spectrum is given by the spectral measure of a function f . One needs here to use the notion of Fourier coefficients of any measure µ on the circle S 1 defined by: 1 µ ˆ(n) = exp(2iπnλ)dµ(λ) (2.2) 0

The Bochner-Hergoltz theorem asserts that any positive definite sequence of numbers is a sequence of Fourier coefficients of some measure uniquely defined by it (we refer to [10, 24] for more informations) . That

18 is the case for the sequence called the auto-correlation coefficients of f , where < f, g > is the scalar product on H = L2 (X, µ). Thus, we associate to each f of H = L2 (X, µ) a unique spectral measure σf on the circle S 1 such that:

1

n

=

exp(2iπnλ)dσf (λ) = σ ˆf (n)

(2.3)

0

In other words, these autocorrelations coefficients of f are the Fourier coefficients of the spectral measure σf . It is possible to show that a function f ∈ Hc if and only if σf (λ) = 0, ∀λ. Then, the measure σf is said to be continuous.The Wiener Lemma asserts that: 1 lim | σˆf (k) |2 = (σf (λi ))2 n→∞ n n−1

(2.4)

i

k=0

where λi are the points of the circle such that σf (λi ) > 0 (for a proof, see [10]). The left hand side of the above equation is called the Cesaro-mean of | σˆf (k) |2 . Thus, the convergence to zero or not of the above Cesaromean gives a characterization of the continuity or not of the spectral measure of f . A sufficient condition for the continuity of the spectral measure is the decay of the auto-correlation coefficients ( for, the convergence to zero of a numerical sequence implies the convergence to zero of its Cesaro-mean), but it is not necessary. The relation of the continuity of the spectral measure to the behavior of the auto-correlations coefficients of f can be further characterized and we refer to [10]. A continuous measure σf is said to be ”absolutely continuous with respect to the Lebesgue measure” if it has a density with respect to the Lebesgue measure, dσf (λ) = ρ(λ)dλ, where: ρ(λ) =

∞

σ ˆf (n)exp(2iπnλ)

(2.5)

n=−∞

ˆf (n) = 0, ∀n = It follows that σf is the Lebegue measure (i.e. ρ = 1) if σ 0. If an absolutely continuous measure has density ρ(λ) which can vanish only on a subset of zero Lebesgue measure, then it is said to be equivalent to the Lebesgue measure. This is the case, for instance, if the Fourier coefficients σ ˆf (n) decay exponentially rapidly as |n| → ∞. In fact, the density ρ(λ) given above as an entire series of z = exp(2iπλ) will be uniformly convergent, so that it has analytic continuation in some annulus around the unit circle. Therefore ρ(λ) has only finite number of zeros.

19


Singular continuous measures σf are such that one can find a set of zero Lebesgue measure which is of measure 1 for σf and vice versa. They are often obtained in DS with zero entropy. Definition 2.4 A DS is mixing if for any A, B ∈ A, lim µ(T −n A ∩ B) = µ(A)µ(B)

(2.6)

n→∞

It is weakly mixing if for any A, B ∈ A, 1 | µ(T −n A ∩ B) − µ(A)µ(B) |= 0 n→∞ n n−1

lim

(2.7)

k=0

We have the following hierarchy of ergodic properties: Mixing ⇒ Weak mixing ⇒ Ergodicity Mixing is equivalent to the decay of correlations of two functions f and g of L2 (X, µ), which means : n | f (T x)g(x)dµ − f (x)dµ g(x)dµ |→ 0 as n → ∞ (2.8) X

X

X

and weakly mixing is equivalent to the decay in Cesaro-mean, i.e. 1 | n→∞ n n−1

k=0

f (T n x)g(x)dµ −

lim

X

g(x)dµ |= 0

f (x)dµ X

(2.9)

X

The following theorem gives a spectral characterization of these properties: Theorem 2.5 Let (X, T, µ) be a DS. • T is mixing iff, for any function f such that f dµ = 0, σˆf (n) → 0 as n → ∞ • T is weakly mixing iff for any function f such that f dµ = 0, σf is continuous. Rapid decay of correlations is the short memory signature of deterministic dynamical systems, among them the class of Bernoulli systems is the most ”stochastic”. Bernoulli systems are systems that are isomorphic to a sequence of independent identically distributed (I.I.D.) random variables. This means that the system (X, T, µ) posseses a generating partition, P = A1 , · · · , Ak (k finite or not), which is independent, Z that is, the map Φ from X to 1, · · · , k given by: Φn (x) = l

if

T n (x) ∈ Al .

20 is one-to-one and this sequence of random variables is independent. The best known example of such system is the baker tranformation, for which the partition P is the division of the unit rectangle into left and right halves. Then, the system is mixing and the entropy of the system is equal to the entropy of the Bernoulli shift (for this notion see [9]). Kolmogorov introduced an intermediate class of transformations, the so-called Ksystems which are in between mixing and Bernoulli systems. A convenient definition is given in terms of the entropy Hµ (T, P). Although the entropy was introduced by Kolmogorov as an invariant quantity preserved by isomorphisms of DS, it can be understood as the measure the non-predictability of the system under observations of the partitition P [22]. The K-systems are characterized as those for which this entropy is strictly positive for any finite partition P. As we mentioned above, Bernoulli systems satisfy the K-property. Thus, we have the following hierarchy: Bernoulli ⇒ K- property ⇒ Mixing ⇒ Weak mixing ⇒ Ergodicity The spectral properties of K-systems are all the same, so we can say that all K-systems are spectrally isomorphic. In particular, there is no spectral difference between those K-systems that are Bernoulli and those which are not. We have the following theorem: Theorem 2.6 Let (X, T, µ) be a K-system. Then, U has countable Lebesgue spectrum as an operator restricted to the orthocomplement of the constant function, {1}⊥ . The theorem means that {1}⊥ decomposes into a countable direct sum of orthogonal invariant subspaces ⊕Hi , i ∈ Z such that U is reduced in each subspace Hi to shift operator acting on an orthonormal basis {ei,j , j ∈ Z}: U ei,j = ei,j+1 . As a consequence, the spectral measure σei,j is the Lebesgue measure. This follows from the computation of the Fourier coefficients: σ êi,j (k) are all zero for any i = j and they are identical to the Fourier coefficients of the Lebesgue measure on the circle.

3.

Dynamical Systems with Zero Entropy

This is a rich family of systems including nondispersive billiards and aperiodic transformations which display divergence of trajectories with a power-law rate and some mildly unpredictabililty in the sense of the linear prediction theory (see references in [12]) among them we shall consider mainly examples of skew products of dynamical systems. We first define the general concept of skew products of DS.


21

Definition 3.1 Let (X, τ, µ) be a DS, (Y, ν) be a probability measure space and for µ-almost all x ∈ X, let Sx be a ν-measure-preserving transformations on Y . The transformation T acting on the product space X × Y defined by : T (x, y) = (τ (x), Sx (y))

(3.10)

preserves the product measure µ × ν. The DS (T, X × Y, µ × ν) is called a skew product of transformations. A particularly important case which occurs in quasi-periodic integrable systems, is the one where Y is a compact group G endowed with the Haar measure m ( a kind of ”Lebesgue measure” on G) and where Sx is given by a ”cocycle”, that is, a function φ : x ∈ X → G and defined by : g ∈ G → Sx (g) = φ(x).g. The skew product T : X × G → X × G defined by: T (x, g) = (τ (x), φ(x).g)

(3.11)

is called a group extension of τ . It is possible to study the ergodic properties of T in terms of the properties of τ and φ . An important property proved by Abramov-Rokhlin is that the metric entropy is zero if τ has zero entropy (see [23]). Here we shall consider two examples of group extensions of the circle rotation on X = S 1 , with τ (x) = x+α, and µ the Lebesgue measure on the circle. I. First Example : G = {±1} We start in showing a relation between these transformations and the plate billiard model with slit introduced by Zaslavsky [27]. A particle moves on an infinite plane among periodically distributed plate obstacles of length β < 1 separated by slits, the spatial period in both (x,y) directions is 1 (see fig.1, in the case β = 1/2 ). The particle reflects elastically at each collision with the obstacle. Let θ be the absolute value of the angle between the outgoing velocity vector and the obstacle. The particle will move either always to the right or always to the left undergoing reflections, according to the value of the angle θ. Let us suppose that 0 < θ < π2 . At time t=0, the coordinates of the particle are given by its position (x0 , 0) and the direction of its ingoing velocity .The position of the particle will increase between time t=0 and time t=1 by α along x-direction and by ±1 along y-direction where α = 1/tg(θ). The trajectory of the particle will be specified, at times t=n, by its x-coordinate xn and the projection of the ingoing velocity vector along the y-direction, n , which will change under a collision. Thus, n = ±1 according as the velocity is upward or downward. We call

22

Figure 1.

Trajectory of particle in the plate billiard with slits

n the ”veolcity direction of the particle”. Let us introduce the ”cocycle ” function: x ∈ [0, 1[→ {±1} as follows : −1 if x ∈ [0, β[ χ(x) = (3.12) +1 if x ∈ [β, 1[ The transformation T maps the state of the particle (x, ) at time t = 0 into the state of the particle at time t=1 according to the formula: T (x, ) = (x + α(mod1), χ(x)ε)

(3.13)

So T maps an ingoing arrow to the next ingoing arrow. When G is endowed with the Haar measure m(±1) = 1/2, T is a two-elements group extension of the rotation. Spectral Properties As well known, the space H = L2 (S 1 × G, µ × m) is a tensor product L2 (S 1 , µ) ⊗ L2 (G, m). It decomposes, by using the orthonormal basis {1, ε} of L2 (G, m), into a direct sum H = H1 ⊕ H2 where : H1 = {g(x) ∈ L2 (S 1 , µ)} H2 = {εh(y), h ∈ L2 (S 1 , µ)}

(3.14) (3.15)

In other words, for any f ∈ L2 (S 1 , µ) ⊗ L2 (G, m), there are g, h ∈ L2 (S 1 , µ) such that : f (x, ε) = g(x) + εh(x). The action of U restricted


23

to H1 (resp. H2 ) is given by: U g(x) = g(x + α) (resp. U (εh(x)) = εχ(x)h(x + α)). It shows that H1 and H2 are invariant. Therefore, the spectrum of U restricted to H1 is reduced to the spectrum of the the Koopman operator of the rotation on H1 . The spectral measure associated to any function from H2 is given through the Fourier coefficients: . It is convenient to introduce the unitary operator Vχ on L2 (S 1 , µL ) given by: Vχ h(x) = χ(x)h(x + α)

(3.16)

Now, using the relation U n (εh) = εVχn (h) we obtain: =< εVχn (h), εh) >=< Vχn (h), h) > which means that the spectrum of U restricted to H2 is reduced to the spectrum of Vχ on L2 (S 1 , µ). The spectrum of this operator have been studied first in the case β = 1/2 and later in more general case . In the first case, it is shown that for any irrational α, Vχ has no eigenfunction in L2 (S 1 , µ), which means that U restricted to H2 has only continuous spectrum. This immediately implies that the system is ergodic. In the general case, the same result holds for some class of α and β with diophantine properties. The spectrum is singularly continuous for every irrational α and almost all β. We refer to [10, 24] for more informations on the ergodic and spectral properties of these transformations. Decay of correlations Now, back to the direction of the (ingoing) velocity at time t = n: εn (x, ε) = χ(x + (n − 1)α) × ....χ(x)ε

(3.17)

If we define the function f (x, ε) = ε, we have εn (x, ε) = f (T n (x, ε)). Clearly the expectation of the velocity directions at all time εi (x, ε) with respect to the invariant measure is zero and its autocorrelation coefficients are < εn , ε0 >=< Vχn (1), 1) >. As the spectrum of Vχ is continuous for α = 1/2, by the Wiener Lemma we have only a decay of correlations in Cesaro-mean: 1 |< εn , ε0 >|2 = 0 n→∞ n n−1

lim

(3.18)

k=0

This result holds also for the absolute value. While the distance traveled by the particle along the x-direction is alway xn = nα, the distance traveled by the particle along y-direction depend on the initial condition (x, ε) and seems as a random walk. This distance, at t=n, is : Sn (x, ε) =

n i=1

εi (x, ε)

(3.19)

24 The mean value of Sn over the initial conditions is zero and the problem is to estimate the asymptotic behavior of the variance of Sn . We shall come back to this question in the next section. II.Second example: G = S 1 A particle moves on an infinite plane among periodically distributed obstacles with spatial period equal to 1 along both (q1 , q2 )-directions. In the q1 -direction, the motion of the particle is uniformly accelerated at each regular time interval by an amount α and has uniform free motion along the q2 -direction. That is, define the velocity p1 (n) = q1 (n + 1) − q1 (n) , then the equations of the projection of the motion in the q1 direction are : q1 (n) = q1 (n − 1) + p1 (n − 1) p1 (n) = p1 (n − 1) + α

(3.20) (3.21)

It is the result of the action of the mapping: T : (p1 , q1 ) → T (p1 , q1 ) given by: T (p1 , q1 ) = (p1 + α, q1 + p1 )

(3.22)

q1 (n) = q1 (0) + np1 (0) + n(n − 1)α/2

(3.23)

Thus, we obtain:

The particle is moreover submitted at the begining of each time interval to a deterministic ”collision” changing the direction of the motion motion up and down along q2 -direction in the following way: the projection of the velocity of the particle p2 at time t = n is given by χ(q1 (n)) where χ(x) is a periodic discontinuous function defined by: −1 if x ∈]0, 1/2] χ(x) = (3.24) 1 if x ∈]1/2, 1] That is, the direction of the velocity at time t = n is : εn (p1 (0), q1 (0)) = χ(q1 (n)) = χ(q1 (0) + np1 (0) + n(n − 1)α/2) (3.25) and the value of the variable q2 at time t = n + 1 is:

q2 (n + 1) = q2 (n) + χ(q1 (n)) =

n

εi (p1 (0), q1 (0))

(3.26)

1

A trajectory is shown in the figure 2. The transformation (3.22) on the Torus T2 is a particular case of the group extension given by a function

25


2

1

25

50

75

100

125

150

-1

-2

-3

-4

-5

Figure 2. Trajectory of a particle of the example II moving with initial null velocity with α = 1/ (7)

φ(x) from S 1 into R and defined as a transformation of the Torus T2 by: T (x, y) = (x + α, y + φ(x))

mod 1

(3.27)

Spectral properties What are the spectral properties of T ? A function f ∈ L2 (T2 ) can be decomposed into a sum of orthogonal functions as: f (x, y) =

∞

e2iπny hn (x)

n=−∞

The subspace of functions depending only of x is invariant and, as above, the restriction of U to this subspace has the discrete spectrum of the rotation on S 1 . All other subspaces of functions of the form e2iπny hn (x) are invariant. Thus the operator U decomposes into direct sum of operators ⊕ Un each acting on these subspaces. The spectrum of Un depends on the properties of φ(x). In the case of equation (3.22), the spectrum is equivalent to the Lebesgue measure. This is also the case for (3.27) if φ(x) = x + g(x) where g has rapidly decreasing Fourier coefficients. But, in general, the spectrum can be purely discrete, or singularly continuous according to the properties of φ(x), we refer to [13] for more detailed

26 results and references. Decay of Correlations If φ(x) = kx, k ∈ Z, k = 0 , there are functions f for which the autocorrelation coefficients decay exponentially rapidly [11]. In particular, the function f (x, y) = χ(y), for example, has vanishing autocorrelations coefficients: σ ˆχ (n) = 0 for any n = 0, so that its spectral measure is the Lebesgue measure. It defines a pairwise independent process with long range dependence. The two-valued function χ(y) on T2 defines a partition {A, AC } of the torus into two regions, the upper half and the lower half of the torus: A = {(x, y) : χ(y) = 1} and AC = {(x, y) : χ(y) = −1} such that: µ(Ai ∩ T −n Aj ) = µ(Ai )µ(Aj ) for any n = 0, where Ai is either A or Ac . We call such partitions pairwise independent. In [11] and [12], it was determined a large family of pairwise independent partitions. That is equivalent to say that for the functions f = 1A − µ(A), the family of functions f ◦ T n are pairwise independent random variables, which does not imply that these variables are jointly independent or Markovian. Actually, the process has infinite memory and its metric entropy is zero. That is on the opposite of the Markov chains of finite memory which have always positive entropy. We conclude that the velocity direction εn is a pairwise independent process with Lebesgue spectrum and zero entropy. Neverthless, this process is unpredictable in the sense of least squares prediction theory [12].

4.

Transport properties

The mechanical origin of transport processes has been discusssed from several points of view ranging from the Boltzmann equation to the derivation of the Brownian motion. As a model of a particle dynamics producing a transport process, the Lorentz gas model of noninteracting particles moving among scatterers have been abundantly studied in order to illustrate these points of view. One classical way consists in the derivation of the Boltzmann equation, either from the Liouville equation or directly under the Grad limit [15], obtaining then a hydrodynamic description. In 1981, Bunimovich and Sinai [5, 6] used methods of probability theory and dynamical systems theory to obtain a diffusion equation for the motion of noninteracting gas of particles elastically scattered among a periodic array of fixed convex discs. The machinery used is that of the central limit theorems in dynamical systems under


27

the invariant measure. This is a generalization of the usual central limit theorem which is formulated in the following terms. If (X, T, µ) is a dynamical system and f ∈ L2 (X, µ) is a function with zero mean and a finite second moment σ 2 , such that {f (T n x)} form an independent sequence of random variables (with respect to µ), then f satisfies the central limit theorem, that is,

n−1 1 1 x2 n lim µ x : √ f (T x) ∈ I = √ exp(− 2 )dx (4.28) n→∞ n 2σ 2πσ I i=0

n converges in law In other words, the normalized sum √1n n−1 i=0 f ◦ T to a gaussian variable of zero mean value and variance σ2 . Moreover, f satisfies to the ”invariance principle” which means that the following rescaled sums converge to the Wiener process, i.e. [t/τ ] √ τ f ◦ T n → Yt n=0

τ →0

(4.29)

where t ∈ [0, 1] and Yt is the Brownian process. The simplest ”toy model” of random walk generated by a deterministic ”chaotic” DS is the following: a particle with an internal state x ∈ X, moves on the lattice Z where it is subject to a deterministic ”collision” with an obstacle at each node of the lattice. The ”collision” is modelled by a Bernoulli DS (X, B, µ). Let m ∈ Z denote the initial position of the particle. The dynamics of the system is defined by a transformation T : X ×Z → X ×Z : T (x, m) = (Bx, m + ξ0 (x))

(4.30)

where ξ0 is a function taking the values ±1. At time t = n, the state of the particle is (B n (x), m + ξ0 (x) + ξ0 (B(x)) + .... + ξ0 (B (n−1) (x))) and the position the particle, for an initial condition m = 0, is given of n−1 i i by Sn ξ0 (x) = i=0 ξ0 (B (x)). If the variables {ξ0 (B (x))} are independent with respect to µ, then this random walk will converge to the diffusion process. For example, in the special case of the baker transform (B, T2 , µ) where µ is the Lebesgue measure on T2 , the independence of ξ0 (B i (x, y) is equivalent to the independence of the traditional partition of the square in two half squares corresponding to ξ0 (x, y) = ±1 according that (x, y) belongs to one or another element of this partition (we refer to most textbooks in ergodic theory and dynamical systems for more mathematical details on the Bernoulli systems).

28 In general, a given two-valued or smooth function will not generate an independent sequence. There were in probability theory several generalizations of the central limit theorem to the case where the sequence is not independent. It was first proved in the case of a markovian sequence (for a proof see in [2]), then extended to the case of a weakly dependent sequence [3] with some mixing properties. Related results in hyperbolic DS go back to the geodesic flow [26] and later to Anosov systems [25]. The ”diffusion coefficient” σ is defined by the convergence of the following limit: 1 n

n−1 ( f (T n x))2 dµ(x) −→ σ 2 n→∞

I i=0

(4.31)

which exists only for some ”large” class of ”smooth” functions f . If the stationary sequence of random variables f (T n x)) are independent, the n variance of the ”Birkhoff sums” Sn f (x) = n−1 i=0 f (T x)) is equal to: Sn f 2 =

n−1 i=0

(f (T n x))2 dµ(x) = nf 2

(4.32)

I

since the independence implies that the autocorrelation coefficients = 0 for any n = 0. This relation holds also if the sequence is pairwise independent, which occurs even with zero Kolmogorov entropy (see explained above). In that case σ = f . For the Lorentz model with finite horizon, studying the hyperbolic properties of collisions of a particle among a periodic array of convex obstacles, Bunimovich and Sinai proved the property (4.31) for the position of the particle and showed a central limit theorem and the convergence to diffusion. For a recent survey see [8]. All these results imply the K-property. Beck and Roepstorff developped an analogous model of the Langevin equation by the convergence of such kind of random walk generated by Bernoulli DS to the Ornstein-Uhlenbeck process [1]. In general nonhyperbolic DS, (X, T, µ), a first question is the convergence of (4.31) to a finite non zero ”diffusion coefficient” σ for some ”large” class of ”smooth” functions f. But, this quantity may diverge or converge to zero.Then the central limit theorem means the convergence of the normalized sums Sn1 f Sn f (x) in distribution to the reduced normal law. Such theorem does not imply necessarly mixing properties. The correlations may not decay at all, as shown in the examples of Burton and Denker [7]. They constructed a class of functions f satisfying CLT for any aperiodic DS (including the irrational rotation). Clearly,

29


the correlation coefficients will not decay when the spectral measure is discrete. These examples have been generalized and there were other results on the convergence of normalized Birkhoff sums to self-similar processes [20]. In our first example of random walk of the previous section, numerical simulations of Zaslavsky and Edelman [27] have shown that anomalous superdiffusion is expected and the property (4.31) is likely not to hold for the sum ni=1 f (T i (x, ε)). Let us compute the second moment of Sn2 in the case where α is a rational number. For the initial condition (x,), we have: (x1 , 1 ) = (x + α, φ(x)) (x2 , 2 ) = (x + 2α, φ(x + α)φ(x)) ... (xn , n ) = (x + nα, φ(x + (n − 1)α) ... φ(x)).

(4.33)

φn (x) = φ(x + (n − 1)α) ... φ(x)

(4.34)

Denote by:

Then:

n

Sn =

φi (x), < Sn >= 0

(4.35)

i=1

We have:
=

n i,j=1

=

i

φi (x)φj (x) dx

0 1

0

= n +2

1

φ2i (x)

dx + 2

(4.36)

φi (x)φj (x) dx

0

i<j

i−1 n

1

1

φi (x)φj (x) dx

(4.37)

0

i=2 j=1

But, φi (x)φj (x) = φ(x + jα) ... φ(x + (i − 1)α)

(4.38)

So < Sn2 > = n + 2

i−1 n i=2 j=1

= n+2

n−1

1

φi−j (x) dx

1

(n − k)

k=1

(4.39)

0

φk (x) dx 0

(4.40)

30 For α = pq . φk+2q (x) = φk (x), for (φ(x))2 = 1. So let n − 1 = 2rq + l, l ≤ 2q − 1. Therefore

= n + 2 (n − k)

φk (x) dx +

0

k=1

2qr+l

1

1

(n − k)

φk (x) dx 0

k=2qr+1

(4.41) However, 2qr

=

r−1

(s+1)2q

(4.42)

s=0 k=s2q+1

k=1

Using m = k − 2qs, the second part of (4.41) could be rewritten as

=

=

=

2q r−1 s=0 m=1 2q r−1

(n − s2q − m)

φs2q+m (x) dx

(4.43)

φm (x) dx

(4.44)

0

(n − s2q − m)

s=0 m=1 2q 1 m=1

1

1

0

φm (x) dx

r−1

0

r(r − 1)2q = rn − 2

(n − s2q − m)

s=0 2q 1 m=1 0

φm (x) dx − r

(4.45) 2q

1

m φm (x) dx

m=1 0

with rn − r(r − 1)q = qr 2 + r(2l + q) ∼ = Cn2 where C is a constant. So this is a balistic motion. In the second example, although moment of Sn f is of the n−1 the nsecond order of n, since Sn f 2 = I ( i=0 f (T x))2 dµ(x) = nf 2 , the behavior of the higher order moments depend on α. As this random walk is symmetrical with respect to zero, all odd moments are zero. Consider, for example, the moment of order 4. It behaves like n3 for rational α, and like n2 for all α having continous fraction expansion with bounded quotients. More generally, the moments of order 2k behave like n2k−1 in the rational case, and like nk for the irrational case with bounded quotients [4]. For all other cases, the behavior of the moments depend on the behavior of the quotients of the expansion of α excluding any asymptotic normal behavior for the normalized sum n1 εi (p1 (0), q1 (0))


31

Acknowledgments We would like to thank G.M. Zaslavsky for fruitful discussions on the plate billards.

32

References

[1] C. Beck and G. Roepstorff, From dynamical systems to the Langevin equation, Physica A, 145, 1-14, 1987. [2] P. Billingsley, Probability and measure, Wiley, NY, 1986 [3] P. Billingsley, Convergence of probability measures, Wiley, 1968 [4] M. Bernardo, M. Courbage , T.T. Truong , Multidimensional gaussian sums arising from distributions of sums of pairwise independent sequence of zero entropy, preprint LPTMC, and , Random walks generated by area preserving maps with zero Lyapounov exponents, Communications in Nonlinear Science and Numerical Simulations, 8, 189-199 (2003). [5] L.A. Bunimovich , Ya. G. Sinai, Statistical properties of Lorentz gas with periodic configuration of scatterers. Comm. Math. Phys. 78, 1981, 479–497. [6] L. A. Bunimovich, Ya. G. Sinai , N. I. Chernov, Markov partitions for twodimensional hyperbolic billiards. (Russian) Uspekhi Mat. Nauk. 45, 1990, no. 3 (273), 97–134; translation in Russian Math. Surveys , 45, 1990, no. 3, 105–152 [7] B.Burton and M. Denker, On the central limit theorem for dynamical systems, Trans. Amer. Math. Soc. 302, 121-134 [8] N.I. Chernov, L.S. Young, Decay of correlations for Lorentz gases and hard balls. Hard ball systems and the Lorentz gas, 89–120, Encyclopaedia Math. Sci., 101, Springer, Berlin, 2000. [9] P. Collet, A short ergodic theory refresher, in this volume. [10] I.P. Cornfeld, S.V. Fomin, Ya.G. Sinai, ”Ergodic theory”, Springer, New York, (1981). [11] M. Courbage, D. Hamdan, Decay of Correlation and mixing properties in a dynamical system with zero entropy. Ergod. Th. Dynam. Syst. 17, no.1, 87-103, 1997. [12] M. Courbage, D. Hamdan, Unpredictability in some nonchaotic dynamical systems, Phys.Rev.Lett. 74, 5166-5169, 1995. [13] G.R.Goodson, A survey of recent results in the spectral theory of ergodic dynamical systems , J. Dynam. Control Systems, 5, 173-226, 1999. [14] P.R. Halmos, Lectures in Ergodic Theory Chelsea Publishing Company, NY, 1956 [15] E.H. Hauge, What can we learn from Lorentz models, in G. Kirczenow and J. Marro ed. , Transport phenomena, p.337, Springer , 1974.

33 [16] E. Hopf, Proof of Gibbs hypothesis Proc.Natl. Acad. Sci. U.S.A. 18 , 333-340, 1932. [17] I.A. Ibragimov, Some limit theorems for stationary processes, Theory Prob. Appl., 7, 349-382, 1962. [18] B.O. Koopman, Hamiltonian systems and transformations in Hilbert space, Proc.Natl. Acad. Sci. U.S.A. 17 , 315-318, 1931. [19] B.O. Koopman and J. von Neumann, Dynamical systems and continuous spectra, Proc.Natl. Acad. Sci. U.S.A. 18, 255-263, 1932. [20] Michael T. Lacey, On weak convergence in dynamical systems to self-similar processes with spectral representation, Trans.Amer. Math. Soc. 328, 767-778, 1991. [21] C. Liverani ”Central limit for deterministic systems. ” in: International Conference on Dynamical Systems (Montevideo, 1995), 56–75, Pitman Res. Notes Math. Ser., 362, Longman, Harlow, 1996. [22] Donald S., Ornstein, Ergodic theory, randomness, and dynamical systems. Yale University Press, New Haven. [23] K.Petersen, Ergodic Theory, Cambridge University Press, 1983. [24] Martine Queffelec, Substitution Dynamical Systems-Spectral Analysis. LNM 1294, Springer -Verlag 1987. [25] M.E. Ratner, A central limit theorem for Y-flows on three-dimensional manifolds, Dokl. Akad. Nauk. SSSR, 186, 1969, 519-521, English translation : Soviet. Math. Dokl., 10, 629-631, 1969. [26] Ya.G. Sinai, The central limit theorem for geodesic flows on manifold with negative constant curvature, Soviet. Math.1, 983-986, 1960. [27] G.M. Zaslavsky and E. Edelman, Weak mixing and anomalous kinetics along filamented surfaces, CHAOS, 11 , 295-305, 2001.

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COMPLEXITY, FRACTAL DIMENSIONS AND TOPOLOGICAL ENTROPY IN DYNAMICAL SYSTEMS Valentin Affraimovich, Lev Glebsky IICO–UASLP Av. Karakorum 1470, Lomas 4a. San Luis Potosı, ´ SLP. M´ ´xico

Abstract

Instability of orbits in dynamical systems is the reason for their complex behavior. Main characteristics of this complexity are –complexity, topological entropy and fractal dimension. In this two lectures we give a short introduction to ideas, results and machinery of this part of modern nonlinear dynamics



37

PART I DYNAMICAL CHAOS IN TERMS OF THE -COMPLEXITY 1.

Definition of the (, n)-complexity.

Instability of orbits in a dynamical system causes complexity of their behaviour. There are some characteristics of instability. Among them are the local ones: for example Lyapunov exponents let measure the local rates of divergence of neighbouring orbits from a basic one; and the global ones: for example topological entropy that measures the number of –different types of segments of orbits of temporal lenghts t as t goes to infinity. Here we shall deal with the –complexity function. It is a global characteristic of instability that shows, roughly speaking, how many from very close to each other at the initial moment orbits became –recognizable at the moment t. It seems to us that the –complexity is worth studying in details because of the following: It is pre–limiting quantity — this function contains an information not only about asymptotics but about fluctuations as well; There is a direct link to fractal dimensions and the topological entropy, main parameters of dynamical chaos; It contains a non-trivial information not only in hyperbolic situations but also in “polynomial” ones.

1.1

(, n)-complexity

The definition of (, n)-complexity will be discussed following mainly Bowen [19]. Let us consider a dynamical system generated by an evolution operator f on the phase space M . Let A ⊂ Mbe a subset of initial points k (it could be invariant or not) and n (x) = n−1 k=0 f x an orbit segment of temporal length n going through an initial point x ∈ A. Two segments n (x) and n (y), x, y ∈ A, are said to be (, n)-separated if there exists k, 0 ≤ k ≤ n−1, such that dist(f k x, f k y) ≥ , where dist means distance in the phase space M . The maximal possible number of distinct segments of orbits with accuracy is defined by C,n (A) = max{#segments in a (, n) − mutually separated set}

(1)

38 and is said to be (, n)-complexity of the set A. One may define it in another way. Given a metric d on M introduce the metric dn as follows dn (x, y) =

max d(f i x, f i y).

0≤i≤n−1

Definition 1.1 1 Given > 0, n > 0, A ∈ M , a set Y ∈ A is (, n)–separated iff for any diferent x, y ∈ Y one has dn (x, y) ≥ . 2 The number C,n (A) = max{|Y |, Y is an (, n)–separated set}, where | · | denotes the cardinality of a set, i.e., the number of points in it, is called the (, n)–complexity of A It was shown in [B] (see also [24] for compact invariant subset of initial points) that h = htop (A) =: lim limn→∞ →0

log C,n (A) n

(2)

is the topological entropy of the dynamical system (f k , M ) on the set A, and in [T] (for Lipschitz–continuous f ) that b =: lim lim→0 n→∞

log C,n − log

(3)

is the upper box dimension of the set A. Thus, we may think that if 0 < b < ∞, 0 < h < ∞ then ¯ n) C,n (A) = −b · ehn · C(,

(4)

¯ n) is a subexponential function of ln and n. where C(, To illustrate the property (2), consider one-dimensional mixing dynamics on the interval [0, ], x ∈ [0, ] with an exponential divergence of trajectories. Let δx0 is the initial distance at t = 0 between two trajectories and δxt ≥ is the distance at time t. Then ≤ δx0 exp ht

(5)

i.e. for τ ≥ t two trajectories are separated. The number of such trajectories is A A ht e C,t (A) = (6) δx0 where A is the length of a small initial interval A. The properties (1) and (2) follow directly from (6) for b = 1. If the set A has the box-dimension b, then (5) should be replaced by (/δx0 )b ≤ eht

(7)

39

Dynamical Chaos in Terms of the -complexity

and correspondingly, instead of (6) C,t (A) (A /δx0 )b = (A /)b eht

(8)

For more general situations we may assume ¯ t) C,t (A) = (A /)b eht C(,

(9)

¯ t) is a slow varying function of ln and t. where C(, The expression (9) shows in an explicit way how the (, n)-complexity depends on the time interval t, accuracy and the domain A of a set of initial conditions. Remark 1.1 Complexity of an orbit. k If one chooses an orbit Γ(x0 ) = ∪∞ k=0 f x0 in the capacity of the set A of initial points, one will arrive to a definition of the -complexity of the orbit Γ(x0 ). It is not difficult to show that for any small δ > 0, C(1+δ),n (clos(Γ(x0 ))) ≤ C,n (Γ(x0 )) ≤ C,n (clos(Γ(x0 ))) , i.e., the complexity of the closure of an orbit asymptotically behaves in the same way as the complexity of the orbit. Furthermore, making use of the definition of complexity for an arbitrary set A, one may introduce the complexity of a measure. Definition 1.2 Given an invariant measure µ, the quantity C,t (µ) := inf {C,t (A) | µ(A) = 1} A

(10)

ia called the complexity of the measure µ.

1.2

Hausdorff and Box dimensions

Let X be a metric space with a distance d(x, y), x, y ∈ X. For any subset Z ⊂ X let {Ui } be a finite or countable collection of open sets of diameter less than such that Ui ⊃ Z; here diamUi := sup{d(x, y) : x, y ∈ Ui }. For any α > 0 we introduce (diam Ui )α , (11) m(α, , Z) = inf {Ui }

i

where infimum is taken over all covers {Ui } with diameter ≤ , and m(α, Z) = lim m(α, , Z), →0

(12)

40 the α–dimensional Hausdorff measure (the limit exists because of monotonicity of m(α, , Z) as a function of ). It is simple to see that m(β, , Z) ≤β−α m(α, , Z),which implies thatthere existsa unique critical value αc of α such that m(α, Z) = 0 if α > αc and m(α, Z) = ∞ if α < αc . The quantity αc =: dimH Z is called the Hausdorff dimension. We already mentioned the upper box dimension. It could be defined in a different way. In the sum 11 one may consider open set of diameter equals , i.e., m(α, , Z) = N · α where N is the number of such sets. Then m(α, Z) = lim sup→0 m(α, , Z), m(α, Z) = lim inf →0 m(α, , Z), and dimB Z = sup{α|m(α, Z) = ∞} is the upper box dimension, and dimB Z = sup{α|m(α, Z) = ∞} is the lower box dimension. If dimB Z = dimB Z = b then b is called the box dimension. Example 1.1 Let J be an invariant set of the map g : [0, λ0 ] ∪ [1 − λ1 , 1] → [0, 1], x/λ0 if x ∈ [0, λ0 ], g(x) = (13) 1−λ1 x if x ∈ [1 − λ1 , 1] λ1 − λ1 where 0 < λ0 < λ1 < 1, λ0 + λ1 < 1, consisting of all points of all orbits belonging to [0, 1]. It is clear that J is a Cantor set. Proposition 1.1 For the set J we have that dimH J = s0 , where s = s0 is the root of the equation λs0 + λs1 = 1.

(14)

To make (14) evident, consider the cover of J by basic sets of the n–th generation. Then the sum i (diam Ui )α in (11), up to a constant, becomes n−1 λαik = (λα0 + λα1 )n . (15) i0 ,...,in−1 k=0

If α > s0 , then (15) goes to zero as n → ∞, that shows us that dimH J ≤ s0 . To get the opposite inequality, people use the technique of so–called Moran covers [34]. Similar formulas could be obtained in the case when not all words are admissible, i.e., in the case of subshifts. In these cases Hausdorff dimensions of invariant sets can be expressed in terms of topological pressure. It was R. Bowen who introduced this quantity in the theory of dynamical systems [34, 20].

2.

Dynamical Chaos

We give a definition of dynamical chaos following mainly Takens’ ideas see [39, 40] and [8].


41

We say that a signal (an observable) φ(t) is generated by a dynamical system (f t , M ) if there exist an initial point x0 belonging to M and the function ψ : M → R such that φ(t) = ψ(f t x0 ). Consider the case of discrete time t ∈ Z. Introduce space of all bounded observables b = {a = (a0 , a1 , . . .), ai = φ(i) ∈ R} and endow it |ai | with the norm ||a|| = ∞ 0 2i , so that B becomes a Banach space. The shift map T a = (a1 , a2 , . . .) generates a “universal” dynamical system that generates all bounded observables. i Fix an observable a = (a0 , a1 , . . .) and denote by M the set clos ∞ i=0 T a , the set of limiting points of the orbit {T i a }. We say that an observable a = ( a0 , a1 , . . . ) is deterministically generated if 1 The fractal dimension dimB M = b < ∞

(16)

2 The topological entropy of f = T |M , h(f ) < ∞.

(17)

Because of the Ma˜ ne Theorem (see for instance[8]), the condition 16 implies that, generally speaking, for any n ≥ 2b + 1 the system (f t , M ) is topologically conjugated to the system generated by the map Tˆ : (ak , . . . , ak+n−1 ) → (ak+1 , . . . , ak+n ). It gives a clue how to “reconstruct” the original system (f t , M ) by knowing the only observable a . According to many specialists begining with Alekseyev [12], we deal with dynamical chaos if 0 < h(f ) < ∞. Thus, in terms of –complexity dynamical chaos means, in fact, the validity of ( 4) with 0 ≤ b < ∞, 0 < h < ∞.

3.

Symbolic dynamics

If one chooses a partition of the phase space by finitely many pieces, labels them by symbols, say {0, . . . , p − 1}, and denotes the position of the point at the moment n by the symbol an ∈ {0, . . . , p − 1}, then one determines a symbolic sequence corresponding to the orbit with the shift map corresponding to the evolution operator. More rigurously, given + p ∈ N, consider the set Ωp = {0, · · · , p − 1}Z endowed with the product topology and a distance compatible with this topology. A subshift is the

42 dynamical system (Ω, σ) where σ is the shift operator and Ω ⊂ Ωp is a closed and σ–invariant subset. A point in Ω is denoted by ω = {ωk }k≥0 . Given the word ω0 · · · ωn , ωi ∈ {0, · · · , p − 1}, the subset [ω0 · · · ωn ] := {ω ∈ Ω : ωi = ωi , 0 ≤ i ≤ n} is called a cylinder. Every cylinder is open and closed. A word ω0 · · · ωn , ωi ∈ {0, · · · , p − 1}, is said to be admissible in Ω if the corresponding cylinder [ω0 · · · ωn ] is not empty. There are many distances on Ω corresponding to the topology in which Ω is compact. For example let (λ0 , . . . , λq−1 ), 0 < λi < 1, be real numbers. Then, for every pair of points ω, ω ∈ Ω define the real non– negative function λω0 · · · λωi−1 , if ωk = ωk , 0 ≤ k < i, ωi = ωi dΩ (ω, ω ) = (18) 1, if ω0 = ω0 and dΩ (ω, ω) = 0. Very often people use this metric with λi = e−1 . A subshift (Ω, σ) is said to be (topologically) mixing if there exists k0 ∈ Z+ (the mixing time) such that for any admissible words ω0 · · · ωn and ω0 · · · ωm and any k ≥ k0 , there exists a word ω0 · · · ωk such that is admissible. the concatenated word ω0 · · · ωn ω0 · · · ωk ω0 · · · ωm The subshift (Ω, σ) is said to be specified (or to have the specification property) if there exists n0 ∈ Z+ such that for any pair of admissible words, ω0 · · · ωn and ω0 · · · ωm , there exists k ≤ n0 and a word ω0 · · · ωk is admissisuch that the concatenated word ω0 · · · ωn ω0 · · · ωk ω0 · · · ωm ble. For very important subshifts, so called topological Markov chains, a set Ω is defined by a transition matrix A, i.e. ω ∈ Ω iff Aωk ,ωk+1 = 1, k ≥ 0. Any mixing topological Markov chain is specified and has positive topological entropy.

3.1

Multipermutative systems

Among non–trivial systems with zero topological entropy the simplest one are multipermutative systems. Let Ω = {0, 1, . . . , q − 1}N0 with the metric (18). Definition 3.1 [2] A map T : Ω → Ω is said to be multipermutative if for every ω ∈ Ω the sequence T ω is given by T ω = (ω0 + p0 , ω1 + p1 (ω0 ), . . . , ωi + pi (ω0 , . . . , ωi−1 ), . . . ) with pi : Ai → A for i > 0 and p0 ∈ A = {0, . . . , q − 1}. At every coordinate addition is understood to be modulo q.


43

Cylinders of length L are denoted by ω L := [ω0 , . . . , ωL−1 ] ⊂ Ω, and i they determine the integer value ||ω L ||q = L−1 i=0 ωi q . Example 3.1 The q–adic adding machine is a multipermutative system (Ω, S) such that Sω = (ω0 + 1, ω1 + s1 (ω0 ), . . . , ωi + si (ω0 , . . . , ωi−1 ), . . . ), with si (ω0 , . . . , ωi−1 ) = 1 if (ω0 , . . . , ωi−1 ) is maximal and si (ω0 , . . . , ωi−1 ) = 0 otherwise. The word (ω0 , . . . , ωi−1 ) is maximal when ωj = q − 1 for j = 0, . . . , i − 1. For every L ≥ 1, a map {0, 1, . . . , qL − 1} → {0, 1, . . . , q − 1}L is well–defined where n → S n 0L = ω L is a bijection, and n = ||ω L ||q . For every L ≥ 1, the set {S n 0L : n = 0, 1, . . . , q L − 1} is a cycle of period qL . The next result is a dynamical characterization of minimal multipermutative systems. Theorem 1 [2] For (Ω, T ), a multipermutative system, the following four statements are equivalent. 1). (Ω, T ) is minimal. 2). For every L > 0 and every cylinder ω L := [ω0 , . . . , ωL−1 ], the sequence ω L , T ω L , T 2 ω L , . . . is periodic with smallest period qL . 3). For every L > 0, the numbers (“integrals”) πL := ω pL (ω L ) and L the constant p0 ∈ A are relatively prime to q. 4). (Ω, T )is topologically conjugate to the q–adic adding machine (Ω, S). The theorem tells us about universality of multipermutative systems in the case when one has the same number of symbols for every level. The next section is devoted to a more general situation.

Polysymbolic generalization. We consider now multipermutative systems with different alphabets Ai = {0, . . . , qi −1} at every coordinate i = 0, 1, . . . . Such systems will be referred to as polysymbolic systems. The set of sequences is Ωq∗ = A0 × A1 × · · · , and the size of alphabets at every coordinate i is denoted by the sequence q∗ = (q0 , q1 , . . .) of positive integers qi = card(Ai ), i ≥ 0. A polysymbolic system (Ωq∗ , T ) is multipermutative if for every ω = (ω0 , ω1 , . . . ) the map T : Ωq∗ → Ωq∗ is defined by T ω = (ω0 + p0 , ω1 + p1 (ω0 ), . . . , ωi + pi (ω0 , . . . , ωi−1 ), . . . ) with pi : A0 × · · · × Ai−1 → Ai and p0 ∈ A0 .

(19)

44 A polyadic adding machine (Ωq∗ , S) (called an odometer, too) is defined as the usual q–adic adding machine, except that a word (ω0 , . . . , ωi−1 ) ∈ A0 ×· · · ×Ai−1 is maximal when ωj = qj − 1 for each j = 0, . . . , i − 1. Theorem 1 is extended to polysymbolic systems as follows. Theorem 2 Let (Ωq∗ , T ) be a multipermutative system which is polysymbolic. Then the following statements are equivalent. (P1). (Ωq∗ , T ) is minimal. (P2). For every L > 0 and every cylinder ω L := [ω0 , . . . , ωL−1 ] the se quence ω L , T ω L , T 2 ωL , . . . is periodic with smallest period L−1 i=0 qi . (P3). For every L > 0 the numbers πL := ω pL (ω L ) and the constant L p0 ∈ A are relatively prime to qL and q0 , respectively. (P4). (Ωq∗ , T ) is topologically conjugate to the polyadic adding machine (Ωq∗ , S). Remark 3.1 If a multipermutative system is not minimal then its phase space can be partitioned into an infinite collection of invariant minimal sets and on each of them the system behaves in the way described in the theorem.

4.

Invariant sets as a result of inductive procedures

The global behavior of orbits could be very complex (here, an (semi– i )orbit through an initial point x0 is Γ(x0 ) := ∞ f x0 ; a union of i=0 orbits Y is an invariant set: f (Y ) ⊂ Y ). Complexity of such a behavior is reflected in the geometry of invariant sets and can be measured by Hausdorff and box dimensions and other dimension–like characteristics. Invariant sets are constructed by using methods of symbolic dynamics.

4.1

Geometric constructions for repellers

Many invariant sets are resulting from so–called geometric constructions [34]. Let (σ, Ω), Ω ⊂ Ωp = {0, . . . , p−1}N , be a subshift, a closed σ– invariant subset of the full shift with p symbols. The word (i0 , . . . , in−1 ) is admissible if the corresponding cylinder [i0 , . . . , in−1 ] has nonempty intersection with Ω. Consider p closed subsets ∆0 , . . . , ∆p−1 ⊂ Rm . Define basic sets ∆i0 ,...,in−1 which satisfy the following assumptions: (A). ∆i0 ,...,in−1 are closed and nonempty if (i0 , . . . , in−1 ) is admissible. (B). ∆i0 ,...,in−1 j ⊂ ∆i0 ,...,in−1 , j = 0, . . . , p − 1.


45

(C). diam∆i0 ,...,in−1 → 0 as n → ∞. We can define now a nonempty set ∞

F =

∆i0 ,...,in−1 .

(20)

n=1 (i0 ,...,in−1 )

The closed set F becomes a Cantor set, provided that the following “separation conditions” hold (D). ∆i0 ,...,in−1 ∩∆j0 ,...,jn−1 ∩F = ∅ whenever (i0 , . . . , in−1 ) = (j0 , . . . , jn−1 ). The coding map χ : Ω! → F is defined as follows: for any ω= (i0 , . . . , in−1 , . . .) ∈ Ω, χ(ω) = x if x ∈ ∆i0 ,...,in−1 . The simplest constructions are of Moran type. In this case Ω = Ωp and basic sets satisfy additional axioms.

(M1). Every basic set is the closure of its interior. (M2). For any n, Int∆i0 ,...,in−1 ∩ Int∆j0 ,...,jn−1 = ∅ if (i0 , . . . , in−1 ) = (j0 , . . . , jn−1 ). (M3). The basic set ∆i0 ,...,in−1 ,j is homeomorphic to ∆i0 ,...,in−1 . (M4). There are numbers 0 < λj < 1, j = 0, . . . , p − 1, such that diam∆i0 ,...,in−1 ,j = λj diam∆i0 ,...,in−1 . Moran proved that in this case dimH F = s0 , where s = s0 is the root of the (Moran) equation p−1 λsi = 1. (21) i=0

Conditions (M1)–(M4) provide a more general geometric scenario than the one presented in [35]. In all these cases the Hausdorff and box dimensions of the set F coincide and are equal to the root sλ of an equation of the Bowen type. For our construction this equation is PS (sλ log λ) = 0, where PS (φ) is the topological pressure of the function φ on the set S with respect to σ (see below). The set J is constructed with the help of the contractions u0,1 : [0, 1] → [0, 1], u0 (x) = λ0 x,

u1 (x) = λ1 x + 1 − λ1 ,

such that g ◦ ui = id on [0, 1]. For every word i = (w0 , · · · , wi−1 ) ∈ {0, 1}i , define the sets ∆w0 ,··· ,wi−1 := uwi−1 ◦ · · · ◦ uw0 ([0, 1]),

46 i.e., the ∆–sets are basic sets of the geometric construction for the set J. Moreover, diam ∆w0 ,··· ,wi−1 = λw0 · · · λwi−1 and dist(∆i0 , ∆i1 ) = (1 − λ0 − λ1 )λw0 · · · λwi−1 > 0

(22)

where dist(x, y) = |x − y|. Thus, J is resulting from a Moran construction.

4.2

Sticky sets in Hamiltonian systems

An area preserving map f of the plane, possessing an infinite hierarchy of islands–around–islands structure, has invariant sets of zero Lebesgue measure on which it behaves similarly to multipermuitative systems [9, 2]. The map T , generating a multipermutative system, is not chaotic and its topological entropy is zero. A set F on which f is topologically conjugate to T , nevertheless, may appear as a result of a Moran type geometric construction. Sticky sets are the sets of all limiting points of infinite hierarchy of islands. A closed topological disk P is said to be an island of stability if f n (P ) = P for some integer n. We now give a definition of infinite hierarchy of islands–around–islands structure (sticky riddle) for the general case when not all words i = (i0 , . . . , in−1 ) might be admissible. A collection P of islands {Pi : i is Ω–admissible} is said to be a sticky riddle if the sets Pi are pairwise disjoint, are contained in a compact set, and (i) for any island Pi ∈ P there is an island Pj ∈ P, |i| = |j|, such that f (Pi ) = Pj ; (ii) if f (Pi ) = Pj then for any admissible ik there is s ∈ {0, 1, . . . , q−1} such that f (Pik ) = Pjs ; (iii) diam(Pi ) → 0 as |i| → ∞; (iv) for any ω = (i0 , i1 , . . .) ∈ Ω, if xn ∈ Pi0 ,...,in−1 , n > 0, then limn→∞ xn exists; (v) if xn ∈ Pi , yn ∈ Pj , |i| = |j| = n, n > 0, and i = j at least for one value of n then limn→∞ xn = limn→∞ yn . These axioms reflect our understanding of an infinite islands–around– islands hierarchy: (i) an island of the n–th generation is mapped into an island of the same generation;


47

(ii) if an island Pik lies in the vicinity of the island Pi then its image Pjs lies in a vicinity of Pj ; (iii) to be packed into a compact set, the islands of the n–th generation should be small if n >> 1; (iv) there should be only one point of accumulation of islands Pi0 ,...,in−1 for any fixed ω = (i0 , . . . , in−1 , . . .); (v) for different points ω = (ω0 , ω1 , . . .), ω = (ω0 , ω1 , . . .) in Ω the corresponding points of accumulation of islands should be different. Let P be a sticky riddle. For any ω = (i0 , i1 , . . .) ∈ Ω and any sequence xn ∈ Pi0 ,...,in−1 , define x = x(ω) := limn→∞ xn . The set Λ = {x(ω) : ω ∈ Ω} is said to be a sticky set. It is well defined thanks to Axioms (iii)–(v). It was shown in [2] that f |Λ is topologically conjugate to a multipermutative system, i.e., f |Λ has zero topological entropy.

4.3

Geometric constructions of sticky sets

Some numerical observation [15] show that sometimes every island of stability Pi , together with all its satellites Pij , belongs to a basic set ∆i of a geometric construction. So, the set Λ can be resulted from this construction. Axiomatically, the conditions for that can be expressed as follows. (P1) There exists a collection of sets {∆i : i is admissible} that are closed, and for each admissible word i, Pij ⊂ ∆i for every admissible word ij. (P2) Pi ∩ ∆ij = ∅ for every admissible i and ij. (P3) ∆ij ⊂ ∆i , for every admissible words i and ij. (P4) diam∆i0 ...in−1 → 0 as n → ∞. (P5) Separation axiom. ∆i ∩ ∆j ∩ F = ∅ if i = j, |i| = |j|, where ∞

n=1

i0 ,...,in−1

F =

∆i0 ...in−1

is admissible

Thus, if these axioms are satisfied, then Λ = F . Let us emphasize that an invariant set with nonchaotic dynamics is resulted from a geometric construction, modeled by a full subshift (σ, Ωp ) or a subshift with positive topological entropy. In other words, we have a big difference between

48 temporal and spatial behavior of a system. To describe such a situation, we need characteristics which could take into account both temporal and spatial behavior. We introduce them in the following Lecture.

Measures of –complexity

5.

To study complexity functions in details, one needs to know more about general properties of the ε-complexity of a metric space (without dynamics). We introduce and describe some quantities which contain an essential information about ε-complexity, the measures of ε-complexity in an “abstract” metric space. The main results will be related to the εcomplexity defined on the base of the notion of ε-separability. The notion was used first by Kolmogorov and Tikhomirov [29] in their study of solutions of PDE and realization of random processes (Shannon suggested to pay attention to this notions in 1949, though). We will also describe ε-complexities based on the notion of ε-nets. We prove that measure of ε-complexity defined on the base of the notion of ε-separability is equivalent to the dual measure that is defined through ε-nets. [11] It appeared naturally that some results and ideas from discrete mathematics are worth to be exploited. We believe that we made the first step in this direction.

5.1

Set-up and definitions

Separated sets and complexity. space with a distance d.

Let X, d be a compact metric

Definition 5.1 1 Given > 0, a set Y ⊆ X is -separated iff for any different x, y ∈ Y one has d(x, y) ≥ . 2 The number C (X, d) = C := max{|Y |, Y is an -separated set}, where | · | denotes the cardinality of a set, is called the -complexity of X. 3 An -separated set Y is optimal iff |Y | = C . Let us show the following natural inequality. Proposition 5.1 Given D1 , D2 ⊆ X and ε > 0 one has Cε (D1 ∪ D2 ) ≤ Cε (D1 ) + Cε (D2 ).

49


Proof 5.1 Let Y ⊆ D1 ∪ D2 be an optimal ε-separated set in D1 ∪ D2 . Then Yi = Y ∩ Di is an ε-separated set in Di and |Y | ≤ |Y1 | + |Y2 | ≤ Cε (D1 ) + Cε (D2 ). Remark 5.1 Invariant sets in some dynamical systems can be treated as results of inductive procedures. Come back to the example 1.1 with λ1 = λ2 = 13 The dynamical system generated by the map f : R → R, 3x, x ≤ 1/2, f (x) = 3x − 3, x > 1/2, has an invariant set K containing all orbits belonging to the interval [0, 1]. One can see that K is the one-third Cantor set, so that ∞

K=

∆i0 ...in−1 ,

n=1 (i0 ...in−1 )

where ij ∈ {0, 1}, ∆i0 ...in−1 are intervals of the length 3−n arising on the n-th step of construction of the Cantor set. Therefore, if ε ≈ 3−n then Cε ≈ 2n = {the number of different words of length n in the full shift with 2 symbols} = ehn , where h = ln 2 is the topological entropy of the full shift. Thus, ln Cε ln 2 h ≈ = = dimH K, − ln ε − ln 1/3 ln λ where dimH K is the Hausdorff dimension of K and λ = 1/3 is the contraction coefficient. We obtained the familiar Furstenberg formula [25]. This example shows that if a subset of a metric space is the result of an inductive procedure governed by a symbolic dynamical system then the ε-complexity contains, in fact, an important dynamical information. In this subsection we give a dual definition ε-nets and complexity. of complexity. Given x ∈ X let O (x) = {y : d(x,y) < }, the ball of O (x). radius centered at x. Given Y ⊆ X let O (Y ) = x∈Y

1 Given > 0, a set Y ⊆ X is an -net iff Oε (Y ) =

Definition 5.2 X. 2 The number

R (X, d) = R := min{|Y |, Y is an -net},

50 is called the dual -complexity of X. 3 An -net Y is optimal iff |Y | = R . The similar results to the one in Proposition 5.1 holds for dual complexities. Proposition 5.2 Given D1 , D2 ⊆ X and ε > 0 one has Rε (D1 ∪ D2 ) ≤ Rε (D1 ) + Rε (D2 ). Proof 5.2 Let Yi ⊆ Di be an optimal ε-net in Di . Then Y = Y1 ∪ Y2 is an ε-net in D1 ∪ D2 and Rε (D1 ∪ D2 ) ≤ |Y | ≤ |Y1 | + |Y2 | = Rε (D1 ) + Rε (D2 ). Any optimal ε-separated set is an ε net, therefore Cε ≥ Rε . On the other hand the following statement holds. Proposition 5.3 Rε/2 ≥ Cε Proof 5.3 It follows directly from the definition that any pair of different points in an ε-separated set Z can not belong to a ball of radius ε/2. Thus we cannot cover Z by less than |Z| balls of radius ε/2. Assuming that Z is optimal we obtain the inequality above. Let us introduce bε = sup Rε/2 (Oε (x)). x∈X

Obviously, for any D ⊆ X one has bε Rε (D) ≥ Rε/2 (D). It is not difficult to check that bε ≤ 2d (2d + 1) for a subset of the Euclidean space Rd .

Ultrafilters. Now we give some known results and definitions that can be found, for instance, in [18]. Definition 5.3 A set F ⊂ 2N is called to be a filter over N iff it satisfies the following conditions: If A ∈ F and B ∈ F, then A ∩ B ∈ F, If A ∈ F and A ⊂ B then B ∈ F, ∅ ∈ F. Let an be a sequences of real numbers, a is called to be a limit of an with respect to a filter F, a = limF an , if for any ε > 0 one has {n | |an − a| < ε} ∈ F. ¿From the definition of a filter it follows that limF an is unique, if exists.


51

Example 5.1 Let FF = {A ⊆ N | N\A is finite }. FF is said to be a Frechét filter. One can check that it is, indeed, a filter. A limit with respect to FF coincides with ordinary limit. Definition 5.4 A filter F is called to be ultrafilter iff for any set A ⊆ N one has A ∈ F or N\A ∈ F. Theorem 3 A bounded sequences has a limit with respect to an ultrafilter. This limit is unique. Example 5.2 For i ∈ N let Fi = {A ⊆ N | i ∈ A}. It is an ultrafilter. Such an ultrafilter is called proper for i. One can check that limFi an = ai . So, limits with respect to a proper ultrafilter are not interesting. Proposition 5.4 An ultrafilter F is proper (for some i ∈ N) if and only if it contains a finite set. This proposition implies that an ultrafilter is non-proper if and only if it is an extension of the Frechét filter FF .

5.2

Measures of complexity

Our goal is to define a measure reflecting an asymptotic behavior of the -complexity as goes to 0. For that we will use the technique of ultrafilters. Given > 0, consider an optimal -separated set A . Introduce the following functional 1 I (φ) = φ(x) C x∈A

where φ : X → R is a continuous function. It is clear that I is a positive bounded linear functional on C(X). Moreover, for any φ ∈ C(X) the family I (φ) is bounded. Fix a sequence E = {εn }, εn → 0 as n → ∞ and an arbitrary non-proper ultrafilter F. Consider I(φ) = lim Iεn (φ). F

I is a positive bounded linear functional on C(X). Theorem 4 The functional I is independent of the choice of an optimal sets Aε . Proof 5.4 The proof is based on the following proposition. Proposition 5.5 Let A and B be optimal -separated sets. There exists a one-to-one map α : A → B such that d(x, α(x)) ≤ for any x ∈ A.

52 Let Aε and Bε be optimal -separated sets, ε ∈ E. Let αε : Aε → Bε be the map from Proposition 5.5. Then 1 1 1 | φ(x) − φ(x)| = | (φ(x) − φ(αε (x))) | ≤ rφ () Cε Cε Cε x∈Aε

x∈Bε

x∈Aε

where rφ () = sup{|φ(x)−φ(y)| : d(x, y) < }, the modulus of continuity of φ. Since X is a compact, rφ () → 0 as → 0. It implies the desired result due to the choice of the ultrafilter F. So, we need only to prove Proposition 5.5; it will be done below. In the proof of Proposition 5.5 we will need the Marriage Lemma of P. Hall, see for instance [37]. Lemma 5.1 For an indexed collections of finite sets F1 , F2 , . . . , Fk the following conditions are equivalent: there exists an injective function α : {1, 2, ..., k} →

k

Fi such that

i=1

α(i) ∈ Fi ; For all S ⊆ {1, 2, . . . , k} one has |

Fi | ≥ |S|.

i∈S

y) < }, the ball of radius centered Recall that O (x) = {y : d(x, at x. Given Y ⊆ X let O (Y ) = O (x). x∈Y

Proof 5.5 Proposition 5.5 For any x ∈ A let Bx = O (x) ∩ B. If we show that for any S ⊆ A the following inequality holds | Bx | ≥ |S|, (24) x∈S

then the proposition follows from Lemma 5.1 due to |A| = |B| = C . To Bx | = |O (S) ∩ B| < |S| for prove inequalities (24), suppose that | some S ⊆ A. Then

x∈S

|S ∪ (B \ (O (S) ∩ B)| = |S| + (|B| − |O (S) ∩ B|) > |B| = C , on the other hand, the set S ∪ (B \ (O (S) ∩ B) is -separated. We have a contradiction with optimality of B. So, we have defined a functional I which may depend on the choice of the sequence E and the ultrafiter F only. Sometimes we will write IE,F to emphasize this dependence. It is well known, that IE,F generate unique regular Borel measure µE,F on X such that µE,F (X) = 1.


53

Definition 5.5 The measures µE,F (X) will be called measures of complexity. We are going to show examples of (X, d) when µE,F = µ is independent on E, F and when µE,F depends on E, F. In the first case IE,F (φ) = I(φ) = lim Iε (φ). ε→0

Of course, it is difficult to find optimal sets and construct directly measures of complexity in real situations. Nevertheless, it is possible to work with them by using some of their intrinsic properties. Let us show now that measures of complexity are invariant with respect to local isometries. Definition 5.6 A homeomorphism τ : X → X is called to be ε-isometry iff d(x, y) = d(τ (x), τ (y)) for all x, y ∈ X, d(x, y) ≤ ε A homeomorphism τ : X → X is called to be local isometry iff it is ε isometry for some ε > 0. It is clear that an isometry is a local isometry. Proposition 5.6 Local isometries with composition form a group. Proof 5.6 It is easy to check that the composition of two ε-isometries is an ε-isometry. Let τ be an ε-isometry. Then τ −1 is uniformly continuous and there exists ε > 0 such that if d(x, y) ≤ ε ,then d(τ −1 (x), τ −1 (y)) ≤ ε.Consequently,if d(x, y) ≤ ε then d(x, y) = d(τ −1 (x), τ −1 (y)), so, τ −1 is an ε -isometry. We do not know if ε-isometries form a group. Proposition 5.7 Let τ be an ε0 -isometry and A be an ε-separated set, ε ≤ ε0 . Then τ −1 (A) is also ε-separated. Proof 5.7 Assume, on the contrary, that τ −1 (A) is not ε-separated, i.e., there are different x, y ∈ τ −1 (A) with d(x, y) < ε ≤ ε0 . Then d(τ (x), τ (y)) = d(x, y) < ε, so A cannot be ε-separated. Theorem 5 Let τ be a local isometry. Then µE,F is invariant, i.e. µE,F (A) = µE,F (τ −1 (A)) for all measurable A. Proof 5.8 It is enough to show that for all φ ∈ C(X) IF (φ ◦ τ ) = IF (φ).

(25)

There exists ε0 > 0, such that τ is an 0 -isometry. Let Aε be an optimal ε-separated set, ε ≤ ε0 . It follows from Proposition 5.7 that τ −1 (Aε )

54 is an optimal ε-separated set. It implies the validity of Equation (25). Indeed, I (φ) =

1 1 φ(x), I (φ ◦ τ ) = C C x∈A

x∈τ −1 (A

φ(x), )

and the result follows from Theorem 4. Corollary 5.1 Let a continuous group operation ∗ be defind on X such that right shifts rg (x) = g ∗x (left shifts lg (x) = x∗g) are local isometries for all g ∈ X. Then µE,F is the normalized Haar measure on (X, ∗). In particular, µE,F does not depend on E, F. Example 5.3 Let X = Ωp , the full shift with p symbols, i.e. Ωp = + {0, 1, ..., p − 1}Z with the distance dq (x, y) =

∞ |xi − yi | i=0

qi

, q > 1.

Ωp can be equipped by the group operation ⊕ as follows: (x ⊕ y)i = xi + yi ,

mod p

It is clear that (Ωp , ⊕) is a continuous group. Moreover, the right translation by any element is an isometry. Therefore µE,F = µ coincides with the Haar measure which, in fact, is the (1/p, ..., 1/p)-Bernoulli measure. Example 5.4 Let X = ΩM be a topological Markov chain, defined by a finite matrix M : {0, 1, ..., p − 1}2 → {0, 1}, i.e. ΩM = {< x0 , x1 , ... > | xi ∈ {1, 2, ..., p − 1} and M (xi , xi+1 ) = 1}. Metric d is the same as in Example 5.3. Cylinder [a0 , a1 , ..., an−1 ] of the length n is the set of all x ∈ ΩM , such that xi = ai for i = 0, 1, ..., n − 1. A word < a0 , a1 , ..., an−1 > is admissible iff [a0 , a2 , ..., an−1 ] = ∅. Let Wn be the set of all admissible words of the length n and α be a permutation of Wn such that (α(w))n−1 = wn−1 for every w ∈ Wn (admissible permutation). Given such an α define gα : X → X as follows gα (x) = (α(x0 , x1 , ..., xn−1 ), xn , xn+1 , ...). It is simple to see that gα is a local isometry. It implies that µE,F ([a0 , a1 ,..., , 0,b1 ...,bn−1 ])if [a0,a1,...,an−1]= ∅, [b0 , b1 , ..., bn−1 ] = ∅ and an−1 ])=µE,F ([b an−1 = bn−1 . Indeed, under these assumptions there exists an admissible permutation α : Wn → Wn such that α(a0 , a1 , ..., an−1 ) = b0 , b1 , ..., bn−1 .


55

So, the measure µE,F of a nonempty cylinder [a0 , a1 , ..., an−1 ] depends only on an−1 and n. Let vi (n) = µE,F ([a0 , a1 , ..., an−2 , i]) for an admissible < a0 , a1 , ..., an−2 , i > (vi (n) = 0 if there is no admissible words of length n ending by i). It is simple to check that vj (n + 1) vi (n) = j,M (i,j)=1

This relation can be rewritten in the matrix form v(n) = M v(n + 1), where v(n) = (v0 (n), v1 (n), ..., vp−1 (n))T is a column vector. If M is a primitive matrix (M p > 0 for some p) then this equation uniquely defines the measure µE,F , which in this case turns out to be independent of E, F. Indeed, by Perron Theorem matrix M has unique positive eigenvector e with eigenvalue λ > 0 (in our case, in fact, λ > 1). Let P be the set of all lines in Rp , generated by non-negative vectors. From the proof of Perron Theorem (see, for example, [28]) M n (P ) = {le }, n∈N

where le is a line, generated by e. Since v(n) > 0 and v(k) = M n v(n+k), one has lv(k) ∈ M n (P ) for any n. Hence, v(k) = ck e. So, v(n) = λ−n c0 e. We have proved the following Proposition 5.8 Let M be a primitive matrix and C ⊂ ΩM is an admissible cylinder of length n, ending by i. Then µE,F (C) = λ−n ei , where (e0 , e1 , ..., ep−1 ) is the positive eigenvector of M , with e0 +e1 +...ep−1 = 1. Remark 5.2 It was constructed in [11] an example where µE,F is not unique.

5.3

Measures of dual complexity

To define measures of dual complexity we proceed in the same way as in Subsection 5.2, just replacing ε-separated sets by ε-nets. Given > 0, consider an optimal -net A . Introduce the following functional 1 I˜ (φ) = φ(x). R x∈A

Consider ˜ I(φ) = lim I˜εn (φ). F

56 Theorem 6 The functional I˜ is independent of the choice of an optimal ε-nets Aε . Proof 5.9 The proof is similar to the one of Theorem 4, just instead of Proposition 5.5 one should use Proposition 5.9, formulated below. Proposition 5.9 Let A be an optimal -net and B be an ε-net. There exists an injective map α : A → B such that d(x, α(x)) ≤ 2 for any x ∈ A. The proof is based again on the Marriage Lemma. Definition 5.7 The measures νE,F (X) corresponding to I˜E,F will be called dual measures of complexity. Proposition 5.10 Let τ be an ε0 -isometry and A be an ε-net, ε ≤ ε0 . Then τ (A) is also an ε-net. Proof 5.10 Given x ∈ X we have to prove that x ∈ Oε (τ (A)). Due to surjectivity of τ there exists y ∈ X, x = τ (y). There exists a ∈ A such that y ∈ Oε (a). By the definition of ε-isometry x = τ (y) ∈ Oε (τ (a)). Using Proposition 5.10, Proposition 5.6 one can prove the following analogue of Theorem 5. Theorem 7 Let τ be a local isometry. Then νE,F is invariant, i.e. νE,F (A) = νE,F (τ −1 (A)) for all measurable A. We don’t know if µE,F and νE,F can be different, but we can prove the following theorem. Theorem 8 If there exists k ∈ N such that for any x ∈ X and any small enough ε > 0 one has Cε (Oε (x)) ≤ k, then µ and ν are equivalent and, moreover, 1 νE,F (A) ≤ µE,F (A) ≤ kνE,F (A) k for any Borel set A ⊆ X. It can be easily seen that if X is a subspace of a finite dimensional Euclidean space then the conditions of the Theorem are satisfied

57

PART II : SPATIO-TEMPORAL CHARACTERISTICS OF COMPLEXITY 1. Generalized Caratheodory construction We describe here a general approach developed by Ya. Pesin on the basis of classical Carath´ éodory results. It is convenient to represent them in the form of tables. The quadruple (F, ψ, ξ, η) is said to be a Carath´ éodory structure where ψ, ξ, η are defined in Table 1.

Table 1 Generalized Caratheodory Construction (Pesin ’86) X–space ≡ set

F –collection of subsets

Example Metric Space Open Subsets; Balls B (x) = {y : dist(x, y) < }

ψ : F → R+ – a function, s.t. Axiom A: For any > 0 there is a finite or countable subcollection Ui } with Ψ(U Ui ) ≤ , s.t. {U U ⊃ X i i ξ, η : F → R+ are functions s.t. ξ(U ) ≥ 0, ∀U ∈ F Axiom B: η(U ) > 0, ∀U = ∅, U ∈ F

ψ(U ) = diam(U ) diam Ui ≤

Haussdorff ξ(U ) η(U )

≡ =

1 diam (U )

Carath´ édory Axiom C: For any δ < 0 ∃ > 0 such that η(U ) ≤ δ provided that ψ(U ) ≤ (for any U ∈ F ).

ξ(U ) η(U )

≡ =

1 φ(diam U )

58 Given Z ⊂ X, let us consider a finite or countable cover G = {Ui } of Z by elements of F, with ψ(Ui ) ≤ . Then, introduce the sum Mξ (α, , G, Z) =

ξ(Ui )η(Ui )α

(1)

ξ(Ui )η(Ui )α ,

(2)

i

and consider its infimum Mξ (α, , Z) = inf

G

i

where the infimum is taken over all cover G ⊂ F of Z with ψ(Ui ) ≤ . The quantity M (α, , Z) is a monotone function in ; therefore, there exists the limit m(α, Z) = lim M (α, , Z). →0

It was shown in [34] that there exists a critical value αc ∈ [−∞, ∞] such that +∞ m(α, Z) = 0, α > αc if αc = . m(α, Z) = ∞, α < αc if αc = −∞ The number αc is said to be the Charathéodory dimension of Z relative the structure (F, ψ, ξ, η). See Table 1

1.1

Examples

If F is the collection of all open balls {B(x, } of all diameters > 0 centered at all points x ∈ X, ξ(B(x, )) ≡ 1, η(B(x, )) = , then αc = dimH Z, the Hausdorff dimension. Another nontrivial example is dimension–like definition of the topological entropy [34]. Assume that X is compact and f : X → X is a continuous map. Given n > 0, 0 > 0, the Bowen set is defined as Bn (x, 0 ) = {y ∈ X : ρ(f i x, f i y) ≤ 0 , 0 ≤ i ≤ n}. Let F be the set of all Bowen sets, ξ(Bn (x, 0 )) ≡ 1, ψ(Bn (x, 0 )) = 1/n and η(Bn (x, 0 )) = e−n . Then the 0 –topological entropy of f on Z, htop (f |Z, 0 ), is the Carathéodory dimension αc and htop (f |Z) := lim sup0 →0 htop (f |Z, 0 ). It was shown in [34] that this entropy coincides with the standard topological entropy if Z is compact and f –invariant set.

59

Spatio-temporal Characteristics of Complexity

Table 2 Carath´éodory dimensions Z⊂X

Example

Mc (Z, α, ) =

Mc (Z, α, )

inf

     

G  ψ(ui )≤   ui ⊃Z   ui ∈G ψ(ui )≤

if

ξ(ui ) · ν(ui )α

     

inf G

diam Ui ≤

    

⇒

∃ lim Mc (Z, α, ) = mc (Z, α)

diam Uiα

or

inf G

φ(diam Ui )α

i

diam Ui ≤

→0

Properties of mc as a function of Z (i) Z1 ⊂ Z2 ⇒ mc (Z1 , α) ≤ mc (Z2 , α) (ii) mc k Zk , α ≤ k (Zk , α)

mH (Z, α) outer Haussdorf measure

(If mc (Φ, α) = 0 then m is outer measure) mc as a function of α

dimH (Z) Haussdorff dimension

αc

=

sup{α : mc (Z, α) = ∞}

=

inf{α : mc (Z, α) = 0}

=

dimc Z –

Caratheodory ´ dim. of Z

60

Table 3 Examples

F

ξ

ψ

ν

1

diam U

diam U

Open sets

diamH (x)

U (or

diamB (x)

balls)

Bowen

1

sets B(u)

same

Balls B (x)

Result

exp

sup x∈B(u)

m|u >−1

1 m(u)

e−m(u)

same

same

htop (if X inv. and compact)

P (φ) – top pressure k φ(f x)

(if x is inv. and comp.)

k=0

dimq (X) (if µ is regular) µ (B (x))q

including HPq (µ) and Correlation dimenssion

Open {U }

e−qτ (u)

diamU

diamU

•

•

•

•

Dimenssion for Poincaré recurrences

•

61


2. 2.1

Topological pressure and Hausdorff dimension Dimension-like definition of topological pressure

Let us remind the definition for subshifts (the definition for arbitrary dynamical systems can be found in [28]). Let ψ be a real–valued continuous function on a subshift Ω. Let   |ω|−1 Zn (ψ, Ω) = exp sup ψ(σ j ω) , (3) ω∈ω

|ω|=n

j=0

where the sum is taken over all cylinders [ω] ⊂ Ω of length |ω| = n. It is proved in [41] that the limit PΩ (ψ) = lim

n→∞

1 log Zn (ψ, Ω) n

(4)

exists. The limit is called the topological pressure of the function ψ on Ω with respect to σ. For every constant c ∈ R, the topological pressure satisfies the property PΩ (c + ψ) = c + PΩ (ψ).

(5)

Consider the potential ψ ≡ 0 then PΩ (0) = htop (σ|Ω), the topological entropy. Roughly speaking, the system (σ, Ω) has ehtop n different paths of temporal length n (with some accuracy), each of them “costs” |ω|−1 j nPΩ (ψ) is the total price for passing exp j=0 ψ(σ ω) units, and e through all of them. It is known that topological pressure is independent of the metric (preserving a given topology) and is invariant under topological conjugacy [28]. Example 2.1 Let us calculate the topological pressure in the case where Ω = ΩA , the topological Markov chain with a p × p transition matrix A, and the function ψ(ω) depends only on the first symbol: ψ(ω) = ψ(ω0 ). In this case Zn (ψ, Ω) =

(i0 ,...,in−1 )

n−1

exp

j=0

ψ(ij )

(6)

62 where the sum is taken over all ΩA –admissible words (i0 , . . . , in−1 ). Set ψ(i) = log ρi , i = 0, . . . , p − 1, then Zn (ψ, ΩA ) =

n−1

ρk .

(7)

(i0 ,...,in−1 ) k=0

It is not a difficult algebraic exercise to show that Zn (ψ, ωA ) = RB n−1 E T

(8)

where R = (ρ0 , . . . , ρp−1 ), E = (1, . . . , 1) and B = A · diag(ρ0 , . . . , ρp−1 ).

(9)

As a corollary of formula (9) we obtain that PΩ (ψ) = log λ0 where λ0 is the spectral radius of the matrix B. For a finite or a countable cover C of Ω by cylinders of lengths greater than n and β ∈ R let   |ω|−1 Z(β, ψ, C, Ω) = exp −β|ω| + sup ψ(σ j ω) . (10) ω∈[ω] j=0

[ω]∈C

It is proved in [34] that the topological pressure PΩ (ψ) coincides with the threshold value ) * PΩ (ψ) = sup β : lim (inf{Z(β, ψ, C, Ω) : |C| ≥ n}) = ∞ . (11) n→∞

2.2

Bowen’s equation

Let us show now how the topological pressure is related to the Hausdorff dimension. Assume that a set F is modeled by a Moran construction and the corresponding subshift is a topological Markov chain (σ, ΩA ). Choose a cover of F by basic sets of the n-th generation. Then,

α diam∆i0 ,...,in−1 =

n−1

λαik

(i0 ,...,in−1 ) k=0

(i0 ,...,in−1 )

=

(i0 ,...,in−1 )

n−1 exp α ϕ(ij ) = Zn (αϕ, ΩA ) (12) j=0

63 where ϕ(i0 , . . . , ) = log λi0 . We know that Zn (αϕ, ΩA ) ≈ exp nPΩA (αϕ) . Hence, Zn (αϕ, ΩA ) 1 if PΩA (αϕ) > 0 and Zn (αϕ, ΩA ) 1 if PΩA (αϕ) < 0. It follows that if α0 is the root of the (Bowen’s) equation Spatio-temporal Characteristics of Complexity

PΩA (αϕ) = 0

(13)

then dimH F ≤ α0 . The opposite inequality can be proven by using the technique of Moran covers and the dimension–like definition of topological pressure [34].  2.2 Let us come back to Example 1.1. In this case A = Example 1 1 , 0 < λ0,1 < 1 are rates of contraction, ψ(i0 , i1 , . . .) = α log λi0 =  1 1   α λα λ 0 1  αϕ(i0 , i1 , . . .). Thus, ρi = λαi , i = 0, 1, B =  and PΩ2 (αϕ) = α λ0 λα1 log λα0 + λα1 . The Bowen’s equation (13) becomes the Moran’s equation (14). Example 2.3 Consider now the mean” topologicalMarkovchain “golden 1 1

. Assume that 0 < λ0,1 < 1 are 1 0 α rates of contraction.  Here again ρi = λi but the matrix B has the form λα0 λα1 . The characteristic equation of matrix B is µ2 −µλα0 − B= α λ0 0 α α (λ0 λ1 ) = 0 and spectral radius is r = (1/2) λα0 + λ2α 0 + 4(λ0 λ1 ) . with the transition matrix A = 

Thus, the Hausdorff dimension of the corresponding set F is the root α of the Bowen’s equation log (1/2) λα0 + λ2α = 0. If 0 + 4(λ0 λ1 ) √ Λ0 = λ1 = λ then the equation becomes α log λ + log (1 + 5)/2 , √ i.e., dimH F = α0 = log (1 + 5)/2 / − log λ. If you take into account √ that log (1 + 5)/2 = htop , the topological entropy of the topological Markov chain (σ, ΩA ), then we obtain the relation ([25]) dimH F =

htop . − log λ

64

2.3

General subshifts

It was shown in [34, 35] that the Bowen’s equation (13) holds not only for topological Markov chains and not only for finitely many values of rates of contraction. Consider Ω ⊂ Ωp an arbitrary subshift with positive topological entropy and let λ : Ω → R+ be an arbitrary H¨ older continuous positive function, such that λ(ω) < 1 for any ω ∈ Ω. We may replace the Moran axiom (M4) by the following assumptions: there are positive constants c and c such that |ω|−1

diam ∆i0 ,...,in−1 ≥ c inf

ω∈[ω]

λ(σ j ω),

(14)

j=0 |ω|−1

diam ∆i0 ,...,in−1 ≤ c sup

λ(σ j ω).

(15)

ω∈[ω] j=0

One can show that the following statement holds. Lemma 2.1 There exists a positive constant d such that |ω|−1

d

λ(σ j ω) ≤ |χ([ω])|,

(16)

j=0

for every ω ∈ [ω]. By using this Lemma, it was shown in [34, 35] that the dimH F = αc , where αc is the root of the Bowen’s equation PΩ (α log λ) = 0. Similar formula was obtained for conformal repellers [36] and in many other situations [34]. The examples of sticky sets in previous sections show us that, in general, we should apply a wider notion than the Hausdorff dimension to describe simultaneously behavior of orbits on invariant sets and their geometric features. The generalized Carathéodory construction allows us to do it.

3.

Spectra of dimension for Poicare recurrences

In the study of spatio–temporal behaviour of orbits, it is natural to take into account that typical orbits in Hamiltonian systems and orbits in attractors in dissipative systems repeat their behavior in time. This repetition can be expressed in terms of Poincaré recurrences. Consider a dynamical system (Rm , f ) where the mapping f : Rm → Rm is continuous. Let F ⊂ Rm be an f –invariant subset. In the framework of the general Carathéodory construction we consider covers by


65

open balls. For each Z ⊂ F , denote by B (Z) the class of all finite or countable covers of Z by balls of diameter less than or equal to . For an open ball B ⊂ Rm let the Poincaré recurrence be defined as τ (B) = inf{τ (x, B) : x ∈ B}, where τ (x, B) = min{t ≥ 1 : f t (x) ∈ B} is the first return time of x ∈ B. Given G ∈ B (Z) and α, q ∈ R, consider the sum ξ(τ (B))q diam B α , (17) Mξ (α, q, , G, Z) = B∈G

where the real nonnegative function ξ : R → R is such that ξ(t) → 0 as t → ∞. Below we will consider the functions ξ(t) = e−t and ξ(t) = 1/t. Next we define Mξ (α, q, , Z) = inf{Mξ (α, q, , G, Z) : G ∈ B (Z)}.

(18)

For fixed q the limit mξ (α, q, Z) = lim→0 Mξ (α, q, , Z) has an abrupt change from infinity to zero as one varies α from minus infinity to infinity. There is a unique critical value αc (q, ξ, Z) = sup{α : mξ (α, q, Z) = ∞}

(19)

such that mξ (α, q, Z) = ∞ if α < αc (q, ξ, Z), provided that αc (q, ξ, Z) = −∞, and mξ (α, q, Z) = 0 if α > αc (q, ξ, Z), provided that αc (q, ξ, Z) = ∞. The function αc (q, ξ) := αc (q, ξ, F ) is said to be the spectrum of dimensions for Poincaré recurrences, specified by the function ξ. The value q0 (ξ) = sup{q : αc (q, ξ) > 0} is said to be the dimension for Poincaré recurrences specified by the function ξ. The definition has been introduced in [6, 7] and [9]. In the case ξ(t) = exp(−t), a quantity similar to q0 was introduced in [33] and was called the AP–dimension. Roughly speaking, q0 is the smallest solution of the equation αc (q, ξ, A) = 0. Notmany specific examples are known where the dimension for Poincaré recurrences has been explicitly computed or estimated ([22], [30]).

3.1

Non-homogeneous Bowen’s equation

Let (F, f ) be a dynamical systems over a Cantor set F ⊂ Rd whose geometric construction is described symbolically a ` la Moran and f has the specification property and positive topological entropy (for which we use ξ(t) = e−t in (17)). For the sake of simplicity, we impose some aditional conditions.

66 Controlled packing of cylinders. Given an open ball B ∈ Rd , a cylinder [ω0 , ω1 . . ., ωn−1 ] is called B–maximal iff χ([ω0 , ω1 . . ., ωn−1 ]) ⊂ F ∩ B and χ([ω0 , ω1 . . ., ωn−2 ]) ⊂ F ∩ B. The set of all B–maximal cylinders is denoted by CMax(B). Let C be a cover of F by sets out := of B . The collection of all B–maximal cylinders in C, CMax(C) d has CMax(B), is a cover of S by cylinders. We say that F ⊂ R B∈C controlled packing of cylinders if there exist positive constants C0 and a, independent of and C, such that for every open ball B ∈ C, every 0 < ρ < 1 and every positive integer N one has + # [ω] ∈ CMax(B) : |χ([ω])| ∈ ρN +1 , ρN ≤ C0 N a .

(20)

A fractal set F resulting from a Moran construction satisfying (20) is said to have the controlled–packing property. We proved in [3] that for dimension d = 1 fractal sets always satisfy the controlled–packing condition (20).

A strong Moran construction. We define a subclass of Moran constructions, satisfying (M1)–(M3), by adding the following “gap condition”. There is a constant (gap) G > 0 such that for all admissible words (ω0 , . . . , ωi−1 , ωi ) and (ω0 , . . . , ωi−1 , ωi ) one has dist(∆ω0 ...ωi−1 ωi , ∆ω0 ...ωi−1 ωi ) ≥ G diam ∆ω0 ...ωi−1 , dist(∆ω0 , ∆ω0 ) ≥ G,

(21) (22)

if ωi = ωi in (21) and ω0 = ω0 in (22). Moran constructions satisfying condition (21) and (22) are said to be strong Moran constructions and the corresponding fractal sets F are said to satisfy a gap condition. Theorem 1 ([3]) Assume that set F either has the controlled–packing property or it satisfies the gap condition (21–22). Let the system (F, f ) be topologically conjugate to a subshift (S, σ) with the specification property and positive topological entropy. Then, for ξ(t) = e−t and the parameter region q ≥ 0 and α ≥ 0, the spectrum αc (q) is the solution of the equation PS (α log λ) = q.

(23)

The dimension for Poincaré recurrences coincides with the topological entropy of the subshift (S, σ), i.e., q0 = htop (σ|S).


3.2

67

Poincare recurrences and Lyapunov exponents; multifractality

When (F, f ) in Theorem 1 is a d–dimensional conformal repeller [34, 14], there exists a relation between the entropy spectrum of Lyapunov exponents and the spectrum of dimensions for Poincaré recurrences. Let us remind that a conformal repeller F is an f –invariant set such that: (i) Df (x) = a(x)Isom(x), x ∈ F , where a(x) > 1 is a scalar and Isom(x) is an isometry and (ii)!F is locally–maximal, i.e., there is a neighborhood U ⊃ F such that i≥0 f i (U ) = F . Let x ∈ F and denote by Λ(x) the expression 11 lim log Df j (x) f , n→∞ n d n−1 j=0

whenever this limit exists, and call it the Lyapunov exponent. We denote by E the set of points x ∈ F where the Lyapunov exponent does not exist. For β ∈ R we set Eβ := {x ∈ F \ E : Λ(x) = β} . We have that F = E ∪ β∈R Eβ . Let us define the entropy spectrum of the Lyapunov exponents by f (β) := htop (σ|Eβ ). Here, htop denotes the topological entropy for non– compact sets as it is defined in [34, 19]. Following the proofs of Theorem 1 and Theorem 2 in [21] one can show that in this case f (β) = inf {PF (αφ) + αβ} α

where φ(x) = −(1/d) log Dx f . In view of Theorem 1 αc (q) fulfills the equation q = PS (αc log(λ(ω))). Since PS (αc log(λ(ω))) = PF (−αc (1/d) log Dx f ) we get

f (β) = min{αc−1 (α) + αβ}. α

The spectrum f (β) is strictly concave and defined on a closed interval. Hence, the entropy Lyapunov spectrum f (β) and the inverse of the recurrence spectrum αc−1 form a Legendre transform pair. In this case the spectrum αc is strictly decreasing. It is strictly convex iff φ(x) = −(1/d) log Dx f is not cohomologous to a constant. In the latter case the support of f (β) reduces to a point and αc (q) is linear: αc (q) =

q − htop (σ|S) . log λ

68 For all the systems we consider we have that αc (htop (f |F )) = 0 and αc (0) = dimH (F ). Hence, for the quasi–conformal repellers considered in this paragraph and for 0 ≤ q ≤ htop (σ|S) the critical value αc (q) lies in the interval [0, dimH (F )]. Let us emphasize that if q = 0 then αc (0) = htop / log λ = dimH ΩA , thus, the spectrum of dimensions can be treated as a family “joining” the extreme values: the topological entropy and the Hausdorff dimension. Another relation between Lyapunov exponents and local rates of Poincaré recurrences was exposed in [38]. Itw assho wn that for a large class of interval maps with an invariant measure µ with possitive entropy, the 1 ratio − lnr τ (Br (x)) goes to λ−1 µ as r → 0 a.e., where λµ is the Lyapunov exponent of µ. A similar relaton was obtained in [4].

3.3

A case of sticky sets

Let us adopt the assumptions of Section 4.2, i.e., a sticky set Λ is a result of a Moran type geometric construction, so that inequalities (14) and (15) are satisfied and f |Λ is topologically conjugate to a multipermutative system (T, Ωp ), see Section 4.2. Since htop (T ) = 0, then the gauge function ξ(t) should be different from e−t . We know that if multipermutative system is minimal, the time needed to come back to a cylinder of the length n is exactly pn . It allows us to guess that the right gauge function is ξ(t) = 1/t. So, we find the spectrum αc (q, ξ, Λ) for ξ(t) = 1/t. Theorem 2 ([3]) Assume that F is modeled by the full shift (Ωp , σ) and satisfies the gap condition. Let the system (F, f ) be topologically conjugate to a minimal multipermutative system (Ωp , T ). Then, for ξ(t) = 1/t and the parameter region q ≥ 0 and α ≥ 0, the spectrum αc (q, ξ) is the solution of the equation PΩp (α log λ) = q log p,

(24)

The dimension for Poincaré recurrences is equal to 1. Thus, we see that again for q = 0, αc (0, ξ, Λ) = dimH Λ, the Hausdorff dimension of set Λ. Moreover, if α = 0, the equation becomes: htop (f |Λ) = q log p, and, since htop (f |Λ) = log p, then q0 (ξ) = 1. This result is completely consistent with the observation that τ ([ω0 , . . . , ωn−1 ]) = pn . There are generalizations of this result that, together with Theorem, show that some features os space–time behaviour of orbits are reflected


69

in spectra of dimensions for Poincaré recurrences, and these spectra can be calculated in terms of a non-homogeneous Bowen equation.

Conclusions and Aknowledgements As one can see, this course of lectures is just an introduction to chaotic dynamics and complexity. The list of references is also far from being complete. A description of the subject from a physical viewpoint can be found in [10, 1, 13, 17, 43, 23] and references therein. Neverteless, I believe that this course may be usefull for people who are interested in nonlinear problems. W would like to thank J.–R. Chazottes, P. Collet, M. Courbage, E. Ugalde, J. Ur´ıas and G. Zaslavsky for valuable disscusions. V.A. was partially supported by CONACyT grant 485100-5-36445-E. L.G. was partially supported by CONACyT-NSF grant E120.0547 and PROMEP grant PTC-62.

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70 [7] V. Afraimovich, Poincaré recurrences of coupled subsystems in synchronized regimes, Taiwanese J. of Math. 3 (1999) 139–161. [8] V. Afraimovich, S.-B. Hsu, Lectures on Chaotic Dynamical Systems, AMS Studies in Advanced Mathematics, 28 International Press, 2003 [9] V. Afraimovich and G.M. Zaslavsky, Sticky orbits of chaotic Hamiltonian dynamics, 519-532 Lecture Notes in Physics, 511 (1998) 59–82 [10] V. Afraimovich, G. M. Zaslavsky, Space–time complexity in Hamiltonian dynamics, Chaos, 13 (2003) 519-532 [11] V. Afraimovich and L. Glebsky, Measures of –complexity, Preprint, math.DS/0312042 in http://xxx.land.gov [12] V.M. Alekseyev, Quasirandom Dynamical Systems I: Quasirandom Diffeomorphisms, MAth. USSR Sbornik, 5, (1968), 73-128. [13] R. Badii and A. Politi, Complexity, Cambridge University Press, Cambridge, 1999. [14] L. Barreira, A non–additive thermodynamic formalism and applications to dimension theory of hyperbolic dynamical systems, Ergod. Theory and Dyn. Syst., 16 (1996) 871–928. [15] S. Benkadda and G.M. Zaslavsky (Eds.) Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasmas, Lecture Notes in Physics, 511 (1998). [16] F. Blanchard, B. Host, A. Maass, Topological complexity, Ergod. Theory Dyn. Syst. 20 (2000), 641-662. [17] G. Boffetta, M. Cencini, M. Falcione and A. Vulpiani, Predictability: A way to characterize complexity, Phys. Rep. 356 (2002), 367-474. [18] N. Bourbaki, Elements of mathematics. General topology. Part 1. Hermann, Paris, 1966. [19] R. Bowen, Topological entropy for non–compact sets, Trans. Amer. Math. Soc. 184 (1973), 125–136. [20] R. Bowen, Hausdorff dimension of quasi–circles. Publ. Math. IHES, 50 (1979), 259–273. [21] T. Bohr and D. Rand, Entropy function for characteristic exponents, Physica 25D (1987), 387–398.


71

[22] H. Bruin, Dimensions of recurrence times and minimal subshifts, in: Dynamical systems: from crystal to chaos, (J.–M. Gambaudo, P. Hubert, P. Tisseur and S. Vaienti, Eds.) World Scientific (2000) 117–124. [23] F. Ceccone, M. Falcione and A. Vulpiani, Complexity Charachterization of dynamical Systems through Predictability. Preprint (2003) [24] E. I. Dinaburg, Relation between diferent entropy characteristics of dynamical systems, Math. Izvestia AN SSSR, 35, (1971), 324-366. [25] H. Furstenberg, Disjointness in ergodic theory, minimal sets and a problem in Diophantine approximation, Mathematical Systems Theory 1 (1967), 1–49. [26] Furstenberg H, Amer. J. Math. 85 (1963), 477. [27] P.R. Halmos, Measure theory, Springer-Verlag, 1974. [28] A. Katok and B. Hasselblatt, Introduction to the modern theory of dynamical systems, Cambridge University Press, London New York (1995). [29] A.N.Kolmogorov and V.M. Tikhomirov, ε-entropy and ε capacity of sets in functional spaces, Usp. Mat. Nauk. 14 (1959), 3-86. [30] P. Kurka and A. Maass, Recurrence dimension in Toeplitz subshifts, in: Dynamical systems: from crystal to chaos, (J.–M. Gambaudo, P. Hubert, P. Tisseur and S. Vaienti, Eds.) World Scientific (2000) 165–175 [31] M. I. Malkin, Rotation intervals and the dynamics of Lorentz-type mappings, Selecta Mathematica Sovietica, 8, (1991) 267-275. [32] J. Milnor, On the entropy geometry of cellular automata, Complex Syst. 2 (1997) 357-385 [33] V. Penné, B. Saussol, S. Vaienti, Dimensions for recurrence times: topological and dynamical properties. Discrete and Continuous Dynamical Systems 5 (1999) 783–798. [34] Y. B. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications. Chicago Lectures in Mathematics, The University of Chicago Press (1997). [35] Ya.B. Pesin and H. Weiss, A multifractal analysis of equilibrium measures for conformal expanding maps and Markov Moran geometric constructions, J. Stat. Phys. 86:1–2 (1997) 233–275.

72 [36] D. Ruelle, Repellers for real analytic maps, Erg. Th. Dyn. Syst. 2 (1982), 99–107. [37] H. J. Ryser, Combinatorial mathematics, The Carus Mathematical Monographs, 15 The Mathematical Association of America, 1963. [38] B. Saussol, S. Troubetzkoy and S. Vaienti, Recurrences, dimensions and Lyapunov exponents, J. of Stat. Physics, 106, (2002), 623–634. [39] F. Takens, Detecting Strange Attractors in Turbulence, Lecture Notes in Maths Springer–Verlag, Berlin, 898 (1980), 336-382 [40] F. Takens, Distinguishing deterministic and random systems, Nonlinear Dynamics and Turbulence, G.I. Barrenblatt, G.Iooss, D.D. Joseph, eds., Pitman, (1983), 314-333 [41] Peter Walters, A variational principle for the pressure of continuous transformations, Amer. J. Math. 97 (1976) 937–971. [42] Peter Walters, An Introduction to Ergodic Theory, Springer Graduate Texts in Math 79, New York (1982). [43] G.M. Zaslavsky Physics of Chaos in Hamiltonian Systems, Imperial College Press, London–Singapore (1998).

WORKING WITH COMPLEXITY FUNCTIONS G. M. Zaslavsky1 V. Afraimovich2 1 Courant

Institute of Mathematical Sciences, New York University 251 Mercer Street, New York, New York 10012 and Department of Physics, New York University, 2-4 Washington Place New York, New York 10003. 2 Instituto

Abstract

de Investigaci´ on en Comunicación Optica UASLP Av. Karakorum 1470, Lomas 4ta sección San Luis Potosı, ´ SLP, 78210, M´ éxico.

Complexity function are described that reflects evolution of instability in dynamical systems. They grow exponentially fast for systems with positive entropy and polynomially for the ones with zero entropy. These quantities can be measured and we show in this lecture notes how to do it in specific situations.

[DOI: 10.1063/1.1566171]

1.

Introduction

The main reason of complexity of behavior of orbits in dynamical systems is their instability, that implies unpredictability. In an oversimplified way one can say that the less predictable is a system, the larger complexity should be assigned to the system. The original version of the dynamics complexity was closely linked to the system’s instability and entropy [KT], [T], [Bo]. A typical situation of the chaotic dynamics could be associated with a positive Kolmogorov– Sinai entropy, or be similar to the Anosov–type systems. This type of randomness and complexity of the systems can be characterized by the 73 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 73–85. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

74 exponential divergence of trajectories in phase space, and because of that the complexity function grows exponentially in time. As long as investigation of chaotic dynamics reveals new and more detailed pictures of chaos, it was found out that typical Hamiltonians do not posses ergodicity, the boundary of islands in phase space makes the dynamics singular in their vicinity, and even zero measure phase space sets in the Sinai billiard are responsible for the anomalous kinetics. There are numerous examples of the Hamiltonian systems, referred as the chaotic ones, that do not have exponential dispersion of trajectories for arbitrary long time intervals. These pieces of trajectories, called flights, appear with a probability that is not exponentially small [Z1], [Z2]. The behavior of systems with zero Lyapunov exponents definitely has some level of complexity that is reflected in polynomial growth of the complexity function. Thus, the complexity function distinguishes strongly chaotic systems and the weakly chaotic ones. Furthermore, the complexity function contains an information not only about asymptotics but about pre–limiting behavior as well. For example, if for some time interval close orbits belong to chaotic sea in the phase space of a Hamiltonian system then they contain a greater amount of instability than for another time interval for which they are close to a sticky set. Therefore the rate of growth of the complexity function on the first interval will be greater than on the second one. In other words, some details about fluctuations can be extracted from the behavior of the complexity function.

2.

Definition of complexity function

What kind of dynamics should be consider as a simple one and what as a complex one? We do not assume that there is the only definition of complexity which particularly depends on how this notion will be applied to the dynamics. A more or less accustomed definition of the complexity depends on how trajectories are mixed in phase space due to the dynamics. The stronger is mixing, the more complex is the dynamics. One can immediately comment on some weak features of this type of approach that deals with global phase space and global mixing. The process of mixing can be non–uniform in time and it can have local space–different rates. We call these features of mixing as space–time non–uniformities

75

Working with complexity functions

and, speaking about the space, we have in mind the full phase space or its part where the dynamics is ergodic. The former comment leads us to a necessity of such definition of the complexity which could embrace non–uniformity of space–time dynamics represented by trajectories. Let Tˆt : M → M be evolution operators of a dynamical system with the phase space M and x(t) = Tˆt x0 be a point on the trajectory going through x0 at time instant t. Then the function d(x(t), y(t)) = dist(Tˆt x0 , Tˆ t y0 )

(1)

defines a distance between two trajectories x(t) = Tˆ t x0 , y(t) = Tˆ t y0 , x0 , y0 ∈ M

(2)

with initial conditions x0 , y0 . For a discrete time t = n, one should make a corresponding replacement in (2). Any segment of a trajectory of the temporal length n can be written as n−1

Sn (x) = ∪ Tˆk x. k=0

(3)

Two segments Sn (x), Sn (y) are said to be (ε, n)–separated if there exist k, such that (4) d(Tˆk x, Tˆk y) ε, 0 k n − 1. Consider a set A ⊂ M of initial conditions x ∈ A and the corresponding bunch of trajectories started at x ∈ A. After time n we have a set of segments, and some of them are mutually ε–separated. Definition 2.1 The magnitude Cε,n = max{# segments of mutually ε, n − separated}

(5)

is said to be (ε, n)–complexity of the set A. As the function of time n it is called the ε–complexity function. The value ln Cε,n (A) is called the ε–capacity of the set A. To get more details on the introduced definitions, let us consider a one–dimensional system with uniform mixing dynamics on the interval [0, l], x ∈ [0, l], with an exponential divergence of trajectories and with the constant rate of divergence h. Let δx0 be the initial distance at t = 0 between two trajectories and δxt = ε be the distance at time t. Then ε = δx0 exp ht,

(6)

76 that is, for τ t two trajectories are ε–separated. The number of such trajectories is A A ht = (7) e , Cε,t (A) = δx0 ε where A is the length of a small initial interval A. For τ > t the complexity grows exponentially. If the set A has the box–dimension b, then (6) should be replaced by (ε/δx0 )b = eht

(8)

and correspondingly, instead of (7), Cε,t (A) = (A /δx0 )b = (A /ε)b eht .

(9)

For more general situations we may assume Cε,t (A) = (A /ε)b eht C(ε, t),

(10)

where C(ε, t) is a slow varying function of ln ε and t compared to the main terms. It also can be written as follows b = lim lim

n→∞ ε→0

ln Cε,n (A) ln (1/ε)

(11)

[Ta]. The expression (10) shows an explicit way how the complexity depends on the time interval t, accuracy ε, and dimension b, and also, the domain A of a set of initial conditions. The equations (7) and (10) can be treated as follows. Let again M be the phase space (we assume here that it has a finite volume). Let Γ = Γ(M ) be its phase volume, and Γ0 be the phase volume of a set of initial conditions A0 ⊂ M at time t0 = 0 and consider their evolution At up to time t. Let Γt be the volume of the minimal convex enveloping set At . Then for systems with exponential divergence of trajectories Γt = Γ0 eht

(12)

To find how many different states can occupy the volume Γ, one should define an “elementary” minimal volume of one state, i.e., εb . Then max {# states in Γt } = (Γ0 /εb ) eht , A

(13)


77

where maximum is considered with respect to different sets A in Γ0 . Expressions (7) and (8) permit an important physical interpretation. Hamiltonian chaotic dynamics preserves the phase volume, i.e. ˆ t = Γ0 = Γ(At ). The enveloped or coarse–grained phase volume Γt Γ grows approximately as in (12). The number of states in Γt depends on the definition of a state in the enveloped phase volume. Let one state occupies an elementary volume ∆Γ. Then the number of states that occupy the volume Γt is simply (14) N (t; ∆Γ) = Γt /∆Γ = (Γ0 /∆Γ) exp (ht) Γ(M )/∆Γ Let us emphasize that this interpretation stops working when t > (1/h) log(Γ(M )/Γ0 ). Remark. The examples and considerations of the previous section where based on the exponentially increasing complexity or, which is the same, on the exponential uniform dispersion of trajectories. Real Hamiltonian dynamics does not possess uniformity. There are singular zones of different types where Lyapunov exponent is close to zero. As a result, for a time that can be astronomically large and not achievable in any practice, the corresponding segments of trajectories have only polynomial in time dispersion. The formal expression (10) for Cε,t (A) has a limitation for the bounded Hamiltonian dynamics: (15) Cε,t (A) Γ(M )/εb The constraint (15) does not depend on time and therefore it implies the existence of tmax = tc (ε) or tmax = tN (∆Γ) (16) such that trajectories or their segments, being non–separated during the tmax , will be indistinguishable.

3.

Local Complexity Functions

As before, we deal with Hamiltonian dynamics of systems in a phase space M endowed with a distance dist and discrete or continuous time t. The dynamical system Tˆt : M → M defines a distance dist (Tˆt x, Tˆ t y) between points on two trajectories at time t with initial states x, y ∈ M . We will need the following definition: Two trajectories with initial points x, y ∈ A will be (ε, t)–indistinguishable if dτ (x, y) = dist (Tˆτ x, Tˆ τ y) < ε, 0 τ t. (17) A useful notion of local complexity is based on the verification of divergence of trajectories from fixed ones.

78 Consider a small domain A ⊂ M with diameter δA , and fix some number ε δA ε δM .

(18)

where δM is the diameter of the phase space M (that is assumed to be bounded). Let us pick a point x0 ∈ A and call the corresponding trajectory the basic one. A set QN = {xk ∈ A}N k=1 is said to be locally (ε, t)–separated if (a) For every xk there is 0 τk t such that dist (xk (τk ), x0 (τk )) ε

(19)

dist (xk (τ ), x0 (τ )) < ε, 0 τ < τk ,

(20)

and

(b) for every pair (k, k ), 0 k, k N , dist (xk (τk ), xk (τk )) ε, 0 τk , τk t.

(21)

The time τk in (19) can be considered as the “first arrival” to separation. If (21) is not valid, then a pair of trajectories corresponding to the pair (k, k ) is ε–indistinguishable, and it should be treated as one trajectory during the time t. Definition 3.1 The number C(ε, t; x0 , A) = max{N |QN is locally (ε, t) − separated }

(22)

is called the local complexity function of t. A set QN is called (ε, t)–optimal if it is locally (ε, t)–separated and N = C(ε, t; x0 , A). It is simple to see that C(ε, t ; x0 , A) C(ε, t; x0 , A) if t t

(23)

From the physical point of view there is a restriction on the growth of this function:

79


C(ε, t; x0 , A) Γ(M )/εb = N .

(24)

So, if we are interested in the separation of a bunch QN of N trajectories with initial points xj ∈ QN , j = 1, . . . , N , and their evolution xj (t) = Tˆt (xj ), then after time t we find out that there are N0 = N0 (ε, t, QN ) ε–separated (from the basic one) trajectories – denote this set by QN0 – and N − N0 indistinguishable (from the basic one) trajectories. If QN0 is locally separated then we obtain an estimate, N0 (ε, t, QN ) C(ε, t; x0 , A)

(25)

If, in addition, QN0 is (ε, t)–optimal, then N0 (ε, t, QN ) = C(ε, t; x0 , A).

(26)

In Definition 3.1 we consider (ε, t)–separation of the set A of trajectories that are close to the basic one started at x0 . That is why the complexity C(ε, t; x0 , A) is called local. Now we can consider a set B (m) (m) of basic trajectories with initial points x0 ∈ B. for each x0 we can (m) attach a set {A(m) } of trajectories with initial conditions xk in the (m) vicinity of x0 . For the sake of definiteness assume that (m)

(m )

dist (x0 , x0

) > 2ε, m = m ,

(27)

so that any two trajectories from the vicinities of different basic trajectories are disjoint at least by ε and thus (ε, t)–separated. Each basic trajectory generates the corresponding local complexity (m) function C(ε, t; x0 , {A(m) }). The set B of initial points of basic trajectories will be called B–ensemble. Definition 3.2 The number C(ε, t; B, {A}) =

(m)

C(ε, t; x0 , {A(m) })

(28)

m

is called complexity function of the B–ensemble. (m)

The B– ensemble is specified by the B–set of x0 and initial con(m) (m) (m) ditions xk ∈ A(m) around each x0 within domain δA (compare to (18)). While the construction of sets A(m) can be done fairly uniformly due to its small size, the choice of B–ensemble is free and makes it flexible

80 depending on the problem to be solved. The complexity functions C(ε, t; x0 , A) and C(ε, t; B, {Ak }) show a level of time–proliferation of ε–separated trajectories from the initial set of a large number of indistinguishable points. If we are interested in the only typical (for some measure) orbits we may adjust definitions above to this situation by assuming that B–set (k) always consists of typical points x0 for this measure. Remark. There is a special measure so called the measure of complexity, for calculation of ε–complexity, that distinguishes important ini tial points from non–important ones, see the lecture notes of V. Afraimovich and L. Glebsky in this volume. Roughly speaking, if the measure of complexity of the set A is positive in definitions above, then the local complexity function or the complexity function of the ensemble behaves asymptotically in the same way as the complexity function of the whole phase space with some slow evoluting positive factor. Numerical simulations show that it often occurs for randomly chosen basic orbits and A–sets.

4.

Probability of ε–Divergence

One of the main points we are interested in is behavior of complexity functions in a neighborhood of the sticky set [AZ98]. It is well–known that a “standard” trajectory in chaotic sea behaves in the intermittent way: after relatively short chaotic burst it is attracted to the sticky set for a long time, then comes back to mixing part of the chaotic sea, etc. If our consideration is restricted to a neighborhood of one (or several) basic orbit, then fast separated pieces of orbits correspond to a mixing type of behavior and their initial points are situated much “far” from the sticky set than initial points of slow–separated pieces of orbits. By using this observation, we may eliminate fast–separated points, that is, in fact we may choose such initial points in B–set which practically belong to the sticky set. In other words, we may in principle calculate local or non local complexity of a measure concentrated on the sticky set. More rigorously, assume that an invariant measure ρ is given. Then in definition 3.1 we consider only sets QN = {xj }N j=1 such that (a) QN is locally (ε, t)–separated;


81

(b) the points xj are ρ–typical points (or points which are very close to the ρ–typical ones). The maximal number of elements in such sets QN will be called the local (ε, t)–complexity of measure ρ. We denote it by Cρ (ε, t; x0 , A). Similarly in definition 3.2 we consider only sets QN containing ρ– typical (or very close to them) points. Then we obtain the (ε, t)– complexity of measure ρ, Cρ (ε, t; B, {A}). It is useful to introduce the following quantity 1 [Cρ (ε, t + ∆t; x0 , A) − Cρ (ε, t; x0 , A)] ≈ pρ (ε, t; x0 )∆t, N (29) where N = Cρ (ε, t; x0 , A). This quantity gives a probability to diverge by distance ε from the basic orbits during the time interval [τ, τ + ∆t], and pρ (ε, t, ; x0 ) is the corresponding probability density function. We call them ε–divergence probability and ε–divergence probability density.

Pρ (ε, t; ∆t, x0 ) =

Similarly, 1 [Cρ (ε, t + ∆t; B, {A}) − Cρ (ε, t; B, {A})] N ≈ pρ (ε, t; B)∆t, (30)

Pρ (ε, t; ∆t, B) =

gives the probability to diverge from basic orbits going through B– ensemble during the time interval [τ, τ + ∆t], and pρ (ε, t, B) is the corresponding probability density function.

5.

Calculation of Local Complexity Function

In this section we would like to show that the above introduced definitions of complexity or ε–divergence probabilities are constructive and fairly simple for the utilization. From now, we will choose initial points xj in A in such a way that the distances between xj and the basic point x0 are the same for all j, and we denote it by δ. Moreover, for the sake of simplicity we omit the argument A in C(ε, t; x0 , A) and {A} in C(ε, t; B, {A}). So C(ε, t; x0 ) = C(ε, t; x0 , A) and C(ε, t; B) = C(ε, t; B, {A}).

82 As usual in numerical simulations we sill assume that randomly cho(k) sen points x0 , x0 are typical with respect to some measure ρ we are interested in. The values of N0 in (25), (26) and in (29), (30) depend on the choice of δ, ε, xo (or B). The smaller is δ, the longer t should be considered until the maximal value of C(ε, t; B) or C(ε, t, x0 ) will be achieved. This makes the limit δ → 0 fairly simple. Understanding a way to work with the parameter ε is more complicated. Consider one trajectory xt that starts at x. Let t be very big and select a set of points xk along a trajectory and, approximately, almost uniformly distributed. We can operate with points xk in the same way as with the basic points x(k) in (27). As a result, we obtain the quantity Γ(A) =N (31) εb where points xk belong to the same trajectory. That means that while C(ε, t; x) characterizes the ε–divergence from x during t, C(ε, t; xk ) characterizes the ε–divergence from xk , points x are taken in different places of the phase space and points xk are taken along the only trajectory. It is natural to believe that for an appropriately typical set of x ∈ M and of xk ∈ St and fairly large t, the equality C(ε, t; xk ) = N0 (ε, t; xk )

C(ε, t; x) = C(ε, t; x), t → ∞

(32)

holds where subscripts k are omitted. This equality can be treated as an analog of the ergodic theorem. A corresponding simulation for C(ε, t; x) was performed in [LZ]. The scheme of calculation of C(ε, t; x) is similar to one used for calculation of the Lyapunov exponents, except for some details: (a) The value of ε is much less than the distance typically used to evaluate the Lyapunov exponents. (b) Trajectories of some tracers are ε–separated after a very long time. Just these trajectories correspond to events of our main interest and their statistics should be collected. (c) The scheme of obtaining of C(ε, t; x) provides the “probability” C(ε, t; x)/N to have ε–separation at time t. Therefore one needs to use more general distribution than C(ε, t; x), [AZ].


Figure 1. time

6.

83

Sticky domain Ac where a pair of trajectories is separated by ε after long

Concluding Remarks 1 The complexity of behavior of trajectories is characterized not only by complexity functions but by a flight complexity function as well that takes into account spacial separation (in addition to the temporal one). We omitted in these lecture notes the discussion of the flight complexity function and its relevance to the description of the non–uniformity of behavior of trajectories in the phase space of the typical Hamiltonian system – see [AZ] for definitions and discussions.

2 The growing interests of specialists to characterize the non–longtime evolution (specially in some biological models) and, in fact, the absence of corresponding machinery allow us to believe that the complexity function very soon will find its application to problems in many areas of nonlinear sciences. In [AZ] we made the first step in this direction by giving rigorous definitions and studying simple examples. Now, it is the time to make real use of these quantities which reflect adequately enough the evolution of instability in Hamiltonian and non–Hamiltonian systems.

84 3 There are different ways how complexity function can be applied to real systems. Here is one of them [ZE]. Consider a trajectory that starts at the domain A and visit a singular zone Ac before the first return to A (Fig. 1). There are many such trajectories if the dynamics is topologically transitive. The singular domain Ac is characterized by a long time during which trajectories are staying in Ac . Moreover, distribution density of recurrences Prec (t) is defined by the following asymptotics [Z2]

Prec (t) ≈ 1/tγ ,

t→∞

(33)

(or more specifically, t 1/h). Assume that there is the only one singular domain Ac . Consider now a large number N of pairs of trajectories with initial distances of the order δ 1, fix ε and calculate for them C(ε, t, x)/N . Conjecture: In the limit δ → 0, N → ∞, t → ∞, we have ∞ Prec (t) → const

dx C(ε, t, x).

(34)

0

Since the probability to enter Ac is small, the probability to enter Ac more than one time can be neglected, and a finite dispersion of a pair by ε during a large t can occur only in the case where both trajectories enter a tube of the diameter ε around Ac . Typically, it can be an annulus of the width ε around a sticky island’s border. More results on this conjecture will appear in [ZE].

7.

Acknowledgments

V. A. was partially supported by CONACyT grant 36445-E. G. Z. was supported by the U. S. Navy Grant N00014-02-1-0934 and the U. S. Department of Energy Grant DE-FG02-92ER54184.

References [AZ] V. Afraimovich and G. M. Zaslavsky, Space-time complexity in Hamiltonian dynamics, Chaos 13 (2003) 519–532.


85

[AZ98] V. Afraimovich and G. M. Zaslavsky , Sticky orbits of chaotic Hamiltonian dynamics, Lecture notes Physics, 511 (1998), 59–82. [Bo] R. Bowen, Topological entropy for non-compact sets, Trans. AMS 84 (1973), 125–136. [KT] A. N. Kolmogorov and V. M. Tikhomirov, ε–entropy and ε–capacity of sets in functional spaces, Usp. Math. Nauk 14 (1959), 3–86. [LZ] X. Leoncini and G. M. Zaslavsky, Jets, stickiness, and anomalous transport, Phys. Review E. 65 (2002), 046216. [T]

V. M. Tikhomirov, , Usp. Math. Nauk 18 (1963) 55.

[Ta] F. Takens, Distinguishing deterministic and random systems in Nonlinear Dynamics and Turbulence, edited by G. I. Barenblatt, G. Iooss, and D. D. Joseph (Pitman, New York), (1983) 314–333. [Z1] G. M. Zaslavsky, Physics of Chaos in Hamiltonian systems , Imperial College Press, London, (1999). [Z2] G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys. rep. 371 (2002), 461–580. [ZE] G. M. Zaslavsky and M. Edelman, to be submitted.


SRB distribution for Anosov maps Giovanni Gallavotti Fisica Roma 1, I.N.F.N. Abstract: Lectures on the construction of the SRB distributions for Anosov maps 1. Anosov maps and Gibbs distributions. Anosov maps are the paradigm of systems with chaotic evolution. Here I consider only smooth maps and in fact I shall illustrate them through the simplest examples which are wo dimensional maps of the torus T 2 . In general a smooth map S of a compact smooth manifold M is a map with the property that around each point x one can establish a system of curvilinear coordinates based on smooth surfaces Wxs and Wxu of complementary positive dimension which is i (1) covariant: S∂W Wxi = WSx , i = u, s; this means that the tangent planes ∂W Wxi to the coordinate surfaces at x are mapped over the corresponding planes at Sx (2) continuous: ∂W Wxi depends continuously on x (3) hyperbolic: the length of a tangent vector v is amplified by a factor Cλ−k for k > 0 under the action of S k if v ∈ Wxu and under the action of S −k if v ∈ Wxs with C > 0 and λ < 1. This means that if an observer moves with the point x it sees the nearby points moving around it as if it was an hyperbolic fixed point. (4) transitivity: there is a point with a dense orbit A key example is the map of the torus T 2 defined by ϕ1 1 1 ϕ1 S = (1.1) ϕ2 1 0 ϕ2 This is a very simple system and one checks immediately the above four properties. The first three are immediate; the existence of a dense orbit follows from the fact that the dϕ map preserves the probability distribution µ0 (dϕ) = (2π)2 (“volume distribution”) and that

T −1 1 f (S i ϕ)g(ϕ)µ0 (dϕ) = f (ϕ)µ0 (dϕ) · g(ϕ)µ0 (dϕ) T →∞ T i=0 lim

(1.2)

for all f, g smooth (say analytic): this means that the dynamical system (S, µ0 ) is mixing hence ergodic and therefore µ–almost all points have a dense orbit. f by Fourier analysis, making use of the fact that the eigenvectors of the matrix proof: 1 1 have components whose ratio is an irrational number.] 1 0 Anosov maps in general do not have an invariant distribution which admits a density with respect to the volume. Very simple examples are small perturbations of the above map, e.g. ϕ1 + ϕ2 + ε sin ϕ1 (1.3) Sε ϕ = ϕ1 which has no invariant distribution with density with respect to the volume i.e. Lebesgue) distribution as we shall see (this is non trivial).

The key question is which is the statistics µ of motions? this means to ask which are the asymptotic properties of a point x chosen randomly with respect to the volume distribution. The question is interesting because it is believed that the properties that 87 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 87–105. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

88 are usually observed when studying the motions of a system started from an initial datum typical for random choices with a distribution continuous with respect to the volume. Precisely this means considering the limit T −1 1 f (S i x) T →∞ T i=0

lim

(1.4)

and to if it exists for all smooth functions f and for almost all points x with respect to the volume and, furthermore, for all such f and for almost all points x it is independent of x. If so this will define a unique probability distribution µ on M such that

T −1 1 f (S i x) = µ(dy) f (y) T →∞ T M i=0 lim

(1.5)

and the distribution µ is naturally called the statistics of the map S. It is also called the SRB distribution or the SRB statistics of the dynamical system (M, S). The lack of a density of µ with respect to µ0 is at the same time a difficulty and a main point of interest: clearly we cannot imagine to write µ(dx) = ρ(x)µ0 (dx) because, in general, ρ(x) does not exist and we must give up the hope of setting up a differential equation determining ρ. Of course the question of existence of a statistics SRB can be raised in much more general contexts: however in the case of Anosov maps it can be solved quite explicitly and completely (in the affirmative). More generally it is believed that in physical applications the systems will behave chaotically enough so that a SRB distribution (i.e. a statistics of almost all points with respect to the volume) exists: for this reason it is important to understand Anosov maps. They play in chaotic dynamics the role plaid by harmonic oscillators in regular dynamics. No system is a harmonic oscillator but one always tries to show that a system can be reduced “for practical purposes” to a system of harmonically coupled harmonic oscillators; and likewise in chaotic dynamics one tries to show that the system behaves as an Anosov map (or its continuous time version which is a “Anosov flow”, which I will not consider here). Some even go to the point of stating a Chaotic hypothesis that essentially says that for practical purposes a chaotic system can be regarded as an Anosov system, [GG99, GG02]. The technical advance on the theory of Anosov systems is easily understood if one first studies the simple map in (1.1). In fact we can imagine to define a partition P = (P1 , . . . , Pn ) of phase space fine enough so that if two points x, y have the same history on the partition, i.e. if for all k ∈ Z it is S k x, S k y ∈ Pσk then x = y; i.e. if two points x, y have the same history σ on P then they coincide. Anosov maps admit special partitions P, called Markov partitions, which have the very special property that the sequences of symbols that can be histories of points in M are simply characterized by a compatibility matrix T : this is a n × n–matrix Tσσ with entries 0, 1. And a sequence σ = (σj )j∈N can be the history on P of a point if Tσi σi+1 = 1 for all i ∈ N , such sequences are called T –compatible; and almost all points (with respect to the volume µ0 ) are represented by a unique T –compatible sequence. A few points may admit several compatible sequences that represent them: however there is a bound on the number of different sequences.1 Note that the above statement is quite non trivial: in general given any partition we can define the histories of points on it: however the set of sequences that one obtains 1

The latter ambiguity is closely related to the ambiguity that one has in representing a real number in basis 10 (say): the number 1 can be represented as 1.00000... or as 0.99999....

89 are subject to subtle compatibility relations rather than the simple nearest neighbor compatibility just mentioned. The compatibility will in general involve conditions on infinitely many symbols in σ. In a more colorful language we can say that via a Markovian partition we turn the phase space M into the configuration space of a one dimensional lattice spin system with a hard core nearest neighbor exclusion. The main result will be that the SRB distribution, regarded as a probability distribution on the set of histories corresponding to the points of the phase space M , will be a Gibbs state with a short range interaction potential on the configurations of the hard core lattice spin system just mentioned. Consider the map in (1.1) or its perturbation in (1.3). Note that the origin is a fixed point and if ε = 0 its stable and unstable manifolds are two (orthogonal) straight lines emerging from the the origin and with quadratic irrational slopes (that can be elementarily computed). We imagine drawing some finite parts of them, see fig. 1.

Fig. 1:

O

Portions of the stable and unstable manifolds of the Arnold’s cat map drawn after representing T as a lattice of copies of the square [0, 2π]2 with opposite sides identified. The heavy dots mark the endpoints. 2

Of course, since the slopes are irrational, the lines will fill densely the torus, if continued indefinitely and drawn in T 2 . We however stop in a strategic way depicted in fig. 2. Namely after reporting inside T 2 the lines in fig. 1 above we realize that we have divided T 2 into rectangles with a few exceptions. These exist because the end points (marked by bullets in fig. 1) may (and will in general) end up in the “middle” of some rectangle. This is not desired as we want to have a partition into exact rectangles: hence we continue the drawn lines beyond their endpoints until there is an intersection with the orthogonal lines and we stop there. We do this first for the expanding line (i.e. the one with positive slope) and then for the contracting line (the one with negative slope). The construction is described in fig. 2.

Since the boundaries of the rectangles are subintervals cut on the drawn portions of stable and unstable manifolds of the origin we call ∂s Pσ and ∂u Pσ the sides of the rectangle Pσ which lie on the drawn stable and unstable lines. Therefore applying S to Pσ it will be stretched along the direction of ∂u Pσ and contracted along the direction ∂s Pσ . Nevertheless the a priori very unlikely event will happen that the S–image of the

90 5

10

11

9

9

4 3

1 7

7

2 6

8 10

5

4

11

8

Fig. 2: The continuation parts are marked as dashed and come out of the points marked in fig. 1; note that the four vertices of the square are in reality the same point (i.e. the origin) because of the periodic conditions that must be imagined at the boundaries.

sides in ∂s Pσ (respectvely the S −1 -image of the sides of ∂u Πσ ) will become subsets of the union of the sides of the same type.

91 s

s ∆

S∆ u

u

Fig. 3:

The figures illustrate, symbolically as squares, a few elements of a markovian pavement. An element ∆ of it is transformed by S into S∆ in such a way that the part of the boundary that contracts ends up exactly on a boundary of some elements of P.

This is the markovian property of the partition P. To it we always add the condition that the rectangles are so small that the streching cannot elongate any of them so that S applied to a rectangle transforms it into a set which intersects twice another rectangle of the partition. The latter smallness property can be achieved simply by choosing the two drawn parts of stable and unstable lines emerging from the prigin long enough (and using the if continued indefinitely they would be dense). Define Tσσ = 1 if the image of the interior of Pσ intersects the interior of Pσ and Tσσ = 0 otherwise; then a moment of reflection shows that if σ is a sequence such that Tσi σi+1 ≡ 1 then there is a point x ∈ M whose history on P is precisely σ and viceversa if x is a point it admits one history which is T –compatible. If the elements of the partition are small enough so that the situation in fig. 3 is realized and no S–stretched rectangle crosses twice or more any other rectangle there is a unique x with history σ. A point x might admit more compatible histories but not more than 4: multiple compatible histories for the same point occur when the point or one of its S k –images falls on the boundary of some of the rectangles: the set of such points has zero volume. Hence apart from points which in their motion visit the boundaries of the rectangles and which, therefore, form a set of zero volume there is a 1-1 correspondence between points and sequences. Remark: Note that the above constructions are really only based on the existence of a fixed point with dense stable and unstable manifolds (i.e. curves, as we consider two dimensional cases). Therefore the construction is not really heavily using the special structure of the map S. Indeed in general one can show that in Anosov systems the stable and unstable manifolds of every point are dense. Furthermore the construction based on the existence of a fixed point can be easily extended to the case in which the fixed point is replaced by a periodic point: hence it is very general because one can prove that Anosov maps admit a dense set of periodic points. The correspondence point–history can be used to transform the volume distribution into a probability distribution on the space of the compatible sequences: this is due to the property that points that have multiple compatible histories form a set of zero volume. Hence the volume distribution µ0 becomes a probability distribution m0 on the configurations σ of a lattice spin system with a nearest neighbor hard core. Very remarkable is that m0 turns out to be a Gibbs state with short range interaction. We shall check that this is true both for the map (1.1) and for (1.3) and is a general property of all Anosov maps. The Gibbs property of the volume distribution once coded into a probability distrifbution on the space {1, 2, . . . , n}Z T of T –compatible sequences can be checked by studying the probability of the set of points that between time −N and time N have preassigned

92 history σ−N , . . . , σN . The property illustrated in fig. 3, i.e. the markovian property, im −k Pσk , whose histories form the set of compatible plies that the latter set of points ∩N −N S ...0 ... N , is a small rectangle whose size decreases exponentially sequences denoted Cσ−N −N ...σ0 ...σN with min(N, N ); we call x , x , x the points in its lower left corner and, resectively, to the right of it or above it (see fig. 4). S N Pσ−N Pσ0

S −N PσN δux

δsx

Fig. 4: The angle ϕ(x ) is marked in the dashed region (in general it is not 90o ) around the corner denoted x in the text; the marked corners represent x and x (with the first up and the latter on the −j P , with σ = σ . left of x ); the shadowed region represents the intersection ∩N σj 0 0 −N S

...0 ... N This is illustrated in the fig. 4. Therefore the m0 –probability of Cσ−N is the −N ...σ0 ...σN −k area of the rectangle ∩N S P which can be read directly from the figure to be simply σk −N λN +N δσuN δσs −N if δσuN is the size of the unstable side of QσN and δσs −N is the size of the stable side of Qσ−N . in formulae

−k m0 (∩N Pσk ) = λN +N δσuN δσs −N −N S

N −1

Tσj σj+1

(1.6)

j=−N

where we have inserted the product to say explicitly that m0 vanishes unless the string σ−N . . . σN is T –compatible. It is a very instructive exercise to check that setting pσσ = (δσu )−1 λ Tσσ δσu ,

pσ = δσs δσu

(1.7)

the (1.6) is the probability of the sequence of events σ−N . . . σN in the stationary Markov probability pσσ . The check is easy chain with single event distribution pσ and transition if one notes that it must be σ δσs δσu = 1 and σ λ Tσσ δσu = δσu (the relations express that the sum of the areas of the squares Pσ is equal to the volume of the torus T 2 i.e. 1 with our normalizations and likewise that the sum over σN of the areas of the rectangles −1 −k −k Pσk equals the area of the rectangle ∩N Pσk ). ∩N −N S −N S Thus we see that the volume distribution becomes a Markov process on the symbolic representation of the points of phase space. This is a special Gibbs state with nearest neighbor potential − log pσσ or equivalently with potential − log Tσσ , i.e. a purely nearest neighbor hard core potential. More generally consider the map (1.3): the stable and unstable manifolds of the origin will be slightly modified and curved. We perform the “same” construction and we see that not only we construct a partition with the markovian property described by fig. 3 but the partition is very close to the one constructed above for the ε = 0 case and in particular it has the same compatibility matrix. This means that the symbolic description

93 of the two systems is exactly the same and the remark could be used to dervive in this case the important Anosov theorem that states that two Anosov diffeomorphisms S and S of M that are close enough as maps (in the C 2 –topology at least) are isomorphic in the sense that there is a homeomorphism h of M into itself such that Sh = hS , i.e. they differ by a change of coordinates (which in general is, however, only H¨ older continuous). For ε small the figure fig. 4 remains the same, with the sides of the rectangles slightly −k Pσk is now curved for ε small and the result is that the area of the “rectangle” ∩N −N S approximately sin ϕ(x)

N

−i λ−1 u (S x)

i=1

N

λs (S i x)δσu−N δσs N

(1.8)

i=1

−k Pσk and the size of the sides has to be computed by compowhere x is a point in ∩N −N S sition rule of differentiations and λu (x), λs (x) denote the expansion and the contraction rates of the length of the unstable manifold and of the stable manifold under S. Proceeding heuristically we express (1.8) in terms of symbolic dynamics. This is possible because the functions λu (x), λs (x), ϕ(x) which are no longer constant can be shown to be Hölder continuous with exponent α < 1:

|λu (x) − λu (x )| ≤ Ld(x, x )α ,

| sin ϕ(x) − sin ϕ(x )| ≤ Ld(x, x )α

(1.9)

if d(x, x ) is the distance between x and x and L > 0 is a constant. Suppose for simplicity that Tσσ ≡ 1,i.e. that there is no compatibility condition to fulfill: this is usually not the case if T arises from an Anosov map (in fact I think that it is possible that this is never the case), nevertheless suppose so for a moment. Then we can consider the sequence 1 with entries 1 and if σ is the history of x write def

log λu (x) = Λu (σ) and Λu (σ ) =Λu (1) + (Λu (1σ0 1) − Λu (1))+ + (Λu (1σ−1 σ0 σ1 1) − Λu (1σ0 1)) + . . . =

∞

Φuk (σ−k . . . σk )

(1.10)

k=0

where 1σ−j . . . σj 1 denotes the symbolic sequence obtained by completing with symbols 1 to the right and to the left the string σ−j . . . σj and def

Φuk (σ) = Λu (1σ−k . . . σk 1) − Λu (1σk+1 . . . σk−1 1)

(1.11)

The property (1.9) implies that |Φuk (σ)| ≤ F e−kκ Φuk

(1.12)

dependsonly on σ−k . . . , σk . Likewise we can introduce for some F, κ > 0; note that Φsk (σ−k . . . σk ), sk (σ−k . . . σk ) to represent − log λs (x) and − log sin ϕ(x). The case of hard core (i.e. nontrivial compatibility matrix) can be treated similarly: instead of continuing with 1 to the right and left we continue in some “standard way” depending on the compatibility matrix: we prefix a compatible sequence of symbols and continue a given finite string to the right with the right half of the fixed string of symbols after interposing a string to allow the matching between the last symbol of the given finite string and the first of the right half of the prefixed string: the interplating string a exists because transitivity implies that there is a power a of T such that Tσσ > 0. Hence for every pair of symbols σ, σ we can find, and fix once and for all, an interpolating

94 string which is compatible and has σ as first symbol and σ as a–th. Likewise we can proceed for the continuation to the left. The expression (1.8) with N = N becomes, if τ denotes the shift operation on sequences σ N N h − Φu Φsk (τ −h σ )+c(σ)+∆N (σ) k (τ σ)− h≥k≥1 h≥k≥1 e (1.13) with ∆N (σ ) depending only on the values of σj with j close to ±N in the sense that |∆N (σ ) − ∆N (σ )| < De−κ if σ and σ differ only at distance ≥ from ±N ; likewise c(σ ) depends only on the values of σ at sites near the origin. One recognizes that (1.13) is the formula that would describe a Gibbs state for the one dimensional lattice spin system in [−N, N ] ∩ Z with a nontranslation invariant potential which is Φu to the right of the origin and Φs to the left in an external field which is appreciably different from 0 only near the origin and near the boundary at ±N . The reason the expression (1.8) is approximate is due to the finiteness of N, N . Therefore taking the limit as N → ∞ in (1.13) we find, at the heuristic level in which we are working, that the volume distribution µ0 is coded via the symbolic dynamics into a Gibbs state fr the one-dimensional lattice with hard core nearest neighbor interaction and with potential Φu to the right of the origin and Φs to the left of the origin, and with a further potential given by a potential c localized around the origin. Few systems are better understood than one dimensional lattice spin systems with short range interaction. Such systems, when translation invariant, are ergodic, mixing and isomporphic to Bernoulli shifts; furthermore their correlation functions decay exponentially fast. The nontranslation invariant case is similar: to the far right of the origin the Gibbs state looks like the translation invariant state with potential Φu and to the far left like the translation invariant state with potential Φs . This means that if F (σ ) is a Hölder continuous function T −1 1 F (τ j σ ) = dm(σ )F (σ ) T →∞ T j=0 lim

(1.14)

with m0 probability 1 if m is the Gibbs state with potential Φu . Hence the SRB distribution will exist and it will be the probability distribution that is coded into m by the correspondence between points and histories on the Markovian partition. Note that changing sign to time we likewise construct the SRB distribution that gives the statistics of the motions observed backward in time: it is the probability distribution that, by the correspondence between points and histories on the Markovian partition, is coded into the Gibbs state m− with potential Φs . Clearly the above discussion solves the problem in a very theoretical form: the real problem remaining is how to control the SRB distribution, i.e. how to compute Φu and, therefore, the time averages of the observables. This will be discussed in the next lecture. Problems. (1) Consider the map S of T 2 in (1.1) into itself. Show that the following partition is not markovian because the corresponsence thatit establishes between points and histories is not one–to–one even excluding a set of zero volume of points. (2) Show that the following refinement of the previous partition is Markovian, (3) Three dimensional Anosov map Consider the torus T 3 and the map on it defined by the matrix

95 2

1

1

3

2 Fig. 5: A pavement with three rectangles whose sides lie on two connected portions of stable and unstable manifold of the fixed point at the origin.

2

23

1

1

3

3 4 43

1 1

3

1 1

4

2 23

2

2 1 1

Fig. 6: A Markovian pavement (left) for the square root S0 of Arnold’s cat map. It is obtained from the partition in Fig. 5 by continuing a little further the stable manifold of the origin breaking into two parts the rectangle labeled 2 (whose large size was responsible for the non-Markovian nature of the pavement in Fig. 5). The images under S0 of the pavement rectangles is shown in the right figure: corresponding rectangle are marked by the same colors.

0 1 0 N =

0 1

0 0

1 0

Nk

,

1 0 0

def

=

p k

def

= vk

qk rk

and show that this is an Anosov map of T 3 . (Hint: check that the eigenvectors have the three components which verify a Diophantine property. See the following problems.) (4) (Rationally independent 3–vectors,) Suppose that the two equations n3 = an ± b, with a, b integers, do not admit integer solutions, and let ω be real and a root of the equation ω 3 = aω + b. Show that the vector w = (1, ω, ω 2 ) has rationally independent components. The case a = b = 1 defines a real root ω which is called the spiral mean. (Hint: If not ν · w = ν1 + ν2 ω + ν3 ω2 = 0 with 0 = ν ∈ Z3 and ω would be a quadratic number, i.e. a √ number of the form ω = (x + y) with x, y rational. Since ω is also a solution of the third order equation √ √ y must be rational: in fact if y is irrational then the equation ω 3 = aω + b implies

x3 + 3xy − ax − b +

√ y (y + 3x2 − a) = 0,

and necessarily y + 3x2 − a = 0 and x3 + 3xy − ax − b = 0, which means that 8x3 − 2ax + b = 0 admits a rational root x = p/q, with p, q relatively prime, i.e. 8p3 − 2apq 2 + bq 3 = 0. This says that (2p)3 is is integer: therefore n3 = an − b admits an integer divisible by q 2 so that q = 1 or q = 2 and n = 2p q solution, in contradiction with the hypothesis. Hence ω has to be rational. If ω = p/q, p, q relatively prime integers, then p3 = a p q2 + q 3 and q 2 divides p3 , hence q = 1 and p is an integer solution of n3 = a n + b, which again contradicts the hypothesis.)

(5) (Tartaglia’s formula and rational independence) Let 3 p, 2 q be integers with q 2 + p3 > 0 and suppose that the equations z 3 = 3pz ± 2q do not admit 1 1 integer solutions. Show that w = (1, ω, ω2 ) is a rationally independent vector if ω = (q + (p3 + q 2 ) 2 ) 3 + 1

1

1

1

(q − (p3 + q 2 ) 2 ) 3 . Show that, therefore, w = (1, 2 3 , 4 3 ) is a rationally independent vector. Check that the spiral mean is , ,

ω=

3

1 2

+

1 2

23

27

+

3

1 2

−

1 2

23

27

.

(Hint: By Tartaglia’s formula, divulged by Cardano, ω is a real root of ω 3 = −3pω + 2q: hence one applies problem (4). Note that 21/3 is a root of ω3 = 2.)

96 (6) (An example of a Diophantine 3–vector,) Let 0 1 0 N = 0 0 1 , 1 1 0

Nk

1 0 0

def

p

=

k

qk rk

def

= vk

Show that ρj = pj , qj , rj verify the recursion ρj = ρj−2 + ρj−3 , j ≥ 3. Let λ > 1, λ+ , λ− be the three eigenvalues of N , |λ+ | = |λ− | = λ−1/2 and let w0 , w + , w − = (1, λj , λ2j ), j = 0, ± be the respective

eigenvectors (note that λ3j = λj + 1). Let N T be the transposed of N . Given ν ∈ Z3 define, if it exists, k so that for 0 ≤ h ≤ k |(N T )h ν · w0 | ≤ max |(N T )h ν · w± |, ±

|(N T )(k+1) ν · w0 | > max |(N T )(k+1) ν · w± |, ±

k

otherwise set k = −1. Check that |ν | > B|λ± |−k = B λ 2 for some B > 0 (which only depends on the matrix N ); deduce that therefore w0 is a Diophantine vector with exponent τ = 2. (Hint: The recursion can be derived from the remark that N 3 v0 = v 0 + N v 0 . To check the Diophantine property remark that 1 ≤ |(N T )k ν| because N has no zero eigenvalue; hence by the definition of k and for a suitable b > 0 depending on the basis w0 , w ± , one has 1 ≤ |(N T )k ν | ≤ b max± |w± · (N T )k ν | ≤ k

b |ν | λ− 2 max± |w ± | (in fact in R3 all metrics are equivalent). Therefore, by the definition of k, one has |w 0 · ν | = λ−(k+1) |w0 · (N T )(k+1) ν | ≥ b λ−(k+1) |(N T )(k+1) ν | ≥ b λ−(k+1) ≥ b |ν |−2 .)

(7) (More examples of Diophantine 3–vectors,) Show that the example of problem (6) can be extended by considering 3 × 3 integer entries matrices N with only one eigenvalue with modulus greater than 1 and two others with modulus less than 1 and with eigenvectors with rationally independent components. Find a few examples by studying matrices whose characteristic equation has roots that can be discussed by the techniques of problems (4),(5). The vectors constructed in this way are all in a class called Pisot–Vijayaraghavan Diophantine 3–vectors. (Hint: For instance consider matrices obtained from N in problem (5) by replacing the last row (0, 1, 0) by (0, p, q).)

Bibliographical note The Markov partitions construction for Anosov maps based on the stable and unstable manifolds of a fixed point is very simple but it applies only to two–dimensional maps with a fixed point or a periodic point: I learnt it from M. Campanino. The general construction goes way back to Adler–Weiss, who gave the first example [AW68], to Sinai [Si68], to Bowen [Bo70], Ruelle [Ru73]. The three dimensional Diophantine vectors theory and construction is taken from C. Chandre, [Ch99],[CM01]. The general theory of Markov partitions is due to Y. Sinai, [Si68a], [Si68b]; Bowen and Ruelle extended it to more general systems. 2. Perturbative construction of the conjugation. The lack of smoothness of the stable and unstabe manifolds, of the isomorphism conjugating close Anosov maps and the lack of absolute continuity of the SRB distribution may induce pessimism about being able to say something concrete about the qualitative properties of such systems. In reality the situation is much better than one might fear. For instance in the context of perturbation theory it is possible to study the perturbations of Arnold map quite easily. In a sense the lack of regularity is only an apparent phenomenon due to a somewhat inappropriate a priori hope of continuity. The change of properties of motions following small changes of the time evolution map are very smooth and regular if seen from the appropriate point of view and this will be the key to the study of the SRB distribution for syste,s close to syste,s ad,itting a si,ple SRB distribution, Like (1.1). To illustrate this I shall discuss the map (1.3) as a function of ε for ε small. The result to be discussd is

97 Let f (ϕ) be a real trigonometric polynomial, f (ϕ) =

the two-dimensional torus T 2 and let

ν∈Z2 ,|ν|≤N

Sε ϕ = S0 ϕ − εf (ϕ),

with

S0 =

1 1

1 0

eiν ·ϕ f ν , defined on

.

(2.1)

For β ∈ (0, 1) there exist C(β) < ∞ and ε0 (β) > 0 such that for |ε| < ε0 (β) the equation H ◦ S0 = Sε ◦ H

(2.2)

defines a unique homeomorphism ϕ → H(ϕ) which is analytic in ε in the complex disk |ε| < ε0 (β) and H¨ older continuous with exponent at least as large as β and with H¨ older continuity modulus bounded by C(β).

This tells us that we can conjugate the map Sε to the map S0 via a function H that is analytic in ε. However, in general it is not true that we can conjugate S0 to Sε via an analytic homeomorphism. Indeed the equation - ◦ Sε = S0 ◦ H H

(2.3)

cannot be studied with the method developed in the proof below because it would require - ) in powers of ψ. an expansion of H(ψ Proof: We shall write ϕ = H(ψ ) = ψ + h(ψ ), ψ ∈ T 2 ; then the relation (2.1) becomes an equation for h, namely S0 h(ψ ) − h(S0 ψ ) = εf (ψ + h(ψ)),

(2.4)

Hence we look for a solution which is analytic in ε: h(ψ ) = εh(1) (ψ) + ε2 h(2) (ψ) + · · · with h(k) an ε–independent function. For instance the equation for the first order is S0 h(1) (ψ) − h(1) (S0 ψ ) = f (ψ).

(2.5)

√ We call v + , v − the two normalized eigenvectors of S0 relative to the eigenvalues (1± 5)/2 √ and we call λ the inverse of the largest one (λ = ( 5−1)/2), so that λ+ = λ−1 , λ− = −λ: in the following the only property that we shall use is λ < 1. The functions f , h can be split into two components along the vectors v ± :

f (ψ ) = f+ (ψ)v + + f− (ψ )v − ,

(2.6)

h(ψ) = h+ (ψ)v+ + h− (ψ )v− , (1)

and the equations (2.5) for h± are (1)

(1)

(1)

(1)

λ+ h+ (ψ ) − h+ (S0 ψ) = f+ (ψ),

(2.7)

λ− h− (ψ) − h− (S0 ψ) = f− (ψ ). The equations (2.7) can be solved by simply setting h(1) α (ψ) = −

p∈Zα

−|p+1|α αλα fα (S0p ψ),

α = ±,

(2.8)

98 where Z+ = [0, ∞) ∩ Z and Z− = (−∞, 0) ∩ Z are subsets of the integers is taken of the inequality λ = λ−1 + = |λ− | < 1 to ensure convergence. (k) Therefore the equations for h± become h(k) α (ψ ) =

·

s

∞ 1 s! s=0

(v αj

k1 +···+ks =k−1, ki ≥0 α1 ,...,αs =±

p∈Zα

Z

and advantage

αλ−|p+1|α · α

(2.9)

s p j) · ∂ϕ ) fα (S0p ψ) h(k αj (S0 ψ) , j=1

j=1

and proceeding as usual in perturbation theory we shall use a graphical representation for the (2.9). (k1 ) α1 (k2 ) α, p α 1 (k) = v v s! r s>0 k1 +...+ks =k−1 (ks−1 ) αs (ks ) Fig. 7: Graphical interpretation of (2.9) for k ≥ 1. (k)

Here the l.h.s. represents hα (ψ). Representing again, in the same way, the graph (k)

elements that appear on the r.h.s. one obtains an expression for hα (ψ ) in terms of trees, oriented toward the root. v5

ν v1 j r

0

v1

ν v0

v6 v3

ν =ν 0 v0

v7 v8 v4

v2 v12

v9 v10

v11 Fig. 8: A tree ϑ with k = 13, and some decorations. Only two mode labels (see below) and two momentum labels (see below) are explicitly marked on the lines 0 , 1 ; the number labels, distinguishing the branches, are not shown. The arrows represent the partial ordering on the tree.

A tree ϑ with k nodes will carry on the branches a pair of labels α , p , with p ∈ Z and α ∈ {−, +}, and on the nodes v a pair of labels αv , pv , with αv = αv and pv ∈ Zαv such that pv , (2.10) p(v) ≡ pv = wv

where the sum is over the nodes following v (i.e. over the nodes along the path connecting v to the root), v denotes the branch v v exiting from the node v, and to each tree we

99 shall assign a value given by

Val(ϑ) =

v αv −|pv +1|αv p(v) ∂αvj fαv (S0 ψ ), λαv sv ! j=1

s

v∈V (ϑ)

(2.11)

def

where ∂α = v α · ∂ ϕ , V (ϑ) is the set of nodes in ϑ, the nodes v1 , . . . , vsv are the sv nodes preceding v (if v is a top node then the derivatives are simply missing). If Θk,α denotes the set of all trees with k nodes and with label α associated to the root line, then one has ∞ hα (ψ) = εk Val(ϑ), (2.12) k=1

ϑ∈Θk,α

and the “only” problem left is to estimate the radius of convergence of the above formal power series. For this purpose it is convenient to study the Fourier transform of the function hα (ψ ). This is easily done graphically because it is enough to attach a label (“mode label”) ν v ∈ Z2 to each node and define the momentum that flows on the tree def branch v v, i.e. ν v = wv ν w . Then (2.12) becomes

hα (ψ ) =

∞

εk

h(k) α,ν =

·

ϑ∈Θk,ν,α pv ∈Zαv

−

v∈V (ϑ)

−p(v ) S0 ν v

eiν ·ψ h(k) α,ν ,

(2.13)

ν∈Z2

k=1

with

αv −|pv +1|αv λαv fαv ,S −p(v) ν · 0 v sv !

· v αv ,

(2.14)

v∈V (ϑ) v =v0

where Θk,ν ,α denotes the set of all trees with k nodes and with labels ν and α associated with the root line. (k) Calling F = maxν |fν | we can estimate ν |ν |β |hα,ν |. The only problem is given by the presence of the factor |ν|β . In fact consider first the case β = 0: since we are assuming that f is a trigonometric polynomial there are only (2N + 1)2 possible choices for each ν v , given pv , such that |S0−pv ν v | ≤ N . Hence fixed ϑ, {αv }v∈V (ϑ) and {pv }v∈V (ϑ) the remaining sum of products in (2.14) is bounded by (if λ ≡ λ−1 + ≡ −λ− )

(3N )2k N k F k

v∈V (ϑ)

λ|pv | . sv !

(2.15)

The sum over the pv is a geometric series bounded by(2/(1 − λ))k . The combinatorial problem is well known: the factor v (1/sv !) becomes, after summing over all the trees, simply bounded by 23k , (22k due to the number of trees for fixed labels and 2k due to the sum of labels αv ). This is nt the best bound but it suffices in our case in which f is a trigonometric polynomial. In fact we have not used the (sv !)−1 and we have bounded them by 1: they would be necessary if we had supposed f to be only analytic rather than a trigonometric polynomial. Therefore for β = 0 we have proved that the conjugating function H exists and that def inside the complex domain |ε| < ε0 (0) = (3N )−3 F −1 2−4 (1−λ) it is uniformly continuous and uniformly bounded with a uniformly summable Fourier transform.

100 Taking β > 0 requires estimating |ν |β : we bound it by v |ν v |β . Then we can make −p(v) use of the fact that |S0 ν v | ≤ N to infer that |ν v | ≤ λ−|p(v)| BN , where B ≥ 1 is a components with ratio a quadratic suitable constant (because the eigenvectors of S0 have β irrational, hence a Diophantine irrational). The sum v |ν v | is over k terms which we estimate separately so that we can write v |ν v |β ≤ k|ν v |β where |ν v | = maxv |ν v |. This can be taken into account by multiplying (2.15) by an extra factor (BN )β λ−β|p( v)| ≤ BN λ−β

v

|pv |

. Therefore if β < 1 the bound that we found for β = 0 is modified into ε0 (β) = (3N )−3 F −1 (1 − λ1−β )2−5 .

(2.16)

This shows that H(ψ ) analytic in ε in the disk with radius ε0 (β). Furthermore, since in (2.16) we inserted (for simplicity) an extra factor 2−1 in excess of the result obtained by the procedure described, the H¨ older modulus is also uniformly bounded by a suitable function C(β) of β. Note that ε0 (β) → 0 for β → 1. The map H is a homeomorphism. In fact it is one-to-one because if ϕ = ψ i + h(ψ i ) for i = 1, 2 and ψ 1 = ψ 2 one would have S0k ψ 1 + h(S0k ψ 1 ) = S0k ψ 2 + h(S0k ψ 2 ) for all k, which is impossible being incompatible with the hyperbolicity of S0 (as a matrix). Furthermore given any ϕ there is a ψ such that H(ψ ) = ϕ: this is a general property of injective maps of the torus T d of the form ϕ = ψ + h(ψ ), with h Hölder continuous and smaller tha π. Indeed if d = 1 we see that as ψ runs around the circle the point ϕ = ψ + h(ψ) follows and must pass through all points of the circle as well: hence if ψ is given there is a ϕ and a function k(ϕ) such that ψ = ϕ + k(ϕ) and the function k is continuous. If d = 2 one repeats twice the argument: first we compute the partial inverse of ϕ1 = ψ1 + h1 (ψ1 , ψ2 ) by fixing ψ2 and determining k (ϕ1 , ψ2 ) such that ψ1 = ϕ1 + k (ϕ1 , ψ2 ), with k continuous; then one considers the map ϕ2 = ψ2 + h2 (ϕ1 + k (ϕ1 , ψ2 ), ψ2 ) and repeats the argument obtaining ψ2 = ϕ2 + k2 (ϕ1 , ϕ2 ). One finally sets ψ1 = ϕ1 + k1 (ϕ1 , ϕ2 ) with k1 (ϕ1 , ϕ2 ) = k (ϕ1 , ϕ2 + k2 (ϕ1 , ϕ2 )).

Problems. (1) (A property of the golden mean) Prove that since λ is Diophantine the correlations of Arnold’s cat map decay superexponentially. (Hint: Given two functions f and g, analytic on T2 , write their correlation in Fourier space, C=

ν

fν g−S −p ν ≤ F G

e−κ|ν | e−κ

|S −p ν |

ν

where F, G, κ, κ are constants depending on f, g. Then use the Diophantine condition to deduce the inequality |S −p ν | > cλp /|ν|−τ , for suitable constants c and τ . If λ is the golden mean one can take τ = 1.)

(2) Show that if |S0p ν | ≤ N , then there is a constant B = O(N ) such that |ν| ≤ λ−|p| B and B = N is a possible choice. (Hint: Call the eigenvectors v α , α = ±, and let 2B = maxµ∈Z2 ,|µ|≤N,α=± |v α · µ|;

then note that, by the spectral decomposition of the matrix S0 , one has |ν| = |S0−p µ| ≤ 2|λ|−|p| |µ| ≤ B|λ|−|p| .)

(3) (Alternative to the convergence proof for H) Show without using the Fourier transform that the series for H defined by (2.11) converges. (Hint: Bound the s-th derivatives by the maximum of F of the Fourier coefficients of f times N s .) (4) (Homeomorphism in the case of analytic f ) Show that the series for H defined by (2.11) converges under the only assumption that f is analytic. (Hint: Bound the m-th derivatives by the maximum F∞ of |f (ϕ)| in a strip |Im ϕj | < ξ (for some ξ > 0) times m!ξ −mv F∞ . The factorial is compensated by the sv !’s in (2.11) and the estimate proceeds as in the trigonometric polynomial case.) (5) (Continuity of Markov pavements in d = 2) Consider a two-dimensional Anosov map (Ω, S) and assume that it admits a fixed point x0 . Let Sε be

101 a small perturbation of S (in class C ∞ ) depending on a parameter ε. Show that the map Sε admits a Markov pavement Pε = {P0ε , . . . , Pnε } with the same compatibility matrix T as that of P0 . (Hint: Construct a Markov pavement P = {P0 , . . . , Pn } for (Ω, S) by the method of Section 1. Note that Sε will have a fixed point xε which merges differentially with x0 as ε → 0 together with any finite portion of its stable and unstable manifolds. Note that the construction of P is based on two finite connected portions of the stable and of the unstable manifolds of x0 .)

Bibliographical note The methods in this section are essentially due to F. Bonetto and P. Falco, see [GBG03] Sect. 10.1; they provide analyticity in ε. They can be extended to cover the case of chains of weakly coupled maps, see [BFG03] and [GBG03]. 3. Perturbative construction of SRB distribution. We have seen that the conjugation H that transforms a perturbed cat map Sε of the torus T 2 into a “free” Arnold’s map S0 is not differentiable (in general) and we could only prove that it can be taken to be H¨ older continuous with a prefixed exponent β < 1 for a perturbation strength that is suitably small. In spite of the lack of regularity we can still use the homeomorphism ϕ = H(ψ ) to construct the dynamics as well as the stable and unstable manifolds of each point. If ϕ = H(ψ ) the latter manifolds are given by parametric equations of the form

ϕ(t) = H(ψ + tvα )

t ∈ R, α = ±,

(3.1)

where t → ψ + t v α is the unstable or stable manifold for the unperturbed map S0 through ψ if α = + or α = −. However the above parameterization is not very useful because the function H(ψ + tvα ) is not regular as a function of t. This can be seen already from the fact that the first order term in its expansion in powers of ε cannot be (in general) differentiated with respect to (1) t at t = 0. Indeed we see from (2.7) that the component h− (ψ) can be differentiated in the direction v+ because term by term differentiation enhances convergence since p < 0; (1) on the other hand the component h+ (ψ ) cannot be differentiated in the direction v + −(p+1)

(unless special cancellations occur) because the convergence factor λ+ in (2.7) is compensated by the λp+ that the differentiation along v + brings out since p ≥ 0. To (2) second order in ε not even the t–derivative of the component h− (ψ + v + t) can be shown to exist.

To construct the stable an unstable manifolds as well as the other necessary ingredients to define the SRB distribution we apply once more the technique discussed in the previous . the (non compact) space T 2 × R2 we define the dynamical system two sections. Calling Ω S.0 (ϕ, v) = (S0 ϕ, S0 v).

(3.2)

This is a system that fails to be an Anosov system because the phase space is not compact. We can nevertheless consider its perturbation S.ε (ϕ, v ) = (S0 ϕ + εf (ϕ), S0 v + ε(v · ∂ϕ )f (ϕ)), - -

(3.3)

and we can attempt at finding an isomorphism between S.ε and S.0 or, since this turns out to be in general impossible (as it will be implicit in what follows), between S.ε and S.0,ε defined by S.0,ε (ϕ, v ) = (S0 ϕ, (S0 + Γε (ϕ))v),

(3.4).

102 with Γε (ϕ) a matrix diagonal on the basis v ± (on which S0 is diagonal too). Therefore . =H . ◦ S.0,ε , i.e. . of a simple form and such that S.ε ◦ H we look for a map H . : (ψ, w) ← H → (ϕ, v) = (ψ + h(ψ), w + K(ψ)w),

(3.5)

as we already know how to conjugate Sε with S0 , from the analysis of Section 2. The equation that the matrix K(ψ) has to verify is (S0 K(ψ) − K(S0 ψ)S0 )ij = −ε∂ϕj fi (ψ + h(ψ))− − ε∂ϕs fi (ψ + h(ψ ))K(ψ )sj + Γε (ψ)ij + (K(S0 ψ )Γε (ψ))ij ,

(3.6)

where ∂ϕ denotes a derivative of f with respect to its original argument and repeated indices mean implicit summation (to abridge notations). We write the above matrix equation on the basis in which S0 and Γε are diagonal, i.e. on the basis formed by the two eigenvectors v ± of S0 in which the matrices K, Γ have been assumed to take the form γ+ (ψ) 0 0 k+ (ψ) Γ(ψ) = , K(ψ) = . (3.7) 0 γ− (ψ) k− (ψ) 0 If α = ± and β = −α (3.6) becomes, by setting ∂α = v α · ∂ϕ ,

0 = −ε∂α fα (ϕ) − εKβα (ψ)∂β fα (ϕ) + γα (ψ), (λα Kαβ (ψ) − λβ Kαβ (S0 ψ )) =

(3.8)

= −ε∂β fα (ϕ) − εKαβ (ψ)∂α fα (ϕ) + Kαβ (S0 ψ )γβ (ψ), where ϕ means ψ + h(ψ), λ+ = λ−1 , λ− = −λ are the eigenvalues of S0 (with λ = √ ( 5 − 1)/2), and γα (ψ) = v α · Γ(ψ)v α . The above equations for K, Γ can be solved uniquely by the same technique employed in Secton 2: we skip the details. The K, Γ matrices turn out as in Section 2 to be H¨ older continuous in ψ and analytic in ε at ε small and given by an explicitly constructed power series in ε. Once the K matrix has been constructed we have access to the vectors tangent to the stable and unstable manifolds. One just notes that if ϕ = H(ψ ) then the unstable direction at ϕ will be w+ (ψ) = v + + K(ψ)v + . Furthermore a Markov pavement for Sε will be the image of a Markov pavement for S0 under the map ψ → H(ψ ); therefore ψ evolved with S0 and ϕ evolved with Sε will have the same symbolic history σ on such pavements. Hence Xε (σ) = ϕ and X0 (σ ) = ψ if ϕ = ψ + h(ψ ). The expansion rate along the unstable manifold will be λu (σ ) = eΛu (σ) with |w+ (S0 ψ )| (3.9) Λu (τ σ) = − log λ+ + γ+ (ψ ) |w+ (ψ)| where |w+ (ψ )| = 1 + k+ (ψ)2 .2 2

. (ψ, v ) = The chain of identities stemming from (3.5), (3.7) and leading to (3.9) is (ϕ, w(ϕ)) = H (H(ψ), (1 + K(ψ ))v) and .ε (ϕ, wu ) = . . , v+ ) = H . (S0 ψ , (λ+ + γ+ (ψ )v + )) = (Sε ϕ, (1 + K(S0 ψ ))(λ+ + γ+ (ψ))v+ ) S Sε H(ψ which says that the vector (1 + K(ψ))v + is mapped by the differential of Sε to (λ+ + γ+ (ψ))(1 + K(S0 ψ))v + , whence (3.9).

103 The functions γ+ , k+ are analytic in ε and H¨ older continuous in ψ with a uniformly bounded modulus C(β) if β < 1 and ε is small enough and since σ fixed means ψ fixed (because σ is the history of a point ψ on a Markov pavement for the ε–independent map older continuous in σ as well as in S0 ) we see that Λu (σ) is analytic in ε at fixed σ and H¨ ψ and therefore in ϕ. Hence the function Λu (σ ) can be written as in (1.10) in terms of a exponentially decaying potential. Therefore we have completed the construction of the potential for the symbolic dynamics whose Gibbs state is the SRB distribution. Since one dimensional short range lattice systems with nearest neighbor hard core are well understood we can proceed to use their theory to compute properties of the SRB distribution in powers of the perturbation parameter ε. For instance the SRB distribution µε will be a Gibbs state for the energy function Λu (σ ) and, therefore, it will mix at an exponential rate any pair of H¨ older continuous function. Note that if G is a Hölder continuous function then G ◦ H −1 has a representation on the Markov pavement Pε that is independent of ε. Moreover Gibbs states with exponentially decaying potential (in the sense of (1.12)) have the property that the expectation values of observables depend analytically on the parameters on which the potential itself depends analytically. This can be summarized in the following corollary.

(Mixing for SRB distributions) Let F and G be two H¨ older continuous “observables” (i.e. functions on T 2 ). Then if µε denotes the SRB distribution for Sε the following properties hold. (i) The expectation values µε (F ) and µε (G) are defined and H¨ older continuous in ε. Moreover the expectation values µε (F ◦ H −1 ) and µε (G ◦ H −1 ) are analytic function in ε for |ε| ≤ ε0 , with ε0 independent of F or G. (ii) If F is an analytic observable then the expectation value µε (F ) is analytic in ε for ε ≤ εF , with εF dependent on F . (iii) The functions F, G mix at an exponential rate in the sense that the difference between the l.h.s. and the r.h.s. of

−−−−→ µε (F )µε (G) µε (dϕ)F (S n ϕ)G(ϕ) −n→±∞

µε ((Sεn F )G) ≡

(3.10)

tends to 0 bounded by a constant (depending of F, G) times e−κF,G n , for a suitable constant κF,G > 0. (iv) The volume distribution µ0 (dϕ) = dϕ/(2π)2 “mixes with the SRB distribution” exponentially fast in the sense that given the functions F, G the difference between the l.h.s. and the r.h.s. of µ0 ((Sεn F )G) ≡

−−−−→ µε (F )µ0 (G) µ0 (dϕ)F (S n ϕ)G(ϕ) −n→+∞

(3.11)

tends to 0 bounded by a constant (depending of F, G) times e−κF,G n . The latter limit has, in general, a different value if n → −∞ and one has to replace the SRB distribution µε with the one for Sε−1 (which essentially has the same properties as µε ). We can therefore appreciate the power of the perturbation expansion method which allows us to obtain rather detailed and precise informations about the Markov partition (which of course can be defined as the H–image of the trivial partition for Arnold’s cat map S0 ); and, what is more important, it provides us with an analytic description of the SRB distribution (in the considered small perturbation problem).

104 In particular it is possible and interesting to compute the derivatives of µε (F ) at ε = 0. One easily finds the following special case of a general resul ∞ µ0 (∂ (F ◦ S0k ) · f ◦ S0−1 ); ∂ε µε (F )ε=0 = −

(3.12)

k=0

This formula was proved, in a much more general setting, by Ruelle, see [Ru97], and is very interesting for applications because it closely resembles the standard Green-Kubo formula, [GR97]. Bibliographical note The methods in this section are also essentially due to F. Bonetto and P. Falco, see [GBG03] Sect. 10.3; they provide analyticity in ε. They can be extended to cover the case of chains of weakly coupled maps, see [BFG03] and [GBG03]; the extension can be remarkably based on classical techniques of Statistical Mechanics (Mayer expansion or cluster expansion): see [GBG03]. Problem. (1) Show that the map in (1.3) does not admit an invariant probability distribution with density with respect to the volume µ0 , for ε small enough and not 0. (Hint: first show that the phase space contractin has negative average with repect to the SRB ditribution, for ε > 0 small using (3.12). This i,plies that the SRB distribution is singular with respect to ν0 . Then let µe be a distribution invariant and with density:

µε (F ) ≡ µε (Sεn F ) ≡

µ0 (dxc)ρε (x)F (S n x) −n→∞ −−−→ µ0 (ρe )µε (F ) ≡ µε (F )

so that µε ≡ µε .)

References [AW68] Adler, R., Weiss, B.: Similarity of automorphism of the tours, Memoirs of the American Mathematical Society, no. 98, 1970. [BFG03] Bonetto, F., Falco, P.L., Giuliani, A.: , Analyticity and large deviations for the SRB measure of a lattice of coupled hyperbolic systems, to appear in Journal of Mathematical Physics, 2004. [Bo70] Bowen, R.: Markov partitions for Axiom A diffeormorphisms, American Journal of Mathematics 92 (1970), 725-747. [Ch99] Chandre, C.: PhD thesis, Physics Dept., Université de Bourgogne, 1999. [CM01] Chandre, C., MacKay, R.S.D.: Approximate renormalization with codimension-one fixed point for the break-up of some three-frequency tori, Physics Letters A 275 (2000), no. 5-6, 394–400. [GBG03] Gallavotti, G., Bonetto, F., Gentile, G.: Aspects of the ergodic, qualitative and statistical theory of motion, p. 1–415, Springer–Verlag., 2004. [Ru73) Ruelle, D.: A measure associated with axiom A attractors, Americqn Journal of Mathematics, 98 (1976), 619–654. [Ru97] Ruelle, D.: Differentiation of SRB states, Communications in Mathematical Physics, , 187 (1997), 227–241. [Si68a] Sinai, Ya.G.: Markov partitions and C-Diffeormorphisms, Functional Analysis and Applications 2 (1968), 64–69.

105 [Si68b] Sinai, Ya.G.: Costruction of Markov partitions, Functional analysis and Applications 2 (1968), no. 2, 70–80.


DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY Pierre Gaspard Center for Nonlinear Phenomena and Complex Systems, Faculte´ des Sciences, Université Libre de Bruxelles, Campus Plaine, Code Postal 231, B-1050 Brussels, Belgium

Abstract

1.

Recent work on the connections between dynamical systems theory and nonequilibrium statistical mechanics is reviewed with emphasis on results which are compatible with Liouville’s theorem. Starting from a general discussion of time-reversal symmetry in the Newtonian scheme, it is shown that the Liouvillian eigenstates associated with the PollicottRuelle resonances spontaneously break the time-reversal symmetry. We explain that such a feature is compatible with the time reversibility of Newton’s equations because of a selection of trajectories which are not time-reversal symmetric. The Pollicott-Ruelle resonances and their associated eigenstates can be constructed not only for decay processes but also for transport processes such as diffusion or viscosity, as well as for reaction-diffusion processes. The Pollicott-Ruelle resonances thus describe the relaxation toward the thermodynamic equilibrium. The entropy production of these relaxation processes can be calculated and shown to take the value expected from nonequilibrium thermodynamics. In nonequilibrium steady states, an identity is obtained which shows that the entropy production directly characterizes the breaking of timereversal symmetry by nonequilibrium boundary conditions. The extension to quantum systems is also discussed.

Introduction

The methods of dynamical systems theory are ubiquitous in nonequilibrium statistical mechanics since the pioneering work by Sinai and coworkers on the ergodic theory of hard-ball systems [1, 2, 3, 4]. The defocusing character of elastic collisions shows that typical many-particle systems have a spectrum of positive Lyapunov exponents and a positive Kolmogorov-Sinai entropy per unit time [5, 6, 7, 8]. This chaotic behavior can be expected from the typical master equations ruling the stochastic processes of nonequilibrium statistical mechanics such as the


108 Boltzmann-Lorentz, the Fokker-Planck, or the Pauli equations, which all develop a positive (and even infinite) Kolmogorov-Sinai entropy per unit time [9, 10]. Moreover, a vanishing Kolmogorov-Sinai entropy would be in contradiction with experimental observation [11]. The connection between dynamical systems theory and nonequilibrium statistical mechanics has been strengthened by the discovery of relationships between the decay or escape rates of a system and the characteristic quantities of its dynamical instability, randomness or fractality [12, 13, 14, 15]. These characteristic quantities are large-deviation properties of the temporal dynamics. These new relationships are connecting the large-deviation properties of the temporal evolution of the probabilities or phase-space volumes to the irreversible properties such as the transport coefficients or the entropy production. All these new relationships from the escape-rate formula [12, 13, 14, 15] to the fluctuation theorem [16, 17, 18, 19, 20, 21, 22, 23] share this very same structure, as will be explained later. Besides, Pollicott and Ruelle have introduced a concept of resonance for Axiom A systems, showing that hyperbolic systems can present exponential decays which are intrinsic to their dynamics and independent of the circumstances of extraneous observation [24, 25, 26, 27]. The concept of Pollicott-Ruelle resonance leads to a spontaneous breaking of the time-reversal symmetry in the statistical description of the time evolution, which provides an explanation of irreversibility in agreement with the microreversibility. This explanation has recently been documented in great detail for deterministic diffusive systems with the verification that the decaying modes associated with the Pollicott-Ruelle resonances have the entropy production expected from the irreversible thermodynamics of diffusive processes [28, 29, 30, 31, 32]. One of the basic difficulties has here been to respect Liouville’s theorem that phase-space volumes are preserved by the microscopic Hamiltonian dynamics. It should here be emphasized that Liouville’s theorem is playing a central role in Boltzmann’s considerations. The violation of Liouville’s theorem by some hypothetical phase-space contraction leads to the catastrophic situation where the differential entropy – which should naturally be associated with the stationary probability distribution of the system itself according to Boltzmann’s statistical interpretation – is equal to minus infinity instead of remaining bounded. The purpose of the present paper is to show that it is possible to develop a theory of irreversibility for an underlying microscopic dynamics which is time-reversal symmetric and moreover preserves the phase-space volumes in conformity with Liouville’s theorem. This allows us to construct, in particular, nonequilibrium steady states with a well-defined

DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY

109

entropy. The general idea of the theory is to consider special solutions of Liouville’s equation, which are decaying in time. These solutions turn out to be concentrated on the unstable manifolds which are distinct from the stable manifolds. This leads to a natural breaking of the time-reversal symmetry in the statistical description. The plan of the paper is the following. Elementary considerations on the time-reversal symmetry and its breaking are developed in Sec. 2. The general theory is presented in Sec. 3. The theory is applied to simple decay processes in Sec. 4. The escape-rate formalism which gives access to the transport coefficients is presented in Sec. 5. The hydrodynamic modes of diffusion are constructed for infinite spatially extended systems in Sec. 6. The entropy production is obtained in Sec. 7. The case of nonequilibrium steady states is treated in Sec. 8. The possible extension to quantum systems is discussed in Sec. 9. Conclusions are drawn in Sec. 10.

2.

Time-reversal symmetry and its breaking

The phenomenon of spontaneous symmetry breaking is well known. Typically, the solutions of an equation have a lower symmetry than the equation itself. This is in particular the case for the time-reversal symmetry of Newton’s equations. The purpose of the present section is to develop this remark, which sheds some light on the apparent dichotomy between the everyday experience of an irreversible world ruled by timereversible laws. If we consider a system of particles of masses ma interacting with electromagnetic or gravitational forces, it is well known that Newton’s equations d2ra (a = 1, 2, ..., N ) (1) = Fa (r1 , r2 , ..., rN ) dt2 ruling the motion of the particles is time-reversal symmetric. Time redra a versal consists in reversing the velocities dr dt or momenta pa = ma dt of the particles while keeping invariant their positions ra : ma

Θ (r1 , r2 , ..., rN , p1 , p2 , ..., pN , t) = (r1 , r2 , ..., rN , −p1 , −p2 , ..., −pN , −t) (2) Time reversal is an involution: Θ2 = 1. We denote by Φt the flow of trajectories induced by Newton’s equations (1) in the phase space of positions and momenta Γ = (r1 , r2 , ..., rN , p1 , p2 , ..., pN ) ∈ M

(3)

The phase space M is made of all the possible physically distinct states of the system. The flow Φt exists and is unique according to Cauchy’s

110 theorem, which is the property of determinism. The time-reversal symmetry of Newton’s equations is expressed by Θ ◦ Φt ◦ Θ = Φ−t

(4)

which means that if the phase-space curve C = {Γt = Φt (Γ0 ) : t ∈ R}

(5)

is a solution of Newton’s equations, then the time-reversed curve ˜ t = Φt ◦ Θ(Γ0 ) : t ∈ R} Θ(C) = {Γ

(6)

starting from the time-reversed initial conditions Θ(Γ0 ) is also a solution of Newton’s equations. The further question is to know if the solution C of Newton’s equations is or is not identical to its time-reversal image Θ(C). If we have a timereversal symmetric solution: C = Θ(C)

(7)

the solution has inherited of the time-reversal symmetry of the equation. However, the solution does not have the time-reversal symmetry of the equation if we have a time-reversal non-symmetric solution: C = Θ(C)

(8)

in which case we can speak of the breaking of the time-reversal symmetry by the solution C. For the one-dimensional harmonic oscillator of Hamiltonian H=

p2 1 + mω2r2 2m 2

(9)

or the particle falling with a uniform acceleration H=

p2 + mgr 2m

(10)

all the trajectories are time-reversal symmetric: C = Θ(C), ∀ C. In contrast, this is no longer the case for the free particle in uniform motion p2 (11) H= 2m because the solution rt = (p0 /m)t + r0 is distinct from the time-reversed solution rt = −(p0 /m)t + r0 (unless the momentum vanishes: p0 = 0).


111

Indeed, the momentum of these solutions points in different spatial directions so that different successions of events happen along these trajectories. For the inverted harmonic potential H=

1 p2 − mλ2r2 2m 2

or, more generally, for the potential barriers p2 r2ν H= + V0 exp − 2ν − V0 2m 2a

(12)

(ν = 1, 2, ...)

(13)

the trajectories CE with a negative energy E are identical to their timereversal image: CE = Θ(CE ) for E < 0. But those with a positive or vanishing energy are not: CE = Θ(CE ) for E ≥ 0. In particular, the trajectories with a vanishing energy are the stable Ws and unstable Wu manifolds of the equilibrium point r = p = 0 and they are distinct. The time-reversal symmetry maps these manifolds onto each other Wu = Θ(Ws )

(14)

and, here, we have that Wu = Ws . In conclusion, the solutions of Newton’s equations typically break the time-reversal symmetry notwithstanding the fact that the equation itself possesses the symmetry. This phenomenon is known as spontaneous symmetry breaking and it manifests itself in different contexts. We owe to Newton – and his contemporaries who cast his laws of motion into differential equations – the separation between the law and its realizations into specific solutions. This is a major historical step with respect to pre-Newtonian science. Newton’s ordinary differential equations leave open the determination of the initial conditions which thus remain arbitrary and out of the law in the Newtonian scheme. The few preceding examples suggest that, in typical systems, many initial conditions lead to solutions which are not time-reversal symmetric, i.e., such that C = Θ(C). The separation between time-reversible laws and its irreversible realizations thus appears at the origin of problems with the arrows of time, which were absent in pre-Newtonian science. The preceding discussion shows that an irreversible world – which is given by a specific trajectory of Newton’s equations – is not incompatible with a time-reversible equation of motion. The phenomenon of spontaneous symmetry breaking should have familiarized us with such an apparent dichotomy. Similar considerations apply to Liouville’s equation ruling the time evolution of statistical ensembles of trajectories in classical statistical

112 mechanics. Liouville’s equation is time-reversal symmetric if the underlying Hamiltonian system is. Similar considerations also apply to Schr¨ odinger’s equation of quantum mechanics, as well as to von Neumann-Landau equation of quantum statistical mechanics. The breaking of the time-reversal symmetry is the feature of transport processes such as diffusion or viscosity. Diffusion is ruled by the irreversible equation (15) ∂t n = D ∇2n for the density n(r, t) of tracer particles which can be so dilute that they do not interact with each other while they diffuse in the surrounding fluid. The previous discussion suggests that the diffusion equation (15) describes the time evolution of ensembles of trajectories which are not time-reversal symmetric. The anti-diffusion equation ∂t n = −D ∇2n

(16)

describes the time evolution of the time-reversed trajectories. The purpose of the following sections is to show how the previous considerations can be developed into a quantitative theory of irreversibility which is not incompatible with the time-reversal symmetry of Newton’s equations.

3.

General theory

A probability density is associated with a statistical ensemble {Γ(i) }∞ i=1 of phase-space points as N 1 δ Γ − Γ(i) N →∞ N

p(Γ) = lim

(17)

i=1

If these points are the initial conditions of trajectories, the probability density evolves in time according to N 1 (i) δ Γ − Φt Γ0 N →∞ N

pt (Γ) = lim

(18)

i=1

The time evolution of this probability density is ruled by Liouville’s equation ˆ ∂t p = {H, p} ≡ Lp (19) given in terms of the Poisson bracket with the Hamiltonian of the system according to Liouville’s theorem, which defines the Liouvillian operator


113

ˆ The time integral of Liouville’s equation is the Frobenius-Perron L. operator ˆ p0 (Γ) = p0 (Φ−t Γ) pt (Γ) = Pˆ t p0 (Γ) = exp(Lt)

(20)

The probability density allows us to calculate the mean values of the observables which are functions of the phase-space variables: (21) !A"t = A(Γ) pt (Γ) dΓ as well as the time correlation functions !A(0)B(t)" or !A(0)A(t)". If the evolution operator is defined on a Hilbert space of phase-space functions, we can define a unitary group of time evolution ˆ t p0 = exp(−iGt) ˆ p0 pt = U

(22)

with the Hermitian generator given in terms of the Liouvillian operator by ˆ = iL ˆ G (23) The resolvent of the Hermitian operator is defined by ∞ 1 ˆ ˆ = −i R(z) = eizt e−iGt dt ˆ z−G 0

(24)

The unitary operator can be recovered by integration in the complex variable z along the contour C+ + C− as ˆ ˆt = 1 e−izt R(z) dz (25) U 2πi C+ +C− (see Fig. 1). The retarded or advanced evolution operator can be obtained by integrating along the contour C± as 1 t ˆ ˆ θ(±t)U = e−izt R(z) dz (26) 2πi C± By deforming the contour of integration C+ to the lower half-plane of the complex variable z, we can pick up the contributions from several complex singularities of the analytic continuation of the resolvent. These complex singularities can be poles, branch cuts, or else. The poles are called the Pollicott-Ruelle resonances [24, 25, 26, 27]. The sum of the contributions from the resonances, the branch cuts, etc... gives us an expansion which is valid for positive times t > 0 and which defines the forward semigroup: ˆ ˜ α |p0 " + · · · !A|Ψα " exp(sα t) !Ψ (27) !A"t = !A| exp(Lt)|p 0" α

114

Re s

anti-resonances

backward semigroup

C+

0

C−

resonances

forward semigroup

0 Im s = ω Figure 1. Complex plane of the variable s = −iz. The vertical axis Re s = Im z is the axis of the rates or complex frequencies. The horizontal axis Im s = −Re z is the axis of real frequencies ω. The contour C+ is slightly above the real-frequency axis and is deformed in the lower half-plane to get the contributions of the resonances for the forward semigroup. The contour C− is slightly below the real-frequency axis and is deformed in the upper half-plane to get the contributions of the anti-resonances for the backward semigroup. The resonances is mapped onto the anti-resonances by time reversal. Complex singularities such as branch cuts are also possible but not depicted here.


115

where the dots denote the contributions beside the simple exponentials due to the resonances (see Fig. 1). In such an expansion, the dependence on the observables appears via the right-eigenvectors of the Liouvillian operator ˆ α " = sα |Ψα " L|Ψ (28) and the dependence on the initial probability density via the lefteigenvectors ˆ = sα !Ψ ˜ α| ˜ α |L !Ψ

(29)

On the other hand, the analytic continuation to the upper half-plane of the complex variable z gives an expansion valid for negative times t < 0, which defines the backward semigroup: ˆ ˜ α ◦ Θ|p0 " + · · · !A"t = !A| exp(Lt)|p !A|Ψα ◦ Θ" exp(−sα t) !Ψ 0" α

(30) The time-reversal symmetry implies that a singularity located in the upper half-plane at −zα = −isα corresponds to each complex singularity in the lower half-plane of the variable zα = isα . The analytic continuation has the effect of breaking the time-reversal symmetry and shows that the semigroups are necessarily restricted to one of the two semi-axes of time. Several important mathematical questions should be answered in order to obtain such expansions: What is the spectrum of complex singularities of the resolvent? ˜ α, What is the nature of the right- and left-eigenvectors, Ψα and Ψ in each term of the expansion? For which class of observables A is defined the right-eigenvector Ψα ? For which class of initial ˜ α? probability density p0 is defined the left-eigenvector Ψ Once each term is well defined, does the whole series converge for some classes of observables A and initial probability densities p0 ? Important results have been obtained for Axiom A systems which is a class of dynamical systems having the properties that: (1) Their non-wandering set Ω is hyperbolic; (2) Their periodic orbits are dense in Ω. For these systems, Pollicott and Ruelle have proved that the spectrum close to the real axis contains resonances (poles) which are independent of the observable and initial probability density within whole classes of smooth enough functions [24, 25, 26, 27]. The resonances are responsible for exponential decays exp(sα t) at rates −sα which are thus intrinsic

116 to the dynamics. If the resonances are degenerate of multiplicity mα , Jordan-block structures may appear which introduce polynomial corrections to the exponential decays, leading to tmα −1 exp(sα t) [10]. In specific systems such as the multibaker maps and the simple examples given in the next section, the whole resonance spectrum as well as the complete expansions of the semigroups have been constructed for very smooth observables and probability density [10, 33, 34]. If the observable and the probability density have a certain degree of regularity, the expansion can be limited to the contributions of the first few resonances plus a rest which decays faster than the decay rate of the last used resonance. This rest becomes negligible after a long enough time. For non-hyperbolic systems such as intermittent maps or bifurcating equilibrium points, branch cuts appear which are associated with powerlaw decays [35, 36, 37, 38]. An important remark is that the eigenvectors (28) and (29) are not given by functions but by mathematical distributions of the Schwartz type. Therefore, their density does not exist as a function and their cumulative function is required for their representation. In bounded systems which are mixing, there exists a unique eigenvalue at s0 = 0. The non-trivial Pollicott-Ruelle resonances of the forward semigroup are then located away from the real-frequency axis with Re sα < 0. However, there exist most interesting situations in which the leading Pollicott-Ruelle resonance has a non-vanishing real part Re s0 < 0. This happens for open systems with escape as shown in Secs. 4 and 5, as well as for spatially periodic systems as shown in Sec. 6. In open systems with escape, the leading Pollicott-Ruelle resonance defines the escape rate (31) γ = −s0 The associated eigenvector Pˆ t Ψ0 = es0 t Ψ0

(32)

is given by the conditionally invariant measure of density Ψ0 = lim e−s0 t Pˆ t Υ t→∞

(33)

where Υ is an arbitrary function. The density Ψ0 is typically singular so that we define the cumulative function ξ F0 (ξ) = Ψ0 (Γξ )dξ (34) 0

where the integral is carried out over a curve {Γξ } in the phase space.


117

In hyperbolic systems, the escape rate can be expressed in terms of the Lyapunov exponents and the Kolmogorov-Sinai entropy per unit time which is defined with respect to the natural invariant measure having the non-wandering set for support [10, 39, 40]: γ= λi − hKS (35) λi >0

If we introduce the partial information dimensions 0 ≤ di ≤ 1 associated with the Lyapunov exponents according to the fundamental work by Young and others [39, 41, 42], the escape rate can be rewritten as γ=

(1 − di )λi

(36)

λi >0

These remarkable formulas generalize Pesin’s identity [43]. The escape implies that some partial dimensions are smaller than unity, which shows that the non-wandering set is fractal in some unstable directions as well as in the corresponding stable directions by time-reversal symmetry. In contrast, the conditionally invariant measure associated with the escape rate γ and defined by the density (33) is absolutely continuous with respect to the Lebesgue measure along the unstable directions but fractal in the stable directions.

4.

Decay processes

In this section, we consider simple systems with escape of particles from a phase-space region where the non-wandering set is located. The number of particles remaining in this region is decaying so that the process is transient.

4.1

Inverted harmonic potential

The first example is the Hamiltonian (12) with an inverted harmonic potential. The equilibrium point at r = p = 0 constitutes the nonwandering set of this system. The equilibrium point is hyperbolic so that we here have a simple example of Axiom A systems. Its Lyapunov exponents are given by (+λ, −λ). The invariant measure is a Dirac delta distribution located at the phase-space point r = p = 0. Its KolmogorovSinai entropy is equal to zero so that the system is nonchaotic as expected. According to Eqs. (31) and (35), the leading Pollicott-Ruelle resonance should here be given in terms of the positive Lyapunov exponent: s0 = −γ = −λ (37)

118 It is straightforward to verify this result and even obtain the full spectrum of the Pollicott-Ruelle resonances together with their associated eigenvectors, as shown here below. After the canonical transformation  1  q = √2mλ (x − y) , (38)  (x + y) p = mλ 2 the Hamiltonian (12) becomes H=λxy

(39)

In the new coordinates, the unstable manifold is the axis of the xcoordinate and the stable manifold is the axis of the y-coordinate. The solutions of Hamilton’s equations   x˙ = + ∂H ∂y = +λ x (40)  = −λ y y˙ = − ∂H ∂x Φt (x, y) = e+λt x, e−λt y

define the flow

(41)

In the long-time limit t → +∞, exp(−λt) is a small parameter in terms of which we can carried out a Taylor expansion of the statistical averages according to [10]

−λt dx dy A e/+λt x , e y p0 (x, y) 01 2 =x −λt = e dx dy A x , e−λt y p0 e−λt x , y

!A"t =

=

(43)

∞ ∞ 1 −λlt l l 1 −λmt m m e e y ∂y A(x , 0) x ∂x p0 (0,y) m! l! m=0 l=0 (44) ∞ 1 −λ(l+m+1)t 1 l m m l e dx x ∂y A(x , 0) dy y ∂x p0 (0, y) (45) m! l!

= e−λt

=

(42)

l,m=0 ∞

dx dy

˜ lm |p0 " e−λ(l+m+1)t !A|Ψlm " !Ψ

(46)

l,m=0

We can therefore identify the right- and left-eigenstates as

Ψlm (x, y) = ˜ lm (x, y) = Ψ

1 l x (−∂y )m δ(y) m! 1 m y (−∂x )l δ(x) l!

(47) (48)


119

The eigenstates are given by the derivatives of the Dirac distribution. The right-eigenstates Ψlm have the unstable manifold y = 0 for support, ˜ lm have the stable manifold x = 0 for support. while the left-eigenstates Ψ We can check that these distributions are respectively the eigensolutions of the Liouvillian operator and of its adjoint: ˆ Ψlm = −λ (l + m + 1) Ψlm L ˜ lm ˆΨ ˜ lm = −λ (l + m + 1) Ψ L

(49) (50)

Accordingly, the Pollicott-Ruelle resonances of the inverted harmonic potential are simply given by the integer multiples of the Lyapunov exponent λ: slm = −λ (l + m + 1) (51) with l, m = 0, 1, 2, 3, ... The right- and left-eigenstates are given by Schwartz distributions defined on smooth enough test functions A and p0 . In order for !A|Ψlm " to be defined, the observable A(x, y) must be m-times differentiable transversally to the unstable manifold y = 0 and integrable with xl ˜ lm |p0 " is defined along the unstable manifold. On the other hand, !Ψ if the probability density p0 (x, y) is l-times differentiable transversally to the stable manifold x = 0 and integrable with y m along the stable manifold. The full series (46) converges if A(x, y) is an entire analytic function of exponential type in y and an infinitely differentiable function of compact support in x, while p0 (x, y) is an entire analytic function of exponential type in x and an infinitely differentiable function of compact support in y. The exponential decay at the escape rate γ = λ proceeds continuously for the conditionally invariant measure given by the leading righteigenstate Ψ00 (x, y) = δ(y): Pˆ t Ψ00 = e−γt Ψ00

(52)

This conditionally invariant measure corresponds to a statistical ensemble of trajectories which are uniformly distributed on the unstable manifold y = 0 and nowhere else. This is an example of selection of trajectories (or initial conditions). The irreversible process of decay is the feature of trajectories located on the unstable manifold y = 0 which is distinct from the time-reversed manifold x = 0. The trajectories have been selected by the time evolution. Indeed, if we start from an ensemble of initial conditions located in the vicinity of the equilibrium point x = y = 0, this cloud of points tends to concentrate along the unstable manifold during the time evolution. In the long-time limit, we obtain

120 with Eq. (33) the leading eigenstate Ψ00 (x, y) = δ(y) which has the unstable manifold for support. This support has a zero Lebesgue measure because the decay process is transient. In this way, we end up in the long-time limit with a time-reversal non-symmetric trajectory associated with the irreversible decay process.

4.2

Generic potential barrier

A generic potential barrier is defined as a barrier with a quadratic maximum. An example of such potential is given by the Hamiltonian (13) with ν = 1. This Hamiltonian system has a unique equilibrium point of saddle type at r = p = 0 and is a further example of Axiom A system. Here, the Lyapunov exponent is given by 3 V0 λ= (53) ma2 The Kolmogorov-Sinai entropy is here again vanishing, hKS = 0, so that this system is nonchaotic. The motion in the vicinity of the saddle equilibrium point r = p = 0 is described by the aforementioned inverted harmonic potential (12). At large distances, the particle is no longer accelerated and moves in free motion contrary to the situation in the inverted harmonic potential, but this difference does not affect the spectrum of Pollicott-Ruelle resonances which is the same in both systems. The spectrum of Pollicott-Ruelle resonances of the generic potential barrier is thus given by slm = −λ (l + m + 1)

(54)

with l, m = 0, 1, 2, 3, .... Accordingly, this system presents a dynamical instability without chaos. Indeed, the particle remains at the top of the hill if it has no momentum, so that r = p = 0 is a stationary solution. However, perturbed trajectories escape to infinity. Figure 2b depicts different trajectories issued from several initial conditions which are very close to the point r = p = 0. Figure 2c represents the fraction of trajectories which are still in the interval −a ≤ r ≤ +a at the current time t. This fraction defines the survival probability. We observe that this survival probability decays exponentially at long times. The rate of decay defines the escape rate γ of the hill. Since the trajectories which escape at very long times are issued from very near the point r = p = 0, we can understand that the long-time decay is controlled by the dynamical instability near r = p = 0 and, thus, that the escape rate is equal to the Lyapunov exponent, γ = λ. In Fig. 2d, the phase portrait is drawn in the plane

121


(a)

(c)

V(r)

P(t)

1

0.1

0

1

2

3

t

4

4

r Ws

(d)

Wu

()

0

1

p

t

2

3

()

Wu

Ws

()

()

0

(b)

r

r

Figure 2. Particle moving down a hill under the Hamiltonian (13): (a) the potential V (r) versus position r; (b) positions r of different trajectories issued from initial conditions very close to the unstable equilibrium point versus time t; (c) a typical survival probability P (t) versus time t; (d) the phase portrait in the (r, p)-plane with (±) (±) the stable Ws and unstable Wu manifolds of r = p = 0. (±)

(±)

Γ = (r, p). The stable and unstable manifolds Ws,u = {Γs,u (t), t ∈ R} are connected to the stationary solution at Γ = 0. The signs denote both branches of these manifolds. For t → +∞, the probability density tends to concentrate along the unstable manifold. We can check that the distribution 4 5 +∞ exp(γτ ) δ Γ − Γu() (τ ) dτ (55) Ψ00 (Γ) = =± −∞

is an exact eigenstate of the Frobenius-Perron operator Pˆ t Ψ00 (Γ) = Ψ00 (Φ−t Γ) = exp(−γt) Ψ00 (Γ)

(56)

corresponding to the leading Pollicott-Ruelle resonance s0 = −γ = −λ. () () The result (56) is a consequence of Φt Γu (τ ) = Γu (τ + t) and of the Liouville theorem [44]. The distribution (55) is the density of the conditionally invariant measure associated with the escape rate. We here have a further example of selection of initial conditions. The irreversible character of the escape is the feature of trajectories located on the unstable

122 manifold. The fact that the unstable manifold is distinct from its time reversal – which is the stable manifold – expresses the irreversibility of the escape process and allows for the unidirectional decay (56) in time. Although the time-reversed process is ideally possible because Hamilton’s equations are time-reversal symmetric there are at least two reasons that prevent the realization of such a time-reversed process. A first reason is that dynamical instability can spoil the preparation of a timereversed state initially prepared in a realistic way. In order to prepare such a state, particles must be placed on their initial conditions. Since the phase space is a continuum, this operation is affected by errors. For the time-reversed state, the initial conditions must be aligned precisely on the stable manifold. But any error transverse to the stable manifold gives to the initial condition a small component in the unstable direction that will be amplified over a time of the order of the inverse of the Lyapunov exponent multiplied by the logarithm of the inverse of the error. Therefore, the cloud of trajectories will be concentrated along the unstable manifold after a time longer than this Lyapunov time and the forward semigroup will anyway control the long-time evolution. This problem of sensitivity to initial conditions does not manifest itself if the eigenstate is prepared on the unstable manifold because the errors are exponentially damped in this case. The second reason is that the eigenstate (55) belongs to the spectral decomposition of the forward semigroup which is only defined for positive times t > 0. On the other hand, the timereversed eigenstate belongs to the backward semigroup which is defined for negative times t > 0. It turns out that the backward semigroup may not be prolongated to positive time because, for typical initial probability densities and observables, the asymptotic series only converges for t < 0 and diverges for t ≥ 0. The origin of time is therefore an unbreakable horizon for the use of the backward semigroup. This horizon is stronger than the horizon due to the positive Lyapunov exponent of a trajectory. Indeed, the Lyapunov horizon can be postponed by taking smaller errors on the initial conditions. In contrast, the horizon of an expansion such as Eq. (46) cannot be postponed.

4.3

Nongeneric potential barriers

The Hamiltonians (13) with ν = 2, 3, 4, ... give examples of nonhyperbolic systems which do not satisfy Axiom A. The system is both nonhyperbolic and nonchaotic. The equilibrium point r = p = 0 is still unstable but the instability is no longer exponential. The decay is here given by a power law. If we prepare a statistical ensemble of trajectories in the vicinity of the top of the hill and we measure the time Tesc taken


123

by these trajectories to escape from the hill, we find that the probability that the escape happens beyond the current time t decays according to the power law 1 (57) Prob{Tesc > t} ∼ 1 t ν−1 The decay is no longer exponential so that the resonances of the previous examples are here replaced by a branch cut {Re s + i Im s : Re s ≤ 0, Im s = 0} in the plane of the complex variable s of the Liouvillian ˆ −1 for the forward semigroup. By time-reversal symresolvent (s − L) metry, the spectrum of the backward semigroup has a branch cut in the upper half-plane. Stable and unstable manifolds are still associated with the equilibrium point r = p = 0 and the decay process forward in time will here also be controlled by trajectories concentrated on the unstable manifold and thus distinct from the time-reversed trajectory. Therefore, there is here also a spontaneous symmetry breaking of the time-reversal symmetry but due to an instability which is weaker than exponential. This shows the great generality of the present theory since it may also apply to nonhyperbolic and nonchaotic systems. The difference with respect to hyperbolic systems is that the spectrum of complex singularities controlling the semigroups may be dominated by branch cuts instead of resonances in nonhyperbolic systems. Branch cuts appear in other examples of nonhyperbolic systems such as bifurcating vector fields and intermittent maps [35, 36, 37, 38]. Only a precise analysis of the spectrum can determine the long-time behavior of the statistical ensembles of trajectories. We should notice that, generically, the potential barrier has a quadratic maximum so that exponential decay controlled by Pollicott-Ruelle resonances is the generic behavior in the family of Hamiltonian systems (13). The robustness of hyperbolicity is an important property which has been pointed out in many circumstances.

4.4

The disk scatterers

The disk scatterers form a family of open dynamical systems with escape which includes chaotic systems. The disk scatterers or disk billiards are Hamiltonian-type systems in which a free particle undergoes elastic collisions on a finite number of immobile hard disks. Energy as well as the phase-space volumes are conserved. Since elastic collisions on circular disks are defocusing these systems are hyperbolic with a positive Lyapunov exponent. Axiom A is satisfied if the disks are sufficiently far apart and do not hide each other. The non-wandering set is composed of

124 0

(a)

-1

1

...12121212...

Re s

2

(b))

-2 -3 -4

0

5

Im s

10

15

Figure 3. Two-disk scatterer: (a) configuration of the system; (b) spectrum of Pollicott-Ruelle resonances.

the trajectories bouncing forever between the disks. The non-wandering set is of zero Lebesgue measure so that almost all the trajectories escape in free motion to infinity. A first example is the two-disk scatterer. Here, the non–wandering set is composed of the unique periodic orbit bouncing between the two disks along the line joined their centers. This periodic orbit is unstable with stable and unstable manifold. The situation is similar to the case of the generic potential barrier: the escape rate is equal to the Lyapunov exponent, γ = λ, and the Kolmogorov-Sinai entropy is vanishing, hKS = 0. The system is hyperbolic but nonchaotic. The spectrum of PollicottRuelle resonance is however more complicated because of the periodic bouncing motion of frequency ω between both disks: slm = −λ(l + 1) + i m ω

(58)

with l = 0, 1, 2, 3, ... and m = 0, ±1, ±2, ±3, ... The resonances form a half periodic array extending toward negative values of Re s and separated from Re s = 0 by a gap given by the escape rate γ = λ (see Fig. 3). An example of chaotic system is given by the three-disk scatterer. The scatterer is made of three disks at the vertices of an equilateral triangle. After each collision on a disk, the particle has the choice to go to one or the other of the two other disks. This generates an exponential proliferation of possible orbits so that the non-wandering set here contains an uncountable fractal set of trajectories. If the disks are sufficiently far apart, these trajectories can be put in correspondence with a symbolic dynamics. The properties of the three-disk scatterer can be analyzed in


125

Figure 4. Three-disk scatterer: (a) configuration of the system; (b) spectrum of Pollicott-Ruelle resonances. A1 , A2 , and E denote the irreducible representations of the group C3v of symmetry of the scatterer with the three disks forming an equilateral triangle. Each resonance belongs to one of these irreducible representations. The A1 and A2 -resonances are simply degenerate and the E-resonances are doubly degenerate.

great detail [10, 45, 46, 47]. The Kolmogorov-Sinai entropy is here positive so that the escape rate is no longer entirely given by the Lyapunov exponent but by the escape-rate formula (35): γ = λ − hKS

(59)

The spectrum of Pollicott-Ruelle resonances can be obtained with great precision by periodic-orbit theory (see Fig. 4) [47]. The resonances which are the closest to the real-frequency axis control the decay of statistical ensembles of trajectories escaping the scatterer. Indeed, the survival probability presents an exponential decay modulated by irregular oscillations. The gross exponential decay is controlled by the escape rate while oscillations appear which are the feature of the other PollicottRuelle resonances. This important observation has been obtained in the three- and four-disk scatterers [47].

5.

Escape-rate formalism

The idea of the escape-rate formalism is to consider first-exit problems in deterministic systems sustaining transport processes such as diffusion or viscosity [12, 13, 14, 15, 48]. First-exit or first-passage problems are

126 very well known in stochastic theory and reaction-rate theory [49]. Firstexit problems have been considered since Kramers’ pioneering work [50]. In Kramers’ problem, the escape of particles over a barrier is a way to obtain the reaction rate. This problem has many applications in a variety of physical, chemical, and astronomical systems [51]. The aforementioned disk scatterers are simple models of unimolecular reactions [45], but the escape here occurs for a deterministic dynamics. That the transport coefficients can also be obtained by setting up a first-exit problem does not seem to have been largely noticed. The advantage of this method is to work with a finite system on which boundary conditions are imposed. Boundary conditions as well as initial conditions can be changed at will although the equations of motion may not.

5.1

Viscosity and other transport coefficients

In order to set up a first-exit problem for a given transport property, absorbing boundary conditions should be imposed in the space of variation of the so-called Helfand moment [52], which is the time integral of the corresponding microscopic current [14, 48]. For viscosity, the Helfand moment is given by G= √

N 1 xa pya V kB T a=1

(60)

where T is the temperature, V the volume of the system, and kB Boltzmann’s constant. This Helfand moment gives the center along the x-axis of the y-component of the momenta of the particles. The shear viscosity coefficient is known to be given by the Einstein type of formula η = lim

t→∞

1 !(Gt − G0 )2" 2t

(61)

There exists a Helfand moment associated with each transport coefficient [52]. Einstein’s formula (61) shows that the Helfand moment has a diffusive motion. The corresponding diffusivity coefficient is simply given by the viscosity coefficient according to Eq. (61). We can set up a first-exit problem by considering the escape of trajectories out of the interval − χ2 < G < + χ2 in the space of variation of the Helfand moment. The escape rate is directly proportional to the viscosity coefficient and inversely propertional to the square of the distance between the absorbing boundaries at ± χ2 : π 2 (62) γη χ


127

We may now consider the same problem for the underlying deterministic dynamics in the phase space of this N -particle system [12, 13, 14, 15]. In the phase space R6N , the absorbing boundaries are imposed on the hypersurfaces N χ 1 xa pya = ± (63) G= √ 2 V kB T a=1

We can consider the set of trajectories forever trapped between these hypersurfaces. We should here point out that, in the three-dimensional physical space, the particles are in a box of finite volume V and they bounce on the walls of the box so that trajectories exist for which the Helfand moment can remain in a compact phase-space region such as the region between the hypersurfaces (63). If the dynamics is hyperbolic, the trajectories which never escape this region are unstable and form a subset of vanishing probability in the phase space. Almost all trajectories escape from this region with an escape rate γ which can be estimated by Eq. (62). On the other hand, the escape rate is given by dynamical systems theory as the difference between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy according to Eq. (35) or in terms of the partial information dimensions by Eq. (36). Combining with the result (62), we obtain a relationship between the transport coefficient and the characteristic quantities of chaos:   χ 2 η = lim lim (1 − di )λi  (64) χ→∞ V,N =nV →∞ π λi >0

V,χ

This formula has recently been used to calculate the viscosity coefficients in a hard-disk model of fluid [48]. An important remark is that irreversible properties such as the transport coefficients already appear in systems with few particles if the dynamics is spatially extended by periodic boundary conditions. This is well known for diffusion in the Lorentz gases but it also true for the other transport coefficients. This is natural because the transport coefficients are routinely computed by molecular dynamics with periodic boundary conditions. It is important to increase the number of particles in the computation in the regimes where the collective effects become essential at high densities n, but the transport properties can already be obtained from the collision between two or three particles at low densities [53]. The escape-rate formalism has the advantage of displaying the irreversibility in the leading Pollicott-Ruelle resonance. In closed systems,

128 the leading Pollicott-Ruelle resonance is vanishing and the interesting behavior is hidden in the next-to-leading resonance. By imposing absorbing boundary conditions, the irreversible behavior is promoted to the leading resonance. For the first-exit problem of viscosity of escape rate (62), the spectrum of Pollicott-Ruelle resonances can be expected to be given by πj 2 (65) sj −η χ with j = 1, 2, 3, ..., if the system is hyperbolic. The associated righteigenstates and, in particular, the conditionally invariant measure associated with the leading Pollicott-Ruelle resonance s1 = −γ, is smooth in the unstable directions but fractal in the stable ones [13]. This measure decays according to Eq. (32) for t → +∞. The time-reversed measure belongs to the backward semigroup and is clearly distinct from the conditionally invariant measure of the forward semigroup. We here have another manifestation of the spontaneous breaking of the time-reversal symmetry: this one explicitly for transport properties.

5.2

Diffusion and mobility

The escape-rate formalism can be applied to a broad variety of situations. For diffusion in a two-degree-of-freedom system such as the Lorentz gas, the Helfand moment is just given by the position of the tracer particle Gt = xt and Einstein’s formula (61) gives the diffusion coefficient D. Alternatively, the diffusion coefficient can be given by Eq. (64) where the partial information dimension can be replaced by the partial Hausdorff dimension 0 ≤ dH ≤ 1 in the unique unstable direction of Lyapunov exponent λ [12] L 2 [λ(1 − dH )]L D = lim (66) L→∞ π which has been applied to the hard-disk Lorentz gas [13]. We can also consider the mobility µ in a conservative Lorentz gas with an external electric field F [54, 10]. On large spatial scales, this process is described by the biased diffusion equation ∂t n = ∇ · (D∇n − µFn)

(67)

where µ is the mobility coefficient. If two absorbing boundaries are placed at a distance L from each other and transverse to the electric field F, the escape rate is given by π µ2F 2 γD (68) 2+ L 4D


129

By combining with the escape-rate formula (35), the diffusion coefficient can be obtained in the limit   L D = lim lim λi − hKS  (69) 2 L→∞ F →0 π λi >0

L,F

and, thereafter, the mobility coefficient in the limit   4D  λi − hKS  µ2 = lim lim F →0 L→∞ F 2 λi >0

(70) L,F

Other problems can be envisaged with similar considerations.

5.3

Reaction-diffusion

Reaction-diffusion processes have also been considered in which a point particle undergoes elastic collisions in a hard-disk Lorentz gas with sinks [55, 56]. The sinks are circular holes replacing some disks in the Lorentz gas. If we denote by X a moving point particle and by ∅ its annihilation, the reaction scheme is given by X + disk X + sink

X + disk → ∅ + sink

(71) (72)

Accordingly, the number of point particles in the Lorentz gas decreases with time because of their annihilation at the sinks. This decrease is characterized by the survival probability of a point particle. The time evolution of this survival probability depends on the geometry of the Lorentz gas. For a Lorentz gas with a periodic lattice of disks and a periodic superlattice of sinks, the same cell containing one sink and several disks repeats itself in a superlattice over the whole system. In this case, the decay is exponential and an escape rate γ can be defined: periodic geometry:

P (t) ∼ exp(−γt)

(73)

If the sinks are sufficiently far apart and form a triangular superlattice, the escape rate can be evaluated in terms of the diffusion coefficient D of the periodic Lorentz gas as γ (14.0 ± 0.4)

D ns ln nnds

(74)

130 where ns is the density of sinks and nd the density of disks [55]. The trajectories which never escape form a fractal repeller in the threedimensional phase space of an energy shell. The Hausdorff dimension in this phase space is equal to DH = 2dH +1 where 0 ≤ dH ≤ 1 is the partial Hausdorff dimension in the stable or unstable directions. This partial dimension is given in terms of the escape rate (74) and the Lyapunov exponent of the Lorentz gas as γ dH 1 − (75) for ns → 0 λ For a d-dimensional Lorentz gas with a random configuration of sinks, the survival probability decays according to a stretched exponential [57] d (76) random geometry: P (t) ∼ exp −Cd t d+2 with a coefficient depending on the concentration of sinks as Cd ∼ | ln(1− 2 ns /nd )| d+2 . Such a stretched exponential finds its origin in a branch cut in the complex plane s of the Liouvillian spectrum. The density of escape rates γ = −s along this branch cut such that ∞ P (t) = dγ ρ(γ) exp(−γt) (77) 0

presents the Lifchitz tail

ρ(γ) ∼ exp −

Ad

d

(78)

γ2

with a coefficient Ad related to Cd . In d = 2, the trajectories which never escape form a set of zero Lebesgue measure but of Hausdorff dimension equal to the phase-space dimension. Indeed, we can here evaluate the number N () of cells of size required to cover the non-escaping set as [56]

3 1 1 1 N () ∼ (79) exp −2C2 ln 3 λ where λ is the Lyapunov exponent. This number increases as → 0 more slowly than for a plain three-dimensional object so that the non-escaping set is a fractal but its dimension is nevertheless equal to DH = 3 [56].

6. 6.1

Hydrodynamic and other relaxation modes Lattice Fourier transform

There is another way to promote the irreversibility to the leading Pollicott-Ruelle resonance. If the dynamics is spatially periodic as it is


131

Figure 5. Periodic hard-disk Lorentz gas: Example of trajectory of a point particle elastically bouncing in a triangular array of immobile hard disks. The distance between the disks does not allow trajectories to cross the lattice without collision so that typical motion is diffusive [3].

the case for the periodic Lorentz gases (see Fig. 5), we may still consider probability densities which are not necessarily periodic in space. The probability density extends over the infinite system and can be decomposed into Fourier components by a lattice Fourier transform. A remarkable property is that each Fourier component obeys a time evolution which is independent of the other components [10]. The FrobeniusPerron operator ruling the time evolution of a Fourier component depends on its wavenumber k. Accordingly, the Pollicott-Ruelle resonances now depend on the wavenumber. If the wavenumber is vanishing, we recover the dynamics with periodic boundary conditions which admits an invariant probability measure describing the microcanonical equilibrium state. In contrast, an invariant probability measure no longer exists as soon as the wavenumber is non-vanishing. Instead, we find a conditionally invariant complex measure which decays at a rate given by a non-trivial leading Pollicott-Ruelle resonance sk . This conditionally invariant measure defines the hydrodynamic mode of wavenumber k and the associated Pollicott-Ruelle resonance sk gives the dispersion relation of the hydrodynamic mode [10].

132 The hydrodynamic mode of wavenumber k is an eigenstate of the operator Tˆl of translation by the lattice vector l ∈ L: Tˆl Ψk = eik·l Ψk

(80)

This condition can be called a quasiperiodic boundary condition because it imposes on the state a periodicity of wavelength 2π/k beside the intrinsic spatial periodicity of the system (which is for instance the size of the lattice cells in the periodic Lorentz gases). The translation operator commutes with the Frobenius-Perron operator 4 5 Tˆl , Pˆ t = 0 (81) so that we may find an eigenstate common to both the spatial translations and the time evolution: Pˆ t Ψk = esk t Ψk

6.2

(82)

Diffusion

For the periodic Lorentz gases, the leading Pollicott-Ruelle resonance is nothing else than the dispersion relation of diffusion given by van Hove formula [58] as sk = lim

t→∞

1 ln!exp [ik · (rt − r0 )]" = −Dk2 + O(k4) t

(83)

where r is the position of the tracer particle in diffusive motion and D is the diffusion coefficient. The Pollicott-Ruelle resonance coincides with the eigenvalue s0 = 0 of the microcanonical equilibrium state if the wavenumber vanishes. However, we observe that the Pollicott-Ruelle resonance depends on the wavenumber and becomes non-trivial for nonvanishing wavenumber. This behavior is in contrast to what happens in closed systems where the interesting Pollicott-Ruelle resonances are next-to-leading beside the zero eigenvalue corresponding to the invariant probability distribution. The present considerations with quasiperiodic boundary conditions greatly simplify the use of the Pollicott-Ruelle resonances. The eigenstate Ψk can be considered as a conditionally invariant complex measure, which is smooth in the unstable directions but singular in the stable ones. In general, this eigenstate is a mathematical distribution of Schwartz type which cannot be represented unless integrated in

133


0.4

Im F k

0.2 kx = 0

0

kx = 0.5

-0.2

kx = 0.9

-0.4 -0.2

0

0.2

0.4 0.6 Re Fk

0.8

1

1.2

Figure 6. Periodic hard-disk Lorentz gas: Curves of the cumulative functions of the hydrodynamic modes of wavenumber kx = 0.0, 0.5, and 0.9 with ky = 0 [59].

phase space with some test function. For instance, we may consider the cumulative function ξ Fk (ξ) = Ψk (Γξ ) dξ (84) 0

where Γξ is a curve of parameter ξ in the phase space. In the case of diffusion in the periodic Lorentz gases, such cumulative functions can be directly defined as [59] θ 0 dθ exp [ik · (rt − r0 )θ ] (85) Fk (θ) = lim 2π t→∞ 0 dθ exp [ik · (rt − r0 )θ ] with integration on initial positions at an angle ξ = θ with the horizontal axis and radial initial velocities around a scattering center. The purpose of the denominator is to compensate the exponential decay exp(sk t) of the numerator in order to define a function in the limit t → ∞. For zero wavenumber k = 0, we recover the microcanonical equilibrium state. Indeed, the cumulative function (85) is now given by F0 (θ) =

θ 2π

(86)

which is the real cumulative function of a uniform probability density 1 . If the wavenumber is non-zero, the cumulative function is Ψ0 (θ) = 2π

134 complex and depicts a fractal curve in the complex plane. The fractal character of the curve tends to increase with the wavenumber as observed in Fig. 6. In order to determine the fractal Hausdorff dimension of this curve, we introduce the Ruelle topological pressure for two-degree-offreedom systems [60] 1 ln !|Λt |1−β " t→∞ t

P (β) ≡ lim

(87)

where |Λt | > 1 is the stretching factor of a given trajectory, by which an error on the initial condition is amplified after a time t. The Ruelle topological pressure is the generating function of the mean Lyapunov exponent and its statistical moments. The Ruelle topological pressure is equivalently given by P (β) = hKS (β) − β λ(β)

(88)

in terms of the mean Lyapunov exponent and Kolmogorov-Sinai entropy of an invariant measure µβ which gives a relative probability weight |Λt |−β to each trajectory [15, 10]. The natural invariant measure of the Liouvillian dynamics is the one with β = 1. It was proved in Ref. [59] that, for Axiom A systems, the Hausdorff dimension DH of the fractal curve corresponding to the cumulative function of a diffusive hydrodynamic mode of wavenumber k is given by the root of P (DH ) = DH Re sk

(89)

where the left-hand side involves the Ruelle topological pressure and the right-hand side the leading Pollicott-Ruelle resonance, i.e., the dispersion relation of diffusion. In the limit of a vanishing wavenumber, the dispersion relation of diffusion vanishes because of Eq. (83) so that the dimension is equal to unity by the property that P (1) = 0 in the microcanonical equilibrium state. If we now expand Eq. (89) in powers of k2 , we obtain the Hausdorff dimension as [59, 61] DH = 1 +

D 2 k + O(k4 ) λ

(90)

in terms of the diffusion coefficient D and the mean Lyapunov exponent λ = −P (1). This relationship is in agreement with numerics [59]. This Hausdorff dimension is larger than unity and increases with the wavenumber. It can be rewritten in the form 1 λ(DH − 1)k k→0 k2

D = lim

which is similar to Eq. (66) where the wavenumber is k = π/L.

(91)


135

Therefore, the diffusive modes are given by singular distributions instead of regular functions. The fractal character is related to the exponential instability in the microscopic dynamics. In nonhyperbolic systems, diffusion may become anomalous in some regimes with a non-quadratic dispersion relation such as sk −C|k|β−1 (2 < β < 3) and the corresponding modes are also singular [62].

6.3

Reaction-diffusion

We may also consider reaction-diffusion processes in the hard-disk periodic Lorentz gas in which the moving point particle carries a color A or B which changes with probability p0 upon collision on some special disks called catalysts [10, 63, 64, 65, 66]. The catalytic disks form a superlattice of the triangular lattice of hard disks. The reaction scheme is here: A + disk B + disk B + catalyst ,

A + disk B + disk A + catalyst

with probability p0

(92) (93) (94)

The Liouvillian dynamics of such reaction-diffusion systems decouples into diffusion for the total density of particles and reaction for the difference between the densities of A and B particles. The diffusive sector precisely obeys the time evolution described in the previous subsection. On the other hand, the dispersion relation of the reactive modes are given by the following generalization of the van Hove formula [66] 1 ln!(1 − 2p0 )Nt exp [ik · (rt − r0 )]" t→∞ t

sk = lim

(95)

where Nt is the number of catalysts met by the point particle during the time interval t. The dispersion relation (95) gives the leading PollicottRuelle resonance of the time-evolution operator in the reactive sector. By expanding in powers of the wavenumber k, we obtain sk = −2κ − D(r) k2 + O(k4)

(96)

1 ln!(1 − 2p0 )Nt " 2t

(97)

with the reaction rate κ = − lim

t→∞

and the reactive diffusion coefficient !(1 − 2p0 )Nt (xt − x0 )2" t→∞ 2t!(1 − 2p0 )Nt "

D (r) = lim

(98)

136 assuming isotropy [66]. There exist two important regimes [65]: The rate-limited regime is the regime where the reaction probability p0 goes to zero. In this regime, the reaction rate to vanish as κ = O(p0 ), while the reactive diffusion coefficient converges to the diffusion coefficient itself: D (r) = D + O(p0 ). The diffusion-limited regime corresponds to the situation where the catalysts are so much far apart that the reaction is controlled by the diffusion time between the catalysts. In this case, the reaction rate behaves as [64] D nc κ (6.8 ± 0.3) nd (99) ln nc in a two-dimensional triangular periodic Lorentz gas with a density nd of hard disks and a density nc of catalysts. The reactive modes are given by singular distributions with complex cumulative functions defined as [66] θ Nt (θ ) exp [ik · (r − r ) ] t 0 θ 0 dθ (1 − 2p0 ) Fk (θ) = lim 2π (100) ) N (θ t→∞ t exp [ik · (rt − r0 )θ ] 0 dθ (1 − 2p0 ) In the complex plane, these functions depict fractal curves of Hausdorff dimension DH given by the root of the equation [66] Q(DH , DH ) = DH Re sk

(101)

where sk is the dispersion relation (95) of the reactive modes while the function in the left-hand side is defined as the following generalization of the Ruelle topological pressure (87): 1 ln !|1 − 2p0 |αNt |Λt |1−β " t→∞ t

Q(α, β) ≡ lim

(102)

Therefore, the reactive modes as well as the diffusive modes present a singular character [66]. In conclusion, the singular character of the relaxation modes is directly related to the fact that these modes describe transient processes. We find here again the selection of trajectories associated with a given semigroup of time evolution and, thus, the breaking of the time-reversal symmetry. Here, the selection consists in weighting the trajectories according to the conditionally invariant complex measure. The relaxation modes of the forward semigroup are smooth in the unstable direction but singular in the stable direction. This singular character is of great importance in order to understand the production of entropy in these modes of relaxation, which is the purpose of the following section.


7. 7.1

137

Entropy production Ab initio derivation of entropy production

In nonequilibrium thermodynamics, irreversibility is expressed by the production of entropy out of equilibrium. In order to verify that the previously constructed hydrodynamic modes are conform to this criterion, we have calculated the entropy production during a relaxation controlled by such a hydrodynamic mode of diffusion and showed that the value expected from nonequilibrium thermodynamics is recovered. This verification has been carried out in a series of papers starting with the multibaker model of diffusion till general diffusive processes in multi-particle deterministic dynamical systems [10, 28, 29, 30, 31, 32]. This work has shown that, indeed, the singular character of the hydrodynamic modes of diffusion leads to a positive entropy production which precisely takes the value expected from nonequilibrium thermodynamics. Therefore, there is consistency between the existence of time-asymmetric singular modes and an irreversibility associated with an entropy production. In order to evaluate the entropy production, we first have to introduce the partition the phase-space region R corresponding to the lattice cell l into phase-space cells {A} of equal volume and define the corresponding coarse-grained entropy: St (R|{A}) ≡ −

A⊂R

νt (A) ln

νt (A) c0

(103)

where Boltzmann’s constant is taken equal to unity, c0 is a constant fixing the constant of entropy at equilibrium, and νt is the nonequilibrium measure at time t. The time variation of the entropy over a time interval τ is given by the difference ∆τ S(R|{A}) = St (R|{A}) − St−τ (R|{A})

(104)

On the other hand, the entropy flow is defined as the difference between the entropy which enters the phase-space region R and the entropy which exits that region: ∆τe S(R|{A}) ≡ St−τ (Φ−τ R|{A}) − St−τ (R|{A})

(105)

Accordingly, the entropy production over a time τ is defined as ∆τi S(R|{A}) ≡ ∆τ S(R|{A}) − ∆τe S(R|{A})

(106)

138 We here consider a large but closed system such as a periodic Lorentz gas in a large box. Since the system is mixing, the nonequilibrium weights of the cells {A} converge to their equilibrium values lim νt (A) = µeq (A)

t→∞

(107)

The knowledge of the Pollicott-Ruelle resonances sk and the associated ˜ k allows us to control the asymptotic approach to eigenmodes Ψk and Ψ the equilibrium value: ˜ k |p0 " + · · · →t→+∞ µeq (A) (108) νt (A) dk !IA |Ψk " exp(sk t) !Ψ where IA (Γ) is the indicator function of the cell A and p0 is the initial density. The integral is performed over the wavenumbers k of the first Brillouin zone of the lattice. By expanding in powers of the wavenumber k, we have been able to show in Ref. [32] that the entropy production takes the expected value 1 τ D ∂n(l, t) 2 (109) ∆ S(R|{A}) τ i n(l, t) ∂l in the long-time limit. In Eq. (109), n(l, t) denotes the mean particle density in the lattice cell l corresponding to the phase-space region R. This mean particle density converges to its uniform equilibrium value in the long-time limit: limt→∞ n(l, t) = neq . In the same limit, the gradient of particle density decreases as the system approaches the thermodynamic equilibrium and the entropy production progressively vanishes according to Eq. (109) as expected by nonequilibrium thermodynamics. A remarkable feature is that the result (109) holds for a partition into arbitrarily small phase-space cells {A}. This is due to the fact that the hydrodynamic modes of diffusion are singular down to arbitrarily small scales, which confers to the entropy production (109) an incomparable robustness. The singular character of the hydrodynamic modes of diffusion is an essential ingredient to obtain the positive entropy production (109). The reason is that the diffusion coefficient in Eq. (109) is given in terms of the nonequilibrium steady states (NESS) of diffusion as ∞ !vx (0)vx (t)"eq dt (110) D = −!vx "neq = −!vx Ψex "eq = 0

where vx denotes the velocity of the tracer particle. The nonequilibrium steady state in Eq. (110) −∞ t v(Φ Γ)dt (111) Ψg (Γ) = g · r(Γ) + 0


139

appears by expanding the hydrodynamic modes of diffusion (82) at small wavenumbers [67]: ∂Ψk (112) Ψg = −i g · ∂k k=0 where g is the gradient of concentration. Finally, the Green-Kubo formula (110) is obtained from Eq. (111) because !vx x"eq = 0. We notice that Eq. (111) is the nonequilibrium steady state corresponding to a mean linear profile of concentration between two particle reservoirs which are arbitrarily far apart [10]. Because of Eq. (112), the singular character of the hydrodynamic modes is therefore transfered to the nonequilibrium steady state. From Eq. (111), it is clear that the NESS is singular because the unbounded random walk of the tracer particle makes the last term infinite. This singular character is hidden in the Green-Kubo formula because of the average with vx , but it can be displayed in deterministic systems if vx is replaced in Eq. (110) by an observable such as the indicator function of the curve Γθ used to define the cumulative function (85) of the hydrodynamic modes. In doing so, we obtain the cumulative functions of the NESS also known as generalized Takagi functions: θ Tg (θ) ≡ Ψg (Γθ ) dθ (113) 0

The cumulative function Ty (θ) − Ty (π/2) corresponding to a gradient in the y-direction in the hard-disk periodic Lorentz gas is depicted versus the cumulative function Tx (θ) corresponding to a gradient in the xdirection in Fig. 7, where we observe the singular self-similar character [44]. These cumulative functions are smooth in the unstable phase-space directions but singular in the stable directions, which comes from the breaking of the time-reversal symmetry in the forward semigroup. As explained in Sec. 2, there is no contradiction with the time-reversal symmetry of Newton’s equations because of the selection of time-asymmetric trajectories in the construction of the semigroup.

7.2

Comparison with the second law of thermodynamics

The previous result shows that the coarse-grained entropy increases from the initial time when the initial probability distribution is prepared, up to its equilibrium value as t → +∞. The approach to the equilibrium value is performed at the rate of entropy production expected from nonequilibrium thermodynamics. The use of the eigenstates of the forward semigroup is essential for this result [28, 29, 30, 31, 32].

140

0.3

T y (T) T y (S/2)

0.2 0.1 0 -0.1 -0.2 -0.3 -0.3

-0.2

-0.1

0

0.1

0.2

0.3

T x (T) Figure 7. Periodic hard-disk Lorentz gas: Curves of the cumulative functions of the NESS for triangular lattices of hard disks of unit radius with an intercenter distance of d = 2.001, 2.1, 2.2, 2.3 [44].


141

The coarse-grained entropy measures the disorder in the probability distribution down to the phase-space scale given by the size of the cells used in the coarse graining. Typically, the initial distribution has heterogeneities on large phase-space scales so that the initial entropy is lower than after some time evolution. Indeed, in mixing systems, the time evolution acts by deforming the initial probability distribution so that the heterogeneities are smoothed out in some phase-space directions but compressed in others. The heterogeneities do not disappear since the convergence toward the equilibrium state is only a weak convergence and not a strong one. Hence, the coarse-grained entropy reaches its equilibrium value after the heterogeneities have become smaller than the size of the cells of the coarse graining. This happens forward and backward in time so that the coarse-grained entropy has a similar increase forward and backward in time. The increase backward in time should be excluded according to the second law of thermodynamics. This exclusion can be understood by referring to the problem of preparation of the initial conditions. As discussed in the introduction, the Newtonian scheme allows the preparation of arbitrary initial conditions in the phase space although the world is following a single trajectory from a unique initial condition in a far remote past. The preparation of specified initial conditions requires the use of a preparing device which is typically surrounding the system under study. This preparing device is not included in the description by Newton’s equations of the subsystem under study. The preparation of the initial conditions involves processes taking in place in the preparing device which are also of relaxation type. We should therefore consider the coarse-grained entropy of the total system composed of the subsystem and the preparing device. The entropy of this total system can be expected to come from even lower values than its value at the instant when the initial condition of the subsystem is launched, in agreement with the second law of thermodynamics. Every time new initial conditions are prepared, the description should be enlarged to include the preparing device together with the subsystem. We have here the following regression: The consideration of a larger system pushes the choice of initial conditions backward in time allowing a lower entropy in a remoter past because the initial conditions of the larger system involve more degrees of freedom and are thus statistically more correlated in the past. This reasoning leads us to impose the condition that the phase-space cells used in the coarse-grained entropy must have a size smaller than the smallest heterogeneities of the initial probability distribution. This condition is such that the coarse-grained entropy should remain constant

142 as long as the time evolution does not refine the heterogeneities below the scale of the used phase-space cells, even for weird initial distributions with high statistical correlations down to some small phase-space scales. At long enough time, the coarse-grained entropy increases in conformity with the second law of thermodynamics and nonequilibrium thermodynamics, as shown here above. We should here mention that the phase-space cells used in the definition of the coarse-grained entropy can also be tailored to the dynamics of the system as proposed in Ref. [68].

8. 8.1

Nonequilibrium steady states Breaking the time-reversal symmetry at boundaries

Stationary states can be maintained out of equilibrium by imposing nonequilibrium constraints at the boundaries of an open system. These constraints induce fluxes of energy or matter across the system, leading to an irreversible entropy production. Such nonequilibrium constraints can be considered as boundary conditions on the solutions p(Γ, t) of Liouville’s partial differential equation (19). These boundary conditions are imposed to the probability density p(Γ, t) on some hypersurfaces in the phase space where the particles are incoming the system. These phase-space hypersurfaces correspond to the physical boundaries of the system in the three-dimensional world. Most of the trajectories of the full system enter in the phase-space domain delimited by these hypersurfaces and they spent in general a finite amount of time inside the domain before exiting. The average value of the time between entrance and exit is of the order of 1/(Avn) where A is the area of the container n is the density of particles, and v is the mean speed of the particles. This time is of the order of 10−29 second for air at room temperature and pressure in a container of 1 m3 . We notice that the number of particles inside the boundaries of the open system may fluctuate so that we may have to generalize the Liouville equation (19) into a hierarchy of Liouville’s equations for the probability density p(N ) of N particles inside the system: ∂t p(N ) = {H (N ) , p(N ) }

(114)

with the global normalization

∞

p(N ) dN Γ = 1

N =0

and appropriate boundary conditions [10, 69].

(115)


143

Typically, the incoming trajectories are statistically uncorrelated while the outgoing trajectories are finely correlated according to the dynamics inside the system. Therefore, the incoming probability distribution is smooth with heterogeneities on spatial scales of the order of the macroscopic distances between the different thermostats or chemiostats at different temperatures or chemical potentials. In contrast, the outgoing probability distribution has extremely fine heterogeneities on tiny phase-space scales, reflecting the fine statistical correlations induced by the microdynamics internal to the system. After some transient time, a nonequilibrium steady state (NESS) establishes itself which is described by an invariant probability distribution. Clearly, this invariant probability distribution is not time-reversal symmetric because the boundary conditions explicitly break the timereversal symmetry [10, 67]. Distinct trajectories C and Θ(C) have different probability weights in this invariant measure µ: µ(A) = µ [Θ(A)]

(116)

for the phase-space cells A ⊂ M such that A = Θ(A). In contrast, the invariant probability distribution µeq describing the thermodynamic equilibrium is time-reversal symmetric: µeq (A) = µeq [Θ(A)]. Therefore, the boundary conditions required to define some NESS typically break the time-reversal symmetry.

8.2

The phase-space structure of NESS

An example of such NESS can easily be obtained for diffusion in an open Lorentz gas between two chemiostats or particle reservoirs at phasespace densities p± separated by a distance L. The phase-space density inside the system can only take either the value p− corresponding to the reservoir on the left-hand side or p+ from the reservoir on the righthand side. In order to determine which value, we have to integrate the trajectory backward in time until the time of entrance in the system, T (Γ) < 0. The value is p− (resp. p+ ) if the particle enters from the left-hand (resp. right-hand) side. If we denote by g=

p + − p− ex L

(117)

the gradient of phase-space concentration, we can write the invariant density of the NESS in the form: ; : T (Γ) t p+ + p− v Φ Γ dt (118) + g · r(Γ) + pneq (Γ) = 2 0

144 Indeed, the integral of the particle velocity v backward in time until the time of entrance gives the position of entrance r(ΦT (Γ) Γ) = ±L/2 minus the current position r(Γ) which cancels the first term in the bracket. With the gradient (117), we end up with the result that pneq (Γ) = p±

(119)

whether the trajectory enters from the left- or right-hand side [10]. The density of the NESS is a piecewise constant function with its discontinuities located on the unstable manifolds of the fractal repeller of the escape-rate formalism of Sec. 5. Indeed, trajectories on the unstable manifold of the fractal repeller remain trapped between both reservoirs under the backward time evolution. The fractal repeller of the escaperate formalism therefore controls the structure of the NESS. Its invariant density is very different from the one of the thermodynamic equilibrium but it remains absolutely continuous with respect to the Lebesgue measure as long as the reservoirs are separated by a finite distance L. In the limit where the reservoirs are separated by an arbitrarily large distance L, the time of entrance goes to infinity: T (Γ) → ∞. If we perform this limit while keeping constant the gradient g, the invariant density (118) is related to the density (111) according to Ψg (Γ) =

lim

L,p+ −p− =gL→∞

p + + p− pneq (Γ) − 2

(120)

Therefore, we reach the conclusion that the discontinuities on the unstable manifolds of the fractal repeller of the escape-rate formalism gives the singular character to the NESS as discussed in the previous Sec. 6. There is thus a deep connection between the fractal repeller of the escape-rate formalism, the fractal structure of the hydrodynamic modes, and the singular character of the NESS. The results mentioned in Sec. 7 show that the entropy production of the NESS (118) takes the value (109) expected from nonequilibrium thermodynamics as long as the phase-space cells chosen to define the coarse-grained entropy (103) remain larger than the phase-space size of the regions between the discontinuities of the density (118). Below this size the entropy production vanishes because of the absolute continuity of the invariant distribution of the NESS. However, in chaotic systems, the crossover size decreases exponentially fast with the separation L between the reservoirs so that nonequilibrium thermodynamics should hold down to extremely small scales in the phase space of diffusive systems only extending over a few dozens of mean free paths [10].


8.3

145

Entropy production and time-reversed entropy per unit time

Interesting relationships can be obtained by assuming that the nonequilibrium system has a (possibly approximate) Markovian description in terms of a master equation such as < + d p(ω, t) = p(ω , t) Wρ (ω |ω) − p(ω, t) W−ρ (ω|ω ) dt

(121)

ρ,ω

where Wρ (ω|ω ) is the rate of the transition ρ between the states ω and ω [70]. A reversed transition −ρ is associated with each transition ρ. We suppose that the master equation admits a unique stationary solution. At the thermodynamic equilibrium, the stationary solution satisfies the conditions of detailed balance: peq (ω ) Wρ (ω |ω) = peq (ω) W−ρ (ω|ω )

(122)

These conditions are in general not satisfied in NESS for which we have the more general conditions (d/dt)pneq (ω) = 0. In a NESS, the entropy production is given by [71, 72] + 1 < pneq (ω ) Wρ (ω |ω) 1 τ ∆i S = pneq (ω ) Wρ (ω |ω) − pneq (ω) W−ρ (ω|ω ) ln τ pneq (ω) W−ρ (ω|ω ) 2 ρ,ω,ω

(123)

On the other hand, we can characterize the dynamical randomness in the NESS by considering the multiple-time probability µneq (ω0 ω1 ω2 ...ωn−2 ωn−1 )

(124)

to observe stroboscopically the system in the states ω0 ω1 ω2 ...ωn−2 ωn−1 at the successive times t = 0, τ, 2τ, ..., (n − 2)τ, (n − 1)τ . The dynamical randomness is characterized by the τ -entropy per unit time corresponding to this partition P into states {ω} and sampling time τ [9]: h(P, τ ) ≡ lim − n→∞

1 nτ

µneq (ω0 ω1 ...ωn−1 ) ln µneq (ω0 ω1 ...ωn−1 )

ω0 ω1 ...ωn−1

(125) The dynamical randomness in the time-reversed path ωn−1 ωn−2 ...ω2 ω1 ω0 can be characterized by the time-reversed entropy per unit time [73] hR (P, τ ) ≡ lim − n→∞

1 nτ

µneq (ω0 ω1 ...ωn−1 ) ln µneq (ωn−1 ...ω1 ω0 )

ω0 ω1 ...ωn−1

(126)

146 The difference between both entropies per unit time gives the entropy production in the NESS: 1 τ (127) ∆ S = hR (P, τ ) − h(P, τ ) ≥ 0 τ i in the limit τ → 0 [73]. The non-negativity is a consequence of the fact that the difference hR − h between Eqs. (126) and (125) is a relative entropy which is known to be non-negative [74]. Equation (127) is a relationship between the entropy production and two characteristic quantities of the underlying dynamics. In particular, for a fine enough partition P and sampling time τ , the entropy per unit time h(P, τ ) converges toward the Kolmogorov-Sinai entropy per unit time if this latter is well defined. Both hR and h are given by large numbers but their difference gives the entropy production which is a quantity of the same order as those characterizing hydrodynamics. The relationship (127) says that the ratio of the multiple-time probabilities of a path and of a time-reversed path over a time interval t = nτ is given in terms of the irreversible entropy production ∆ti S over the time t = nτ → ∞: µneq (ω0 ω1 ...ωn−1 ) t e∆i S µneq (ωn−1 ...ω1 ω0 )

(128)

for µneq -almost all paths in the NESS. This relation explicitly shows that the breaking of the time-reversal symmetry in the NESS is directly related to the entropy production.

8.4

Entropy production and the fluctuation theorem

A fluctuation theorem can also be derived for Markovian processes described by the master equation (121) [19, 20, 21, 22, 23]. The breaking of detailed balance in the NESS is measured by the fluctuating quantity [20] µneq (ω0 ω1 ...ωn−1 ) Z(t) (129) µneq (ωn−1 ...ω1 ω0 ) The statistical moments of this quantity can be derived from the generating function 1 (130) Q(η) ≡ lim − ln!e−ηZ(t) " t→∞ t The use of such generating functions have been proposed for transport coefficients and other nonequilibrium properties in Refs. [15, 10, 75]. The fluctuation theorem says that the generating function (130) obeys the symmetry: Q(η) = Q(1 − η) (131)


147

The quantity Z(t) increases on average with a rate equal to the mean entropy production in the NESS so that !Z(t)" dQ di S (132) (0) = lim = t→∞ dη t dt neq We can introduce the Legendre transform of the generating function as R(ζ) = maxη [Q(η) − ζ η]

(133)

which satisfies the identity ζ = R(−ζ) − R(ζ)

(134)

as a consequence of the symmetry (131). Since the function R(ζ) is Z(t) the decay rate of the probability that t ζ, we find that the ratio Z(t) between the probabilities that Z(t) t ζ and t −ζ behaves as 5 4 µneq Z(t) ∈ (ζ, ζ + dζ) t 5 eζt , 4 for t → +∞ (135) Z(t) µneq t ∈ (−ζ, −ζ + dζ) Here, we see that the time-reversal symmetry is explicitly broken by the invariant measure µneq of the NESS. These results apply to nonequilibrium reactions [22]. Recently, the Schnakenberg analysis [71] of the graph of the Markovian process has allowed us to identify the affinities of the macroscopic nonequilibrium constraints on the system and to define the generating function of the nonequilibrium currents between the thermostats or chemiostats [23]. The Green-Kubo or Yamamoto-Zwanzig formulas as well as the Onsager and higher-order reciprocity relations can be derived from the fluctuation theorem for this generating function [23].

9. 9.1

Quantum systems Quantum Liouvillian resonances in infinite quantum systems

The idea that the Pollicott-Ruelle resonances can describe transient irreversible processes extends to many-body quantum systems which have a continuous spectrum. Examples of such systems includes the spinboson systems and other system where a quantum subsystem is coupled to an infinite thermal bath [76]. Other examples are given by systems with many coupled spins [77]. In the case of a quantum subsystem coupled to a thermal bath such as the spin-boson model, the observables of the subsystem have a time

148 evolution of relaxation type. It is possible to set up a quantum Liouvillian description in much the same spirit as the one described in Sec. 3 and to analyze the analytic continuation of the resolvent of the quantum Liouvillian operator, i.e., von Neumann’s operator in the infinite-system limit. The resolvent may have different kinds of complex singularities including poles, which can thus be identified as quantum Liouvillian resonances analogue to the classical Pollicott-Ruelle resonances [76]. This programme has been successfully carried out for the spin-boson model of Hamiltonian: ˆ ˆ = −∆ σ ˆb + λ σ ˆz + H H ˆx B 2

(136)

where σx , σy , and σz are the Pauli matrices. The parameter ∆ is the energy splitting between the two levels of the subsystem in absence of coupling to the reservoir, and λ is the perturbation parameter. The reservoir or thermal bath is a collection of harmonic oscillators of Hamiltonian ˆb = 1 pˆ2α + ωα2 qˆα2 (137) H 2 α while the coupling between the two-level subsystem and the bath is described by the operator ˆ = B cα qˆα (138) α

and characterized by the so-called spectral strength J(ω) =

c2 α δ(ω − ωα ) 2ω α α

(139)

In the weak-coupling limit, the quantum resonances can be obtained by perturbation theory and they are given by the eigenvalues of the Redfield operator of the forward semigroup [78]. We obtain four eigenvalues s = 0 s = −g(∞) + O(λ4) h(∞) g(∞) s = ±i ∆ + + O(λ4) − 2 2

(140) (141) (142)

The first eigenvalue corresponds to the invariant equilibrium state. The other ones describe exponential decays, exp(st). The second one is the decay of the populations of the two levels. The third and fourth eigenvalues describe the damped oscillations of the quantum coherences. The


149

coefficients are given by with ∞ ∆ cos ∆t Re C(t) dt = 2πλ2J(∆) coth (143) g(∞) = 4λ2 2k BT 0 ∞ sin ∆t Re C(t) dt (144) h(∞) = 4λ2 0

in terms of the time correlation function of the bath ∞ ω ˆ ˆ dω J (ω) coth C(t) ≡ !B(t)B"eq = cos ωt − i sin ωt 2kB T 0 (145) The resonance spectrum of the backward semigroup is given by {−s}. If a n-level quantum subsystem is coupled to a thermal bath, the Redfield operator should have n2 eigenvalues s describing a time evolution exp(st). One of them corresponds to the equilibrium invariant state, n − 1 to the decay of the populations of the n levels, and n2 − n to the damping of the quantum coherences. The n − 1 decays of the population are real eigenvalues, while the n2 − n eigenvalues for the coherences form n2−n pairs of complex conjugated eigenvalues s = −γlm ± iωlm . In the 2 weak-coupling limit, the imaginary part are given by the Bohr frequencies ωlm = (El − Em )/2 of the unperturbed systems up to corrections of order λ2. Diffusive behavior of an electron moving on a chain coupled to a thermal bath can similarly be described in terms of quantum Liouvillian resonance depending on the wavenumber k of the diffusive modes [79]. Quantum Liouvillian resonances corresponding to exponential decays have also been observed in systems with many coupled quantum spins [77]. In such systems, the continuous spectrum comes from the infinitely many spins in the system in place of an external thermal bath.

9.2

Emergence of relaxation behavior in finite quantum systems

We may wonder under which conditions relaxation behavior which is the feature of a continuous spectrum emerges in finite quantum systems. The problem is here that finite quantum systems have necessarily a discrete energy spectrum and, as a consequence, a discrete Liouvillian spectrum given by the Bohr frequencies. There is here a great difference with respect to classical systems where the Liouvillian spectrum can be continuous even in finite mixing systems with two degrees of freedom such as the chaotic Sinai or Bunimovich billiards [1, 2]. In finite quantum systems, the discreteness of the spectrum implies the presence of almost-periodic oscillations after an early decay in the

150 time evolution of the mean value of some observable starting from a nonequilibrium initial density matrix. The almost-periodic oscillations manifest themselves beyond the Heisenberg time which is proportional to the level density: tHeisenberg = nav (E). If the spectrum is quasicontinuous, the level density can be dense enough so that the Heisenberg time is postponed after a very long time. In a recent study of a system where the thermal bath of bosons is replaced by a finite system defined in terms of Gaussian random matrices of N levels, it has been shown that the early decay before the Heisenberg time can be well described in the weak-coupling regime by a quantum master equation obtained by averaging over the Gaussian random matrix ensemble [80, 81]. The condition of validity is that the quantum system playing the role of the thermal bath should after a dense enough spectrum with a number of levels N > 10/λ2 for λ < 0.3. Below this value, the trajectories of the individual systems in the ensemble fluctuate too much for the average behavior to be representative. For λ = 0.1, a few thousands of energy levels is enough to have an excellent description in terms of the quantum master equation [80, 81]. The classical relaxation behavior can also be reached in the semiclassical limit without coupling the system to a thermal bath [82]. This has been shown experimentally by Sridhar and coworkers who extracted the Pollicott-Ruelle resonances by statistical analysis of the scattering of microwaves on disk scatterers [83, 84, 85]. Indeed, the autocorrelation function in energy of the wave scattering cross-section has poles at the classical Pollicott-Ruelle resonances of the scatterer [86, 87, 88, 89]. We here have an example of emergence of classical relaxation behavior out of the wave-mechanical underlying dynamics.

9.3

NESS in quantum systems

A NESS can be obtained if a quantum subsystem is coupled to several thermal baths at different temperature. Recently, it has been shown that such quantum NESS are singular in the sense that they belong to classes of states which are not equivalent to the state of thermodynamic equilibrium [90]. This singular character which appears in quantum NESS is very much reminiscent of the singular character we have described in the previous sections.

10.

Conclusions

In this paper, we have tried to show that the thermodynamics of irreversible processes can be understood in terms of expansions which are asymptotic in time and valid for either t > 0 (forward semigroup)


151

or t < 0 (backward semigroup). In classical dynamical systems, these asymptotic expansions use the Pollicott-Ruelle resonances and other singularities at complex frequencies. These resonances and other complex singularities are obtained by analytic continuation toward complex frequencies of the spectral functions given by the Fourier transform of the time correlation functions of the observables. For sufficiently unstable dynamics, the Pollicott-Ruelle resonances and other singularities are independent of the particular observables provided they belong to some classes of smooth enough test functions so that we can say that they are intrinsic to the dynamics of the system. They can be conceived as some kind of generalized eigenvalues of the Frobenius-Perron operator or of its generator, the so-called Liouvillian operator. The imaginary part of the complex frequency z = is leads to an exponential time behavior exp(st) = exp(−izt). The analytic continuation toward the lower half of the complex plane with Re s = Im z < 0 defines the forward semigroup, while the backward semigroup is obtained by continuation toward the upper half-plane with Re s = Im z > 0. Starting with the unitary group evolution, we thus obtain a description which is splitted into two semigroups valid on distinct time semi-axes. In the case of diffusive processes, the diffusion equation is valid only for positive times and the anti-diffusion equation only for negative times. Therefore, the two equations never coexist and, moreover, the anti-diffusion equation can be excluded because of the amplification of errors in the preparation of a state violating the second law, which is related to the non-convergence of the expansion of the backward semigroup to positive times. This mechanism makes precise Boltzmann’s explanation of irreversible processes as processes for which the time-reversal history is highly improbable. We thus find a spontaneous breaking of the time-reversal symmetry in the statistical description of the transient relaxation toward the state of thermodynamic equilibrium. This breaking of the time-reversal symmetry can be interpreted by a selection of trajectories which are not time-reversal symmetric. This selection occurs for instance in the case of decay processes where the eigenstate associated with the leading Pollicott-Ruelle resonance is concentrated on the unstable manifolds of the non-wandering subset. Therefore, the decay at positive times uses an eigenstate or conditionally invariant measure made of trajectories which are selected by the dynamics and which are moreover non-symmetric under time reversal. Indeed, the time-reversal symmetry maps the unstable manifolds onto the stable manifolds which are physically distinct trajectories. A similar selection of trajectories occurs for the eigenstates describing the transient processes of the escape-rate formalism.

152 In the case of the hydrodynamic and other relaxation modes in infinite spatially periodic systems such as diffusion in the periodic Lorentz gases, the eigenstates of the forward semigroup are smooth along the unstable manifolds but singular in the stable directions. Here, the eigenstates are constructed by weighting selectively the different trajectories in order for the eigenstate to be conditionally invariant. The time-reversed eigenstates of the backward semigroup are singular in the unstable directions but smooth in the stable ones and are therefore qualitatively different from the eigenstates of the forward semigroup. Therefore, we have here also the breaking of the time-reversal symmetry, which can be connected to the thermodynamic criterion of irreversibility by calculating the entropy production and showing that it conforms to the value expected from nonequilibrium thermodynamics. The remarkable observation is here that this value is obtained because of the singular character of the hydrodynamic modes in agreement with the idea of selection of time-reversal non-symmetric trajectories. As discussed in Sec. 9, the concept of resonance can be extended to quantum systems. Moreover, in several diffusive deterministic systems, we have shown in great detail that the leading Pollicott-Ruelle resonance gives the dispersion relation of diffusion. In hyperbolic systems, the leading PollicottRuelle resonance is related to quantities of the underlying dynamics such as the Lyapunov exponents, the Kolmogorov-Sinai entropy per unit time, as well as the fractal dimensions. Relationships have thus been obtained between the transport coefficients and the characteristic quantities of chaos. In the escape-rate formalism, the leading Pollicott-Ruelle resonance is the escape rate which is proportional to the transport coefficient and given by the difference between the sum of positive Lyapunov exponents and the Kolmogorov-Sinai entropy. For diffusion, we get [12, 13] D

π L

 2γ= 

λi >0

 λi − hKS 

(146)

L

and for viscosity or other transport coefficients [14, 48]   π 2γ=  λi − hKS  η χ λi >0

(147)

χ

Very similar relationships (88)-(89) are obtained for the hydrodynamic modes of diffusion in the two-dimensional Lorentz gases where the dis-


153

persion relation sk is also the leading Pollicott-Ruelle resonance [59] Dk2 −Re sk = λ(DH ) −

hKS (DH ) DH

(148)

For NESS, there are the boundary conditions which explicitly break the time-reversal symmetry by weighting selectively the incoming trajectories by statistically uncorrelated probabilities. Here, the entropy production can also be given in terms of the difference between two quantities characterizing the underlying dynamics [73] 1 τ ∆ S = hR (P, τ ) − h(P, τ ) τ i

(149)

The similarity with the relationships of the escape-rate formalism is clear. In the right-hand side, the Kolmogorov-Sinai entropy is simply replaced by the τ -entropy per unit time of the partition P, while the role of the sum of positive Lyapunov exponents is played by the time-reversed entropy per unit time. In the left-hand side, we recover an irreversible quantity which is here the entropy production of the NESS. Finally, it is worthwhile to point out that the fluctuation theorem in the form (134) ζ = R(−ζ) − R(ζ)

(150)

has again the same structure as Eqs. (146)-(149) with an irreversible quantity in the left-hand side and the difference between two decay rates of probabilities in the right-hand side [16, 17, 18, 19, 20, 21, 22, 23]. All the relationships (146)-(150) are indeed large-deviation formulas for the statistical time evolution in nonequilibrium conditions. They all have the feature of giving an irreversible property as the difference of two large-deviation quantities such as the decay rates of multiple-time probabilities or the growth rates of phase-space volumes. Moreover, they are compatible with Liouville’s theorem. The discovery of these new dynamical large-deviation properties is a major advance in nonequilibrium statistical mechanics during the last fifteen years. Acknowledgments. The author thanks Professor G. Nicolis for support and encouragement in this research. The author and this research are financially supported by the National Fund for Scientific Research (F. N. R. S. Belgium).

154

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ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS Decoherence and a stochastic Schr¨ odinger equation approach Walter T. Strunz Theoretische Quantendynamik, Physikalisches Institut, Universit¨ at Freiburg, Hermann-Herder-Str. 3, 79104 Freiburg, Germany [email protected]

Abstract

The interaction of a quantum system with a many degree of freedom environment entails damping and the loss of quantum coherence. In these two lectures we focus on two aspects of open quantum system dynamics. First, we present decoherence as a dynamical phenomenon and determine the corresponding time scales, discuss an experiment and the nature of robust states, and mention universal properties of shorttime decoherence of macroscopic superpositions. Secondly, in analogy to the classical description of open systems in terms of Langevin equations, we present a stochastic approach to open quantum system dynamics in terms of a stochastic Schr¨ odinger equation. We give a microscopic ¨ derivation that allows us to generalize the stochastic approach to nonMarkovian situations. We discuss a convolutionless formulation of our theory and finally, we focus on the quantised version of Brownian motion as an often-employed example.

Keywords: Open quantum systems, decoherence, stochastic Schrödinger equations, non-Markovian dynamics

Introduction These are lecture notes on two aspects of open quantum system dynamics. First, we present decoherence, i.e. the loss of quantum coherence due to the interaction with a “heat bath” or “environment”. As current experiments and technologies aim to push quantum coherent phenomena to more and more macroscopic scales, a clear understanding of the decohering mechanisms is crucial. As we will see, decoherence for a quantum system with a meaningful classical limit should be dis159 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 159–187. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

160 tinguished from ordinary dissipation, or damping, due to possibly vast differences in time scales. Typically, probabilities (the diagonal elements of a density operator) are affected by the environment on a dissipation time scale τdiss , while coherences (the off-diagonal elements) may decay more rapidly on a new time scale τdec . For the ratio one finds scaling laws τdiss /τdec ∼ D δ with some exponent δ, where D is a measure for the “distance” covered by the underlying quantum state: the further coherent components of the quantum state are apart, the more drastic is the difference between fast decoherence (τdec ) and relatively slow damping (τdiss ). For truly macroscopic sizes D, decoherence may well appear instantaneous. In our first lecture we determine the relevant time scales, based on simple models. Most notably, we use the damped harmonic oscillator whose decoherence has been investigated in a beautiful experiment involving a damped electromagnetic field mode of a microwave cavity. The precise form of the interaction between the quantum system and its environment determines which states are most vulnerable to decoherence, or, conversely, those states that are most robust. We discuss a short-time approach to decoherence and explain how the universally observed absence of quantum coherent superpositions (“Schr¨ odinger cats”) on macroscopic scales is a direct consequence of these results. The second lecture is devoted to recently established stochastic methods for the description of the dynamics of open quantum systems. In analogy to the classical description of open systems in terms of Langevin equations, stochastic Schr¨ odinger equations for pure states may replace master equations for the ensemble evolution. The reduced density operator is recovered by sampling over many realizations of the stochastic pure states. We give a microscopic derivation of the stochastic Schrödinger equation that allows us to generalize the stochastic approach to nonMarkovian situations. Such “memory” effects usually make themselves felt through integrals over the past. However, as we will see, in some cases it is possible to revert to a convolutionless formulation of our theory, leading to time-local evolution equations, despite the non-Markovian nature of the dynamics. Finally, we mention results on the quantised version of Brownian motion as an often-employed example: it can be shown that each solution of the stochastic Schrödinger equation is a localized wave packet that follows the classical Langevin equation as ¯h → 0. In this way, the dynamics of classical point particles may be derived from a stochastic approach to open quantum system dynamics.

Aspects of open quantum system dynamics

1.

161

Decoherence in open quantum systems

The superposition principle states that for any two states |ψ1 " and |ψ2 ", the sum |ψ" ∼ |ψ1 " + |ψ2 " is a valid state of the quantum system. Sometimes, due to environmental influences, such superpositions are not dynamically robust. They decay into a mixture ρ = 12 (|ψ1 "!ψ1 |+ |ψ2 "!ψ2 |). These lectures are concerned with the nature of robust states and the time scales involved during the transition from the coherent superposition to the mixture, i.e the time scale of decoherence. Several mechanisms contribute to the loss of quantum coherence: The influence of some fluctuating, external classical field corresponds to unitary, yet stochastic dynamics. Certain interferences will be washed out by averaging over these fluctuations. Moreover, uncertainties in initial conditions will also lead to a suppression of interference phenomena resulting from the average over a distribution of initial states. Finally, entanglement between the quantum system and its environment causes decoherence. Due to entanglement, the state of the open system may no longer be described by a single state vector, even if both, system and environment are in a single pure state. Instead, the reduced density operator of the open system will generally be a mixture. It is this third, genuinely quantum origin of decoherence that will be our major concern in these lectures, see also [1, 2, 3, 4, 5].

1.1

Quantum optical damped harmonic oscillator

The state of an electromagnetic field mode inside a cavity provides a vivid example of an open quantum system and leads to a clear understanding of the decoherence phenomenon. Losses through the cavity mirrors means a decay of the number of photons inside. The field modes outside the cavity thus play the role of the “environment” to which energy is dissipated. This standard model is discussed in all quantum optics text books [6]. It reveals important aspects of decoherence that will later on be addressed in a more general framework. Most importantly, there is a beautiful experiment confirming the findings, as explained below. The single electromagnetic field mode inside the cavity is governed by the Hamiltonian Hsys = ωa† a. We use annihilation and √ creation operators a, a† . For a material oscillator, a = (q + ip)/ √ 2 with the position and momentum operators q = mω/¯h Q, p = P/ ¯hmω made dimensionless with the help of the oscillator ground state dimensions. Note that these expressions involve Planck’s constant ¯h such that q and

162 p become very large numbers for macroscopic values of the physical quantities Q and P . The exponential decay of the number of photons inside the cavity !N (t)" = N0 e−γt introduces a relaxation or dissipation time τdiss = γ −1 in addition to the free system time scale τsys = ω −1 .

1.1.1 Master equation. The master equation of the damped field mode takes the Lindblad form [6, 7, 8] γ [a, ρ(t)a† ] + [aρ(t), a† ] . (1) ∂t ρ(t) = −i[ωa† a, ρ(t)] + 2 Here, the standard Born-Markov approximation is made [8] and we assume zero temperature T = 0. In the second part of these lectures we will explain in more detail how to arrive at an evolution equation of this type. A pure initial state ρ0 = |ψ0 "!ψ0 | will in general evolve into a mixed state ρ(t) as time evolves in (1). This holds true even in our case of zero temperature due to entanglement between “system” and “environment”. Yet in this simple example, we make a remarkable observation con1 2 † cerning coherent states |z" = e− 2 |z| +za |0" of the oscillator. With ρ0 = |z0 "!z0 | a pure coherent state, we find that (1) is satisfied by the pure ρ(t) = |z(t)"!z(t)| for all times. Here, z(t) = e−iωt e−γt/2 z0 describes the damped, oscillating motion of the field amplitude. The important observation is that the master equation (1) allows for very specific pure state solutions. Let us now investigate the fate of an initial coherent superposition of √ Strictly coherent states, ρ0 = |ψ0 "!ψ0 | with |ψ0 " = (|z1 " + |z2 ") / 2. √ speaking, the normalization factor is more involved than just 1/ 2 - we will, however always assume |z1 − z2 | 1 so that !z1 |z2 " ≈ 0 and the deviation from our choice is negligible. In its full length, the initial reduced density operator consists of four terms, ρ0 = 12 |z1 "!z1 |+ 12 |z2 "!z2 |+ 1 1 1 1 2 |z1 "!z2 | + 2 |z2 "!z1 |, with the off-diagonal terms 2 |z1 "!z2 | and 2 |z2 "!z1 | the coherences of ρ0 . These are responsible for any interference pattern emerging from the two superposed states. For the initial state just mentioned, the solution of the master equation (1) reads 1 1 (2) |z1 (t)"!z1 (t)| + |z2 (t)"!z2 (t)| 2 2 1 1 + f (t)|z1 (t)"!z2 (t)| + f ∗ (t)|z2 (t)"!z1 (t)|, 2 2 where z1 (t) and z2 (t) again follow the oscillating, damped motion. There is an amplitude f (t) appearing in front of the cross terms (the “coherρ(t) =


163

ences”) with 1 |f (t)| = exp − |z1 (0) − z2 (0)|2 1 − e−γt . (3) 2 Apparently, whenever the initial “distance” between the superposed coherent states is large, i.e. D2 ≡ |z1 (0) − z2 (0)|2 1, their mutual coherence becomes small for γt > 0. Thus, the initially fully coherent reduced density operator becomes a mixed state under time evolution, ρ(t) = 12 |z1 (t)"!z1 (t)| + 12 |z2 (t)"!z2 (t)|, if only D2 = |z1 (0) − z2 (0)|2 1. The larger D, the more rapidly the coherences will disappear. The precise (initial) time scale of their disappearance is of particular interest: Assume γt 1, so that almost no “damping” or even “motion” (due to γ ω in typical experiments) has occurred yet. Still, decoherence with 2 (4) |f (t)| = e−γD t/2 = e−t/τdec will be effective if only the initial distance D is large. A huge separation of time scales between decoherence τdec ∼ (γD2 )−1 and damping τdiss ∼ γ −1 arises for macroscopically distinct coherent states {|z1 ", |z2 "}

relaxation time ∼ D2. (5) decoherence time √ For a material oscillator, upon the identification z = (q + ip)/ 2, the distance D = |z1 − z2 | is just a “phase space distance” D 2 = (|q1 − q2 |2 + |p1 − p2 |2 )/2. The distance squared can easily be much, much larger than unity for macroscopically distinct states, since the phase space coordinates (q, p) are measured in quantum oscillator units involving Planck’s constant h ¯ . Thus, values of astonishing magnitude of the order of D2 ∼ 1040 may appear (which nevertheless have to be taken very cautiously, as we will explain in more detail later). We see that decoherence for superpositions of macroscopically distinct states (D 1) may be very effective, even though damping for one of its components is hardly noticeable. This separation of time scales between damping and decoherence is the reason why decoherence deserves special attention in open quantum system dynamics. For two-level systems, in contrast, the decay of the off-diagonal terms due to damping differs by a factor of two only from the decay of the diagonal terms and no special distinction between decoherence and dissipation seems necessary. As soon as a quantum system allows for a meaningful classical limit, however, superpositions of then macroscopically distinct states decohere towards mixtures almost instantaneously. This may be seen as the reason why it is so difficult to extend coherent quantum phenomena to the macroscopic domain, and is the reason for the absence of quantum superpositions (“Schr¨ odinger cat states”) from the macroscopic world.

164 1.1.2 Experimental verification. Decoherence dynamics was first investigated in a very controlled way by S. Haroche’s group at the ENS in Paris [9], based on a damped field mode just described. A coherent initial state |z" of an electromagnetic field mode inside a microwave cavity was prepared and single Rydberg atoms were sent through the cavity in order to manipulate the field. Due to losses through the cavity mirrors, the dynamics of the field mode is nothing but a realization of the damped harmonic oscillator discussed earlier. The interaction between the atoms (described effectively by two electronic states |g" and |e") and the field mode was made purely dispersive. Thus, the atomic state does not change while flying through the cavity, yet the field amplitude is rotated by an angle ±φ, depending on the state of the atom: |e"|z" −→ |e"|zeiφ " |g"|z" −→ |g"|ze−iφ ".

(6)

The angle φ = Ω2 ti /δ is determined by the Rabi frequency Ω, the detuning δ between cavity frequency and transition frequency between |g" and |e", and the interaction time ti , which can be adjusted through the velocity of the atoms. Before and after the atoms pass the cavity, they may interact with a π/2 pulse such that superpositions of the two atomic √ √ states may be produced: |e" −→ (|g"+|e")/ 2, or |g" −→ (|g"−|e")/ 2. In the actual experiment, a coherent superposition of two coherent states of the field is prepared. For that, an atom is subjected to a π/2 pulse (R1 ), sent through the cavity (C), subjected to the second π/2 pulse (R2 ), and finally detected in the ground state |g", in half of the cases. The state of atom and field is thus transformed according to |e" ⊗ |z"

R1 → C → R2 →

1 √ |g" + |e" ⊗ |z" (7) 2 1 √ |g" ⊗ |ze−iφ " + |e" ⊗ |zeiφ " 2 1 1 |e" ⊗ |zeiφ " − |ze−iφ " + |g" ⊗ |zeiφ " + |ze−iφ " . 2 2

Finally detecting the atom in state “g”, the field state is indeed given by the superposition (“cat state”) of two coherent states |ψfield " = |zeiφ " + √ −iφ |ze " / 2 with distance

√ Ω2 ti ¯ sin D = |z1 − z2 | = |z||e − e | = 2 n δ iφ

iφ

,

(8)

where n ¯ = |z|2 is the mean initial photon number of the field state |z".

165


Decoherence will be effective on a time scale τdec = γ −1 D−2 , i.e. the more rapid, the larger D. Experimentally, this has indeed been confirmed by sending a second atom through the cavity after a controllable time delay τ and going through a similar sequence of manipulations as for the first atom. Measuring the difference between conditional detection probabilities in the |g" and |e" states as a function of the time delay τ between the two atoms, the authors monitor the decay of the coherence between the two superposed field states and confirm its dependence on the size of the superposition D.

1.1.3 Schr¨ odinger equation for “system” and “environment”. Further insight is gained by discussing decoherence in this simple example from the point of view of the full dynamics of system and environment. The underlying master equation (1) may be derived from the total Hamiltonian (see also the second part of these lectures) Htot = Hsys ⊗ 11env + Hint + 11sys ⊗ Henv = h ¯ ωa† a ⊗ 11env + h ¯

gλ a ⊗ b†λ + a† ⊗ bλ + 11sys ⊗

λ

†

†

†

= h ¯ ωa a ⊗ 11env + h ¯ a ⊗ B + a ⊗ B + 11sys ⊗

(9) ¯hωλ b†λ bλ

λ

¯hωλ b†λ bλ

λ

Here, we denote with B = λ gλ bλ the environmental part of the interaction. We can easily explain the meaning of the three terms in the Hamiltonian: while the “system” and “environmental” part describe the free harmonic evolution of the respective modes, the interaction describes the processes of annihilating a “system” photon and creating an environmental photon, and vice versa. In order to arrive at the evolution equation (1), starting from model (9) one has to assume that the bath correlation function is delta-like, α(t − s) ≡ !B(t)B † (s)" ≈ γδ(t − s),

(10)

where B(t) = eiHenv t/¯h Be−iHenv t/¯h = λ gλ bλ e−iωλ t . Instead of determining the reduced state, however, we here want to focus on the total state Ψ(t) of system and environment. With our choice of the initial condition as before, i.e. assuming a coherent state |z" as system part and a zero temperature bath, |Ψ0 " = |z" ⊗ |0"env ,

(11)

the Schr¨ odinger equation i¯ h∂t Ψ(t) = Htot Ψ(t) has a product state solution (12) |Ψ(t)" = |z(t)" ⊗ |β(t)"env ,

166 where all states are coherent states of the corresponding oscillators, with the notation |β(t)"env = |β1 (t)"⊗|β2 (t)"⊗· · ·⊗|βλ (t)"⊗· · ·. Remarkably, (12) represents a particular solution of Schr¨ odinger’s equation without entanglement between system and environment, for all times. Again, we find z(t) = e(−iω−γ/2)t z0 , and βλ (t) = −igλ

t

ds e−iωλ (t−s) z(s)

(13)

0

= −i

gλ 1 − e−(γ/2+i(ω−ωλ )) z0 e−iωt . γ/2 + i(ω − ωλ )

The second expression highlights how the motion of the environmental oscillators depends on the initial state |z0 " of the central oscillator. Linearity demands that an initial superposition of coherent states, 1 |Ψ0 " = √ (|z1 " + |z2 ") ⊗ |0"env , 2

(14)

evolves into an entangled state, |Ψ(t)" =

1 √ |z1 (t)" ⊗ |β 1 (t)"env 2 1 + √ |z2 (t)" ⊗ |β 2 (t)"env . 2

(15)

Thus, some initial states of the open system lead to entanglement between system and environment, while others (very special ones) do not. This observation leads to the notion of robust (or preferred) states of the open system: robust states are those system states that lead to little (or no) entanglement. From the total state (15) we may determine the reduced density operator, ρsys (t) = Trenv [ρtot (t)] = Trenv [|Ψ(t)"!Ψ(t)|] 1 1 1 |z1 (t)"!z1 (t)| + |z2 (t)"!z2 (t)| + !β 2 (t)|β 1 (t)"|z1 (t)"!z2 (t)| = 01 2 2 2 2/ f (t)

1 + !β 1 (t)|β 2 (t)"|z2 (t)"!z1 (t)|; 2

(16)

which allows to identify the coefficient f (t) in (2) as the overlap of the corresponding environmental states |β 1 (t)" and |β 2 (t)". As they evolve differently according to (13) (since |z1 (0) − z2 (0)| 1), the overlap f (t) = !β 2 (t)|β 1 (t)" approaches zero rapidly and the coherences in the


167

reduced density operator disappear. We see that as soon as there is enough information in the environment to be able to deduce the system state, i.e. !β 1 |β 2 " ≈ 0, coherence is lost. In fact, this simple observation allows us to give a nice simple argument for the decoherence time scale: consider the special case of an initial superposition of the vacuum z1 = 0 and a coherent state z2 = z. As the environmental part of the total state corresponding to the vacuum remains in its initial vacuum state (βλ1 (t) ≡ 0 for all λ), the coherences are weighted by a factor with f (t) = !0|β 2 (t)". Decoherence is thus lost as soon as the environmental state |β 2 (t)", corresponding to the coherent state |z", is orthogonal to the vacuum state. In other words, coherence between the states |0" and |z" is lost as soon as a single photon has escaped into the environment. How long does it take to lose a single photon? The damping rate γ determines the time scale τdiss = γ −1 for the loss of all photons, initially of the order of n ¯ = |z|2 . Thus, a single photon is lost after a fraction τdec = τdiss /¯ n = τdiss /|z|2 = τdiss /D2 , with D = |z| = |z − 0| the initial distance between the two superposed states. Thus, this simple argument underlines the general findings (3), based on the master equation (1).

1.2

Interaction determines robust states

Coherent states are singled out for the damped harmonic oscillator, because the interaction between “system” and “environment” is mediated by the annihilation operator: the loss of a photon to the environment doesn’t change coherent states, due to a|z" = z|z". More generally, it is the interaction Hamiltonian Hint (in combination with the “system” Hamiltonian Hsys ), that primarily determines robust states. If the coupling to the environment is mediated by the system’s energy, energy eigenstates will be robust. In general, no exact robust states will exist, since Hsys and Hint will not commute. Let us consider the case of coupling to energy in more detail, sometimes referred to as “dephasing”: a quantum system with Hamiltonian Hsys is coupled to the environment through its energy Hsys , i.e.the interaction Hamiltonian takes the form Hint = Hsys ⊗B with B = λ Bλ a bath operator consisting of many degrees of freedom. The system energy is a conserved quantity, and there is no dissipation, i.e. no energy lost to the environment. Nevertheless, phase coherences between different energy eigenstates of the “system” are destroyed. For an environment of harmonic oscillators, the total Hamiltonian of system and environment

168 reads Htot = Hsys ⊗ 11env + Hsys ⊗

gλ bλ + b†λ + 11sys ⊗

λ

¯hωλ b†λ bλ . (17)

λ

We may either determine the total state as in the first example, or derive a master equation for the evolution of the reduced density operator using standard methods [8]. Doing the latter, one arrives at a master equation ∂t ρ(t) = −

5 4 5 i4 Hsys , ρ(t) − κ Hsys , [Hsys , ρ(t)] . h ¯

(18)

Energy eigenstates of the system Hamiltonian, Hsys |φn " = En |φn " are robust: if ρ0 = |φn "!φn |, then [Hsys , ρ0 ] = 0 and ρ(t) = ρ0 = |φn "!φn | for all times. Energy eigenstates remain pure for all times, these states do not lead to entanglement between system and environment. In general, we can expand ρ0 = nm ρnm |φn "!φm |, and find ρ(t) =

ρnm e−κ(En −Em )

2 t/¯ h

e−i(En −Em )t |φn "!φm |.

(19)

nm

As in the first example of the damped cavity mode, coherences between 2 different robust states are lost: |ρnm (t)| = |!φn |ρ(t)|φm "| =e−κ(En −Em ) t= 2 e−κD t → 0. Again, the decoherence time scale is inversely proportional to the square of a “distance” D = |En − Em | between the superposed robust states. The further they are apart with respect to this distance, the more rapidly their coherence is lost, just as in the case of the damped harmonic oscillator. Here, however, due to the difference in the coupling between system and environment, energy eigenstates rather than coherent states are singled out to be robust. An experiment highlighting the dependence of robust states on the interaction Hamiltonian and investigating decoherence time scales of superpositions with variable “distance” has been performed recently in Boulder [10].

1.3

Limits of the “decoherence formula”

The “decoherence formula” exp{−γD2 t} for the loss of coherences is often used to explain why “the world behaves classically”. As an example, consider a damped harmonic oscillator with D = |Q1 −Q2 |/0 where |Q1 − Q2 | is the physical distance in space and 0 = ¯h/mω the reference quantum length scale of the oscillator. In fact, at finite temperature, 0 has to be replaced by the de Broglie wave length dB = , 2 ¯h /mkT [1, 4], such that D = |Q1 − Q2 |/dB . In order to get a feeling for the numbers involved, consider a superposition of two wave packets with macroscopic distance of |Q1 −Q2 | = 1cm,

169


room temperature T = 300K and macroscopic mass m = 1g: we find D 1020 . Thus, even without noticeable damping (choose, for instance, γ = 10−17 s−1 ), the coherence between the two wave packets is lost after a decoherence time τdec = (γD2 )−1 = 10−23 s. Hence, for macroscopic superpositions decoherence appears instantaneous, even if damping is not noticeable. These impressive numbers should be taken with care, however: in deriving the “decoherence formula”, certain approximations have to be made whose validity has to be questioned. Most notably the Markov approximation is doubtful as soon as the predicted time scales become short; also, the rotating wave approximation, implicit in the underlying Hamiltonian (9), becomes questionable. For the remainder of this section, let us show in detail [11] how the coherence between wave packets of distance D = |Q1 − Q2 |/dB decays as a function of time for a soluble model (without Markov and without rotating wave approximation). We choose a famous model of open system dynamics [12, 13, 14, 15, 16, 17, 18, 19, 20, 21]: an oscillator of mass M and frequency ω, coupled to an environment of harmonic oscillators through its position Q:

Htot =

1 P2 gλ (Q ⊗ qλ ) + + M ω 2 Q2 + 2M 2 λ λ

1 p2λ + mλ ωλ2 qλ2 . 2mλ 2 (20)

Introducing a spectral density of bath oscillators J(ω) =

λ

gλ2 δ(ω − ωλ ) 2mλ ωλ

(21)

one usually writes α(t − s) = h ¯

∞

dω J(ω) coth 0

¯ω h 2kB T

cos ω(t − s) − i sin ω(t − s)

(22) for the relevant bath correlation function. We choose the standard case of so-called Ohmic damping with J(ω) = M γωfc (ω/Λ) and a cutoff function fc (x) = 1/(1 + x2 )2 . Relevant time scales we select are the damping rate γ = ω/105 (small), a cutoff frequency Λ = 100ω (high) and the thermal time kT /¯ h = 20ω (moderately high). The model (20) allows us to derive an exact master equation [16, 19, 20, 22] of the reduced density operator ρ(t), whose linear time evolution we denote with the superoperator ρ(t) = Lt [ρ0 ].

(23)

170 As initial state we choose a superposition of two Gaussian wave packets |φ1 ", |φ2 " with zero momenta and a spatial distance |Q1 − Q2 | such that the dimensionless D reads D = |Q1 − Q2 |/dB .

(24)

As we are interested in the case D 1, the two wave packets can be considered to be orthogonal. Thus, the initial density operator consists of four terms, 1 (25) (|φ1 "!φ1 | + |φ2 "!φ2 | + |φ1 "!φ2 | + |φ2 "!φ1 |) 2 1 = (ρ11 + ρ22 + ρ12 + ρ21 ) . 2 The coherences present in this density operator make themselves felt through the cross terms ρ12 , ρ21 . Defining ρ12 (t) = Lt [ρ12 ] = Lt [|φ1 "!φ2 |] as the temporal successor of the initial coherence ρ12 (0), we measure coherence by evaluating its norm [23] ρ0 =

4

5

n12 (t) = Tr ρ12 (t)ρ†12 (t) .

(26)

A value of n12 approaching zero means loss of coherence. For the model of harmonic oscillators, this norm may be obtained analytically (involving complicated functions of time as they appear in the exact master equation, see [11]). Let us compare these exact results with the decoherence formula (4) for the quantum optical damped harmonic oscillator discussed earlier. We stress that the latter is based on a “rotating-wave”-type coupling (a ⊗ b†λ + a† ⊗ bλ ) in the interaction Hamiltonian (9). In the current case, by contrast, we keep the full Q ⊗ qλ -interaction in (20), i.e. we keep “rotating” terms a ⊗ bλ and a† ⊗ b†λ . For the rotating wave approximation to be valid, one has to assume that the system may undergo many oscillations before damping becomes relevant, i.e. we have to assure ω −1 = τsys τdiss = γ −1 . As we will see, in order to describe decoherence correctly with the simple “decoherence formula” (4), we have to assure not only τsys τdiss but more severely τsys τdec . This puts a constraint to the size of the initial superposition D for the simple exponential decay formula (4) to be valid. In Figures 1 and 2 we show the decay of the coherence norm n12 (t) of (26) for two fairly “small” initial separations D = 16 and D = 80. In Figure 1 (“small Schrödinger cat”, D = 16) the exact decay (full line, “exact”) follows the “decoherence formula” exp{−γD2 t} (dashed line, “exponential”), apart from tiny oscillations. For a larger initial superposition with D = 80, however, τsys ≈ τdec τdiss and we clearly see

171

Aspects of open quantum system dynamics D=16

D=80

1

1 exact exponential

0.8

0.6

0.6

n12(t)

n12(t)

exact exponential

0.8

0.4 0.2 0 0

0.4 0.2

50

Zt

100

Figure 1. Decoherence of a microscopic “Schr¨ odinger cat” state with a distance D = 16. Decoherence is slow: τsys τdec τdiss . Apart from tiny oscillations, the decay follows the “decoherence formula” exp{−γD 2 t}.

0 0

2

4

Zt

6

8

10

Figure 2. Same as Figure 1 for a larger initial “Schr¨ odinger cat”, D = 80. Now τsys ≈ τdec τdiss . The decay is no longer exponential; it shows accelerated decay in time spans when the two superposed wave packets have opposing positions and slowered decay as soon they overlap in space.

the system’s oscillations. The true decay (full line, “exact”) is no longer adequately described by the exponential decay (dashed line, “exponential”). It shows accelerated decay in time spans when the two superposed wave packets have different positions (ωt ≈ 0, π, 2π, . . .) and slower decay in those time spans ωt ≈ π/2, 3π/2, . . ., when the two wave packets have the same position, yet different momenta. Since the coupling is mediated through position Q, in the latter case the environment is unable to “distinguish” between the two wave packets, thus decoherence is less efficient.

1.4

Short time approach to decoherence

Increasing the distance even further, deviations from the exponential decay law are even more pronounced: here we meet the border of the validity of the Markov assumption: it is valid only as long as relevant system time scales are long compared to bath correlation times. As soon as D becomes large, however, decoherence may be so rapid as to compete with short bath correlation times. In Figures 3 and 4 we clearly see the inadequacy of the exponential decay of coherences for still larger initial separations D = 400 and D = 4000. Here, decoherence is faster than any system time scale and thus is independent of the system Hamiltonian. In the regime shown in Figure 3, τdec ≈ τres τsys . The precise temporal decay is thus determined by

172 D=400

D=4000

1

1 ex Gauss

0.8

0.6

0.6

n12(t)

n12(t)

exact exponential

0.8

0.4 0.2 0 0

0.4 0.2

0.1

0.2

Zt

0.3

0.4

Figure 3. Same as Figs 1 and 2, with still larger initial separation D = 400. Decoherence is faster than any system time scale and thus is independent of Hsys : here τdec ≈ τres τsys and the precise temporal decay is determined by the bath correlation function (see text).

0 0

Zt

Figure 4. Same as Figs 1-3, for an even more macroscopic initial distance D = 4000. Decoherence time scale outruns both system and bath correlation times, τdec τres τsys , the decay becomes Gaussian (see text).

the bath correlation function, which will be explained below. In Figure 4 we choose an even more macroscopic initial distance D = 4000. Here, the decoherence time scale outruns both system and bath correlation times, τdec τres τsys , and the decay is well described by a simple Gaussian, as will be derived shortly. The failure of the simple exponential decay law for a more and more macroscopic distance between the superposed states calls for a different analytical approach. The rapid decay displayed in Figures 3 and 4 may be derived by exploiting the fact that decoherence is faster than any time scale emerging from the system Hamiltonian Hsys [11, 24]. This simple observation allows us to determine such rapid decoherence quite universally, without referring to the precise nature of Hsys . Consider a general model of the form Htot = Hsys ⊗ 11env + Hint + 11sys ⊗ Henv

(27)

with unspecified system and environment Hamiltonian and an interaction Hint = Q ⊗ B.

(28)

In the following, we will omit writing out the trivial operators 11 of the total Hamiltonian for brevity. Naturally, we assume the bath coupling

173


operator to consist of many degrees of freedom,

B=

Bλ .

(29)

λ

Decoherence of an initial superposition of two Gaussian wave packets with a spatial separation so large that τdec τsys has to be independent of Hsys . We will see that this assumption is justified by self consistency. In order to determine coherence between far apart wave packets, we evaluate the appropriate matrix elements of the reduced density operator 4

5

ρ(t) = Trenv e−iHtot t/¯h ρtot (0)eiHtot t/¯h .

(30)

We neglect the system Hamiltonian, Htot ≈ Hint + Henv = Q ⊗ B + Henv and further assume initial statistical independence of system and bath, ρtot (0) = ρsys (0) ⊗ ρenv (0), to find 4

5

!Q|ρ(t)|Q " = Trenv e−i(Q⊗B+Henv )t/¯h ρenv (0)ei(Q ⊗B+Henv t/¯h !Q|ρsys (0)|Q ".

(31) This expression is further simplified by introducing the time-ordered exponential

−iQ

e

t 0

dsB(s)/¯ h

= eiHenv t/¯h e−i(Q⊗B+Henv )t/¯h .

(32)

+

We make use of the cyclic property of the trace to arrive at !Q|ρ(t)|Q " = Trenv =

=

e

:

iQ

eiQ t 0

t 0

dsB(s)/¯ h

dsB(s)/¯ h

−

−

e−iQ

e−iQ

t 0

t 0

;

ρenv (0) !Q|ρsys (0)|Q "

dsB(s)/¯ h

dsB(s)/¯ h

> +

+

!Q|ρsys (0)|Q ",

(33)

env

where the large angular brackets denote the environmentalaverage!. . ."env = Trenv [. . . ρenv (0)] and (. . .)− denotes anti-time ordering. Next recall that we assume the environmental part of the interaction to be additivelycomposed of contributions from independent degrees of freedom: B = λ Bλ . Assuming the central limit theorem applies, we may think of B(s) to be a Gaussian process and replace the average in (33) by the Gaussian expression

|!Q|ρ(t)|Q "| = exp −(Q − Q )2

×|!Q|ρsys (0)|Q "|.

t

s

ds 0

0

?

ds {B(s), B(s )}

@

h2 ) env /(2¯ (34)

174 Here, the anti-commutator ¯ {B(s), B(s )} = h

∞

dωJ(ω) coth 0

¯ω h 2kB T

cos ω(s − s )

(35)

is given by the real part of the bath correlation function and we take the absolute value on both sides since an irrelevant additional phase appears in the full expression. Hence, the decay of the coherences is governed by the symmetric part of the bath correlation function. Its decay defines the bath correlation time τenv . Depending on its functional behaviour, this decay need not even be proportional to some power of t so that a “decoherence time scale” may not be easily defined. Still, it is clear that if only |Q − Q | is large enough, the exponent may become so large that indeed, coherences between far apart wave packets will have disappeared before any system time scale becomes relevant. This proves the consistency of our result (34) with the underlying assumptions of neglecting Hsys . In Fig. 3, the exact result (full line, “exact”) is indistinguishable from the general result (34). If the distance |Q − Q | becomes very large, decoherence may even be faster then the decay of the bath correlation function {B(s), B(s )}, i.e. even faster than τenv . In this case we can replace the full expression (34) by its early-time approximation, which to lowest order in t leads to the Gaussian decay 4

5

h2 ) |!Q|ρsys (0)|Q "|. (36) |!Q|ρ(t)|Q "| = exp −(Q − Q )2 !B 2 "env t2 /(2¯ Thus, in this extreme limit, the decay is quadratic (see Fig. 4) and defines the decoherence time scale h ¯ 1 ¯h ∼ . (37) τdec = ∼ |Q − Q | D |Q − Q | !B 2 "env /2 Clearly, unlike the Markov (and rotating wave) result (4), the decoherence time in this limit scales linearly with inverse distance. In Fig. 4 we compare the exact result (full line, “exact”) with the simple Gaussian decay of equ. (36) (dashed line, “Gauss”) and see almost perfect agreement. We stress that for the above results (34) and (36) to be valid, the underlying system Hamiltonian Hsys is entirely irrelevant. The extreme short-time result (36) is even independent of the bath Hamiltonian Henv . In this sense, the daily observable universality of decoherence is proven: as we have shown, for macroscopic superpositions, decoherence is independent of the detailed nature of both the system and the reservoir Hamiltonians.


2.

175

Stochastic Schr¨ odinger equation for open quantum system dynamics

This second lecture is concerned with a stochastic approach to the dynamics of open quantum systems. They are traditionally described in terms of their density operator ρ. Accordingly, investigating the dynamics of an open quantum system demands the determination of ρ(t). In this lecture we present an approach to open quantum system dynamics based on a stochastic Schr¨ odinger equation for pure states such that upon taking the ensemble mean over many realizations, the reduced density operator is recovered. We start with the classical analogy first.

2.1

Ensemble versus stochastic description

Consider a classical Brownian particle with Hamiltonian Hsys = p2 /2m+ V (q), subjected to the influence of some dissipative environment. Its dynamics, based on the phase space distribution ρ(t, q, p), may be described by a Fokker-Planck equation [25] ∂t ρ(t) = {Hsys , ρ(t)} + γ

∂ ∂2 (pρ(t)) + mγkT 2 ρ(t). ∂p ∂p

(38)

The first term on the right hand side indicates the Hamiltonian evolution in terms of a Poisson bracket, the second term represents a friction force with damping constant γ, and finally a diffusion term with diffusion constant mγkT with k Boltzmann’s constant and T the temperature. An alternative dynamical description is based on the stochastic evolution equation of a single Brownian particle, i.e. a Langevin equation q˙ = p/m p˙ = −V (q) − γp + F (t),

(39)

or equivalently, m¨ q + mγ q˙ + V (q) = F (t), highlighting the friction force mγ q˙ and the stochastic force F (t) with zero mean and correlations M [F (t)F (s)] = 2mγkT δ(t − s) (we use M [. . .] to denote ensemble means in order to clearly distinguish them from quantum expectation values denoted by angular brackets !. . ."). The two dynamical descriptions (38) and (39) are equivalent in the sense that the phase space distribution ρ(t) is recovered from the Langevin equation as the ensemble mean over many realizations ρ(t)(q, p) = M [δ(q − q(t))δ(p − p(t))] .

(40)

176 The quantization of such dissipative, diffusive dynamics may be performed in several ways [21]. The microscopic approach, adopted later on in this lecture, takes into account the environment as part of a total Hamiltonian, whereupon Schr¨ odinger’s equation is applied to the unitary dynamics of both, system and environment. Alternatively, an axiomatic approach to open quantum system dynamics is fruitful. In the standard Markov case one finds an evolution equation of the so-called Lindblad form [7]. In the simplest case it reads i 1 [Lρ(t), L† ] + [L, ρ(t)L† ] . ∂t ρ(t) = − [Hsys , ρ(t)] + h ¯ 2

(41)

In general, the single non-unitary contribution in (41) involving an operator L in the Hilbert space of the open system has to be replaced by a sum or integral over such terms. Such master equations for the evolution of density operators ρ(t) can be seen as representing the Schrödinger picture quantum analogue of a classical diffusion equation (38). Sometimes, but not always, master equations derived from a microscopic approach turn out to be of the mathematically desirable Lindblad class (41). Either due to approximations made in their derivation, or through dropping the semigroup property which is requested for (41), it may well happen that useful and even exact master equations are not of Lindblad class - an example of this phenomenon may be encountered in the quantized version of Brownian motion, discussed below. This lecture is concerned with a stochastic description of open quantum system dynamics. We establish the quantum analogue of the classical Langevin equation (39), in the Schr¨ odinger picture. Such stochastic Schr¨ odinger equations or quantum trajectories [8, 26] for system states ψ(t) recover the reduced density operator in analogy to (40) as an ensemble mean over stochastic pure states, ρsys (t) = M [|ψ(t)"!ψ(t)|] .

(42)

We will give a microscopic derivation of our stochastic equation that will turn out to be the non-Markovian generalization of the quantum state diffusion equation of Gisin and Percival [27], 1 † ∗ † † ˜ ˜ ˜ = − i Hsys ψ(t)+(L−!L" ˜ ∂t ψ(t) t )(zt +!L "t )ψ(t)− (L L−!L L"t )ψ(t). ¯h 2 (43) We here use its Stratonovich version with a complex white noise zt such that M [zt zs ] = 0 and M [zt zs∗ ] = δ(t − s). The quantum state diffusion stochastic Schrödinger equation (43) is nonlinear due to the appearance

177


˜ ˜ of expectation values !L"t = !ψ(t)|L| ψ(t)" on the right hand side. Im˜ portantly, (43) preserves the norm of the stochastic states ψ(t). Dropping all non-linear terms in (43), we find its linear version, i 1 ∂t ψ(t) = − Hsys ψ(t) + Lzt∗ ψ(t) − L† Lψ(t) h ¯ 2

(44)

for unnormalized stochastic states ψ(t). Both stochastic equations (43) and (44) recover the reduced density operator according to the crucial property (42). As we will show in the subsequent sections, these stochas˜ are the system part in an expansion of the total tic states ψ(t) or ψ(t) state of system and environment, and thus correspond to (are relative to) a certain environmental state. The stochastic states may thus be seen as conditional states of the open quantum system: they are the system states given a certain state of the environment. The central subject of this lecture is to describe a stochastic approach to open quantum systems beyond the class of Lindblad evolution. It is based on a microscopic description of open system dynamics, taking into account the environment explicitely. In fact, by solving Schr¨ odinger’s equation for both, system and environment in a particular representation, we are able to derive a stochastic Schrödinger equation description of open quantum system dynamics in a very general setting. It reduces to the quantum state diffusion equations (43) and (44) in the limit of Markovian (Lindblad) evolution for the reduced density operator as in (41). Our microscopic approach, however, enables us to deal with general, non-Markovian dynamics.

2.2

Microscopic approach and non-Markovian dynamics

Schr¨ odinger’s equation applies to closed systems only. If an open quantum system is to be described, a standard approach we follow here is to take into account the environment and solve i¯ h∂t Ψ(t) = HΨ(t)

(45)

for both, system and environment. The total Hamiltonian is thus of the form H = Hsys ⊗ 11env + 11sys ⊗ Henv + Hint (46) and consists of the Hamiltonian of the open system Hsys , the Hamiltonian of the environment (alias reservoir or heat bath) Henv and, crucially, the interaction between the two, Hint . To save space, we will drop the unit operators appearing in (46) in the following. Clearly, for a

178 mixed initial state, (45) should be replaced by the von Neumann equation ∂t ρtot (t) = − ¯hi [H, ρtot (t)]. For simplicity, we here assume that system and environment are in a pure joint initial state Ψ0 . Sometimes a more realistic choice is a thermal state for the environment which poses no fundamental difficulty for what we are going to develop, see [28, 29, 30]. However, that choice may lead to somewhat clumsier expressions which is why we stick for the most part of this lecture to the pure Ψ0 . Moreover, we simplify further by assuming that system and environment are uncorrelated initially. The total state is thus a product of system and environment state, the latter is taken to be the vacuum, |Ψ0 " = |ψ0 " ⊗ |0"env .

(47)

Due to the generally growing entanglement between system and environment, solving Schr¨ odinger’s equation (45) is a formidable task and if one is interested in properties of the open system only, it traditionally appeared appropriate to try to derive an effective evolution equation for the reduced density operator ρsys (t) = Trenv [ρtot (t)] = Trenv [|Ψ(t)"!Ψ(t)|] .

(48)

If that turns out to be possible, the problem is reduced to a (non-unitary) evolution equation for ρ(t) in the Hilbert space of the open system only. Alternatively, as mentioned in the last Sect. 2.1, stochastic Schrödinger equations may be employed to describe the dynamics. As soon as the Hilbert space dimension N of the system is very large, this approach is often more efficient since the propagation of an N × N matrix ρ(t) is replaced by the stochastic propagation of an only N -dimensional state ψ(t). We here develop such a stochastic theory based on the solution of the Schr¨ odinger equation (45). The environment is modeled by the standard collection of harmonic oscillators, Henv =

hωλ b†λ bλ ¯

(49)

λ

and the coupling is taken to be of the form ¯ (L ⊗ B † + B ⊗ L† ) Hint = h

(50)

with an arbitrary (hermitian or non-hermitian) system operator L and a bath operatorB consisting of contributions of all environmental oscillators, B = λ gλ bλ , with gλ a coupling constant. It turns out to

179


be convenient to change to an interaction representation with respect to the free environment evolution, such that instead of (46) we use

Htot (t) = eiHenv t/¯h Htot e−iHenv t/¯h = Hsys + h ¯ L ⊗ B † (t) + L† ⊗ B(t) (51) for the total Hamiltonian with B(t) = eiHenv t/¯h Be−iHenv t/¯h =

gλ bλ e−iωλ t .

(52)

λ

The dynamics of the open system is thus influenced by the bath operator B(t), whose statistical properties are captured in its correlation function †

α(t−s) = !B(t)B (s)"env =

λ

2 −iωλ (t−s)

|gλ | e

=h ¯

∞

dωJ(w)e−iω(t−s) ,

0

(53) here at zero temperature. The last equality in (53) defines the spectral density J(ω) of bath oscillators. If the dynamics of system and environment is such that the bath correlation function α(t − s) in (53) may be replaced by a delta function, the reduced dynamics is Markovian and ρ(t) evolves according to the Lindblad master equation (41). A general correlation function α(t − s), however, describes memory effects of the environment and matters become exceedingly more difficult. Such non-Markovian effects are known to be relevant in many situations, in particular at low temperatures and as soon as narrow energy splittings occur, as in tunneling processes [21]. They are also relevant whenever the environment is structured, as for instance the electro-magnetic vacuum in the presence of a photonic band gap material [31]. Moreover, non-Markovian effects play a role in output coupling dynamics of atoms from a Bose Einstein condensate with the aim to build an atom laser [32]. It may also happen that the dynamics appears to be Markovian, yet the evolution equation is not of the standard Lindblad class (41). This is the case, for instance, in the hightemperature limit of the standard quantum Brownian motion model [33] to be discussed in Sect. 2.5. In such cases, transient effects (initial slips, see also [34]) are important and no contradiction with the axioms for Lindblad evolution occurs. Rather than trying to derive an evolution equation for the reduced density operator for general bath correlation α(t−s), we here follow a different route to capture non-Markovian effects. We solve the Schr¨ odinger equation (45) for both, system and environment with initial state (47) and total Hamiltonian (51). For that we expand the total state Ψ(t) in a coherent state basis for the environmental degrees of freedom [29].

180 2.2.1 Expansion in coherent states. Coherent states |z" are minimum uncertainty wave packets and remain coherent under harmonic evolution. We here use unnormalized Bargmann coherent states [35], †

|z" ≡ ezb |0"

(54)

with !z|z " = exp(z ∗ z ) such that the resolution of the identity reads 11

=

d2 z −|z|2 e |z"!z|. π

(55)

Importantly, Bargmann coherent states |z" are analytical in z. We now expand the environmental part of the total state Ψ(t) of (45) in a fixed Bargmann coherent state basis, using z = (z1 , . . . , zλ , . . .) to notate the vector of all environmental coherent state labels, and write |Ψ(t)" =

d2 z −|z|2 |ψ(t, z ∗ )" ⊗ |z". e π

(56)

Here, |z 2 | is to be read as λ |zλ |2 , for instance. The system states |ψ(t, z ∗ )" = !z|Ψ(t)" relative to |z" are analytical in z ∗ = (z1∗ , . . . , zλ∗ , . . .), and they have been assigned all the time dependence of the total state. Note that we refrain from introducing an extra amplitude for the product state |ψ(t, z ∗ )"|z", that being included in the definition of ψ(t, z ∗ ), which will therefore not remain normalized under time evolution. We plug (56) into the Schr¨ odinger equation (45) with total Hamiltonian (51) and find :

;

1 ∂ −iωλ t i∂t ψ(t, z ∗ ) = gλ∗ zλ∗ eiωλ t + L† gλ e ψ(t, z ∗ ). Hsys + L ∗ ¯h ∂ z λ λ λ (57) The crucial next step is to identify the combination

zt∗ ≡ −i

gλ∗ zλ∗ eiωλ t

(58)

λ

appearing in (57) and regard the states ψ(t, z ∗ ) no longer as a function of the vector z ∗ , but as a functional of zt∗ . Then (57) turns into

δψ(t, z ∗ ) , δzs∗ 0 (59) where α(t − s) is nothing but the (zero temperature) bath correlation function (53). Equation (59) was first derived in [29] and is the central i ∂t ψ(t, z ) = − Hsys ψ(t, z ∗ ) + Lzt∗ ψ(t, z ∗ ) − L† ¯h ∗

t

ds α(t − s)

181


starting point for a stochastic description of open system dynamics beyond the standard Markov Lindblad class. In the next subsection we see that zt∗ can be assigned the role of a stochastic process and thus, (59) turns indeed into a stochastic Schr¨ odinger equation for the states ∗ ψ(t, z ) of the open quantum system.

2.2.2 Reduced dynamics. According to expansion (56), the solution of (59) for all vectors z ∗ = (z1∗ , . . . , zλ∗ , . . .), or equivalently, for all functions zt∗ , amounts to determining the solution Ψ(t) of the Schr¨ odinger equation (45), and thus total knowledge about system and environment. We may, for instance, determine the reduced density operator (48) for the open system and find

ρsys (t) =

d2 z −|z|2 |ψ(t, z ∗ )"!ψ(t, z ∗ )|. e π

(60)

The reduced density operator is the mixture of all states ψ(t, z ∗ ), dis2 tributed with the Gaussian probability e−|z| . We conclude that ρ(t) may indeed be determined as an ensemble mean according to (42) through a Monte-Carlo integration in (60), here taken over the Gaussian distribu 2 2 tion M [. . .] = dπz e−|z| [. . .] of z-vectors. On these grounds, equation (59) for the system part ψ(t, z ∗ ) of the total state Ψ(t) turns indeed into a non-Markovian stochastic Schr¨ odinger equation for the determination of the reduced density operator. The stochastic process zt∗ from (58) becomes a Gaussian colored process with zero mean M[zt∗ ] = 0 and correlations M [zt zs∗ ] = α(t − s)

and

M [zt zs ] = 0.

(61)

Here α(t − s) is again the (zero temperature) bath correlation function (53). What remains is the task to solve the central equation (59) which is difficult due to the appearance of a functional derivative under the memory integral. We overcome this problem for a wide range of open system dynamics.

2.3

Convolutionless formulation of non-Markovian dynamics

Memory effects of non-Markovian evolution clearly make themselves felt through the integral over the past in equation (59), involving the bath correlation function and a functional derivative of the current state ψ(t, z ∗ ) with respect to earlier noise zs∗ . In many relevant cases [30], it is possible to replace that functional derivative by some time dependent

182 operator O,

δψ(t, z ∗ ) = O(t, s, z ∗ )ψ(t, z ∗ ), δzs∗

(62)

acting in the Hilbert space of the open system on the current state ψ(t, z ∗ ). We indicate that O may depend on the times t and s, and possibly on the (entire history of the) stochastic process zt∗ . Relevant examples of this replacement will be given shortly. One way to determine the operator O(t, s, z ∗ ) in actual applications [30] is to insert the Ansatz (62) in (59) and use the consistency condition ∂t

δψ(t, z ∗ ) δ = ∗ ∂t ψ(t, z ∗ ). δzs∗ δzs

(63)

Alternatively, a Heisenberg operator approach to determine O(t, s, z ∗ ) may be useful, as shown in [36]. Once the replacement (62) of the functional derivative by an operator is known – sometimes only approximately – the evolution equation (59) takes the more useful convolutionless form [30, 37]

i ¯ z ∗ ) ψ(t, z ∗ ), ∂t ψ(t, z ∗ ) = − Hsys + Lzt∗ − L† O(t, h ¯

(64)

¯ z ∗ ) as the integral over the action of where we defined the operator O(t, the functional derivative over the whole past [38], ¯ z∗) = O(t,

0

t

ds α(t − s)O(t, s, z ∗ ).

(65)

The determination of ψ(t, z ∗ ) is now reduced to solving the simple stochastic Schr¨ odinger equation (64). We recall that (64) does not preserve the norm of the states ψ(t, z ∗ ), thus, if only a numerical solution is possible, it is most often more advisable to use its nonlinear, norm-preserving version, which is derived towards the end of this section. The crucial task remains the determination of the operator O(t, s, z ∗ ) in (62). We give two relevant results, for exactly soluble models see [30].

Weak coupling. The action of the functional derivative may be expanded in powers of the interaction Hint . To 5lowest order, one finds 4 ∗ ∗ −iHsys (t−s)/¯ h LeiHsys (t−s)/¯h + . . . ψ(t, z ∗ ) and thus δψ(t, z )/δzs = e O(t, s) ≈ e−iHsys (t−s)/¯h LeiHsys (t−s)/¯h or ¯ ≈ O(t)

t 0

ds α(s)e−iHsys s/¯h LeiHsys s/¯h .

(66)

(67)

183


A weak coupling stochastic Schrödinger equation equivalent to (64) with the replacement (67) was derived in [39] in a formulation that kept the memory integral over the bath correlation function.

Near Markov. If the bath correlation function α(t − s) falls off rapidly under the memory integral in (59), an expansion of the functional derivative in terms of the time delay (t − s) is sensible [33, 38]. With An (t) = 0t ds sn α(s) and n = 0, 1, 2, . . ., we find for the relevant integrated operator (65) to first order,

¯ ≈ A0 (t)L + A1 (t) − i [H, L] + A0 (t)[L, L† ]L , O(t) h ¯

(68)

neglecting contributions An (t) with n ≥ 2. Of particular interest is the standard Markov limit. As soon as α(t − s) = γδ(t − s) with some constant γ, only A0 is relevant and may ¯ = γ L. In this case, indeed, be replaced by the constant γ/2, thus O 2 the stochastic Schrödinger equation (64) (or equivalently, (59)) reduces to the linear version of the Markov quantum state diffusion equation (44). The next order term in expansion (68) turns out to be relevant for the high temperature limit of the quantum Brownian motion model mentioned in Sect. 2.5. We remark that very often, Markov and weak coupling approximation are only meaningful in combination, referred to as Born-Markov approximation.

2.4

Nonlinear non-Markovian stochastic Schr¨ odinger equation

The linear non-Markovian convolutionless stochastic Schr¨ odinger equation (64) has the drawback of not preserving the norm of the states ψ(t). This may cause problems in simulations (importance sampling) and is overcome by a time-dependent shift of the probability distribution of the noise zt∗ , leading to the non-linear, non-Markovian stochastic Schr¨ odinger equation i ˜ ˜ + (L − !L")˜ ˜ ∂t ψ(t) = − Hsys ψ(t) zt∗ ψ(t) (69) h ¯ ˜ ¯ z˜∗ ) − !(L† − !L† "t )O(t, ¯ z˜∗ )" ψ(t) − (L† − !L† "t )O(t, ˜ for normalized states ψ(t), as shown in [30]. Here z˜t∗

=

zt∗

t

+ 0

ds α∗ (t − s)!L† "s

(70)

184 is a shifted noise with the original zt∗ from (61). The crucial property (42) to recover the reduced density operator is preserved for the ensemble of solutions. In the case of Markov (Lindblad) evolution, the above reduces to the non-linear quantum state diffusion equation (43).

2.5

Quantum Brownian motion

Of particular interest is the stochastic approach to the quantized version of Brownian motion (38,39), which may be derived from the general model (46) with the choices Hsys = p2 /2m+V (q) and a coupling L = q/¯h [12, 15]. In this case the general linear stochastic Schr¨ odinger equation (59) is valid for arbitrary temperature [30, 33]. One simply has to replace the zero temperature bath correlation function (53) by its finite temperature expression, α(t − s) = h ¯

dωJ(ω) coth

¯ω h 2kT

cos ω(t − s) − i sin ω(t − s) . (71)

Typical choices for the spectral density J(ω) is a linear dependence for ω → 0, and a cutoff at some high frequency Λ, as mentioned earlier. The replacement of the functional derivative by the operator O(t, s, z ∗ ) in (62) which is necessary in order to derive the convolutionless version (64) of the non-Markovian stochastic Schr¨ odinger equation (or its nonlinear version (69)) cannot be performed in general. We focus on the high temperature case where the thermal frequency kT /¯h is assumed the largest frequency involved and the bath correlation function may be replaced by [15] ˙ − s). α(t − s) ≈ 2mγkT ∆(t − s) + imγ ∆(t

(72)

Here, ∆(t − s) is a delta-like function, decaying on the bath correlation time scale Λ−1 , where Λ is a cutoff frequency of the spectral density J(ω) in (71). With (72), in the limit of high cutoff frequency Λ → ∞, the functional derivative may be determined from the expansion (68), including that first order. We find ¯ = 1 (A0 (t)q − A1 (t)p/m) , O(t) h ¯

(73)

with the functions An (t) as in (68). After an initial slip on the inverse ¯ becomes O ¯ = mγkT q + cutoff frequency time scale, the asymptotic O γp/2m. With (73) we are in the position to determine single runs of equations (64) or its nonlinear version (69) numerically for an arbitrary potential V (q). For applications to a driven, nonlinear system see [33]. It can be shown explicitely that the ensemble mean evolves according


185

to the well known high temperature master equation of quantum Brownian motion [15] which reduces to the classical Fokker-Planck equation (38) upon taking the limit h ¯ → 0. As shown in [33], single runs of the nonlinear stochastic Schr¨ odinger equation (69) for the Brownian motion case approach minimum-uncertainty wave packets as ¯h → 0, whose centroids follow the classical Langevin equation (39). Thus, (69) may be seen as the quantized version of the classical Langevin equation in the Schr¨ odinger picture, on the level of individual realizations.

3.

Conclusions

Two aspects of open quantum system dynamics are discussed in these lectures. First, decoherence, i.e. the loss of quantum coherence as a dynamical phenomenon is presented in detail. Crucially, the decoherence time scale may differ drastically from dissipation or relaxation time scales: even without noticeable dissipation, coherences may disappear rapidly. For more and more macroscopic superpositions, decoherence becomes faster than any relevant system time scale. The theoretical analysis may be based on a short-time approach. In this limit, decoherence shows universal properties: it is independent of any system Hamiltonian and the detailed nature of the environment. In the second lecture we establish a framework how to describe the dynamics of an open quantum system in terms of stochastic pure system states ψ(t, z ∗ ). We derive the corresponding non-Markovian stochastic Schr¨ odinger equation. The usual reduced density operator is recovered as the ensemble mean over many runs of the stochastic equation. In the standard Lindblad Markov limit, our findings reduce to the quantum state diffusion equation (43). The class of open system dynamics captured by this approach extends far beyond the standard class of Lindblad evolution. The derivation shows that our approach amounts to a solution of the Schr¨ odinger equation for both, system and environment in a particular representation. Remarkably, for the Brownian motion model, as ¯h → 0, solutions of the stochastic Schr¨ odinger equation are localised wave packets whose centroids follow the classical Langevin equation of a Brownian particle.

Acknowledgments Financial support from the Deutsche Forschungsgemeinschaft through SFB276 is gratefully acknowledged.

186

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ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS. Instabilities in the forced truncated NLS. Eli Shlizerman Faculty of mathematical and computer science Weizmann Institute, Rehovot 76100, Israel

Vered Rom-Kedar Faculty of mathematical and computer science Weizmann Institute, Rehovot 76100, Israel

Abstract

A two-degrees of freedom near integrable Hamiltonian which arises in the study of low-amplitude near-resonance envelope solutions of the forced Sine-Gordon equation is analyzed. The energy momentum bifurcation diagrams and the Fomenko graphs are constructed and reveal the bifurcation values at which the lower dimensional model exhibits instabilities and non-regular orbits of a new type. Furthermore, this study leads to some new insights regarding the hierarchy of bifurcations appearing in integrable Hamiltonian systems and the role of global bifurcations in the energy momentum bifurcation diagrams.

Keywords: Parabolic resonance, Near-integrable Hamiltonian systems, homoclinic chaos.

1.

Introduction

Despite a century long study of near integrable Hamiltonian systems, our qualitative understanding of inherently higher dimensional (nonreducible to smooth symplectic two dimensional maps) dynamics is lacking. Recently we proposed a framework for obtaining such qualitative information for a class of near-integrable Hamiltonian systems [25, 26, 27]. Here we provide a first example of a physical system which is studied using this framework, demonstrating that some new insights are gained regarding the physics and adding some new ingredients to the general framework which is still under construction.


190 Qualitative understanding means here that the effect of small perturbations on different unperturbed orbits may be a-priori predicted for some non-trivial time scales. For example, such a qualitative understanding exists for generic near integrable one-and-a half degrees of freedom systems; We know that for such systems most of the phase space will be filled by KAM tori, KAM Cantori and resonance bands whereas homoclinic loops of the integrable system will create homoclinic chaotic zones. While there are some long standing open problems regarding the asymptotic behavior of such systems (notably the decay rate of averaged observables in the chaotic zone and the measure of the chaotic zone [20, 36]), the basic transport and instability mechanisms are well understood on time scales which are logarithmic in the perturbation parameter [29, 33]. Another example is the behavior of orbits of near-integrable n degrees of freedom systems in a neighborhood of an unperturbed compact regular non-degenerate level set with Diophantine frequency vector; While the asymptotic behavior of the solutions in such regions is still unknown (the famous Arnold diffusion conjecture addresses this question), it is known that for extremely long time (at least exponential in the perturbation parameter [31, 15]) the orbits will hover near the preserved KAM tori. In both examples, while the asymptotic behavior is unknown, there is a good understanding of the characteristic behavior of all orbits in a given neighborhood for a long transient time. Recently it has been proposed that a similar qualitative understanding of the behavior of orbits near non-regular level sets in some higher dimensional systems may be achieved by studying the structure of the energy surfaces and their bifurcations using the energy momentum bifurcation diagrams and branched surfaces (see [25, 26, 27] and references therein, including the important previous works [35, 24, 13, 12]). So far, this theory was applied to Hamiltonian systems which are polynomial in their arguments and are built to satisfy some specific bifurcation scenarios via a normal-form type of construction ([25, 26, 27]). In particular, for all these examples, the unperturbed energy surface topology and the foliation of the level sets on them were trivial. In general, the classification of all possible topologies and level sets foliations of an energy surface of integrable systems is unknown: it had been completed for the two degrees of freedom case [14, 24], whereas in higher dimensions even the characterization of specific systems like the Lagrange top turns out to be extremely difficult (see [14, 12, 23] and references therein). Here we apply the general framework of ([25, 26, 27]) to equations which have not been previously analyzed from this point of view - a truncated two mode model of the perturbed Sine-Gordon equation. This is a two-degrees-offreedom model, and, as expected, no new topological structure of the

Energy surfaces and Hierarchies of bifurcations.

191

energy surfaces are discovered. However, a comprehensive way of studying the dependence of these structures on the energy level and on the parameters is presented, leading to identifying new dynamically interesting regimes. This reduced model was suggested by Bishop et al. and collaborators [4, 2, 8, 5, 3, 6, 7, 11] who investigated analytically and numerically the damped and driven Sine-Gordon equation. Their numerical studies have revealed chaotic behavior and the existence of low dimensional chaotic attractors of the PDE. Using the near resonance envelope expansion for the Sine-Gordon model they have transformed the problem to the investigation of the perturbed NLS model. Introducing a two-mode symmetric Galerkin approximation for the solutions of the perturbed NLS they have derived the corresponding truncated equations for the complex amplitudes. Finally, they have shown that the unperturbed truncated system is a two degrees of freedom Hamiltonian system with an additional integral of motion, and, hence, is integrable. They found numerically that in some parameter regime this low dimensional model supplies qualitative behavior which is similar to the observed behavior of the PDE. Investigation of the truncated system lead to the discovery of a new mechanism of instability - the hyperbolic resonance - by which homoclinic solutions to a lower dimensional resonance zone are created [22, 16, 19, 18]. The unperturbed structure of the truncated model which is responsible to this behavior is a circle of fixed points which is hyperbolic in the transverse direction (see section 3 for a precise definition). New methodologies and tools introduced to this PDE-ODE study have finally lead to a proof that the homoclinic resonance dynamics, and in particular the birth of new types of multi-pulse homoclinic orbits which is associated with it, has analogous behavior in the PDE setting (see [17, 11, 28] and references therein). The appearance of a hyperbolic circle of fixed points in the truncated model is not a special property of the NLS model - investigation of the structure of low-dimensional near-integrable Hamiltonian systems (see [26]) shows that hyperbolic resonances are persistent1 phenomenon in n degrees of freedom systems with n ≥ 2; Among such integrable Hamiltonian systems there are open sets of Hamiltonians which have an n − 1 dimensional torus of fixed points which is normally hyperbolic. Furthermore, it is established that parabolic resonant tori (here, the ap-

1 The

existence of such tori may be formulated as the existence of transverse intersection of some finite dimensional manifolds. Hence, using the Transversality theorem, one proves that hyperbolic resonant tori exist for a C 1 −open set of integrable Hamiltonians, which we take hereafter as the definition of persistence.

192 pearance of an unperturbed circle of fixed points which are parabolic in the transverse direction, more details appear in section 3) are persistent in families of near-integrable Hamiltonians if n + p > 2 where n is the number of the degrees of freedom and p is the number of the control parameters [26]. Hence, including one parameter in a two degrees of freedom model naturally leads to a persistent appearance of a parabolic resonance. The truncated NLS model depends on one parameter - the dimensionless wave number of the first symmetric mode. We show that as this parameter is varied, parabolic resonant tori are created. In fact, we are able to provide a complete and novel description of the structure of the energy surfaces as a function of the energy and the parameter, discovering two phenomena which have not been observed nor analyzed yet for this model - the appearance of parabolic resonances and the appearance of resonant global bifurcation. Furthermore, since under conservative perturbations, the solutions remain on a given energy surface, our characterization supplies a priori bounds to the perturbed motion. The paper is organized as follows. In the second section we present the derivation of the truncated model of the NLS (following [4, 22]). In section 3 we describe the structure of regular and singular level sets for a general integrable Hamiltonian (following [27]). We use the truncated NLS model as a particular example for the theory, and describe the structure of the energy surfaces for this model using the EMBD and the Fomenko graphs. The fourth section lists the hierarchy of bifurcations in general and for the truncated model in particular, finding the energy bifurcating values, their bifurcation diagram and the dynamical significance of these bifurcations. We present a few simulations showing the instabilities encountered at such bifurcation values. The short discussion is followed by appendix A, where we prove that the energy surfaces of this model are bounded and that the perturbed surfaces are close to the unperturbed surfaces. Appendix B includes the energy-momentum bifurcations diagrams and Fomenko graphs for several parameter values.

2.

The two-mode truncation model of the perturbed NLS

Here we recall the relevant results of Bishop et al. [4, 2, 8, 5, 3, 6, 7] followed by Kovacic and Haller [22, 16, 19, 18, 17] results who developed the two-mode truncation model and studied some of its features. Bishop et al. investigated the chaotic attractor of the damped driven Sine-Gordon Equation (SGE) with even spatial symmetry and periodic boundary conditions:


ˆ txx + Γ ˆ cos(ωt)), utt − uxx + sin u = δ(−ˆ αut + Λu u(x) = u(x + L), ux (0) = 0.

193

(1)

ˆ is the driving ampliω is the driving frequency, L is the box size, δ Γ ˆ is an additional wave-number dependent tude, δ α ˆ is the damping and δ Λ damping term which was introduced in [22]. The NLS approximation for the SGE is obtained by developing a small amplitude envelope approximation for the near resonance frequency (ω = 1 − δ ω ˜ ) case. More precisely, one looks for solutions of the SGE of the form: √ ˜ [Bδ (X, T )eiωt + Bδ (X, T )∗ e−iωt ], u = 2 δω

(2)

where Bδ (X, T ) is assumed to be analytic in δ, and √ ω = 1 − δω ˜ , X = 2δ ω ˜ x, T = δ ω ˜ t. Introducing a small parameter ε such that ˆ = εΛ, Γ ˆ = ε8δ 3/2 ω ˜ 3/2 Γ, α ˆ = ε2˜ ω α, Λ with all other parameters of order one, one finds that provided2 δε1 the leading order term in δ, B(X, T ) = B0 (X, T ), satisfies the following forced and damped NLS equation: −iBT + BXX + (|B|2 − 1)B = iε(αB − ΛBXX + Γ).

(3)

Consider a two mode complex Fourier truncation for equation (3): 1 B(X, T ) = √ c(T ) + b(T )coskX. 2

(4)

The periodic boundary conditions imply that k= √

2π 2δ ω ˜L

n, n ∈ Z+ .

(5)

By substituting this solution to the NLS equation and neglecting (see [4, 2, 8, 5, 3, 6, 7, 11] for discussion of this step) higher Fourier modes, one obtains the following two ODE’s: 2 This

consistency conditions has not been set explicitly in previous publications.

194 1 1 −ic˙ + ( |c|2 + |b|2 − 1)c + 2 2 1 3 2 2 −ib˙ + [ |c| + |b| − (1 + k 2 )]b + 2 4

√ 1 ∗ (cb + bc∗ )b = iε(αc + 2Γ) 2 1 ∗ (bc + cb∗ )c = iεβb 2

where β = α + Λk2 . By setting α = Λ = 0, one discovers that this is a two d.o.f. Hamiltonian system, which, at Γ = 0 possesses two integrals: 1 I = (|c|2 + |b|2 ), 2 1 1 3 1 1 1 H = |c|4 + |b|2 |c|2 + |b|4 − (1 + k 2 )|b|2 − |c|2 + (b2 c∗2 + b∗2 c2 ). 8 2 16 2 2 8 hence, at ε = 0 the system is integrable. The Hamiltonian part of the perturbation is: ε H1 = −i √ Γ(c − c∗ ). 2 Introducing generalized action angle coordinates (x, y, I, γ) [22] (see next section for more details): c = 2I − x2 − y 2 exp (iγ) , b = (x + iy) exp (iγ) Kovacic and Haller [22, 16, 19, 18, 17] have investigated the corresponding two d.o.f. system. In particular, they have realized that a specially interesting phenomena occurs when the circle b =√0 is a normally hyperbolic circle of fixed points (this occurs for k < 2 at I = 1, see next section for details). Then, at ε = 0, pairs of fixed points on this circle are connected by heteroclinic orbits. This realization lead to a beautiful theoretical study of the behavior of integrable systems with such a normally hyperbolic circle of fixed points under conservative [19, 18, 17] and dissipative [22, 16] perturbations. Kovacic [22] includes in his study global analysis of the integrable system in which, after identifying the critical I values, the level sets H = h are plotted for typical values of I. We present the integrable system using Energy momentum bifurcation diagrams and Fomenko graphs, which corresponds to investigating the surfaces H = h and their extent in the I direction. We propose that these representation are more adequate for understanding the behavior under small perturbations since the total energy H is preserved (for α = 0).

195


3.

Level sets and energy surfaces structure

The structure of energy surfaces and level sets for n degrees of freedom integrable Hamiltonian systems in general (following the formulation of [27]) and their structure for the two mode truncation of the conservatively perturbed NLS in particular are now described. Consider a near integrable Hamiltonian H(q, p; ε) = H0 (q, p) + εH1 (q, p; ε), ε 1, (q, p) ∈ M ⊆ Rn × Rn , where M is a 2n-dimensional, analytic symplectic manifold and H0 and H1 are3 C ∞ and H1 and its derivatives are bounded. H0 represents the completely integrable part of the Hamiltonian (the unperturbed system) and its structure is described below. For any ε, a perturbed orbit with energy h resides on the energy surface H0 (·) = h − εH1 (·; ε). Hence, by assumption, the structure of the unperturbed energy surfaces and their resonance webs in an O(ε)-interval of energies near h supplies global information on the allowed range of motion of the perturbed orbits (see below for a more precise formulation). The two mode truncation of the conservatively perturbed NLS (hereafter, since only Hamiltonian perturbations are considered, we set ε = εΓ and drop the over bar): √ 1 1 2 1 2 (6) −ic˙ + |c| + |b| − 1 c + (cb∗ + bc∗ )b = i 2ε 2 2 2 1 2 3 2 1 2 ˙ −ib + |c| + |b| − (1 + k ) b + (bc∗ + cb∗ )c = 0. 2 4 2 where |b| is the amplitude of the first symmetric mode and |c| is the amplitude of the plane wave, is a two degrees of freedom near integrable Hamiltonian of the form: H(c, c∗ , b, b∗ ; ε) = H0 (c, c∗ , b, b∗ ) + εH1 (c, c∗ , b, b∗ ) with the Poisson brackets {f, g} = −2i
, where

1 1 3 1 1 1 H0 = |c|4 + |b|2 |c|2 + |b|4 − (1 + k 2 )|b|2 − |c|2 + (b2 c∗2 + b∗2 c2 ). 8 2 16 2 2 8 (8) i (9) H1 = − √ (c − c∗ ). 2

At ε = 0 the system is integrable, having the second integral of motion: 1 I = (|c|2 + |b|2 ). 2 3 One

may relax these requirements to the C r case, but this is left for future studies.

196 It can be shown that the unperturbed energy surfaces are bounded (see appendix A), and that the perturbed and unperturbed energy surfaces are close to each other as long as the level sets belonging to the unperturbed energy surface are bounded away from neighborhoods of fixed points, so ∇H0 > const ε. More precisely, using the implicit function theorem (see appendix), it can be shown that under these conditions any cε (hε ), bε (hε ) satisfying H(cε (hε ), cε∗ (hε ), bε (hε ), bε∗ (hε ); ε) = hε = h+ O(ε) are ε−close to c0 (h), b0 (h) values. Since there are a finite number of fixed points of the system (6) and these appear at bounded energy values (see appendix), one can conclude that for most part of phase space the unperturbed and perturbed surfaces are close to each other and that the behavior near fixed points requires further analysis, as expected (roughly, if higher order derivatives of order m are nonsingular one expects that the perturbed surfaces are order ε1/m close to the unperturbed ones). Notice that the closeness of the perturbed and unperturbed energy surfaces does not imply that they are topologically conjugate - counter examples may be easily constructed. Nonetheless, this closeness is sufficient to obtain a-priori bounds on the motion. The integrable n d.o.f. Hamiltonian, H0 (q, p); (q, p) ∈ M ⊆ Rn × Rn , has n integrals of motion: H0 = F1 , F2 , ..., Fn ∈ C ∞ (M ), which are functionally independent at almost all points of M and are pair wise in involution: {Fi , Fj } = 0; i, j = 1, ...n. For simplicity, we assume that the Hamiltonian level sets, Mg = {(q, p) ∈ M, Fi = gi ; i = 1, ..., n}, are compact, so that the integrals of motion are complete. Here, the two integrals of motion are H0 and I and since the level sets of I are 3−spheres the Hamiltonian level sets Mg are clearly compact. By the Liouville-Arnold theorem (see [30] and [1, 21]), the connected compact components of the level sets Mg , on which all of the dFi are (point wise) linearly independent, are diffeomorphic to n-tori and hence a transformation to action-angle coordinates (H0 = H0 (J)) near such level sets is non singular. Here, 1 dI = (c∗ , c, b∗ , b) 2  c∗ ( 14 |c|2 + 12 |b|2 − 12 ) + 14 b∗2 c)  c( 14 |c|2 + 12 |b|2 − 12 ) + 14 b2 c∗ dH0 =   b∗ ( 1 |c|2 + 3 |b|2 − 1 (1 + k 2 )) + 1 bc∗2 2 8 2 4 b( 12 |c|2 + 38 |b|2 − 12 (1 + k 2 )) + 14 b∗ c2

   

thus these are linearly independent for most values of c and b. The values at which dI and dH0 are linearly dependent (e.g. the plane c = 0) are discussed next.


197

Consider a neighborhood of a level set Mg0 which possibly contains a singularity set at which the rank of the dFi ’s is n − 1 (here, dI and dH0 are linearly dependent, but do not vanish simultaneously, so the origin is excluded). Then, on each connected and closed component of such a Hamiltonian level set there is some neighborhood D, in which the Hamiltonian H0 (q, p) may be transformed by the reduction procedure to the form (see [24], [30]): H0 (q, p, J),

(q, p, φ, J) ∈ U ⊆ R1 × R1 × Tn−1 × Rn−1

(10)

which does not depend on the angles of the tori, φ. The symplectic structure of the new integrable Hamiltonian (10) is dq ∧ dp + n−1 i=1 dφi ∧ dJi , where (q, p, φ, J) are the generalized action-angle variables. The motion on the (n − 1) dimensional family (parameterized by the actions J) of (n − 1)-tori is described by the equations: dφi = ωi (q, p, J), dt

dJi = 0. dt

The geometrical structure of the new Hamiltonian, H0 (q, p, J), is such that for any fixed J (J = (J1 , ..., Jn−1 )) an (n − 1)-torus is attached to every point of the (q, p) plane. The (q, p) plane is called the normal plane [1, 32, 9] of the (n − 1)-tori, and defines their stability type in the normal direction to the family 4 of tori. For our example, the transformation to the generalized action angle co-ordinates for c = 0 was found in Kovacic [22]; taking c = |c| exp (iγ) ,

b = (x + iy) exp (iγ)

so that

1 I = (|c|2 + x2 + y 2 ) 2 one obtains the generalized action angle canonical coordinates for the Hamiltonian: H(x, y, I, γ) = H0 (x, y, I) + εH1 (x, y, I, γ), where (I, γ) ∈ (R+ × T ), (x, y) ∈ BI = {(x, y)|x2 + y 2 < 2I} 4 Notice

that a single torus belonging to this family has neutral stability in the actions directions. The normal stability referred to in the Hamiltonian context ignores these directions, see [10, 9] and references therein.

198 and 1 1 7 3 1 1 H0 (x, y, I) = I 2 − I + (I − k 2 )x2 − x4 − x2 y 2 + y 4 − k 2 y 2 , 2 2 16 8 16 2 (11) √ (12) H1 (x, y, I, γ) = 2 2I − x2 − y 2 sin γ. The transformation to these variables is singular at c = 0, namely on the circle 2I = x2 + y 2 , where the phase γ is ill defined and the perturbation term has a singular derivative. In Kovacic and Haller [22, 16, 19, 18] the analysis was performed for phase space regions which are bounded away from this circle. Here we would like to include some information regarding the dynamics near this circle as well. We thus use a similar transformation which is valid as long as b = 0: 1 b = |b|eiθ , c = (u + iv)eiθ , I = (u2 + v 2 + |b|2 ) 2 and obtain the equation of motion in the canonical coordinates (u, v, I, θ) from the Hamiltonian: 3 2 3 2 1 2 3 7 2 H0 (u, v, I) = I + −1 + u − v − k I − u4 − u2 v 2 4 4 4 16 8 1 1 1 + k 2 u2 + k 2 v 2 + v 4 2√ 2 16 H1 (u, v, I) = 2(v cos θ + u sin θ). When both γ and θ are well defined, namely for cb = 0, the two sets of coordinates are simply related: x = |b| cos(θ − γ) |c| u = x, |b|

y = |b| sin(θ − γ) |c| v = − y. |b|

It follows that for x2 + y 2 < 2I we have: ∂H0 (x, y, I) dγ = ω(x, y, I) = I − 1 + x2 = dt ∂I

(13)

and for u2 + v 2 < 2I we have: dθ 1 ∂H0 (u, v, I) 3 3 = ω(u, v, I) = I − 1 + u2 − v 2 − k2 = . dt 2 4 4 ∂I

(14)

Consider now lower dimensional invariant tori which correspond to isolated fixed point(s) of the normal plane. These appear on an n − 1

199


dimensional manifold of J values, the singularity manifold. Locally, one may choose the (q, p, J) coordinate system so that for these tori: ∇(q,p) H0 (q, p, J)|pf = 0,

pf = (q f , pf , Jf ).

(15)

For our example, there are six families of such tori as listed in the following table:

Invariant circle:

θ, γ ∈ T 1

Exists For

Description

1. ppw = (x = 0, y = 0, I, γ),

I≥0

Plane wave (b = 0)

2. psm = (u = 0, v = 0, I, θ), , 4 (−k 2 + 2I), y = 0, I, γ), 3. p± pwm = (x = ± , 7 = (u = ± 67 I + 47 k 2 , v = 0, I, θ),

I≥0

Symmetric mode (c = 0)

I ≥ 12 k2

PW mixed mode (bc = 0)

4. p± smm = (x = 0, y = √±2k, I, γ), = (u = 0, v = ± 2I − 4k 2 , I, θ),

I > 2k 2 I ≥ 2k2

I>

1 2 k 2

” SM mixed mode (bc = 0) ”

Table 1 Following [24] terminology, the invariant tori on which equation (15) is satisfied are called here singular tori and the surfaces of energy and action values on which this equation is satisfied are called singularity surfaces. We will see that the structure of these singularity surfaces serves as an organizing skeleton of the energy surfaces. The invariant (n − 1)-tori have an (n − 1)-dimensional vector of inner ∂H0 (pf ) . The normal stability type of such frequencies, θ˙ = ω(pf ) = ∂J families of (n − 1)-tori is determined by the characteristic eigenvalues (respectively, Flouqet multipliers for the corresponding Poincaré map) of the linearization of the system about the tori; generically, these tori are either normally elliptic5 or normally hyperbolic6 . If the torus has one pair of zero characteristic eigenvalues in the direction of the normal (q, p) plane, it is said to be normally parabolic. 5 If the characteristic eigenvalues of an invariant lower dimensional torus (with respect to its normal (q, p) plane), are purely imaginary (and do not vanish), it is said to be normally elliptic. 6 If all the characteristic eigenvalues of an invariant lower dimensional torus (with respect to its normal (q, p) plane) have a nonzero real part, it is said to be normally hyperbolic.

200 Locally, in the (q, p, J) coordinate system, the normal stability of the invariant torus is determined by:

∂ 2 H0 (16) det = −λ2pf ∂ 2 (q, p) pf where pf satisfies (15). Indeed, when λpf is real and non-vanishing the corresponding family of tori is said to be normally hyperbolic, when it vanishes it is called normally parabolic and when it is pure imaginary it is normally elliptic, see the detailed references in [27] and the discussion of the Hamiltonian case in [9]. For our example such calculation shows that the first and third families of invariant circles become parabolic at I = 12 k 2 whereas the second and fourth families are parabolic at I = 2k2 : Jacobian Eigenvalues

Elliptic For

Hyperbolic For

Parabolic For

1. (λpw )2 = k 2 (−k2 + 2I)

I < 12 k2

I > 12 k2

Ippw = 12 k 2

I < 2k 2

I > 2k 2

Ipsm = 2k2

3. (λpwm )2 = 47 (2k4 − k 2 I − 6I 2 )

I > 12 k2

-

Ippw = 12 k 2

4. (λsmm )2 = 4k2 (2k2 − I)

I > 2k 2

-

Ipsm = 2k2

2. (λsm )2 =

3 2

I + k2

1 2

I − k2

Table 2

3.1

Energy momentum bifurcation diagrams and the branched surfaces

The energy-momentum bifurcation diagram (EMBD) supplies global information on the bifurcations of the energy surfaces structure and their relation to resonances; Consider an integrable Hamiltonian system H0 (q, p) in a region D ⊆ M at which a transformation to the local generalized coordinate system H0 (q, p, J) is non singular. The energymomentum map assigns to each point of the phase space (q, p, J) a point in the energy-momentum space (h = H0 (q, p, J), J). The EnergyMomentum bifurcation diagram (EMBD) is a plot in the (h, J) space (for (h, J) in the range of D) which includes: The region(s) of allowed motion (the closure of all regions in which the energy-momentum mapping is a trivial fibre bundle, see [1], [35]).

201


The singular surfaces (h, J) = (H0 (pf ), Jf ) (see equation (15)) where the normal stability of the corresponding singular (n − 1)tori, defined by equation (16), is indicated. The strongest resonance surfaces on which the inner frequencies of the tori vanish, ωi = 0 (and possibly the regions in which back-flow occurs, where dφi (q,p,J) changes sign along the level set dt (h = H0 (q, p, J), J) for some i = 1, ..., n − 1). The energies atwhichtopologicalbifurcations occur and the branched surfaces (Fomenko graphs if n = 2) in the intervals separated by these bifurcation points. The energy-momentum bifurcation diagram depends on the choice of the generalized action-angle co-ordinates (q, p, J), see [27] for discussion. In particular, the form of the perturbation determines what are the strongest resonant directions, and the actions in the EMBD are chosen accordingly. Here, eq. (9) implies that the dominant resonant direction is indeed the conjugate angle to I, in both the (x, y, I) and the (u, v, I) co-ordinate systems. Hence, the convenient coordinates for the energy-momentum bifurcation diagram is (h, I). Notice that while the momentum variable I is globally defined, the associated angle coordinate is defined differently near the plane wave circles (b = 0) and near the symmetric mode circles (c = 0). The EMBD contains the resonance information for both representations.

3.2

Construction of the EMBD

Calculation of the singular surfaces and the normal stability of the lower dimensional tori are the first steps in depicting the global structure of the energy surfaces. We begin the construction of the EMBD by plotting the singular surfaces (H0 (pf (I)), I) in the (h, I) plane, where (pf (I)) are given by the six families of Table 1: H0 (xf , yf , I) Evaluation

Exist For 2

1. H(xpw , ypw , I) = H(0, 0, I) = ( I2 − I)

I≥0

2. H(usm , vsm , I) = H(0, 0, I) = 34 I 2 − Ik 2 − I

I≥0

± 3. H(x± pwm , ypwm , I) =

15 2 I 14

I ≥ 12 k2

± 4. H(u± smm , vsmm , I) =

I2 2

− (1 + 47 k 2 )I + 17 k4

− I − k4

Table 3

I ≥ 2k 2

202 In figure 1 we plot these curves for the non-dimensional wave number k = 1.025, which is the value used in previous works [4, 2, 8, 5, 3, 6]. Other values of k are presented in Appendix B. We use the usual convention in bifurcation diagrams by which normally stable circles are denoted by solid lines whereas normally hyperbolic circles are denoted by dashed lines (see Table 2). Different colors are used for the different families of invariant circles (Thick and thin black line7 for the plane wave and its bifurcating branch and thick and thin grey line8 for the symmetric mode and its bifurcation branch). The allowed region of motion is shaded - for each point (h, I) in this shaded region there are (c, b) values satisfying H0 (c, b) = h, I = 12 (|c|2 + |b|2 ). An energy surface in this diagram is represented by the intersection of a vertical line with the allowed region of motion. The topology of the level sets for different I values on a given energy surface is represented by the Fomenko graphs as described next.

Figure 1. EMBD graph for k = 1.025. Thick black line- ppw , Thin black line -ppwm Thick grey line-psm , Thin grey line -psmm

3.3

Fomenko graphs.

The Fomenko graphs are constructed by assigning to each connected component of the level sets (on the given energy surface) a point on the graph, so there is a one-to-one correspondence between them. In this way the topological changes of the level sets are discovered and the energy surface may be reconstructed from these graphs. 7 blue

and red and green

8 magenta


Figure 2.

203

Fomenko graphs for k = 1.025.

Consider for example figures 1, 2 and 3; Figure 1 is the energymomentum bifurcation diagram for the truncated NLS model at k = 1.025. The numbered vertical lines on this figure indicate energy values for which the Fomenko graphs where constructed as shown in figure 2. Finally, the third figure shows the energy surface (modulus the angle circle) at the energy level corresponding to line 5 in figure 1 and to diagram 5 in figure 2. The energy surface is plotted twice; it is the two dimensional surface in the (x, y, I) space (respectively (u, v, I) space) multiplied, for all c = 0 (for all b = 0), by the circle γ ∈ S 1 (θ ∈ S 1 ). The redundant presentation in the (u, v, I) space is shown to better explain the level sets topology near the circle c = 0 where the transformation to the (x, y, I) co-ordinates is singular. On the Fomenko graphs, we denote the invariant circles corresponding to the plane wave family (ppw ) and the invariant circles which emanate from them (p± pwm ), by open and full triangles respectively (for clarity, the boundary of the triangle is dashed when it is normally hyperbolic and full when it is normally elliptic). The invariant circles corresponding to the symmetric mode family (psm ) and the invariant circles which emanate from them (p± smm ), are denoted by open and full circles, again with the usual convention for the stability. Let us describe in details the level sets topology on this energy surface as the action I is increased. The lowest I value corresponds to the intersection of line 5 with the solid grey line in figure 1, to the open circle on the Fomenko graph and to the lowest level set in figure 3√- the symmetric mode circle. The level set here is the circle c = 0, |b| = 2Ism which is normally elliptic. It is represented in the (u, v) plane as a point - the origin - which is multiplied by the circle in θ (the representation in the (x, y) plane is singular here). For a bit larger I values each level set is composed of one torus - the Fomenko graph has a single edge for

204 such values of I and indeed we see that in both the (x, y) plane and the (u, v) plane a single circle, corresponding to a torus, appears. When the I value reaches the dashed black line in figure 1 the level set becomes singular - it is composed of the plane wave circle and its homoclinic surfaces, shown as a figure-eight level set in the (x, y) plane. On the three dimensional energy surface this figure eight is multiplied by the circle γ ∈ S 1 . This singular level set is denoted by the open triangle with dashed boundary in the Fomenko graph. As I is increased beyond this I value, each point in the (h, I) plane has two tori associated with it - in the Fomenko graph we see that there are two edges for these values of I, and the corresponding level sets at the (x, y) plane have two disconnected circles. These circles shrink to two points which are two normally elliptic invariant circles of the plane-wave-mixed-mode type at a critical I value at which line 5 is tangent to the curve corresponding to ppwm in figure 1; For energies above line 5 the energy surface splits to two due to this curve. Thus this value of the energy is an energy bifurcation value - the level sets for lower energies (diagram 4) and higher energies (diagram 6) undergo different topological changes as the action is increased along the energy surface (graphs corresponding to such energy bifurcation values are denoted here by * in the Fomenko graph sequences). The two circles p± pwm are denoted by the solid triangles in the Fomenko graph. Further increase of I leads us again to the two tori situation until the curve ppw is intersected again. Then the two tori coalesce at the singular level set of the plane wave and its homoclinics which is denoted as before by an open triangle. Further increase of I leaves us with one connected component of the level sets until the dashed grey line in the EMBD, which denotes the normally hyperbolic circles psm , is intersected. This singular level set is again, topologically, a figure-eight times a circle, but now it is represented in the u, v plane (since the (x, y) coordinates are singular here). It is denoted in the Fomenko grpah by an open circle from which, for larger I values, two edges emanate. These correspond to the two tori which oscillate near the two symmetric-mode-mixed-mode circles. The upper boundary of the energy surface is reached when these two tori shrink to the corresponding invariant circles - when line 5 intersected the thin grey line - when the two solid circles in the Fomenko graph are reached. This rather lengthy explanation can be now repeated for each Fomenko graph without the explicit computation of the corresponding energy surfaces. Namely, these graphs encode all needed information for the reconstruction of the energy surfaces. We note that a similar construction for some n d.o.f. systems has been recently suggested (see [27] and references therein).


205

Figure 3. Corresponding energy surface for Fomenko graph 5 of figure 2. Level sets marked by black thick lines correspond to vertices in the graph.

4.

Hierarchy of bifurcations

Given an integrable family of Hamiltonian systems H0 (q, p, J; µ) depending on the vector of parameters µ there is a hierarchy of bifurcation scenarios: Single energy surface. The first level consists of the values of the constants of motion across which the topology of the level sets on a given energy surface H0 (q, p, J; µ) = h is changed. These are the values at which the singularity surfaces cross the vertical hyperplane H0 = h on the EMBD, and correspond to the vertices in the Fomenko graphs. Energy bifurcation values. The second level consists of the energy bifurcation values hb at which the form of the Fomenko graph changes, namely across which the energy surfaces are no longer equivalent. Thus, it describes how the energy surface geometry is changed with h. Parameter dependence of the energy bifurcation values. The third level consists of the bifurcating parameter values µb at which the bifurcation sequence of the second level changes (by either chang-

206 ing the order of the energy bifurcating values or by adding/substracting one of the energy bifurcation values). Most previous works have concentrated on the first level alone, by which the topology of level sets on a given energy level are studied. For a large class of systems the Fomenko graphs (and the corresponding branched surfaces in higher dimensions) provide a full description of this level. Appendix B includes a few more representative cases for the truncated NLS model. The second level, which is represented by the EMBD, is discussed next.

4.1

Bifurcating Energy values.

Intersecting the energy-momentum bifurcation diagrams with a vertical line (hyper-surface in the n d.o.f. case) and constructing the corresponding Fomenko graphs (branched surfaces) leads to a full description of a given energy surface. It follows that changes in the topology of the energy surfaces can be easily read off from these diagrams - folds, branchings, intersections and asymptotes of the singularity surfaces may all result in such changes. The dynamical phenomena associated with each of these simplest geometrical features of the singularity surfaces are listed below. We note that a complete classification of all the possible singularities of these singularity surfaces and their dynamical consequences has not been developed yet.

Folds in the singularity surfaces and Resonances. Clearly (see for example figure 1) the energy surfaces change their topology whenever there is a fold in the singularity surfaces. Furthermore, it was established in [27] that folds of non-parabolic singularity surfaces correspond to strong resonance relations for the lower dimensional invariant tori: · dH ∗ (p ) = 0 ⇔ φi = 0, i = 1, ..., n − 1. dJi f p∗ f

In particular, a fold of the singularity surface H0 (q f , pf , Jf ) at the nonparabolic torus (q f , pf , Jf ) implies that this n − 1 dimensional torus is n − 1 resonant, namely it is a torus of fixed points. The normal stability of this torus may be elliptic or hyperbolic. To find a set of bifurcating energies we need to list the extremum of the surfaces H0 (q f , pf , Jf ) for the various singularity manifolds and verify that these are non-degenerate. In Table 4 we list the I values for which folds are created for the six singular surfaces of Table 3. The values of I for which the singular circles are parabolic are listed as well.

207

Energy surfaces and Hierarchies of bifurcations. dH (xf , yf , If ) dI

=0

1. (I − 1) = 0 − k2 − 1 = 0

2.

3 I 2

3.

15 I 7

− 1 − 47 k 2 = 0

4. (I − 1) = 0

I-resonance

I-parabolic

Parabolic Resonance

Irpw = 1

Ippw = 12 k2

kpr−pw =

Ipsm = 2k2

kpr−sm =

Ippw = 12 k2

kpr−pw =

Ip4 = 2k2

kpr−sm =

Irsm =

2k 2 +2 3

Irpwm =

4k2 +7 15

Irsmm = 1

√ 2 ,

1 2

√ 2 ,

1 2

Table 4 Using the resonant I values of Table 4 in Table 3 we conclude that the following energy values correspond to bifurcations due to the resonances/folds: 1 hpw r =− , 2 1 4 2 2 1 hsm r =− k − k − , 3 3 3 √ 1 4 4 2 7 pwm hr = k − k − , for k < 2 15 15 30 1 1 hsmm = − − k4 , for k < √ r 2 2

(17)

At each of these energies the corresponding family of circles (for example ppw ) has a circle of fixed points (the open triangle in diagram 3 * in figure 2, which corresponds to a normally hyperbolic circle of fixed points giving rise to hyperbolic resonance under perturbations); for energies below the bifurcating energy (say for h < hpw r , see diagrams 1 and 2 there) the energy surfaces do not include any circle of this family whereas for energies beyond this value (say for h > hpw r , diagram 4-10 there) two circles of this family appear on the same energy surface.

Branching surfaces and parabolic circles. Another source for bifurcations in the energy surface structure appears when the singularity surface splits. For the two degree of freedom case such a splitting is associated with the appearance of a parabolic circle (for the n d.o.f. case we look for a fold in the surface of parabolic tori, namely we look for an n − 2 resonant parabolic n − 1 tori, see [27] for precise statement). Thus, the appearance of the parabolic circle psm at h = hsm p from which ± the branches of circles psmm emerge implies that for energies below this value (graph 1 in figure 2) no such circles appear, and the Fomenko graph

208 has no splitting to two branches whereas larger energies have these two circles as the upper boundary of the energy surface (diagrams 2-10). In Table 4 we list the parabolic values of I. Plugging these values in Table 3 we find two additional values of energy bifurcations which appear due to singularity surface branchings: 1 2 1 2 hpw p = k ( k − 1), 2 4

4 2 hsm p = k − 2k

(18)

Singular surfaces crossings and Global bifurcation. A third possible source for topological changes in the energy surface is the crossing of singular surfaces. Such an intersection of singular surfaces of n − 1 dimensional invariant tori can be a result of one of the following phenomena: 1 Appearance of a higher dimensional singularity, namely an n − 2 invariant torus. 2 Appearance of a global bifurcation - e.g. the creation of heteroclinic connection between two families of n−1 normally hyperbolic families. 3 Appearance of two disconnected and unrelated singular level sets for the same action and energy values. Each one of these phenomenon appears to be persistent under C r integrable perturbation with r > 2. The first and second cases imply that the corresponding energy is a bifurcating energy value, whereas the third does not. For our example, it follows from Table 3 that the two curves (H(ppw (I)), I) and (H(psm (I)), I) cross at I = 0 and at I = Igb = 4k2 and that no other singularity curves cross. I = 0 corresponds to the trivial solution c = b = 0, at which both singularity curves are normally elliptic and at their intersection we have a 4d elliptic point - an n − 2 singular level set, as in the first case above. Thus the corresponding energy, h0 = 0, is an energy bifurcation value. Igb = 4k 2 corresponds to the intersection of the two singularity curves (of ppw and psm ) at a value for which both families are normally hyperbolic (since Igb > Ipsm = 2k2 > Ippw = 12 k2 for all k > 0, see Table 2). Indeed, at this value our system admits four heteroclinic connections between the plane wave circles and the symmetric mode circles (see [22]). The energy-momentum bifurcation diagrams (see figure 1) show the intersection between the corresponding singularity curves (dashed grey and dashed black). The Fomenko graphs (graphs 8,9,10 in figure 2) demonstrate that a global bifurcation must occur - the solid circles (denoting p± smm ) are connected to the open circle (denoting psm and its


209

homoclinic orbits) in graph 8 and to the open triangle (denoting ppw and its homoclinic orbits) in graph 10. Hence, this intersection corresponds to a global bifurcation and the corresponding energy is an energy bifurcation value. Summarizing, we find two additional energy bifurcation values which appear from singularity surfaces crossings: h0 = 0,

hgb = 4k 2 (2k2 − 1)

(19)

Unbounded singularity surfaces. Another possible source for an energy bifurcation value is the appearance of a critical energy at which one of the singularity surfaces tends to infinity (i.e. (H0 (pf ), J1 , .., Jn−1 ) → c c c (hc , J1c , ..., Jj−1 , ∞, Jj+1 , ..Jn−1 ), with possibly more than one infinite direction, see [35]). In this case energy surfaces become unbounded in the Jj direction after this critical energy value. This possibility does not appear in our model and may be associated with the existence of unbounded level sets.

4.2

Bifurcation diagram of the EMBD.

We are now ready to describe the third level of the bifurcation hierarchy - the dependence of the EMBD’s on the parameter of the problem, the wave number k. Equations (17,18,19) include the eight energy bifurcation values for our model. At these values of energies the energy surface structure changes. Hence, the order of these bifurcation values determines the sequence of changes of the energy surface topology. In particular, at k values for which pairs of these curves cross, degenerate bifurcation occur, and the bifurcation sequence of the Fomenko graphs changes. Figure 4 shows the graph of the eight curves pw sm sm pwm smm hpw , hr , hgb , h0 as a function of k. This is a bir , hp , hr , hp , hr furcation diagram of the energy bifurcation values - crossings of curves in this diagram correspond to bifurcations of the EMBD’s. The emerging picture is complicated - there are 13 intersections of these curves, so a complete description of the truncated NLS model consists of 14 different EMBD figures and their corresponding Fomenko graph sequences. A few representative ones are shown in Appendix B. Let us discuss the dynamical consequences of a few of these crossings.

Parabolic Resonances. When the curve corresponding to a fold (circle of fixed points), and the curve corresponding to the parabolic circles intersect, a parabolic circle √ of fixed points is created. Indeed, at the critical value k = kpr−pw = 2 (respectively at k = kpr−sm = √12 ) the plane wave family, b = 0, (respectively the symmetric mode family, c = 0) possesses a parabolic resonant circle at Ipr = 1; at this value of the

210

Figure 4.

Bifurcation diagram of the energy bifurcation values.

pw pwm parameter three bifurcating energy curves intersect hpw r = hp = hr sm sm smm (similarly, at k = kpr−sm , hr = hp = hr ), see figure 4. The corresponding EMBD has therefor a fold occurring exactly at the point at which the singularity curve changes from solid to a dashed line. The appearance of parabolic resonances gives rise to trajectories which have different characteristics than trajectories appearing in 1.5 d.o.f. systems as shown below for our model. In general, it is observed (see [34]) that large instabilities occur near parabolic resonances when additional degeneracies occur - when the curvature of one of the branches at the parabolic resonant points approaches zero and a near flat PR appears (see [26, 25] for the higher dimensional formulation and examples): d2 H0 (x(J), y(J), J) → 0. 2 dJ pf →ppr f

211


Here, we find that

d2 H (x(I), y(I), I) dI 2 0 {p

± ± pw ,psm ,ppwm ,psmm }

= {1, 32 , 15 7 , 1},

namely these are fixed non-vanishing numbers. Hence, we conclude that the instability mechanism associated with the near-flat resonance does not exist in this model. It follows that an introduction of additional parameter which controls, for example, the mixed terms in the Hamiltonian H0 (x, y, I) can alter this property and induce strong instabilities.

Figure 5.

Parabolic resonance near b = 0 with H1 ∝ c − c∗ , k =

√

2

We present simulations near the two parabolic √ √ showing the instabilities resonant circles at k = 2 and k = 1/ 2 respectively. The instabilities are similar to those observed for other PR systems, in the regime which is non-degenerate. Notice that near the symmetric mode circle the perturbation term is small since the perturbed Hamiltonian is proportional to c. In figures 6 and 8 we add a perturbation which is proportional to b, and then one observes a much more pronounced instability. Such a perturbation corresponds to adding a first Harmonic term to the perturbation term, namely adding to the Sine-Gordon equation a perturbation of the form εΓ1 cos( 2πn L x) cos(ωt) which corresponds to considering in the truncated two mode Hamiltonian model the perturbation: i Γ1 i H2 (c, c∗ , b, b∗ ) = − √ (c − c∗ ) − √ (b − b∗ ). 2 2

(20)

In the figures we use the appropriate co-ordinate system for each case - the (x, y, I, γ) system near the b = 0 circle (figures 5 and 6 and the

212

Figure 6.

Parabolic resonance near b = 0 with H1 ∝ (c − c∗ ) + (b − b∗ ), k =

√ 2

(u, v, I, θ) system near the c = 0 circle. The projections of the trajectory on the energy-momentum bifurcation diagram are shown as well these demonstrate that the singularity surfaces dominate the perturbed motion.

Figure 7.

√ Parabolic resonance near c = 0 with H1 ∝ (c − c∗ ), k = 1/ 2


Figure 8.

213

√ Parabolic resonance near c = 0 with H1 ∝ (c − c∗ ) + (b − b∗ ), k = 1/ 2

Resonant global bifurcation. When the global bifurcation curve and the curve corresponding to a fold (circle of fixed points) intersect, a heteroclinic connection between an invariant hyperbolic circle of fixed points and an invariant hyperbolic circle is created. Such intersections sm occur when: hgb = hpw r and when hgb = hr . Simple calculation shows 1 that these scenarios occur at k = 2 and k = √15 respectively: Irpw = 1 = Igb = 4k 2 ⇔ k = Irsm =

1 2

2k 2 + 2 1 = Igb = 4k 2 ⇔ k = √ = .4472... 3 5

In fact, as is seen from figure 4, and may be easily verified at k = 12 sm (respectively at k = √15 ) the curves hgb and hpw r (respectively hr ) are tangent. It implies that for k values in the range (.4472, .5) near resonant behavior of both circles involved in the global bifurcations are expected if is not too small. Geometrically, at these values of k the unperturbed system has a circle of fixed points (at ppw (Irgb−pw ) and psm (Irgb−sm ) respectively, see Table 5) which has four families of heteroclinic connections to a periodic orbit (at psm (Irgb−pw ) and ppw (Irgb−sm ) respectively). The behavior of such a structure under small perturbations has not been analyzed yet to the best of our knowledge. Simulations near these two values reveal

214 an intriguing picture of instability which are not well understood yet. Below a representative simulation is presented.

Figure 9.

Global resonant bifurcation - k =

1 2

Other crossings. Notice that several other crossings exist - these do imply topological changes on the sequences of the Fomenko graphs but do not imply that the local qualitative behavior of solutions will be altered. For example, the global bifurcation energy and the parabolic bifurcation energy of the two corresponding circles cross when hgb = hpw p and when hgb = hsm p . However, it is immediately seen that the I values at which the global bifurcations occurs (Igb = 4k2 ) and the I values at which parabolicity appears (Ippw = 12 k2 , Ipsm = 2k 2 ) are well separated for all k values which are bounded away from 0. Hence the dynamics associated with these two phenomena appear on separate phase space regions and the coincidence of these two energy bifurcation values is not dynamically significant. Finally, at k = 0 many of the energy bifurcation curves cross. Indeed, in the limit of small k we expect quite a complicated behavior as many of the bifurcations occur for very close by I values and the curvature of all the curves in the EMBD are quite small. As we have mentioned small curvature means degeneracies and strongest possible instabilities. However, it appears that here all these phenomena are associated with small I values.


5.

215

Conclusions.

The analysis of the truncated model introduced by Bishop et al. two decades ago revealed two types of instabilities which have not been previously observed nor analyzed in this context - the parabolic resonance and the global resonance circles. It is still unknown whether and how these scenarios will appear in the PDE model - hopefully these will turn out to produce infinite dimensional analogs as did the hyperbolic resonance scenario. While investigating this particular model we have realized that a complete bifurcation analysis of a family of near integrable Hamiltonian system involves a hierarchy of three levels - the bifurcation of level sets on a given energy surface, the bifurcation of the energy surfaces structure which may occur via several mechanisms (resonances, global bifurcations, resonant parabolic circles and the appearance of unbounded energy surfaces) and finally the bifurcating energies dependence on the parameters. The list of bifurcation values at each level may be quite long. It appears that some are more dynamically significant than others, yet a quantifier for such qualitative observations is still lacking. The appearance of global bifurcations as a persistent phenomenon in higher dimensional system is another intriguing direction of research which we hope to develop in the future.

Acknowledgment Support by the MINERVA foundation is greatly appreciated. This study has started while VRK visited the Courant institute. It is a pleasure to thank David McLaughlin, Jalal Shatah and George Zaslavsky for the stimulating discussions and their hospitality during that visit.

216

Appendix: A It is proved that the perturbed energy surfaces of the truncated NLS model are bounded and that they are close to the unperturbed surfaces. First, notice that the unperturbed energy surfaces are bounded in c, b since all the quartic terms have positive coefficients: H0 =

1 4 3 4 1 2 2 1 1 1 |c| + |b| + |b| |c| (1 + cos(2 arg(bc∗ )) − (1 + k 2 )|b|2 − |c|2 (A.1) 8 16 2 2 2 2

Since H = H0 + εO(|c|, |b|) it follows immediately that the perturbed surfaces are h,ε bounded as well. Denote by ch,ε max , bmax the maximal amplitude of c, b on the energy ∗ ∗ surface H(c, c , b, b ; ε) = h: ∗ ∗ ch,ε max = max{|c| : H(c, c , b, b ; ε) = h} ∗ ∗ bh,ε max = max{|b| : H(c, c , b, b ; ε) = h}

√ 4 h,0 it follows from (A.1) that for h 1, ch,0 max , bmax = O( h). Furthermore, it can be shown, using the form of eq. (6), that for large values of h the system cannot have fixed points. In fact, one can prove the following: C C C C Lemma. There exists an h∗ (k) such that for all h > h∗ (k) C ∇H0 |H0 (c,c∗ ,b,b∗ )=h C = 0. C C C C Proof. Let us find all solutions to C ∇H0 |H0 (c,c∗ ,b,b∗ )=h C = 0. Clearly at c = b = 0 ∇H0 |H0 (c,c∗ ,b,b∗ )=h so h∗ (k) > 0 = H0 (0, 0, 0, 0). Using the non-singular transformation to the (x, y, I, γ) co-ordinates for c = 0, and the non-singular transformation C C C C to the (u, v, I, θ) co-ordinates when b = 0, it follows that C ∇H0 |H0 (c,c∗ ,b,b∗ )=h C = 0 only when the invariant circles of table 1 are circles of fixed points, namely at the resonant I values, I = Ir , of table 4. Plugging these resonant I values in table 3, we find that circles of fixed points appear at the following h values: H0 (pf −res ) = 1 4 4 2 7 1 4 k − 15 k − 30 , − 12 − k4 }. It follows that for all h > 15 k {− 12 , − 13 k4 − 23 k2 − 13 , 15 there are no fixed points on the energy surfaces. In fact, it follows from (6) that for h sufficiently large, for all (c, b) satisfying H0 (c, c∗ , b, b∗ ) = h we have: ∂H0 (c, c∗ , b, b∗ ) ∂H0 (c, c∗ , b, b∗ ) , } ≥ Ch3/4 . max{ ∂b∗ ∂c∗ It follows from the implicit function theorem and the form √ (namely √ of the perturbation since Hi (i = 1, 2) are linear in c, b so that |Hi | < O( 4 h)), that for ε = o( h) ε h,0 ch,ε max = cmax + O( √ ) h ε h,ε h,0 bmax = bmax + O( √ ) h Similarly, by the implicit function theorem, when ∇H0 > const, the solution to the equation H(cε (hε ), cε∗ (hε ), bε (hε ), bε∗ (hε ); ε) = hε = h + O(ε)


217

is ε−close to the solution to this equation with the ε = 0 values. Near the fixed points the perturbed surface is expected to be O(ε1/α ) close to the unperturbed one, where α is the order of the tangency of the singularity curve at the resonance. Note that formally the larger the h the larger is the extent of the energy surface and the larger is the range of unperturbed energy surfaces which we need to consider. However, if h is very large the structure of H0 remains asymptotically unchanged and one can verify that in fact this limit may be studied by rescaling; substituting c b c= √ ,b = √ 4 4 h h leads to:

1 ε ∗ ∗ H0 (c, c∗ , b, b∗ ) = h H0 (c, c∗ , b, b ) + O √ + 3/4 Hi (c, c∗ , b, b ) h h

namely to the near-integrable motion with finite h.

218

Appendix: B A few representative EMBD and Fomenko graphs are presented. In the EMBD the thick (thin) black line corresponds to the plane wave family ppw (the mixed mode emanating from it, ppwm ). The thick (thin) grey line corresponds to the symmetric mode family psm (the mixed mode emanating from it, psmm ). These curves are dashed (full) when the corresponding circle is hyperbolic (elliptic). On the Fomenko graphs, we denote the invariant circles corresponding to the plane wave family (ppw ) and the invariant circles which emanate from them (p± pwm ), by open and full black triangles respectively (for clarity, the boundary of the triangle is dashed when it is normally hyperbolic and full when it is normally elliptic). The invariant circles corresponding to the symmetric mode family (psm ) and the invariant circles which emanate from them (p± smm ), are denoted by open and full grey circles, again with the usual convention for the stability.

, Figure B.1.

EMBD graph for k =

1 . 10

, Figure B.2.

Fomenko graphs figure for k =

1 . 10

219


, Figure B.3.

EMBD graph for k =

9 . 40

, Figure B.4.

Fomenko graphs figure k =

9 . 40

220

, Figure B.5.

EMBD graph for k =

3 . 8

, Figure B.6.

Fomenko graphs figure for k =

3 . 8

221

References

[1] V. I. Arnol’d. Dynamical Systems III, volume 3 of Encyclopedia of Mathematical Sciences. Springer-Verlag, second edition, 1993. [2] A. Bishop, R. Flesch, M. Forest, D. McLaughlin, and E. Overman II. Correlations between chaos in a perturbed sine-gordon equation and a trunctaed model system. SIAM J. Math. Anal., 21(6):1511–1536, 1990. [3] A. Bishop, M. Forest, D. McLaughlin, and E. Overman II. A quasi-periodic route to chaos in a near-integrable pde. Physica D, 23:293–328, 1986. [4] A. Bishop, M. Forest, D. McLaughlin, and E. Overman II. A modal representation of chaotic attractors for the driven, damped pendulum chain. Physics Letters A, 144(1):17–25, 1990. [5] A. Bishop, D. McLaughlin, M. Forest, and E. I. Overman. Quasi-periodic route to chaos in a near-integrable pde: Homoclinic crossings. Phys. Lett. A, 127:335– 340, 1988. [6] A. R. Bishop, K. Fesser, P. S. Lomdahl, W. C. Kerr, M. B. Williams, and S. E. Trullinger. Coherent spatial structure versus time chaos in a perturbed sineGordon system. Phys. Rev. Lett., 50(15):1095–1098, 1983. [7] A. R. Bishop and P. S. Lomdahl. Nonlinear dynamics in driven, damped sineGordon systems. Phys. D, 18(1-3):54–66, 1986. Solitons and coherent structures (Santa Barbara, Calif., 1985). [8] S. Bishop and M. Clifford. The use of manifold tangencies to predict orbits, bifurcations and estimate escape in driven systems. CHAOS SOLITONS & FRACTALS, 7(10):1537–1553, 1996. [9] S. V. Bolotin and D. V. Treschev. Remarks on the definition of hyperbolic tori of Hamiltonian systems. Regul. Chaotic Dyn., 5(4):401–412, 2000. [10] H. W. Broer, G. B. Huitema, and M. B. Sevryuk. Quasi-periodic tori in families of dynamical systems: order amidst chaos, volume 1645 of LNM 1645. Springer Verlag, 1996. [11] D. Cai, D. W. McLaughlin, and K. T. R. McLaughlin. The nonlinear Schrödinger equation as both a PDE and a dynamical system. In Handbook of dynamical . systems, Vol. 2, pages 599–675. North-Holland, Amsterdam, 2002. [12] R. H. Cushman and L. M. Bates. Global Aspects of Classical Integrable Systems Birkhauser Verlag AG, 1997.

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[28] D. W. McLaughlin and J. Shatah. Homoclinic orbits for PDE’s. In Recent advances in partial differential equations, Venice 1996, volume 54 of Proc. Sympos. Appl. Math., pages 281–299. Amer. Math. Soc., Providence, RI, 1998. [29] J. Meiss. Symplectic maps, variational principles, and transport. Rev. of Modern Phys., 64(3):795–848, 1992. [30] N. N. Nehoroˇsev. Action-angle variables, and their generalizations. Moscow Math. Soc., 26:181–198, 1972.

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[31] N. Nekhoroshev. An exponential estimate of the time of stability of nearintegrable Hamiltonian systems. Russian Math. Surveys, 32(6):1–65, 1977. [32] J. P¨ oschel. On elliptic lower dimensional tori in Hamiltonian systems. Math. Z., 202:559–608, 1989. [33] V. Rom-Kedar. Homoclinic tangles—classification and applications. Nonlinearity, 7(2):441–473, 1994. [34] V. Rom-Kedar. Parabolic resonances and instabilities. Chaos, 7(1):148–158, 1997. [35] S. Smale. Topology and mechanics. I. Invent. Math., 10:305–331, 1970. [36] G. M. Zaslavsky. Chaos, fractional kinetics, and anomalous transport. Phys. Rep., 371(6):461–580, 2002.


Phase-Space Semiclassical Analysis. Around Semiclassical Trace Formulae Monique Combescure Laboratoire de Physique Th´ éorique, CNRS - UMR 8627 Universit´ é de Paris XI, Bˆˆ atiment 210, F-91405 ORSAY Cedex, France and IPNL, Bˆâtiment Paul Dirac 4 rue Enrico Fermi,Universit´ é Lyon-1 F.69622 VILLEURBANNE Cedex, France email [email protected] Abstract

(1) (2) (3) (4)

What is semiclassical Analysis ? What is a Trace Formula ? Physical Motivations : “Quantum Chaos” Gutzwiller Trace Formula . (Rigorous Approach)

CLASSICAL WORLD Z = Rn × IRn PHASE SPACE z := (q, p) classical state of a particle q : position or coordinate p : momentum Hamiltonian H =

p2 2m

+ V (q)

Real Phase-Space Evolution : (q, p) → (qt , pt ) = φtH (q, p) φtH preserves Phase-Space volumes Classical Observables A : Z → IR Evolution of classical Observables 225

P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 225–238. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

226 A ◦ φtH Geometrical Tranformations • Phase-space translations 0 1l represents rotations of •, J = −1l 0

π 2

• General symplectic tranformations, ie

QUANTUM WORLD H = L2 (R I n ) Hilbert space of quantum states

ϕ ∈ H “wavefunction” ˆ Q Pˆ := −i∇ self-adjoint Operators ˆ := − 2 ∆ + V (q) Quantum Hamiltonian H 2m Self-Adjoint ˆ

Quantum evolution : UH (t) := e−itH/ ϕ → ϕt := UH (t)ϕ Unitary Operator Quantum Observables Aˆ : Weyl Quantization of A Evolution of Quantum Observables ∗ (t)AU ˆ H (t) Aˆt := UH

227 Unitary Transformations in H ˆ • Tˆ(z) := exp(iZ.Jz/) Weyl-Heisenberg operator • F Fourier Transformation • General operators of metaplectic Group

CLASSICAL WORLD (continued) Preserve symplectic form σ(z, z ) = z.Jz M : symplectic matrix 2n × 2n γ e 1l 0 Example M = 0 e−γ 1l dilation/squeezing transformation Multiplicative Group : M = M1 M2 symplectic CLASSICAL STATE Point z ∈ Z

Probability Distribution in PHASE SPACE : f : Z → IR+ f (q, p) ≥ 0

Z

dzf (z) = 1

228

QUANTUM WORLD (continued) ˆ z) generator of Weyl-Heisenberg Group iσ(Z, B : 2n × 2n real symmetric ˆ ˆ ˆ M = eJB → R(M ) = eiZ.B Z/2

iγ ˆ ˆ ˆ ˆ (Q.P + Pˆ .Q)) D(γ) := exp 2

ˆ 1 )R(M ˆ 2 ) is Group ˆ 1 M2 ) = R(M R(M

COHERENT STATE |z"

WIGNER, HUSIMI distributions ϕ ∈ H, ϕ = 1 Wigner Distribution → Wϕ (q, p) not ≥ 0 in general Husimi Distribution Hϕ (z) := |!ϕ|z"|2 ≥ 0 Z

dzWϕ (z) =

Z

dzHϕ (z) = 1

229

1.

What is Semiclassical Analysis ?

Semiclassical Analysis is the passage to the Limit → 0 in the Quantum description in terms of the Evolution of states or Observables. It allows to better understand how the classical world description arises from a Quantum Reality. However the limit → 0 is a SINGULAR one, and in particular doesn’t commute with the limit of large times t → ∞, where classically long term UNPREDICTIBILITY (“chaos”) manifests itself. It will appear in what follows that Semiclassical Analysis (mathematical/physical) has to deal with estimates of Highly Oscillating Integrals with Small Parameter . In what follows we shall present an illustration of Semiclassical Analysis which is simply an estimate of the Semiclassical Propagation of the so-called coherent states using the material presented in the comparative tableau above. The coherent states, usually denoted ϕz in the mathematical notation, or |z" in the Dirac notation familiar to physicists are : |z" := Tˆ(z)|0" (1.1) where :

|0"(x) := (π)−n/4 exp(−x2 /2)

is the (normalized ) ground state of the Harmonic Oscillator ˆ2 ˆ2 ˆ0 = P + Q H 2 ˆ be the Weyl quantization of a classical symbol, possibly depending Let H on time H(z, t) which satisfies the following assumption : ∃m, M, K > 0 :

(1 + z 2 )−M/2 |∂zγ H(z, t)| ≤ K

∀|γ| ≥ m

(1.2)

uniformly for (z, t) ∈ [−T, T ] × Z such that the classical and quantum evolutions respectively (for the classical symbol and its Weyl quantization resp.) exist for t ∈ [−T, T ]. Then it is well known that the stability around a trajectory of the classical flow z → zt is governed by the following symplectic matrix Ft obeying the linear equation : F˙ = JMt F

(1.3)

230 where Mt is the 2n × 2n Hessian matrix of H at point zt of the classical trajectory : 2 ∂ H (zt , t) (1.4) (Mt )j,k := ∂zj ∂zk j,k is symmetric real, and the initial datum is F (0) ≡ 1l

(1.5)

To this symplectic matrix can be associated a unitary operator in H of ˆ t ). We denote by St the metaplectic group (see tableau above) R(F the classical action along the classical flow z → zt : t ds(q˙s .ps − H(zs , s)) (1.6) St (z) := 0

and we define :

qt .pt − q.p 2 ˆ t )|0" Φ(z, t) := eiδt /Tˆ(zt )R(F δs := St (z) −

(1.7) (1.8)

which is, up to the phase, a squeezed state located around the phasespace point zt with a dispersion governed by the matrix Ft . We can prove the following estimate : Theorem 1 Let H be an Hamiltonian satisfying the assumptions (1.2) and the existence of classical and quantum flows for t ∈ [−T, T ]. Then we have, uniformly for (t, z) ∈ [−T, T ] × Z : √ UH (t, 0)ϕz − Φ(z, t) ≤ Cµ(z, t)P |t| θ(z, t)3 (1.9) P being a constant only depending on M and m, and µ(z, t) := Sup0≤s≤t (1 + |zs |) θ(z, t) := Sup0≤s≤t (trFs∗ Fs )1/2 The estimate (1.9) contains the dependance in t, , z of the semiclassical error term. One hopes that this error remains small when → 0, provided that z belongs to some compact set of phase-space, and |t| is not too large. Typically θ(z, t) etλ

231 where λ is some Lyapunov exponent that expresses the “classical instability” near the classical trajectory. The RHS of equ. (3.36) is therefore O(/2 ) provided 1− |t| < (1.10) log −1 6λ which is typically the Ehrenfest time, up to a factor 1/6 that is probably inessential. For more details and prior references see ?.

2.

What is a Trace Formula ?

ˆ Let H := L2 (R I n ) be the Hilbert space of Quantum States, and H be a selfadjoint quantum Hamiltonian acting on it. It genarates via Schr¨ odinger Equation an unitary quantum evolution operator ˆ U (t) := exp(−itH/)

(2.11)

ˆ is pure point, namely : We assume that the spectrum of H ˆ = {λk } sp(H) k∈N

(2.12)

ˆ is a Then for f sufficiently smooth and decreasing at infinity, f (H) Trace Class Operator in H. (it is enough for this that {f (λk )}k∈N∈ l1 ), and we have : ˆ := f (H) f (λk ) (2.13) k∈N

The Trace Formula is simply : ˆ = something else trf (H)

2.1

(2.14)

First Prototype : The Poisson summation Formula

Under suitable definition of the Fourier Transform f → f˜, we have : +∞ n=−∞

f (n) =

+∞

f˜(2kπ)

(2.15)

k=−∞

which holds true for any f ∈ S(R I n ) (that implies of course that also n f˜ ∈ S(R I )), and which is therefore perfectly symmetric between f, f˜.

232 Why do we mean that it is a Trace Formula ? d Consider the quantum operator in dimension 1 : Pˆ = −i dx acting 2 on L ([0, 2π]), with periodic boundary conditions u(0) = u(2π). It is an unbounded operator, whose spectrum if purely discrete :

sp(Pˆ ) = ∠ Z Therefore the LHS of equ. (4.1) is nothing but trf (Pˆ ). What does the RHS represents physically ? Imagine a classical Hamiltonian H(q, p) := p, where q ∈ [0, 2π]. (Pˆ is actually the Weyl quantization (for = 1) of H in L2 ([0, 2π]) with Periodic Boundary conditions). The associated Hamilton’s equations are q˙ = 1 , p˙ = 0 anf thus : q = t

(mod 2π)

The classical trajectories are thus all closed ie periodic in phase space, and are k-repetitions of the primitive orbit of period 2π, ∀k ∈ ∠ Z . This means that the periods of the classical flow are of the form 2kπ, k ∈ ∠ Z. The summation Poisson Formula thus expresses that the trace of a function of a quantum Hamiltonian of a very peculiar form is the sum on the periodic orbits of the corresponding classical flow of the Fourier Transform of that function taken at the periods of the classical flow.

2.2

Second Prototype : The Harmonic Oscillator in dimension 1 ˆ2 ˆ2 ˆ 0 := P + Q H 2

I ) has spectrum n + acting in L2 (R n ∈ N.

1 2

(2.16)

(here again we let = 1), where

Take f ∈ S([0, +∞]). Then obviously ˆ0) = trf (H

∞ n=0

1 f (n + ) 2

(2.17)

233 Replace f by Tˆ(q, 0)f = f (. + q) in equ. (5.1). It becomes f (n + q) = e2iπkq f˜(2kπ) n∈Z ∠

Thus taking q = 1/2 ∞ n=0

k∈Z ∠

eikπ = (−1)k which yields : 1 f (n + ) = (−1)k f˜(2kπ) 2

(2.18)

k∈Z ∠

Since the classical trajectories of the classical Harmonic Oscillator are of the form q(t) = Asin(t + α) every orbit is therefore periodic, with a period which is a k-repetition of the primitive orbit of period 2π. The periods of the closed orbits of the classical flow are thus {2kπ}k∈Z ∠. We see here a factor (−1)k ≡ e(2k)iπ/2 which is here the first manifestation of a “Maslov Index” 2k. Here again, equ. (2.18) expresses that the trace of the function of a Quantum Hamiltonian can be written as a sum over the periodic orbits of the corresponding classical flow of the Fourier Transform of that function, taken at the periods of the classical flow, up to some factor of the form eσk iπ/2 where σk is the Maslov index of the orbit. The important fact to notice is that “someting else” in the RHS of (2.4) is a sum over the periodic orbits of the classical flow, and that the miracle obtained above in the two Prototypes has a Prolongation in the semiclassical limit, as we shall see in the last Section.

3.

Physical Motivations

Consider as an illustration the case of billiards in IR2 which are bounded domains Ω ∈ IR2 with a boundary denoted ∂Ω which is piecewise regular. Then two a priori not connected sets of problems can be addressed : -on the Classical Side consider a point particle moving inside Ω with constant velocity, and specular reflections on the boundary ∂Ω. We interest ourselves to the “spectrum” of lengths of periodic orbits inside the billiard. How are they distributed ?

234 -on the other hand the equivalent of a Quantum Problem is the study of the Dirichlet Laplacian in Ω. What is its spectrum and how is it distributed for large energies ? Is there a link between these two Problems ? The answer is YES, semiclassically (that means asymptotically in the quantum spectrum). It somehow provides an answer of the famous V. Kac Question : CAN WE HEAR THE SHAPE OF A DRUM ? or of a paraphrased problem raised by C.A. Pillet : CAN WE SEE THE SOUND OF A DRUM ? The energy level density (defined in a distributional sense on suitable test functions) is the following : δ(E − λk ) (3.19) d(E) := k∈N

Coming back to the traditional Schr¨ odinger Hamiltonian, given E we can perform an average of d(E) in a small neighborhood, which leads ¯ to the so-called “mean level density” d(E), and the first question to be addressed is that of finding a semiclassical approximation of it. According to H. Weyl, we get : vol(energy shell ΣE ) ¯ d(E) (3.20) hn where as usual : h := 2π → 0

(3.21)

and where the energy shell is : ΣE := {z : H(z) = E} ˆ where H is the classical symbol of H. The physical intuition for it is that the LHS of (3.20) “counts” the average number of states at energy E, and that one quantum state occupies a volume of phase-space of size hn The next section to be raised is the nature of the quantum fluctua¯ tions of d(E) around d(E). These appear to be different according to whether the classical dynamics generated by H is regular, (integrable),

235 or chaotic. -in the integrable case, trace formulae can be obtained and the semi¯ classical behavior of d(E) − d(E) can be expanded as a “sum” over the invariant tori of the classical flow, as obtained heuristically by BerryTabor (1976), and rigorously by Colin de Verdière (1973) in the framework of compact manifolds. -in the chaotic case as a sum over the unstable periodic of the classical flow, as suggested by Balian-Bloch (1972), and Gutzwiller (1971). Notice that in the two last works cited, the proof is heuristic, the sum over the periodic orbits is divergent, and the correction terms in are omitted. Rigorous proofs have been established and are due to Chazarain 1974, Duistermaat-Guillemin1975, Paul-Uribee 1996, Meinrenken 1992, and Combescure-Ralston-Robert1999.All these proofs, as an assump -tion, take the Gutzwiller Hypothesis that the unstable Periodic Orbits are non-degenerate, or a weaker assumption of “clean flow”, and furthermore violate the beautiful classical/quantum duality of the exact Trace Formulae, in that the test functions ϕ considered are assumed to have compact support in Fourier variable, which automatically truncates the sum over periodic orbits to those for which the period is in the support of ϕ. ˜

4.

Gutzwiller Trace Formula (Rigorous Proof )

The proof we want to present here is the simplest one which makes use of semiclassical estimates of the evolution of coherent states, as presented in Section 1. (Theorem 1.1). The main mathematical trick is the following : −n ˆ ˆ dz!z|A|z" (4.22) trA = h Z

ˆ and thus the following : for any trace class operator A,

ˆ −E H ˆ it(E−H)/ ϕ(t) ˜ dte = tr trϕ ˆ = h−n dteitE/ϕ(t) ˜ dz!z|e−itH/|z" Z

We shall now make precise our assumptions on H :

236 Assumptions (H1) H(, z)

∞

j=0

j H (z) j

Hj : Z → IR ∈ C ∞ (Z)

(H2) H0 bounded from below and 0 ≤ H0 (z) + γ0 ≤ C(H0 (z ) + γ0 )(1 + |z − z |)M |∂zγ Hj (z)| ≤ C(H0 (z) + γ0 )

(H3) ∀j ∈ N, ∀γ ∈ N2n (H4) |∂zγ (H(, z) − formly for z ∈ Z

∀z, z ∈ Z

N 0

j Hj (z))| ≤ C(N, γ)N +1

∀ ∈ (0, 1) uni-

(H5) Let Icl :=]λ− , λ+ [ be an interval of classical energy, then H0−1 (Icl ) is bounded in IR2n (the energy surfaces for E ∈ Icl are all compact) (H6) E is a noncritical value for H0 namely H0 (z) = E ⇒ ∇z H0 = 0 We can prove the following result : ˆ its Weyl quantization. Theorem 2 Let H satisfy (H1)-(H6), and H Consider the classical flow φtH0 of H0 on ΣE : {H0 (z) = E}, and denote by γ the periodic orbits of this flow. We assume in addition (Gutzwiller Hypothesis) that the Poincaré map Pγ dosn’t have 1 as eigenvalue, namely that the orbits γ are nondegenerate. Then for any ϕ ∈ S : ϕ˜ ∈ C0∞ , we have, asymptotically as → 0 :  

ˆ H −E trϕ (4.23) j c0,j (ϕ) ˜ h−n ϕ(0)|Σ ˜ E | + j≥2

  ∗ eiSγ / + iσγ π/2 Tγ ϕ(T ˜ γ ) e−i γ H1 + + j cγ,j (ϕ) ˜ 1/2 2π | det(1 l − P ) | γ γ j≥1 where : γ ∗ is the primitive orbit corresponding to γ Tγ (resp. Tγ∗ ) is the period of γ (resp. γ ∗ ) σγ is the Maslov index of γ Sγ is the classical action along γ Pγ is the Poincaré map of γ c0,j are distributions supported in {0} cγ,j are distributions supported in {Tγ }

237 Note that the duality Quantum/Classical is violated by the condition that ϕ˜ is of compact support. The First line of (5.7) corresponds to the “regular part” in , and we recognize the dominant Weyl term (with corrections in that Gutzwiller had “omitted”, or neglected...) The Second line corresponds to the “oscillating part” in , which is, as expected, a sum over the periodic orbits of the classical flow (here truncated since ϕ(T ˜ γ ) = 0 for γ large).

238

References

Balian R., and Bloch C. ; Distribution of eigenfrequencies for the wave equation in a finite domain, Ann. Phys. 69, vol.1, 76-160, (1972) Berry M. and Tabor, M. : Closed orbits and the regular bound spectrum , Proc. Roy .Soc. Lond, A349, 101-23, (1976) Chazarain J. : Formule de Poisson pour les variétés Riemanniennes. Inv. Math. 24,6582, (1974) Colin de Verdière Y. : Spectre du Laplacien et longueurs des géodésiques périodiques I, Compos. Math. 27, 83-106, (1973) Combescure M., Ralston J., and Robert D. : A Proof of the Gutzwiller Semiclassical Trace Formula using Coherent States decomposition, Commun. Math. Phys. 202, 463-480, (1999) Combescure M., and Robert D. : Semiclassical spreading of quantum wavepackets and applications near unstable fixed points oif the classical flow, Asymptotic Anal. 14, 377-404, (1997) Duistermaat J. J., and Guillemin V. : The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29, 39-79, (1975) Gutzwiller M., Periodic orbits and the classical quantization condition, J. Math. Phys. 12, 343-358 (1971) Meinrenken E. : Semiclassical principal symbols and the Gutzwiller’s trace formula, Reports on Math. Phys. 31, 279-295, (1992) Paul T. and Uribe A. : The semiclassical trace formula and propagation of wavepackets, J. Funct. Anal. 132, 192-249 (1996)

ATOM-OPTICS BILLIARDS Non-linear dynamics with cold atoms in optical traps Ariel Kaplan, Mikkel Andersen, Nir Friedman and Nir Davidson Department of Physics of Complex Systems Weizmann Isntitute of Science, Rehovot, Israel [email protected]

Abstract

We developed a new experimental system (the “atom-optics billiard”) and demonstrated chaotic and regular dynamics of cold, optically trapped atoms. We show that the softness of the walls and additional optical potentials can be used to manipulate the structure of phase space.

Keywords: Dipole traps, Cold atoms, Chaotic dynamics, Soft walls

1.

Introduction

The possibility of using lasers to manipulate, cool and trap neutral atoms has revolutionized many areas of atomic physics, and opened new research fields. In particular, cold atomic samples trapped in far-detuned optical traps are an ideal starting point for further experimental work, since the atoms can be confined to a nearly perturbation-free environment, Doppler shifts are almost suppressed, and long interaction times are possible (Grimm et al., 2000). Cold atoms have been used before in non-linear dynamics experiments. First, by confining atoms to the minima of a time-dependent amplitude (or phase) periodic optical potential, effects such as dynamical localization were observed for ultra-cold sodium atoms (Raizen, 1999). More recently, a similar system was used to observe chaos-assisted tunnelling (Steck et al., 2001; Hensinger et al., 2001). The “billiard” (a particle moving in a closed region of coordinate space and scattering elastically from the surrounding boundary) is a well known paradigm of non-linear dynamics and much theoretical work has been done about its properties. An interesting experimental manifestation of classical billiard dynamics is the observation of conductance fluctuations in semiconductor microstructures in the coherent ballistic 239 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 239–267. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

240 regime (Jalabert et al., 1990; Marcus et al., 1992; Baranger and Westervelt, 1998). The structure of this fluctuations is affected by the classical dynamics of the electrons. Chaotic dynamics were also exploited to construct lasers with a very high power directional emission (Gmachl et al., 1998). We present a new experimental system, the “atom-optics billiard” in which a rapidly scanning and tightly focused laser beam creates a time-averaged quasi-static potential for ultra-cold Rb atoms (Friedman et al., 2000). By controlling the deflection angles of the laser beam, we create various billiard shapes. As opposed to textbook billiards and other experimental realizations, the particles in our system have an internal quantum structure. The internal degrees of freedom are coupled to the external ones, and spectroscopy can serve as a sensitive probe for the dynamics (Andersen et al., 2003). In this lectures, we review several classical non-linear dynamics experiments in atom-optics billiards. The dynamics of the atoms confined by such billiards is studied by measuring the decay in the number of confined atoms through a small hole on the boundary. We demonstrate the existence of regular and chaotic motion, and study the effects of scattering by impurities on the dynamics (Friedman et al., 2001). Next, we study the changes in the structure of phase-space induced in a chaotic billiard when the billiard’s boundaries are soft and find that in a chaotic Bunimovich stadium soft walls induce islands of stability in phase-space and greatly affect the system as a whole (Kaplan et al., 2001; Kaplan et al., 2004). Finally, we find that introducing additional potentials can stabilize an otherwise chaotic billiard, or alternatively destabilize an otherwise regular one, by inducing curvature in the particles trajectories between bounces from the billiards walls (Andersen et al., 2002).

2.

Optical Dipole Traps

The interaction of a neutral atom and a light field is governed by the dipole interaction, and is usually separated into two terms which correspond to a reactive force and a dissipative force. When an atom is induces a dipole moment exposed to light, the electric field component E d in the atom, oscillating at the driving light frequency. The amplitude of the dipole moment is related to the amplitude of the field by d = αE, where α is the atomic complex polarizability, which is a function of the driving frequency. The interaction of the induced potential with the driving field gives rise to the potential:

241

Atom-Optics Billiards

Udip = −

1 D E 1 d·E =− Re(α)I(r), 2 20 c

(1)

where I = 20 c |E|2 is the light intensity. The reactive “dipole” force is a conservative one, and is equal to the gradient of the above potential. The dissipative force is related to the power the oscillating dipole absorbs from the field, which is given by: D E ω ˙ = Pabs = d · E Im(α)I(r ). 0 c

(2)

In a quantum picture, the dipole force results from absorption of a photon from the field followed by stimulated emission of a photon into a different mode of the laser field. The momentum transfer is the vector difference between the momenta of the absorbed and emitted photons. The dissipative component has its origin in cycles of absorption of photons, followed by spontaneous emission, in a random direction. Using Eq. 2 we can write then an equation for the rate of spontaneous photon scattering:

Γscatt =

1 Pabs = Im(α)I(r). ω 0 c

(3)

The atomic polarizability can be calculated by using the solutions of the optical Bloch equations, while translational degrees of freedom are taken into account (Cohen-Tannoudji, 1990). Using this result for a two-level atom, introducing the “Rotating Wave Approximation”, and assuming δ γ, the dipole potential and spontaneous photon scattering rate can be written as:

Udip (r) = Γscat (r) =

3πc2 γ I (r) 2ω03 δ 3πc2 γ 2 I (r) 2ω03 δ

(4) (5)

where ω0 is the energy separation of the atomic levels, δ ≡ ω − ω0 the laser detuning and γ the natural linewidth of the transition. The dissipative part of the atom-light interaction is used for laser cooling of atoms (Cohen-Tannoudji, 1990), a pre-requisite for optical trapping. However, spontaneous photon scattering is in general detrimental to trapped atoms, mainly since it can induce heating and loss.

242 Comparing the expressions for the dipole force and scattering rate, under the above approximations, yields the relationship

Udip δ = , Γscat γ

(6)

which indicates that a trap with an arbitrarily small scattering rate can be achieved by increasing the detuning while maintaining the ratio I/δ. Equations 4 and 5 indicate that if the laser frequency is smaller than the resonance frequency, i.e. δ < 0 (“red-detuning”) the dipole potential is negative and the atoms are attracted by the light field. The minima of the potential is found then at the position of maximum intensity. In the case δ > 0 (“blue detuning”) the minima of the potential is located at the minima of the light intensity. Trapping atoms with optical dipole potentials was first proposed by Letokhov (Letokhov, 1968) and Ashkin (Ashkin, 1970) . Chu and coworkers (Chu et al., 1986), were the first to realize such a trap, trapping about 500 atoms for several seconds using a tightly focused red-detuned beam. Later, a far-of-resonant-trap for Rb atoms was demonstrated (Miller et al., 1993), with a detuning of up to 65 nm, i.e. δ > 5 × 106 γ. In this case, the potential is nearly conservative and spontaneous scattering of photons is greatly reduced. A comprehensive review of the different schemes and applications of such optical dipole traps is presented in (Grimm et al., 2000). In the limiting case where the frequency of the trapping light is much smaller then the atomic resonance, trapping is still possible, practically with no photon scattering (Takekoshi et al., 1995). Such a quasi-electrostatic trap, formed by two crossed CO2 laser beams, was used to create a Bose-Einstein condensate without the use of magnetic traps (Barrett et al., 2001). Apart from using far-off-resonance lasers, the interaction between the light field and the atoms can be reduced by the use of blue-detuned traps, in which atoms are confined mostly in the dark. In the first dark optical trap cold sodium atoms where trapped using two elliptical light sheets which where intersected and formed a ”V” cross-section (Davidson et al., 1995). Confinement was provided by gravity in the vertical direction and by the beams’ divergence in the longitudinal direction. Many other configurations where proposed since (Friedman et al., 2002). Of special importance are traps formed with a single laser beam, which are particularly simple to align, hence it is easier to optimize their properties. The simplicity of these traps also enables easy manipulation and dynamical control of the trapping potential, its size and its shape.

243


Figure 1. Examples of atom-optic Billiards of different shapes. Shown are CCD camera images at the rotating beam focus. From left to right: A circular billiard, an elliptical billiard, a tilted Bunimovich stadium with a hole, and a Sinai billiard with a hole.

3.

Experimental Realization of Atom-Optic Billiards

Friedman and coworkers (Friedman et al., 2000) developed a new dark optical trap for neutral atoms: the ROtating Beam Optical Trap (ROBOT). In the ROBOT a repulsive optical potential is formed by a tightly focused blue-detuned laser beam which is a rapidly scanned using two perpendicular acousto-optic scanners (AOS’s). The instantaneous potential is given by the dipole potential of the laser Gaussian beam: U (x, y, t) = k

4 5 2P 2 2 2 exp −2 (x − x (t)) + (y − y (t)) /w 0 0 0 , πw02

(7)

where (x0 (t) , y0 (t)) is the curve along which the center of the Gaussian laser beam scans, w0 is the laser beam waist, P is the total laser power, 2 γ and k = 3πc . When the x and y scanners perform a sinusoidal and 2ω03 δ cosinusoidal scan, respectively, the time-averaged intensity in the focal plane is a ring with a Gaussian cross-section. The radial potential barrier is then given by 1/2 P 2 , U =k π 2πrw0

(8)

where r is the scanning radius. Several measurements were performed to prove that the potential can be regarded as a time averaged potential (e.g. measurements of the oscillation frequencies of atoms in the trap, and the stability as a function of the scanning frequency).

244 We used the ROBOT to create what we denoted “Atom-Optics Billiards” 1 . By controlling the deflection angles of both AOS’s using arbitrary waveform generators, arbitrary scans are performed independently in the x and y direction, thus creating an arbitrary 2D scan of the beam in the focal plane. We create various billiard shapes, such as an ellipse, a circle and a tilted Bunimovich stadium that confine the atoms in the transverse direction (See Fig. 1), and probe their dynamical properties by measuring their decay curve through a hole in their boundary. The departure of our system from the mathematical definition of a billiard due to interactions, softness of the walls, additional forces acting on the atoms, etc., is probably what makes it interesting, and we will devote most of these lectures to these effects. But first, we would like to explain why, in spite of all the latter, we can treat our optical traps as textbook billiards. Three dimensional structure: The effects we study are two dimensional effects. Fortunately, a large difference exists in timescales for the dynamics in the radial and longitudinal dimensions and enables us to treat the billiards as 2D billiards, provided than our experiments are done in a short enough time. Alternatively, to better approximate a true two-dimensional system, we confine the atoms by a stationary blue-detuned standing wave along the optical axis. Here, the atoms are tightly confined near the node-planes of the standing wave, but move freely in these planes, forming “pancake” shaped traps separated by ∼ 400 nm (half the wavelength of the standing wave laser). Velocity distribution: The trap is loaded from a thermal cloud of atoms, and the trapped atoms have a nearly thermal distribution of velocities with a root-mean-square (RMS) width of ∼ 11vrecoil , and a typical velocity similar to this width (vrecoil is the velocity acquired by an atoms that recoils after absorbing a single photon, 6 mm/s for the D2 line in 85 Rb atoms). To better approximate a mono-energetic case, where all atoms have identical velocity, we expose the atoms, after loading into the billiard, to a short (1.5 µs) pulse of a strong, on-resonance, “pushing” beam perpendicular to the billiard beam and at 45◦ to the vertical axis. Following this pushing beam, the center of the velocity distribution is shifted to 20vrecoil , while the RMS width grows to 12vrecoil . After an additional 50 ms of collisions with the billiard’s boundaries, the direction of the transverse velocity distribution is completely randomized by the time the hole is opened.

1 Optical billiards for atoms were developed also in the Raizen group, simultaneously with our work (See (Milner et al., 2001))


245

Figure 2. Numerical simulation of two dimensional trajectories of Rb atoms in different atom-optics billiards. (a) Elliptical billiard: Only one of the two existing types of trajectories is shown, for which atoms are confined by a hyperbolic caustic, and thus excluded from a certain part of the boundary. (b) Circular billiard: Nearly periodic trajectories demand an increasingly long time to sample all regions on the boundary. (c) “Tilted” stadium: Every atomic trajectory would reach a certain region in the boundary with a comparable time scale.

Gravitation: Typically, the mean gravitational energy in the traps is 12 mgh ≤ 70Erecoil , where h is the vertical size of the billiard and 2 is the recoil energy. After the pushing beam, the Erecoil = 12 mvrecoil atom’s kinetic energy is ∼ 400Erecoil and therefore gravity can be considered a small perturbation. In some cases, this perturbation can be of great importance (as seen in 10.2) but in general it can be neglected. Collisions: In the range of atomic densities realized in the billiard, the mean collision time between atoms is much longer than the experiment time, hence the motion of the atoms between reflections from the walls can be regarded as strictly ballistic. Spontaneous Photon Scattering: Scattering events involve a change in momentum and can cause a diffusion in phase space which is destructive to any dynamical effect. To minimize photon scattering and thus ensure ballistic motion of the atoms and elastic reflections from the billiard potential walls, we use far-detuned laser beams to form the billiard (δ = 0.5 − 1.5 nm > 4 · 104 γ) and standing wave (δ = 4 nm > 3 · 105 γ). We performed also numerical simulations in order to check the above assumption of billiard-like dynamics. The simulations include all our experimental parameters: a Gaussian beam scanning at a finite rate, a thermal ensemble, 3D structure of the beam and atomic cloud, the loading process. Shown in Fig. 2 are typical trajectories of a single atom (the simulation shown here is, for illustrative purposes, a 2D one). The motion of the atom shows the same features of the motion in an

246 ideal billiard: Periodic or quasi-periodic for the elliptical (Fig. 2a) and circular (Fig. 2b) billiard and chaotic for the Bunimovich “stadium”(Fig. 2c). Our experimental procedure is described in (Friedman et al., 2001). Briefly, ∼ 3 × 108 atoms are loaded, during 700 msec, from the vapor cell into a magneto-optical trap (MOT) (Raab et al., 1987). Next, after 47 ms of “weak” and “temporally dark” MOT (Adams et al., 1995), and 3 ms of polarization gradient cooling (PGC) (Dalibard and CohenTannouji, 1989), during which the trapping beam is on, we end up with 0.5 − 1×106 atoms in the billiard with a temperature of ∼ 10µK and a peak density of ∼5×1010 cm−3 . After all the laser beams are shut off (except for the rotating beam), a hole is opened in the billiard potential, by switching off one of the AOS’s for ∼ 1 µs every scan cycle, synchronously with the scan. The number of atoms remaining in the trap is measured using fluorescence detection, with a 100µs pulse resonant with |5S1/2 , F = 3" → |5P3/2 , F = 4". The ratio of the number of trapped atoms with and without the hole, as a function of time, is the main data of our experiments.

4.

Integrable and Chaotic Systems

The billiard is a conservative Hamiltonian system. We discuss here some general features of this class of systems. The equations of motion for an N -dimensional Hamiltonian system are: q˙k =

∂H( q , p) ∂pk

q , p) , p˙k = − ∂H( ∂qk

(9)

with k = 1 . . . N . If there are N independent, “well-behaved” integrals Fi (q, p), that are constant along each trajectory, then the system is called Integrable. For a conservative system, one of this constants is the energy. The trajectory of an integrable system in phase-space lies q , p) = fi where fi are constants. Hence, the in the intersection of Fi ( existence of N integrals of motion implies that the trajectory is confined to an N -dimensonal manyfold, M, in the 2N -dimensonal phase-space. This N -dimensional manyfold does not have necessarily a simple structure, but if the integrals of motion ) are “in * involution”, i.e. their Poisson brackets with each other vanish Fj , Fk = 0, then M has the topology of an N-dimensional torus (Schuster, 1984; Gutzwiller, 1990). A natural way to describe the dynamics is by using a set of coordinates that define (i) on which torus the trajectory takes place, and (ii) a coordinate on the torus. The standard way to do that is by using θ), a canonical transformation to “Action and Angle” coordinates, (I, which are defined as those variables that transform H(q, p) into H(I),

247


i.e. a Hamiltonian which does not depend on one of the new variables. Such a transformation always exists for an integrable system (Gutzwiller, 1990). The equations of motion will be then: I) I˙k = − ∂H( ∂θk = 0

θ˙k =

∂H(I) ∂Ik

, = ωk (I)

(10)

where ω is a constant. The equations are then trivially integrated giving: = I(0) I(t)

θ(t) = ω t + θ(0) .

(11)

I are the actions of the torus, and θ the angles on the torus. Hence, ˙ ω = θ are the “frequencies” on the torus. Close (“periodic”) orbits occur only if the frequencies have a rational ratio, i.e. ωωji = m n with m, n ∈ N, for any i, j = 1 . . . N . For irrational frequency ratios the orbit never repeats itself but approaches any point on the manyfold infinitesimally close in the course of time (we will call this trajectories “quasi-periodic”). Irregular or non-integrable trajectories fill a subspace with grater dimensionality, possibly with the same dimensionality as the whole phase space. Some of this irregular regions of phase space are “stochastic”, meaning that, although they are deterministic, they display an extremely sensitive dependence on the initial conditions and are unpredictable in the long time. The field of nonlinear dynamics is rich in ways to characterize a system which exhibits stochastic dynamics. For example, an “ergodic” system is one in which time averages (over a single trajectory) are equal to ensemble averages. This implies that every single trajectory “samples” the entire phase space. The extreme case of stochasticity are K-systems or systems exhibiting “strong chaos”, which are defined as systems in which initially close orbits separate exponentially. Terms such as Mixing and positive K-entropy are also commonly used. We will not be concerned with this fine-tuned hierarchy and will denote our irregular systems, in a loose way of speaking, “chaotic”. It should be noted, that circular, rectangular and elliptical billiards (which exhibit integrable motion), and Sinai and Bunimovich billiards (in which the dynamics is completely chaotic), are the exemptions in Nature. Most systems have a “mixed” phase space in which regular and chaotic regions coexist.

5.

Poincar´ e Surface of Section

Let us restrict ourselves to a 2-dimensional system. Phase space is four-dimensional, and the H = E surface has three dimensions. A convenient way to visualize the dynamics is by using a “Poincaré Surface

248

s2

ψ2 α2

x2

T2

T1 x1

s=0

α1

s3

ψ1

s1

Figure 3. Two ways to construct a Poincare´ surface of section. (a) Phase space is represented using x (the points at which the orbit crosses the y = 0 line), and px = v cos θ. (b) Bouncing Map: the usual coordinates are the arc length, s, and the tangential momentum p = cos α.

of Section” obtained by recording the successive intersections of a trajectory with an arbitrary plane of section. For example, the x surface of section Sx is the intersection of the energy surface with y = 0 and has (px , x) as coordinates (see Fig. 3a). The value of py is determined by the condition H(x, y = 0, px , py ) = E, up to a sign. If we define it as positive, we see that specifying the position of a system on Sx completely specifies its state. An initial point X0 = (x0 , px0 ) will cross Sx again at X1 = (x1 , px1 ), X2 = (x2 , px2 ) and so on. The resulting map Xi = T (X Xi−1 ) of Sx onto itself (the Poincar´ é map) is a simplified picture of the dynamics. Another common way to construct a Poincare´ section for a billiard (see Fig. 3b), is by looking at its bouncing map, which specifies the evolution of position and momentum from one collision with the boundary to the next one. The position on the boundary can be parameterized either by the arc length s or the direction of the tangent, ψ, and the momentum can be specified using the particle’s direction with respect to the tangent, α, or the tangential momentum, i.e. p ≡ cos α (the coordinates (s, p) are called Birkhoff coordinates). Repeated iterations of the Poincar´ é map reveal whether or not the motion is integrable. In the case of integrable motion, the system explores only a 2D torus, and its intersection with S is a smooth, closed curve, which becomes apparent after enough iterations. If the orbit closes (i.e. the motion is periodic) then Xn = X0 for some n, depending of the ratio ω1 ω2 . The torus in this case is full of such curves, and the curve will appear when the iterations are made for many different starting points X0 . In the more common case of irrational ωω12 the curves will be generated by iterating any initial point X0 a large enough number of iterations. For non-integrable motion tori do not exist, and the system explores a three

249


dimensional region. Its crossings with S will not cover a curve, but a two dimensional region of S.

6.

Decay through a hole

One way to characterize the dynamics in a billiard is by looking at the escape probability of a particle from it through a small hole in its boundary or, alternatively, at the decay in the number of particles in it as a function of time. This decay rate is strongly related to the decay in the system correlations, which has been a topic of numerous studies (See for example (Friedman and Martin, 1984)). A different functional dependence is expected for chaotic and regular billiards. Assume that a billiard with area A contains an ensemble of particles with momentum in the range [p, p + δp], and that there is a small hole, of length l, in the billiard’s boundary. It can be shown (Bauer and Bertsch, 1990) that if the billiard is completely ergodic, the number of particles in it, as a function of time, obeys N˙ (t) = N (t)/τc ,

τc =

πA . lp

(12)

Hence, the decay from a chaotic and ergodic billiard exhibits a universal (i.e. similar for all chaotic billiards) exponential behavior, with a decay time constant τc . The decay from an integrable billiard does not have a universal functional dependence, and differs for different billiard shapes. Nevertheless, power-law decays are very common. It is important to note already at this point that even for “completely chaotic” systems there can be deviations, in the long time tail of the decay curve, from the above exponentiallity. For the Bunimovich billiard, for example, an algebraic tail has been observed, when the hole in not infinitesimally small (Legrand and Sornette, 1990; Alt et al., 1996). This tail is induced by the marginally stable “bouncing ball” orbits, where a particle bouncing back and forth between the opposite straight lines. An orbit near one of these closed orbits, will perform a “zig-zag” path for many bounces (a “resonance”) before getting lost in the chaos and the particles can “stick” for a long time to the vicinity of the boundary of the marginally stable region. The phenomena of “stickiness” will play an important role in 9. Note that in our experiments, we choose to used a “tilted” Bunimovich stadium, in which the two semi-circles have different size and the straight lines are not parallel, in order to preclude this strong position-dependent correlations between bounces (Vivaldi et al., 1983).

250

7.

Chaotic and Integrable Dynamics

The radius of curvature of a circular billiard is constant along the boundary, and the orbit of a particle in it is composed of a succession of chords making angles α with the boundary’s tangent (Berry, 1981). The tangential momentum, p ≡ cos α, is then conserved and the system is integrable. The well-known Bunimovich stadium is composed of two semicircular arcs with radius R joined by straight lines with length L. For any 0 < L < ∞ the system is chaotic (Bunimovich, 1974). As already mentioned above, we use a “tilted” Bunimovich stadium, in which the two semi-circles have different size and the straight lines are not parallel. Comparing the atomic trajectories for the circular billiard (Fig. 2b) and the tilted-stadium billiard (Fig. 2c) illustrates what can be denoted a microscopic effect in phase space. Neither of these shapes has a macroscopic stable region in phase space for a hole at any point on the boundary. Nevertheless, as explained in 6, differences in the decay rate are expected. For the circular billiard nearly periodic trajectories exist that require an arbitrarily long time to sample all regions on the boundary (the exactly periodic trajectories that are completely stable have only a zero measure, and hence can be neglected). This yields many time scales for the decay through a small hole on the boundary, and results in an algebraic decay (Bauer and Bertsch, 1990). For the tilted-stadium billiard, phase space is chaotic resulting in a pure exponential decay, with a characteristic decay time τc (Bauer and Bertsch, 1990). The experimental decay curves for the circular billiard and the tiltedstadium with identical potential height, area and hole length, and with the hole located at the bottom, are shown in Fig. 4. The area of the circular billiard and the stadium were set to be equal, with a precision of ∼ 10%. The initial decay is nearly identical for the two shapes. This is expected since it takes a certain time for the atomic ensemble to “feel” the shape of the billiard. However, for longer times the decay curve for the circular billiard flattens, indicating the existence of nearly stable trajectories, whereas that of the stadium remains a nearly pure exponential. Also shown in Fig. 4 are the results of full numerical simulations that contain no fitting parameters. The simulations include the measured three dimensional atomic and laser-beam distributions, atomic velocity spread, laser beam scanning, and gravity. As seen, the simulations fit the data quite well. For long times the circular billiard simulations predict greater stability than the experimental data. A possible explanation for this difference are imperfections of the billiard shape. Including im-


251

Figure 4. Decay of atoms from circular and stadium billiards: The decay from the stadium billiard (•) shows a nearly pure exponential decay. For the circle () the decay curve flattens, indicating the existence of nearly stable trajectories. The full lines represent numerical simulations, including all the experimental parameters, and no fitting parameters. The dashed line represents exp(-t/τc ), where τc is the escape time calculated for the experimental parameters.

perfections in the beam-shape in the simulations yielded similar effects of reducing the long-time stability of the circular billiard. The deviation of the stadium decay curve from the simulation results, can be explained by the existence of a small amount of atoms trapped in areas far from the focus, where the real intensity distribution deviates from the simulated one to a greater extent. Finally, Fig. 4 also shows a pure exponential decay curve with a time constant τc = 4.9 ms, calculated with the measured billiard parameters. As seen, it closely resembles the full simulation results, indicating that for this configuration, the stadium behaves nearly like an ideal chaotic billiard (two-dimensions, no gravity, mono-energetic, zero wall thickness). Note, that deviations from a pure exponential decay, representing correlations due to the finite hole size, are expected at long times (Alt et al., 1996; Vivaldi et al., 1983) but occur below the noise level of our experiment.

8.

Elliptical Billiard and the Effect of Scattering by Impurities

The motion in an elliptical billiard is integrable (the integral of motion is the product of the angular momenta about the two foci (Berry, 1981)). It can also be shown (Koiller et al., 1996) that if one segment of the trajectory crosses the line connecting the two foci, then every segment will cross it, and hence there are two families of trajectories: First,

252

Figure 5. Decay of the number of atoms from an elliptical billiard: The full symbols denote the unperturbed case, in which the surviving fraction for the ellipse with the hole on the long side () decays much faster than for the hole on the short side (•). The insets show CCD camera images of the billiards’ cross-sections at the beam focus. The empty symbols show the case in which a 10 µs velocity randomizing molasses pulse is applied every 3 ms (see text).

“external” trajectories are those that never cross the line between foci. They bounce all round the billiard, and are confined outside elliptical caustics, smaller than the billiard itself but with the same focal points. The second family, “internal” trajectories, are those that cross the line connecting the two focii. They explore a restricted part of the boundary and are confined by hyperbolic caustics, again with the same focal points. Phase space is then macroscopically divided into two separate regions corresponding to the types of orbits described above (Koiller et al., 1996). The decay in a system with such a non-uniform phase space, is very sensitive to the position of the hole. If it is located at the short side of the ellipse, particles in internal trajectories will remain confined and never reach the hole. Alternatively, all trajectories, excluding a zero-measure amount, reach the vicinity of a hole on the long side of the ellipse and hence the number of confined particles decays indefinitely. Figure 5 shows the measured survival probability for the elliptical billiard with a 60 µm hole on the long side and on the short side. These measurements are taken without a pushing beam. To minimize the effect of gravity, the ellipse is rotated such that the hole’s direction is perpendicular to gravity for both cases (see inset). The results reveal that the initial decay rate (for the first few points) is identical for both cases, as expected, confirming the experimental accuracy of our shapes and holes sizes. However, at longer times, the survival probability for the hole on the short side becomes much higher than for the hole on the long side,


253

Figure 6. The surviving fraction of atoms 150 ms after the hole was opened on the long (2) and short (◦) side of the elliptical billiard of Fig. 5, as a function of the repetition rate of the velocity randomizing molasses pulses. At a rate comparable to the escape rate, τc−1 , the stability of the trajectories was significantly reduced.

as expected from the discussion above. In general, the stability due to such a macroscopic phase-space separation is very robust and remains nearly unchanged with or without the standing wave, for a large variety of hole sizes, for different orientations of the ellipse relative to gravity, and with small distortions of the ellipse’s potential shape. Next, we introduce a controlled amount of scattering to the atomic motion. We expose the confined atoms to a series of 10 µs pulses of PGC (using the six MOT beams) every 3 ms. During each pulse, each atom scatters ∼30 photons, and hence its direction of motion is completely randomized, whereas the total velocity distribution remains statistically unchanged. Moreover, the atoms barely move during each short pulse, and maintain ballistic motion between the pulses. Thus, the effect of the pulses resemble the effect of random scattering from fixed points (impurities) or of short range binary atomic collisions. The measured decay curves for this case are also shown in Fig. 6, for the two hole positions of the ellipse. As seen, for the hole on the long side the randomizing molasses pulses cause little change. However, for the hole on the short side, a complete destruction of the stability occurs, and the decay curves for the two hole positions approximately coincide. This effect is further illustrated in Fig. 6, where the measured survival probability (150 ms after the hole was open) as a function of the repetition rate of the velocity-randomizing pulses, is shown for the two hole positions in the ellipse. As seen, the pulse rate required to significantly reduce the stability is approximately one pulse per τc , calculated as 12 ms here, when averaging over the thermal velocity distribution of the atoms.

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9.

Billiards with Soft Walls

Kolmogorov-Arnold-Moser (KAM) theory provides a framework for the understanding of an integrable Hamiltonian system which is subject to a perturbation (Gutzwiller, 1990). As the strength of the perturbation is increased areas of phase-space become chaotic and eventually only islands of stability, surrounded by a chaotic “sea”, survive. The opposite question is of interest as well: What will happen to a completely chaotic system when a perturbation is applied? will it remain chaotic or will islands of stability appear?. For the billiard system, one such perturbation is making the walls “soft”, as opposed to the infinitely steep potential of the ideal case. This kind of perturbation is also interesting in the context of understanding the origins of statistical mechanics, for which the billiard problem is a widely used paradigm (Zaslavsky, 1999). The Sinai billiard (Sinai, 1970) is mathematically analogous to the motion of two disks on a two-dimensional torus, an approximation for gas molecules in a chamber, and has been proved to be ergodic. However, if the potential between the two disks is smooth, as is the actual potential between gas molecules, it was shown that elliptic periodic orbits exist, hence the system is not ergodic (Donnay, 1999). From the practical point of view, physically realizable potentials are inherently soft, and the softness of the potential may result in a mixed phase-space with a hierarchical structure of islands. This structure greatly affects the transport properties of the system (e.g. induces nonexponential decay of correlations), since trajectories from the chaotic part of phase-space are trapped for long times near the boundary between regular and chaotic motion (Zaslavsky, 1999). As an important example, these considerations were studied in the context of ballistic nanostructures, and found to be the cause for the fractal nature of magnetoconductance fluctuations in quantum dots (Ketzmerick, 1996; Sachrajda et al., 1998). However, in these systems the wall softness is often accompanied by non-ideal effects (such as scattering from impurities) and hence its role is still controversial. For a certain kind of dispersing billiards it was theoretically proven that when the wall becomes soft, an island appears near a singular, tangent trajectory (Turaev and Rom-Kedar, 1998; Rom-Kedar and Turaev, 1999). More recently, and partly inspired by our numerical results, the appearance of an additional island, this time near a corner, was also shown (Turaev and Rom-Kedar, 2003). The first billiard that we study is a tilted Bunimovich stadium (see insets in Fig. 7), composed of two semicircles of different radii (64 µm and 31 µm), connected by two non-parallel straight lines (192 µm long).


255

Figure 7. Experimental results for the decay of cold atoms from a tilted-stadium shaped atom-optics billiard, with two different values for the softness parameter: w0 = 14.5µm (◦), and w0 = 24µm (+), and for two different hole positions. (a) The hole is located inside the big semicircle. The smoothening of the potential wall causes a growth in stability, and a slowing down in the decay curve. (b) The hole includes the singular point, no effect for the change in w0 is seen. Also shown are results of full numerical simulations, with the experimental parameters (see text) and no fitting parameters. The dashed line is e−t/τc , the decay curve for an ideal (hardwall) billiard.The insets show measured cross-sections of the (averaged) intensity of the laser creating the soft-wall billiards, in the beam’s focal plane. The size of the images is 300 × 300µm.

When the potential of the wall becomes softer, a stability region appears around the singular trajectory which connects the points where the big semicircle joins the straight lines. We use the laser beam 1/e2 radius, w0 , to set the softness of the billiard’s walls (See Eq. 7), and experimentally control it by the use of a telescope with a variable magnification, located prior to the AOS’s such that w0 could be changed without affecting the billiard’s size and shape. As seen from Eq. 8, the time averaged potential is kept constant for different values of w0 by changing the laser power. In Fig. 7, experimental results for the decay from a tilted stadium with two different values of the softness parameter (w0 = 14.5 µm and w0 = 24 µm), are presented. It can be seen that when the hole is located entirely inside the big semicircle (Fig. 7a), the soft wall causes an increased stability, and a slowing down in the decay curve. When the hole includes the singular point where the semicircle meets the straight line (Fig. 7b), no effect for the change in w0 is seen. We show below that these results can be explained by the formation of a stable island around the singular trajectory, and a sticky region around it. Figure 7 includes also the results of numerical simulations, which include the three dimensional atomic and laser-beam distributions, atomic velocity spread, laser beam scanning and gravity, and no fitting parameters. As

256

Figure 8. Poincaré surface of section for monoenergetic atoms confined in a tiltedstadium atom-optics billiard with parameters specified in text, and three different values of the softness parameter w0 . (a) w0 = 18µm. A small elliptic island appears close to the trajectory which connects the two singular points. Upper inset: a trajectory in the island. Lower inset: a typical chaotic trajectory in this billiard. (b) w0 = 27µm. Three additional islands appear around the central one, and correspond to the periodic trajectory shown in the upper inset. Around these islands there is a large area of ”stickiness”, where the trajectories spend a long time (see trajectory in lower inset). (c) w0 = 30µm. The three previous islands merge into one large elliptic island (see trajectory in upper inset), with some stickiness around it (lower inset).

can be seen, there is a very good agreement between the simulated and measured decay curves. Similar decay measurements and simulations for a circular atom-optics billiard show no dependence on w0 in the range 14.5 − 24 µm, and no dependence on the hole position. To understand these observations, it is useful to look on how the phase-space of the system changes with changing w0 . In Fig. 8, results of numerical simulations for classical trajectories of Rb atoms inside the tilted-stadium billiard are shown. For clarity, we assume a monoenergetic ensemble (with v = 20vrecoil ), a two-dimensional system, and no gravity. The dimensions of the billiard are equal to the experimental ones. Phase-space information is presented using a Poincaré surface of section, showing vx versus x at every trajectory intersection with the billiard’s symmetry axis (y = 0), provided that vy > 0. In Fig. 8(a), the w0 = 18 µm case is shown. A small elliptic island appears around the trajectory which connects the two singular points, as can be seen also from the upper inset, which shows a trajectory in the island. In the lower inset, a typical chaotic trajectory is shown. For w0 < 12 µm, no islands with area larger than 10−4 of the total phase-space (the resolution of the simulations) are observed. In general, the island size increases with the increase of w0 , as can be seen from Fig. 8(b),(c) which correspond to w0 = 27, 30 µm, respectively. For w0 = 27 µm, three additional islands appear around the central one, and correspond to the periodic trajectory shown in the upper inset.


257

Figure 9. Fraction of remaining atoms at 5τc after the hole is open, as a function of the softness parameter (w0 ), for a hole in the big semicircle. (+): experimental results, (): numerical simulation. The dashed line is e−5 , the expected value for small w0 . Also shown are values for the island () and island + stickiness (•) sizes as a fraction of the phase-space area, calculated from the two-dimensional phase-space simulations.

Around these islands there is a large area of ”stickiness” 2 , where the trajectories spend a long time. Such a ”sticky” trajectory is presented in the lower inset of Fig. 8(b). The exact structure of the island and its vicinity depends on the softness parameter w0 in a sensitive way, as can be seen from the w0 = 30 µm case (Fig. 8(c)), where the three previous islands merge with the central one into one big elliptic island, with some stickiness around it. In Fig. 9, the measured fraction of remaining atoms at 5τc (= 42.5 ms) after a hole is opened in the big semicircle is plotted as a function of the softness parameter, w0 , together with the results of numerical simulations. A very good agreement exists between the decay simulations and the measured data, and both converge to the expected value of e−5 for small w0 . To intuitively understand the origin of the increased stability, the results of numerical simulations, showing the relative area in phase space of the island and the combined area of the island and the “sticky” regions, are also presented. The simulations reveal that the total area size of the island and sticky regions grows monotonically with w0 , in a similar way to the decay results. The island size itself is much smaller, and its size has a non-monotonic dependence on w0 . These facts suggest that the remaining atoms in the experiment, at 5τc , are mainly

2 We

define a ”sticky” trajectory as one which spends inside a box surrounding the island (shown in Fig. 2) a time which is more than ×3 longer than expected for a random trajectory.

258 due to stickiness and demonstrate the important effect of the stickiness on the dynamics of the billiard.

10.

Billiards with Curved Trajectories

If particles in the billiard move in straight lines between bounces from the wall, then the movement between them is neutral, and the dynamics is determined solely by the shape of the boundary. Exponential separation of close trajectories (chaotic dynamics) can only occur if the boundary introduces sufficient instability. But if particles move along curved trajectories between bounces from the wall the dynamics crucially depends on the specific properties of this motion (we call these billiards “curved-trajectory billiards”). Curved trajectory billiards have been found relevant for micro devices (Takagaki and Ploog, 2000), and much effort have been put into studying them. In these studies the curvature of the trajectory arises from a magnetic field (Tasnadi, 1997; Robnik and Berry, 1985; Gutkin, 2001; Dullin, 1998) or from the curvature of the surface on which the billiard is made (Gutkin et al., 1999). Billiards where gravity provides the confining force along its axis have been studied experimentally, theoretically and numerically (Lehtihet and Miller, 1986; Wallis et al., 1992; Hughes et al., 2001; Milner et al., 2001). Theoretical work on billiards where the curvature is caused by additional forces, both a constant force field and a parabolic potential, have been done (Dullin, 1998), and the stability conditions of one-periodic and symmetric two-periodic orbits were found. We investigate classical dynamics of curved-trajectory billiards that would have been hyperbolic if the particle had moved in straight lines. We introduce the curvature by applying external force fields on the particle. We investigate two special cases: (a) a uniform force field (trajectories are sections of parabolas), and (b) a parabolic potential (trajectories are sections of ellipses if the potential is attracting, and sections of hyperbolas if the potential is repulsive). These perturbations can lead to the formation of elliptical orbits in hyperbolic billiards. We experimentally demonstrate these effects using atom-optics billiards.

10.1

Periodic Orbit Stability

A periodic trajectory exists in the billiard if there is a “fixed point” in its Poincaré map, T. In general, the map is nonlinear and cannot be written as a 2×2 matrix. In order to analyze the stability of an orbit, we introduce the “stability matrix”, M , which is a locally linearized version of T , and controls the evolution of an initially infinitesimal deviation from the starting point. Periodic orbits can be either stable or unstable,

259


in the sense that an initial orbit, starting at (s0 + δs0 , p0 + δp0 ), with δs0 and δp0 infinitesimally small, will remain close to the periodic orbit starting at (s0 , p0 ) or become distant from it, when the map is iterated. After N iterations, the periodic orbit returns to its initial value, and the stability of the orbit is determined by the eigenvalues of L ≡ M N , which are given by: 4 51/2 2 1 λ± = 2 tr L ± (tr L) − 4 . (13) If |tr L| < 2 then both eigenvalues have absolute value equal to unity and are complex conjugates of the form λ± = exp [±iβ]. The deviations from the periodic orbit oscillate around zero, but remain bounded. Hence, the orbit is stable or “elliptical”. If |tr L| > 2 then both eigenvalues are real, positive and reciprocals of each other, |λ± | = exp [±γ]. Because of the positive exponent deviations will grow exponentially and the orbit is unstable or “hyperbolic”. We limit ourselves to a two-bounce orbit, with impacts on both sides at normal incidence. In this case, the trace calculation leads to the well known geometrical stability conditions, in terms of ρ, the distance between the two scattering points, and R1 and R2 , the corresponding radii of curvature, defined as positive if the particle scatters from a concave surface: ρ2 1 1 tr L = 4 ρ + 2. (14) −4 + R1 R2 R2 R1 Note, that these are the same condition as geometrical optics gives for the stability of a cavity.

10.2

A Constant Force Field

We first see how the presence of a uniform force field (constant acceleration g) affects the stability. The motion in between bounces is no longer in straight lines, but along sections of a parabola. For simplicity we assume a two-periodic orbit along the force field. We assume that the particle possesses enough kinetic energy to access the entire billiard and further limit ourselves to an orbit which is in the direction of the force field F . In this case the linearized Poincaré map is given by: tr L = 2 − 4

p1 p2 + R2 R1

t+4

p 1 p2 2 t . R1 R2

(15)

where p1 and p2 are the magnitudes of the velocities at encounters with the wall and t is the time it takes the particle to move from one bounce

260

Figure 10. Poincaré surface of sections for classical trajectories of Rb atoms inside the half Bunimovich billiard, with the parameters used in the experiment. (a) with arc up: a large island of stability appears around the vertical two-periodic orbit. (b) with arc down: completely chaotic phase-space. Similar phase-space diagrams with no gravity reveal a similar completely chaotic phase-space. Insets show typical trajectories

to the next. This trace is very similar to the trace without a force field. There are in general four solutions to the equation |tr L| = 2 (just as R1 R2 R1 2 without the force field), namely t = 0, t = R p2 , t = p1 and t = p2 + p1 . Since ρ = − 12 g t2 + p1 t, an interpretation identical to the one without the force field is also valid, provided that instead of the geometrical radii of curvature R, we define a pair of “effective” radii given by: 2 Fi = Ri + 1 g Ri R 2 p2i

i = 1, 2

(16)

Using the effective radius the stability can be found from Eq. 14. Note that these effective radii correspond to a new pair of center points, which are now no longer the geometrical centers but are shifted by the 2 Fi = C + 1 g R2i . The shifted centers correspond to points force field to: C 2 pi where orbits starting on the osculating circle, and perpendicular to it, will cross. In the shifted center picture it is now clear that a uniform force field can make stable islands in a hyperbolic billiard of the Bunimovich kind. We demonstrate this experimentally in half a Bunimovich stadium which is obtained by cutting it with a straight wall along its short symmetry axis (see insets in Fig. 11a). This preserves the hyperbolicity of the billiard without a force field. As a force field we use gravity. The half Bunimovich stadium is placed with gravity along its symmetry axis and the arc up. The expression for the gravity-shifted centers indicates that,


261

Figure 11. (a) The fraction of remaining atoms as a function of time for Half a Bunimovich stadium in a constant force field (gravity). : With the arc up. : Arc down. The existence of a stable island in phase-space causes a slowing down in the decay. Inset: Measured light distribution of the billiard (205 µm high), with side sections removed. (b) Fraction of remaining atoms in a dispersing billiard as a function of time when a nearly-parabolic attractive potential is induced by a standing wave beam of variable power. : 0 mW, : 10 mW, : 20 mW and : 30 mW. The formation of a stable island is seen as a slowing down in the decay. Inset: The billiard.

for atoms with low enough mechanical energy, the center of the upper arc is shifted to outside the billiard, and thereby the orbit becomes elliptical, and an island of stability appears (the center of the lower line is always at infinity). Alternatively if the billiard is turned upside down, i.e. the sign of g is changed, the gravity-shifted center is inside the billiard and it stays hyperbolic. We stress that although the stability of this orbit is determined in the shifted center picture, the dynamics of the entire billiard is not equivalent to the dynamics of a billiard without a force field constructed according to the shifted centers. Indeed, we will see new effects such as mixed phase-space and energy dependent stability, which would not occur in equivalent billiards without a force field. Our half Bunimovich stadium is composed of a semicircle of radius R1 = 158 µm connected to a straight section (R2 = ∞) by two 47 µm straight sections. The velocity regime for which the vertical twoperiodic orbit is stable is 10.5 vrecoil < p2 40 ms. This we attribute to ∼ 1% fraction of slow atoms that do not reach the upper section of the billiard, and are trapped in the lower arc by the principle described in (Wallis et al., 1992). This is verified by observing the same fraction of remaining atoms at t > 40 ms without the upper straight section. When the same was done with the arc up, leaving only the lower straight section, no stability was detected, as expected.

10.3

Parabolic Potential

We next consider what happens if the atoms move in an attractive harmonic potential, so the trajectories in-between scattering from the walls are sections of ellipses. For simplicity we view only what happen to orbits along the radial direction of the potential. The motion in between bounces follows the equation of motion x ¨ = −ω x, and together with the initial conditions x(t = 0) = x1 and px (t = 0) = p1 , we calculate for the trace: 1 p2 p1 p 1 p2 2 − 1 sin ωt − + sin 2ωt . (17) tr L = 2 1 + 2 ω 2 R1 R2 ω R2 R1 The stability can again be found from Eq. 14 using shifted centers or effective radii. The center of the coordinate system is at the center of the potential, and the shifted center will be at3 : , Fi = ±Ci / 1 + (ωRi /pi )2 , C

(18)

where the minus refers to Ri < −p2i / ω 2 |xi | . xi is the position of the scattering point. From the shifted centers we see that, in contrast to the uniform force field, a two-periodic orbit can become elliptical even if R1 and R2 are both negative, since the R corresponding to the shifted centers be can positive even though the geometrical R is negative 2 2 (Ri < −pi / ω |xi | ). This means that islands of stability can occur 3 Defining the shifted center of the osculating circle as the point where orbits starting perpendicular to it and displaced an infinitesimal distance along it cross, leads to that every circle have two centers (orbits are ellipses so they cross the x-axis twice). The stability of the orbit can be determined from one of the centers, and the ± makes sure that it is the correct one that is used.


263

around two-periodic orbits in dispersing billiards and Sinai billiards. It is also easily seen that a potential placed asymmetric on the long symmetry axis in a Bunimovich stadium can change the stability of the orbit. We observed islands of stability induced by an attractive parabolic potential experimentally in a dispersing billiard. The billiard is 220 µm high and consists of four convex arcs with the same curvature for each opposite pair (see inset of Fig. 11b). Two of the arcs have a small radius of R = −130 µm and the other two are very weakly curved with a radius of R = −1 cm. The parabolic potential is generated by overlapping a standing wave with the billiard (an additional retroreflected laser beam). The standing wave is red-detuned 2.5 nm from resonance and has a radial Gaussian profile with dimensions larger than the dimensions of the billiard (w0 = 244 µm), so the potential inside the trap can be approximated with a parabolic potential (the effect of gravity is then simply to shift the center of the potential). The decay curve was measured for several powers of the red-detuned laser beam. The results, presented in Fig. 11b, clearly show a slowing down of the decay as the power grows, indicating that an island of stability is formed and grows in size for stronger attractive potentials. When the hole was placed in the bottom of the billiard, we observed a fast decay independent of the strength of the parabolic potential, indicating that the island is indeed formed around a vertical trajectory. It was also verified that no atoms were trapped in the red detuned standing wave by itself for the powers we used. We performed numerical simulations that confirmed that the island is centered around the two-periodic orbit we predict should become elliptical.

11.

Summary

A new experimental system, the “atom-optics billiard”, was introduced in this lectures, and several non-linear dynamics experiments were reviewed. First, the validity of the system as an experimental realization of the billiard model was assessed by the observation of integrable and chaotic motion of cold atoms. Next, we introduced a controlled amount of scattering to the atomic motion by exposing the confined atoms to a series of resonant light pulses. The measured decay curves for an originally stable case show that a complete destruction of the stability occurs. Next, a numerical and experimental observation of the formation of islands of stability in a billiards with soft walls was presented. An island of stability was demonstrated in a tilted Bunimovich stadium billiard,

264 around the periodic trajectory that connects the points where the big semicircle joins the straight lines. Our results show that the appearance of a stable island in a soft-wall billiard is in general accompanied by areas of “stickiness” surrounding it. Although the size of the stable island is a sensitive function of softness, not always showing a monotonic behavior, the “sticky” trajectories modify this behavior and, in general, introduce a monotonic slowing down in the decay of the number of trapped particles, as the walls are made softer. Finally, we analyzed the effect of adding potentials on billiards, and found that they can cause elliptical orbits in otherwise hyperbolic billiards. We experimentally demonstrated these effects and also found that an unstable cavity placed in gravity can serve as a velocity selective cavity which is stable only for a narrow velocity group. Our research provides a well-controlled system for investigations of classical and quantum chaos, and a great degree of control on the structure of phase space. Combined with the ability to perform precision spectroscopy, our system may have both fundamental and practical importance: From the fundamental point of view, the spectroscopy can serve as a very sensitive probe for the dynamics (Andersen et al., 2003), and in particular shed some light on the quantum-classical crossover (Andersen et al., 2004a; Andersen et al., 2004b). From the practical point of view, our system provides a playground for studying the connection between dynamics and decoherence, which is of outmost importance in the emerging field of Quantum computation. In addition, understanding this connection will help develop methods to increase the coherence time for trapped atoms, and thereby further improve the precision and accuracy of spectroscopic measurements.

Acknowledgments The authors wish to thank U. Smilansky and V. Rom-Kedar for usefull discussions. This work was supported in part by the Israel Science Foundation, the Minerva Foundation, and Foundation Antorchas.

265

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CONTROL OF CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS Fereydoon Family Department of Physics, Emory University, Atlanta, GA 30322, U.S.A. phyff@emory.edu

C. Miguel Arizmendi Depto. de Fısica, ´ Facultad de Ingenier´ á, Universidad Nacional de Mar del Plata, 7600 Mar del Plata, Argentina, and the Department of Physics, Emory University, Atlanta, GA 30322, U.S.A. arizmend@fi.mdp.edu.ar

Hilda A. Larrondo Depto. de Fısica, ´ Facultad de Ingenier´ ´ıa, Universidad Nacional de Mar del Plata, Av. J.B. Justo 4302, 7600 Mar del Plata, Argentina larrondo@fi.mdp.edu.ar

Abstract

We have studied the deterministic dynamics of underdamped single and multiparticle ratchets associated with current reversal, as a function of both the amplitude and the frequency of an external driving force. We show that control of current reversals in deterministic inertia ratchets is possible as a consequence of a locking process associated with different mean velocity attractors. Control processes employing small perturbations on the frequency and the amplitude of the external force may be designed in view of the intermixed fractal nature of the domains of attraction of the mean velocity attractors. The range where each control parameter is capable to reverse the current is determined. The presence of quenched noise and the influence of the mass of the particle is also considered in order to design control techniques capable of separating particles of different masses.

Keywords: Ratchets, Chaos, Control of Transport, Particle Separation


270

Introduction The goal of this work is to understand how the transport properties of a particle or a group of particles diffusing under non-equilibrium conditions in an asymmetric potential can be controlled. This work is based on some recent studies of the dynamics of a single particle in an asymmetric potential, in the presence of nonequilibrium fluctuations. In fact, recently considerable attention has been paid to thermal ratchets or correlation ratchets [1] where a nonzero net drift current may be obtained from time correlated fluctuations interacting with ratchetlike broken symmetry structures. The impetus behind these studies is due to the general interest in understanding transport in nonequilibrium processes, particularly molecular motors in biological systems [2], well approximated by overdamped ratchet dynamics. More technological applications of the ratchet model are separation of microscopic and mesoscopic objects [3], surface smoothening [4], and the building of micron-scale devices [5]. In these applications, the mass of the particle may be important and the appropriate model to be used is the inertial ratchet model [6]. Inertial ratchets are characterized by a more complex dynamics than overdamped ratchets, including chaotic motion and multiple reversals in the current direction [6, 7], even in the absence of noise. The problem of controlling the current reversal in inertia ratchets [8, 9, 10, 11], is important for technological applications, such as designing new particle separation techniques. In particular, our goal is to develop a model for the control of transport and separation of nanoparticles diffusing in nonequilibrium conditions under the influence of a ratchet potential. This scenario can be tested using an experiment with nanoparticles moving in an asymmetric ratchet potential. In this paper we investigate different strategies to control the current of an inertial ratchet. The control parameters analyzed are: the strength and frequency of the periodic external force, the strength of the quenched noise that models a non perfectly periodic potential, and the mass of the particles. For each control process the approach presented here is valid in regions in the control parameter space where the ratchet has only two mean velocity attractors. Only noninteracting particles are considered. The success of each control mechanism is guaranteed by the fractal nature of the basins of attraction of these two mean velocity attractors. Then small perturbations of the control variable are enough to produce a jump between solutions with different net drift characteristics.

Control of Chaos and Separation of Particles in Inertia Ratchets

1.

271

Model and Theoretical Background

The system under study in this paper is the same deterministic inertia ratchet investigated in our previous work [9, 10, 11],

¨ x + γ x˙ = cos(x) + µ cos(2x) + Γ sin(Ωt) + αξ(x),

(1)

where is the mass of the particle, γ is the damping coefficient, Γ and Ω are respectively the amplitude and the frequency of an external oscillatory forcing. The unperturbed ratchet potential, with period λ = 2π, is given by, U (x) = − sin(x) −

µ sin(2x), 2

(2)

and has been extensively used in the study of models both without disorder [6, 7, 14] and with disorder [9, 12, 13]. The addition of a quenched disorder term αξ(x) gives a more realistic representation of the substrate. The coefficient α ≥ 0 is the strength of this quenched disorder and ξ(x) are independent, uniformly distributed random variables with no spatial correlations, corresponding to a piecewise constant force on the period of the potential. The bifurcation diagram of a system with 5 parameters is too hard to explore and, on the other hand, usually only one parameter is used to control the system. Let us start then with the case of a one particle system ( = 1.1009) in a perfectly periodic potential ( α = 0), with µ = 0.5 and damping coefficient γ = 0.1109. The amplitude Γ and the frequency Ω of the external oscillatory forcing are the remaining control variables. There are two different experimental strategies to study the dynamics in the parameter plane (Ω, Γ) [9, 10]. In the first experimental method (Method I) the values of Γ and Ω are increased in small steps. The same initial condition is used for each new value of the control variables. In the second method (Method II) that is history dependent, the last position and velocity of the particle evolving under an amplitude Γ and a frequency Ω, are the new initial conditions when the amplitude Γ or the frequency Ω are changed. In both cases the trajectory is discarded for each initial condition during a transitory time taken as 400T and then the permanent regime is studied during 100T with T = 2π/Ω. In all cases the first initial condition is x = 0, with v = 1. The numerical solution of Eq. (1) is obtained with a variable step Runge-Kutta-Fehlberg method [15].

272

Figure 1. (a) Regions of different normalized mean velocities v/vΩ in the parameter space (Ω, Γ) obtained using method I. (b) Enlargement of the rectangle Γ = [0.83, 1.03], Ω = [0.6, 0.7]. Intersection of the surface of Fig. 1(a) with the plane (c) Γ = 0.9, and (d) Ω = 0.67.

2. 2.1

Results Single Particle Transport

Figure 1a shows, the regions of different normalized mean velocities !v"/vω (with vω = λ/T ), as a function of both the external force strength Γ and frequency Ω, using method I. Each value of the normalized mean velocity corresponds to a different color. Inside the rectangle Γ ∈ [0.83, 1.03], Ω ∈ [0.6, 0.7] , there are transitions between normalized mean velocities +1 and -1. This is precisely a region where two mean velocity attractors coexist. The transitions may be more clearly seen in the enlargement of the rectangle, shown in Figure 1b, and they can also be seen in Figures 1c and 1d, where the intersections of the surface in figure 1a, with the planes Ω = 0.67 and Γ = 0.9 are respectively shown. Figure 2a shows the normalized mean velocity of a particle with initial condition x0 = 0, v0 = 1 as a function of Γ in the range Γ between [1,1.05], using method I, where the initial condition is reset whenever Γ

273

Control of Chaos and Separation of Particles in Inertia Ratchets 2.5

2 (a)

(b)

2

1.5

1.5 1

1 0.5 /vω

/vω

0.5 0

0 0.5

0.5

1 1

1.5 1.5 2

2 1

1.01

1.02

Γ

1.03

1.04

1.05

2.5 1

1.1

1.2

1.3

Γ

1.4

1.5

1.6

1.7

Figure 2. (a) Normalized mean velocity of a particle with initial condition x0 = 0, v0 = 1 as a function of Γ using method I. (b) Normalized mean velocity of a particle with initial condition x0 = 0, v0 = 1 as a function of Γ starting at Γ = 0.89665 , using method II.

changes. This figure shows that with method I, the normalized mean velocity has only two values: +1 and -1. A number of transitions take place between the two velocity as Γ increases showing that the system is extremely sensitive to the value of Γ. On the other hand method II produces a very different result. Figures 2b represents the situation with Γ as control parameter. In this case the possible values for the normalized mean velocity in the same region of Γ are again +1 or -1 but the particle remains locked to one of them as Γ is changed. The locking value depends on the starting Γ. For example in Figure 2b, corresponding to a starting value of Γ = 0.89665, the particle remains locked to +1; but if we use the starting value of Γ = 0.89666, the mean velocity of the particle remains locked to -1. Therefore, method I is better than method II for detecting the coexistence of attractors and consequently the most adequate range of the control parameters. Extensive numerical simulations demonstrate that this behavior in the parameter space is obtained when the domains of attraction of the coexisting mean velocities attractors are intermixed fractals. Consider the case of Γ as control parameter, in the above mentioned range. We fixed = 1.1009, γ = 0.1109, µ = 0.5, Γ = 0.89665 and Ω = 0.67. The initial conditions are selected on a grid of 512 × 512 points in the rectangular region limited by x0 ∈ [−3.14, 3.14] and v0 ∈ [0, 2]. These ranges are chosen to cover a whole spatial period for the potential (in the position x0) , and kinetic energies extending from 0 up to the height of the potential barrier (in the velocity v0 ). As far as only two types of solutions exist as discussed above, we denote with a black dot an initial condition (x0 , v0 ) that leads to a trajectory with a normalized

274

Figure 3. (a) Basins of attraction of −vω (solid circles) and +vω (open circles) without quenched disorder (α = 0) with Γ = 0.89665. The solid lines are equipotential. Figure (b) is an enlargement of figure 3a to show the fractal nature of the basins of attraction.

mean velocity −1, and we denote with a white dot an initial condition that leads to a trajectory with a normalized mean velocity +1. In this way, the basins of attraction for both mean velocity attractors were obtained, as shown, for example, in figures 3a and 3b. Figure 3b is an enlargement of figure 3a. Further enlargements of this figure gives evidence of a typical fractal behavior. The solid lines in these figures are equipotential curves with the corresponding potential value printed on them. We used a box counting method to study the fractal nature of the mean velocity domains of attraction. Our results show that the mean velocity attractors are fractal with a dimension d = 1.87 for the negative mean velocity domain of attraction.

2.2

Control with the Driving Amplitude

The practical consequence of the fractal nature of the basins of attraction is that most of the initial conditions belonging to the positive mean velocity domain of attraction are surrounded by initial conditions belonging to the negative mean velocity domain of attraction. Then it is possible to produce a jump from one attractor to the other using only small perturbation in the value of Γ as is shown in Figure 4. Several trials may be required to produce the desired jump but the important point is that only small perturbations are required. In a similar way the direction of movement may be reversed using a small change in Ω.


275

495 490 x

485

Tx 480

t=499T

475 470 465

t=498.7T

460 455 450 490

495

500

505 t/T

510

515

520

Figure 4. Example of control of current direction by selecting the time when Γ is changed from Γ = 0.89665 to Γ = 0.89666. The bold curve corresponds to a changing time t = 498.7T . The thin curve corresponds to a changing time t = 499T .

2.3

Collection of Particles

The above results were obtained by studying the trajectory of only one particle, but ratchet transport is essentially stochastic. To study the transport phenomena from a statistical viewpoint a collection of particles with different initial conditions is used. The collection of particles is represented by an ensemble consisting of hundred particles having identical initial velocities v0 but initial positions equally distributed in the range [xmin , xmax ] . The initial probability density is given by, ρ(x, v, 0) = δ(v − v0 ) [H (x − xmin ) − H(x − xmax )] ,

(3)

where H(x) is the step function. The normalized mean-velocity of the ensemble is: >=

N 1 < v >i . N i=1

As for the one particle case we first show the results of simulations with method I, which for the packet of particles means returning to the initial condition ρ(x, v, 0) when Γ changes. Taking into account the locking effect of the trajectory of each particle when method II is used, it is clear that any desired value of current may be obtained by selecting a region in Figure 3a, with the desired number of black points and white points. As an example, the case of a narrow initial packet consisting of N = 200 particles with uniformly distributed initial positions over the region [xmin ,xmax ] = [5.08, 5.09] having all the same initial velocity

276 1.5

vω 0.2

1

vω 0.5

0 (b)

(a) 0.4

0 0.6

0.5 0.8

1 1.5 0.83

1 1.013

0.93

1.03

Γ

1.023

1.13

1.033

1.043

1.053

Γ

Figure 5. (a) Mean velocity of a narrow set of particles centered at the maximum of the potential and initial velocity v0 = 0.01 as a function of Γ using method I. (b) Example of control of mean velocity for a narrow distribution of particles centered at the maximum of the potential and initial velocity v0 = 0.01 as a function of Γ starting at Γ = 1.013 using method II.

v0 = 0.01 is shown in figure 5a. For Γ = 1.013, about three quarters of the particles in the ensemble have mean velocity < v >i = −vw and the remaining quarter have < v >i = vw giving a normalized ensemble mean velocity > /vω −0.5. If the value of Γ is increased using method II the current remains locked to the value -0.5 initially chosen as it is shown in figure 5b.

2.4

Control with Quenched Noise

The effect of the variation of the strength of quenched noise on the ratchet potential has been suggested [13] as a possible control parameter. Previous results [9, 10, 11] on the variation on the strength of quenched noise on the domains of attraction show that the global shape of the domains is not modified by increasing the quenched noise strength, but the number of points in the negative mean velocity attractor diminishes as α increases: the negative domain of attractions dissolves as α increases. Correspondingly the probability that a particle reaches a positive direction of movement increases with α. Therefore particles initially going into the negative direction may be driven to the positive direction of movement making them enter a region with a small amount of quenched noise. An example is shown in Figures 6a to 6d where a packet of identical noninteracting particles is considered. The particles move in an inhomogeneous media with two regions separated by an interface located at x = 0. the particles separate in two bunches moving with positive and negative mean velocities respectively. As the bunch going to the left enters into the disordered medium it continues without

277

Control of Chaos and Separation of Particles in Inertia Ratchets 600

600

400

200

200 x(nT)/λ

x(nt)/λ

(a) 400

0

0

200

200

400

400

600 0

100

200

300

400

600 0

500

(b)

100

200

300

400

500

300

400

500

t/T

t/T 600

600

400

400

200

200 x(nT)/λ

x(nT)/λ

(c)

0

0

200

200

400

400

600 0

100

200

300

400

500

t/T

(d)

600 0

100

200 t/T

Figure 6. Stroboscopic positions of an ensemble of 100 particles moving in an inhomogeneous media. All the particles start with velocity v0 = 1 and initial positions between xmin = 50λ and xmax = 51λ. The region x > 0 has no spatial disorder. The region x < 0 has quenched disorder with strength α. (a) α = 0.001, (b) α = 0.005, (c) α = 0.008, and (d) α = 0.6.

any important effect. As α increases more and more particles initially going to the left are dispersed and reflected to the right, diminishing the negative current (figures 6b and 6c). It implies that α may be used as a control parameter of the positive and negative currents. For α ∼ = 0.008 all the particles are finally moving to the right. The depth of penetration decreases as α increases over this value. Whenever a disorder threshold that depends on the mass of the particle is reached the localization effect [13] sets in. The localization may be seen in Fig. 6d where a high disorder strength is used and some particles are localized near 0.

2.5

Particle Separation

Finally the possibility of separating particles of different mass was investigated. The case of a distribution of particles of two different

278

/ vω

2

ε=0.99 (solid) ε=1.10 (dotted)

1.5

1

0.5

0

0. 5

1 0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Γ

Figure 7. Mean velocities of two distributions of particles of different mass, as a function of Γ. The solid line corresponds to particles with 1 = 0.99. The dotted line corresponds to particles with 2 = 1.10. All the particles start at x0 = 5.087, with initial velocities between vmin = −0.02 and vmax = −0.01. Method II is employed.

masses is considered in Figure 7. The initial density of particles is given by ρ(x, v, , 0) = δ(x−x0 ) [δ( − 1 ) + δ( − 2 )] [H (v − vmin )−H (v − vmax ) (4) where 1 = 0.99, 2 = 1.10, vmin = −0.02, vmax = −0.01, x0 = 5.087. The initial value of Γ was 0.85 and method II was employed to increase Γ. For Γ ≥ 0.9 particles with mass 1 reversed their motion to reach normal mean velocity -1; but particles with 2 continued with the same normal mean velocity +1.

3.

Discussion and Conclusions

In summary, we presented several strategies for controlling the current of a deterministic inertial ratchet. The method is valid in regions of the selected parameter space where only two mean velocity attractors coexist. The intermixed fractal structure of the domains of attraction of these two attractors assures the possibility of getting a control action using small perturbations. The regions in the parameter space where the method may be successfully applied are detected by means of the experimental method I. Experimental method II is used in the control action to lock the particles to a given mean velocity attractor. Quenched disorder has significant effect on the basins of attraction and consequently current reversals may be produced by inhomogeneous media. In the case of multiparticle systems the desired value of current may be obtained by an appropriate selection of the initial conditions of the particles. If particles of different masses are mixed they may be separated by increasing the strength of the periodic force using method II.


279

In conclusion, we would like to point out that inertial ratchets, in the absence of noise, may be related to the damped-driven pendulum, a chaotic system studied several years ago, among others, by Grebogi, Ott and Yorke [16]. In the case of the pendulum the state variable under study is the angle between the pendulum arm and the rest position, i. e. a S 1 type variable. In the case of a deterministic inertia ratchet the position of a particle in the periodic potential is a R1 state variable. Then a periodic (chaotic) oscillation of the pendulum around the minimum corresponds to a regular (chaotic) movement of a particle in the periodic potential with zero mean velocity. A counterclockwise (clockwise) rotation of the pendulum corresponds to a movement of the particle in the ratchet potential with positive (negative) mean velocity. The main difference between a deterministic inertia ratchet and a damped pendulum is the asymmetry in the potential. This asymmetry gives rise to a preferred direction of movement in the ratchet potential. Nevertheless an important feature persists when the symmetry of the potential is broken: the fractal nature of the different coexisting attractors.

Acknowledgment This research was supported by a grant from the Office of Naval Research.

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280 [11] F. Family, H. A. Larrondo, and C. M. Arizmendi, in Mathematical Modelling and Computing in Biology and Medicine, 5th ESMTB Conference 2003, edited by V. Capasso, (ESMTB, Milan), pp. 36 (2003). [12] M. N. Popescu, C. M. Arizmendi, A. L. Salas-Brito and F. Family, Phys. Rev. Lett. 85, 3321-3324 (2000) [13] C. M. Arizmendi, F. Family and A. L. Salas-Brito, Phys. Rev. E 63 061104 (2001). [14] M. Barbi and M. Salerno, Phys. Rev. E 62, 1988-1994, (2000). [15] W. H. Press, S. A. Teikolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, (Cambridge University Press, Cambridge, 1995), pp. 704-716. [16] C. Grebogi, E. Ott, J. A. Yorke, Science 238 585 (1987)

FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE MEAN WAITING TIMES F. Bardou IPCMS, CNRS and Universit´ ´ Louis Pasteur 23 rue du Loess, BP 43, F-67034 Strasbourg Cedex 2, France [email protected]

Abstract

There exist important stochastic physical processes involving infinite mean waiting times. The mean divergence has dramatic consequences on the process dynamics. Fractal time random walks, a diffusion process, and subrecoil laser cooling, a concentration process, are two such processes that look qualitatively dissimilar. Yet, a unifying treatment of these two processes, which is the topic of this pedagogic paper, can be developed by combining renewal theory with the generalized central limit theorem. This approach enables to derive without technical difficulties the key physical properties and it emphasizes the role of the behaviour of sums with infinite means.

Keywords: fractal time, anomalous diffusion, laser cooling, renewal processes, Lévy flights, central limit theorems

Introduction The fractal time random walk [16, 15] has been developed in the 1970’s to explain anomalous transport of charge carriers in disordered solids. It describes a process in which particles jump from trap to trap as a result of thermal activation with a very broad (infinite mean) distribution of trapping times. It results in an unusual time-dependence of the position distribution which broadens while the peak remains at the origin. The method of choice to study the fractal time random walk is the continuous time random walk technique. Subrecoil laser cooling [1, 4] has been developed in the 1990’s as a way to reduce the thermal momentum spread of atomic gases thanks to momentum exchanges between atoms and laser photons. It is a process in 281 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 281–301. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

282 which, as a result of photon scattering, atoms jump from a momentum to another one with a very broad distribution of waiting times between two scattering events. It results in an unusual time dependence of the momentum distribution which narrows without fundamental limits hence giving access to temperatures in the nanokelvin range. The method of choice to study subrecoil laser cooling is renewal theory [6]. Fractal time random walks and subrecoil cooling seem at first sight very dissimilar. The first mechanism generates a broader and broader distribution, while the second generates a narrower and narrower distribution. Nevertheless, inspection of the theories of both phenomena reveals strong similarities: the continuous time random walk and the renewal theory are two closely related ways to tackle related stochastic processes. Physically, the two mechanisms share a common core, a jump process with a broad distribution of waiting times. The aim of this pedagogic paper is to bridge the gap between fractal time random walk and subrecoil laser cooling. We show that the essential results of the two theories can be obtained nearly without calculation by combining the simple probabilistic reasoning underlying renewal theory and the generalized central limit theorem applying to broad distributions. This provides more direct derivations than in earlier approaches, at least for the basic cases considered here. In the first part, we describe the microscopic stochastic mechanisms at work in the fractal time random walk and in subrecoil cooling and relate them to renewal processes. In the second part, we explain elementary properties of renewal theory and derive asymptotic results using the generalized central limit theorem and L´ évy stable distributions. In the third part, we draw the consequences for the fractal time random walk and subrecoil cooling. The fourth part contains bibliographical notes.

1. 1.1

Fractal time random walk and subrecoil cooling: microscopic mechanisms Fractal time random walk

The notion of fractal time random walk emerged from the observation of unusual time dependences in photoconductivity transient currents flowing through amorphous samples. It can be schematized in the following way. Consider first a one dimensional situation called the Arrhenius cascade [3] in which the charge carriers are placed in a random potential with many local wells and barriers and can jump from one well to another one thanks to thermal activation (Fig. 1a). The Arrhenius cascade potential presents two features: a global tilt representing the effect of

283

Fractal time random walk, laser cooling and renewal processes

the electric field on the carriers and local random oscillations creating metastable traps separated by barriers representing the disorder created by the amorphous material. Thus the potential seen by the carriers is a kind of random washboard with a discrete number of metastables states. τ1

τ2

i

x

i

Ei

0

x

Figure 1. Biased random walks in a disordered system (a) Arrhenius cascade. Each carrier is put in random potential with a global tilt. It undergoes jumps over barriers of random heights Ei from a metastable well to the next one on the right thanks to thermal activation. (b) Photoconductivity setup. At time t = 0, a light pulse creates carriers in the immediate vicinity of the left electrode of an amorphous sample. The carriers of one sign then move through the sample thanks to an applied electric field. (The carriers of the opposite sign are immediately absorbed by the left electrode.)

The mean lifetime of state i, i.e., the mean waiting time before the occurrence of a thermal jump, is given by the Arrhenius law: τ¯i = τ0 eEi /kT

(1)

where τ0 is a time scale, Ei is the height of the energy barrier separating state i from state i+1, k is the Boltzmann constant and T is the temperature. The potential global tilt is assumed to be large enough to neglect backward jumps from i to i − 1. The random walk we consider is thus completely biased. The time spent between the metastable states is neglected. For a given barrier height Ei , the lifetime distribution ψ(τ |E Ei ) is exponential with mean τ¯i : ψ(τ |E Ei ) =

1 −τ /¯τi e . τ¯i

(2)

In the photoconductivity experiments (Fig. 1b), a light pulse creates at time t = 0 carriers localized near the surface of the sample. The carriers then move through the disordered sample thanks to an applied electric field. Thus, this situation can be modelled by a large number of Arrhenius cascade in parallel, each electron path being associated to one cascade.

284 One may (wrongly) expect that the transient current flowing through the sample is quasi-constant at the beginning, while the bunch of carriers propagates through the sample, before decreasing rapidly to zero when the carriers leave the sample after reaching the end electrode. But what is observed is quite different. The current decreases as a power law ∼ 1/t1−α while the carriers are still in the sample, then as ∼ 1/t1+α when some carriers start leaving the sample. For simplicity, we assume here that the sample is semi-infinite so that the carriers never leave the sample. The explanation of this anomalous behaviour will be shown to be related to the distribution of lifetimes τi . The randomness of the τi ’s results from the combination of the exponential statistics of jump times for a given barrier height Ei (eq. (2)) with the barrier height statistics Ei ), conveniently described by an exponential distribution P (E P (E Ei ) =

1 −Ei /E0 e E0

for

Ei ≥ 0,

(3)

where E0 is an energy scale related to the sample disorder. The waiting time distribution ψ(τ ) is then

∞

ψ(τ ) = 0

where γ(α , x) =

x 0

dE Ei P (E Ei )ψ(τ |E Ei ) = αγ(1 + α, τ /ττ0 )

τ0α τ 1+α

(4)

e−u uα −1 du is the incomplete gamma function and α=

kT . E0

(5)

At long times, ψ(τ ) tends to a power law, hence the term “fractal time random walk”: τ0α ψ(τ ) α2 Γ(α) 1+α , (6) τ with Γ(α) = 0∞ uα−1 e−u du. If α ≤ 1, states i have an infinite mean lifetime !τ " = 0∞ τ ψ(τ ) dτ . However, they are unstable since they all ultimately decay to the next state (i+1). Usually, unstable states have a well defined and finite mean lifetime. Here, the somewhat paradoxical presence of unstable states with infinite mean lifetimes is at the origin of the striking properties of the fractal time random walk.

1.2

Subrecoil laser cooling

Laser cooling of atomic gases consists in reducing the momentum spread of atoms thanks to momentum exchanges between atoms and


285

photons. Subrecoil laser cooling consists in reducing the momentum spread to less than a single photon momentum, denoted h ¯ k . This paradoxical goal is achieved by introducing a momentum dependence in the photon scattering rate (see Fig. 2a) so that it decreases strongly or even vanishes in the vicinity of p = 0, where p denotes the atomic momentum, taken in one dimension for simplicity. The mechanism of subrecoil cooling is explained in Fig. 2. Any time a photon is absorbed and spontaneously reemitted by an atom, the atomic momentum undergoes a momentum kick on the order of ¯hk , which has a random component because spontaneous emission occurs in a random direction. Thus, the repetition of absorption-spontaneous emission cycles generates for the atom a momentum random walk (see Fig. 2b), with momentum dependent waiting times τ ’s between two kicks. When an atom reaches by chance the vicinity of p = 0, it tends to stay there a long time. This enables to accumulate atoms at small momenta, i.e., to cool. τp

p

0

-ptrarap -p ap 0

ptrarap ap

p

Figure 2. Subrecoil laser cooling (a) The mean sojourn time at momentum p, τ¯(p), becomes very large for small atomic momenta. (b) Photon scattering creates a momentum random walk with an accumulation in the vicinity of p = 0 due to the momentum dependence of the mean sojourn time τ¯(p).

For a quantitative treatment, we introduce τ¯(p), the mean sojourn time at momentum p (also the mean waiting time between two spontaneous photons for an atom at momentum p). For a given p, the distri-

286 bution ψ(τ |p) of sojourn times at momentum p is ψ(τ |p) =

1 e−τ /¯τ (p) . τ¯(p)

(7)

We need to characterize the distribution of “landing” momenta π(p) after a spontaneous emission. Under favourable but often realistic assumptions, atoms spend most of the time around the origin in the interval [−ptrap , +ptrap ], because they diffuse fast outside this interval and thus come back to it rapidly after leaving it. If ptrap < hk ¯ , then after a spontaneous emission, the distribution of atomic momenta can be considered as uniform: 1 π(p) = . (8) 2ptrap The distribution of sojourn times after a spontaneous emission is thus

ψ(τ ) =

ptrap −ptrap

dp π (p)ψ(τ |p). d

(9)

We consider the physically relevant case of power law τ¯(p), τ¯(p) =

τ0 pβ0 , |p|β

(10)

where β > 0, and τ0 and p0 are time and momentum scales, respectively. Then, one finds, just as in the fractal time random walk, a waiting time distribution with a power law tail: :

αp0 ptrap ψ(τ ) = γ 1 + α, ptrap p0

1/α

;

τ τ0α p0 τ0α 2 −→ α Γ(α) , τ0 τ 1+α τ →∞ ptrap τ 1+α (11)

where α=

1 . β

(12)

If β < 1, the mean waiting time is finite and simple integration gives 1 !τ " = 1−β

p0 ptrap

β

τ0 .

(13)

If β ≥ 1, on the contrary, the mean waiting time is infinite. This divergence of the mean has dramatic (and positive in terms of cooling) consequences (see §3.2).


1.3

287

Connection with renewal theory

Renewal processes are stochastic process in which a system undergoes a sequence of events (denoted by • in Fig. 3) separated by independent random “waiting times” τ1 , τ2 , ... The term “renewal process” comes from engineering. Assume that, at time t = 0, one installs a machine in a factory. When, after being operated for a random lifetime τ1 , the machine breaks down, it has to be replaced by a new one, which will work till it breaks down at τ1 + τ2 and has to be replaced ... If, instead of a single machine, one has installed a large number of identical machines, then, to decide how many replacement machines must be stored at a given time, one needs to know the replacement rate, which we call hereafter the renewal density. τ1

τ2

τ3

τ4

τ5 t

0

Figure 3. Renewal processes. The system undergoes a sequence of events (jumps from trap to trap, momentum kicks ...) at random times separated by waiting times τ1 , τ2 , ...

To understand the statistical properties of renewal processes, various quantities are introduced. The most detailed information is provided by the distribution of the number of renewals, ft (r), i.e., the probability distribution for the system to undergo r events in time t. One also introduces derived quantities, the mean number of renewals at time t, !r"t , and the mean renewal rate at time t, denoted R(t) and called the renewal density. Mathematical expressions for these three quantities will be given in §2.1. Here we show the role they play in fractal time random walk and in subrecoil laser cooling. In the biased (fractal time or non fractal time) random walk, the discretized positions n at time t correspond directly to the number of jumps performed between time 0 and t. Hence the position distribution ρ(n, t) in the fractal time random walk is the renewal number distribution: ρ(n, t) = ft (n).

(14)

The mean position of the carriers is !r"t . The current i(t) measured in photoconductivity experiments before the carriers get out of the sample is proportional to the mean carrier velocity d!r"t /dt, which is the renewal density (see §2.1). Thus, one has i(t) ∝ R(t).

(15)

288 In subrecoil cooling, the momentum distribution ρ(p, t) can be written in the following form:

t

ρ(p, t) = π(p) 0

dtl R(tl )P Ps (t − tl |p),

(16)

where tl is the time of the last jump occurring between 0 and t, R(tl )dtl is the probability that a jump occurs during the interval [tl , tl + dtl ), π(p) is the uniform probability distribution for a jumping atom to land at momentum p and Ps (t − tl |p) is the survival probability for an atom landing at momentum p at time tl to stay there till at least time t. Using eq. (7), one has trivially Ps (t − tl |p) =

∞ t−tl

dτ ψ(τ |p) = e−(t−tl )/¯τ (p) .

(17)

The non trivial physical information is contained in the renewal density R(t). The height h(t) of the momentum distribution peak, h(t) = ρ(p = 0, t),

(18)

is proportional to !r"t , the mean number of jumps between 0 and t. Indeed, for any jump, there is a probability π(p)2dp d =d dp/ptrap to fall in the vicinity [−dp, d dp d ] of the origin and to stay there indefinitely since states in [−dp, d dp d ] have arbitrarily long lifetimes in the limit d dp → 0 (¯ τ (p) −→ ∞, see eq. (10)). Thus the height writes p→0

h(t) =

2. 2.1

!r"t . 2ptrap

(19)

Renewal theory and L´ ´ evy stable laws General formulae

The number of renewals rt in a time t is defined as the number of jumps having occurred before time t. It satisfies Srt ≤ t < Srt +1 rt

(20)

where Srt = i=1 τi is the sum of the first rt waiting times. The relationship between the renewal number distribution ft (r) and the waiting time distribution ψ(τ ) can be obtained from the following simple reasoning.


289

Note first that the distribution, denoted ψ r∗ (S Sr ), of the sum Sr of r independent identically distributed waiting times is the rth convolution product of ψ(τ ) with itself. Moreover, from the definition of the number rt of renewals, one has obviously Pr(rt < r) = Pr(S Sr > t) = [1 − Ψr∗ (t)] ,

(21)

Sr

where Ψr∗ (S Sr ) = 0 ψ r∗ (u) du denotes the distribution function of Sr (in spite of its notation, Ψr∗ (S Sr ) is not the rth convolution product of the waiting time distribution function Ψ(τ ) = 0τ ψ(u) du). The probability distribution ft (r) of the number of renewals at time t is thus finally ft (rt = r) = Pr(rt < r + 1) − Pr(rt < r) = Ψr∗ (t) − Ψ(r+1)∗ (t).

(22)

This expression relates the distribution of a discrete random variable, r, to the distribution fonctions of continuous random variables t. Important quantities derived from the renewal number distribution ft (r) are the mean number of renewals at time t: !r"t =

∞

rfft (r)

(23)

r=0

and the renewal density, i.e., the mean number of renewals per unit time1 : d!r"t R(t) = . (24) dt Usual theoretical treatments of renewal problems are based on Laplace transforms of the waiting time distribution that are indeed well suited to handle the convolution products ψ r∗ (t). Here, we prefer a different approach based only on the generalized central limit theorem. This approach requires nearly no calculation. Moreover, it stresses the role of the behaviour of sums of many random variables. Indeed, most useful distributions tend, under repeated convolution, to a L´ évy stable law given by the generalized central limit theorem. Thus we can obtain from eq. (22) analytical expressions for ft (r) and for related quantities in the limit of large r and hence large t. Depending on the finiteness of the first two moments of the waiting time distribution ψ(τ ), three cases can be distinguished. For brevity, we treat only the two most striking cases: first and second moment both finite in §2.2; first and second moment both infinite in §2.3. renewal density is a rate of events. It has the same dimension as a probability ∞ distribution of times, 1/time, but it is not a probability density. Thus, its integral dt R(t), 0 representing the average total number of events occurring between t = 0 and t = ∞ does not have to be normalized. It is actually often infinite. 1 The

290

2.2

Case of waiting time distributions with finite first and second moment

When both the mean µ = !τ " and the variance σ 2 of the waiting time are finite, the (usual) central limit theorem applies and ψ r∗ (t) tends to a Gaussian distribution :

1 (t − µr )2 ψ (t) −→ √ exp − r→∞ 2σr2 2πσr

;

r∗

(25)

with mean µr = rµ and variance σr2 = rσ 2 . Thus, after changing variables in eq. (22), one has ft (r) −→

r→∞

(t−µr )/σr (t−µr+1 )/σr+1

1 2 √ e−u /2 du. 2π

(26)

Fix (t−µr )/σr and take t → ∞. One expects intuitively that the number of renewals is approximately given by t/µ 1 at large t (this is validated a posteriori by eq. (28)). Thus (t −,µr )/σr (t − µr)/(σ t/µ) and

3

(t−µr+1 )/σr+1 (t−µr)/(σ t/µ)− σµ2 t . This leads for the number of renewals to a Gaussian distribution with mean t/µ and variance σ 2 t/µ3 : ft (r) −→ √ t→∞

1 2πσ 2 t/µ3

4

2

exp − (r−t/µ) 2σ 2 t/µ3

5

.

(27)

Although well known in renewal theory, this result is non trivial: the distribution of the sums of r (→ ∞) terms and of the number of renewals at large times are found to have the same (Gaussian) shape. This is grossly violated with infinite mean waiting times (see §2.3). As direct consequences of eq. (27), the mean number of renewals !r"t tends to t/µ at large times: !r"t −→

t→∞

t µ

(28)

and, using eq. (24), the renewal density tends to the reciprocal of the mean waiting time: 1 R(t) −→ , (29) t→∞ µ in agreement with intuition. Physical comments and an example of a renewal process with finite mean waiting time will be presented in §3.1.

291


2.3

Case of waiting time distributions with infinite first and second moment

We consider now the case of waiting time distributions with infinite first two moments, focusing on the canonical example of distributions with (Pareto) power law tails of index α, αττ0α , τ 1+α

ψ(τ ) −→

τ →∞

(30)

with 0 < α < 1 (slightly more general cases can be handled using the theory of regular variations). The distributions ψ r∗ of the sums Sr no longer tends to Gaussians but, according to the generalized central limit theorem, to L´évy stable laws: ψ r∗ (t) −→

r→∞

1 r1/α

Lα,B

t

(31)

r1/α

where Lα,B (u) is a one-sided (u ≥ 0) L´évy stable law given by its Laplace transform, ∞

0

e−su Lα,B (u) du = e−Bs , α

(32)

and B is a scale parameter given by B = Γ(1 − α)ττ0α .

(33)

After changing variables in eq. (22), one obtains

ft (r) =

t/r1/α

t/(r+1)1/α

Lα,B (u) du,

(34)

which leads to the following asymptotic expression2 : ft (r) −→

t→∞

t αr 1+1/α

Lα,B

t

r1/α

(35)

In sharp contrast with the finite moments case, the renewal number distribution ft (r) differs strongly from the distribution of the sum (eq. (31)) even though both distributions involve L´ évy stable laws. As will be shown in §3.1, ft (r) has a slow decay at small r and a fast decay at large r unlike ψ r∗ (t) which decays as 1/t1+α . The mean number of renewals !r"t is finite and can be related to a negative moment of a Lévy stable law: !r"t

2 As

t Sr =

r

τ i=1 i

0

∞

rfft (r) dr = tα

∞ 0

u−α Lα,B (u) du.

(36)

scales as r1/α , the limit r → ∞ corresponds to the limit t → ∞.

292 Hence, using finally

∞ 0

u−α Lα,B (u) du = 1/(BαΓ(α)) and eq. (33), one has

sin(πα) t α !r"t −→ . (37) t→∞ πα τ0 Finally, using eq. (24), the renewal density tends asymptotically to an ever decreasing power law: R(t) −→

t→∞

sin(πα) 1 . π τ0α t1−α

(38)

Physical comments and an example of a renewal process with infinite mean waiting time will be presented in §3.1.

3.

Application to fractal time random walk and subrecoil cooling

We have seen in §1.3 that the most important quantities appearing in biased random walks and in subrecoil cooling are directly related to renewal theory: renewal number distribution ft (r) (position distribution in biased random walks), mean number of renewals !r"t (peak height in subrecoil cooling) and renewal density R(t) (current in the biased random walk, momentum distribution in subrecoil cooling). In this section, we thus apply the results on renewal processes obtained in §2 to biased random walks and subrecoil cooling. We emphasize the physics consequences of the divergence of the mean waiting time in renewal processes by opposing, for each application, one example with finite mean waiting time and one example with infinite mean waiting time. These examples reveal the generic features of the cases with finite or infinite mean waiting times.

3.1

Biased random walks

As an example of waiting time distribution with finite mean and standard deviation generating a (non fractal time) random walk, we consider an exponential distribution ψ(τ ) =

1 −τ /µ e µ

for τ ≥ 0

(39)

with mean and standard deviation both equal to µ. In this case, the convolutions of ψ(τ ) have the simple explicit forms of the Gamma (or Erlang) distributions ψ n∗ (τ ) =

τ n−1 e−τ /µ . (n − 1)!µn

(40)

293


Thus, applying eq. (22), the renewal density ft (r) and hence the position distribution ρ(n, t) are exactly known: e−t/µ ρ(n, t) = n!

n

t µ

.

(41)

(One recognizes the Poisson distribution of mean !r"t = t/µ, as expected: in this case, the renewal process is a Poisson process.) As shown in Fig. 4, this exact position distribution rapidly tends to the Gaussian distribution given by eq. (27): :

1 (n − t/µ)2 ρ(n, t) −→ exp − t→∞ 2t/µ 2πt/µ

0.4

t =µ

;

.

(42)

exact (Poisson) asymptotic expression

ρ(n,t)

0.3 t = 2µ

0.2

t = 5µ t = 10µ t = 20µ

0.1 0 0

10

20

n

30

40

Figure 4. Time evolution of the position distribution ρ(n, t) for a biased random walk with an exponential waiting time distribution √ (finite τ ). The distribution propagates at constant speed 1/µ and spreads as t. The exact result (eq. (41)) and the asymptotic one (eq. (42)) agree even at short times.

We thus recover the intuitive picture of normal transport: a position √ distribution propagating at a constant speed 1/µ and spreading as t.

294 The current i(t) resulting from such a distribution in photoconductivity experiments is related to the renewal density (eq. (15)) which leads, using eq. (29), to i(t) ∝ 1/µ. (43) As an example of waiting time distribution with infinite mean and standard deviation generating a fractal time random walk, we consider a Pareto distribution3 of index α = 1/2, 1/2

ψ(τ ) =

τ0 2τ 3/2

for

τ ≥ τ0

(44)

with τ0 > 0. As there is no simple analytic form for ψ n∗ (t) and thus for the position distribution ρ(n, t) = Ψn∗ (t) − Ψ(n+1)∗ (t) (eq. (22)), we obtain ρ(n, t) through numerical simulation (see Fig. 5). On the other hand, using eq. (35) and the known form for the asymmetric Lévy stable law of index 1/2 called the Smirnov law (or using eq. (32)), the asymptotic distribution is found to be a half-Gaussian 3

ρ(n, t) =

t→∞

τ0 πττ0 2 exp − n . t 4t

(45)

The fact that a half-Gaussian is obtained should not give the impression that fractal time transport is similar to normal transport described by full Gaussians (eq. (42)). First, the half-Gaussian is specific to Pareto waiting time distributions with α = 1/2 (see below for other α’s). Second, the properties of the half-Gaussian in fractal time transport are completely different from those of the full Gaussian of the normal transport. Indeed, instead of propagating, the distribution peak remains at the origin n = 0 at all times. Only the tails spread to the right, more and more slowly as times goes by. This is due to the fact that the carriers statistically tend to be trapped into deeper and deeper traps at long times, which slows down their motion. The resulting current is thus a decreasing function of time, at all times. Using eq. (15) and eq. (38), one obtains: 1 i(t) ∼ √ . (46) τ0 t It is worth examining the general case of waiting time distributions with power law tails like eq. (30) and infinite means (α < 1). From the 3 For

a Pareto distribution of index α > 2 (finite standard deviation), the normal transport case is recovered.


295

1

ρ(n,t)

0.8 t = τ0

exact (simulation) asymptotic expression

0.6 0.4 t = 5τ0

0.2

t = 20τ0 t = 100τ0

0 0

5

10

20

15

n

25

Figure 5. Time evolution of the position distribution ρ(n, t) for a biased fracta al time random walk constructed from a Pareto distribution of index α = 1/2 (infinite (infinit τ ). The distribution spreads slowly towards positive values but its maximum always remains at the origin. Note the smaller position scale and the longer time scales compared to Fig. 4, which emphasizes the slowness of transport in the fractal time random walk. The exact (simulated) and asymptotic results (eq. (45)) are in good agreement. (They have seemingly different norms at short times because the exact result is a discrete distribution while the asymptotic one is a continuous distribution.)

following expansion of asymmetric Lévy stable laws with α < 1, Lα,B (x)

x→∞

αB , Γ(1 − α)x1+α

(47)

one finds the behaviour close to the origin (for t → ∞):

ρ(n, t)

n→0

τ0 t

α

.

(48)

Hence, the position distribution is flat (n independent) at small n, as already observed for the special case α = 1/2. Moreover, using Lα,B (x) Ax x→0

α−2 2(1−α)

exp −

C xα/(1−α)

,

(49)

296 where A and C are constants, one obtains

Cr1/(1−α) ρ(n, t) ∝ exp − α/(1−α) t

,

(50)

up to power law corrections. As 1/(1 − α) > 1, the transport front always decreases faster than exponentially and presents a well defined characteristic position. Consequently, the mean carrier position yielding the current, is well defined, unlike the mean waiting time which is infinite. Using eq. (15) and eq. (38), one finds that the current decays as a power law: 1 i(t) ∝ 1−α , (51) t as already observed for the special case α = 1/2 and in agreement with photoconductivity transient experiments.

3.2

Subrecoil cooling

Consider first the case of waiting time distributions with α = 1/β > 2 (eq. (11)) ensuring a finite mean waiting time !τ " (and a finite !τ 2 "). According to eq. (29) and eq. (13), one has R(t) −→

t→∞

1−β τ0

ptrap p0

β

(52)

and thus, applying eq. (16) with eq. (8) and eq. (17), one finds the momentum distribution 

− t 1−β  ρ(p, t) = 1 − e τ0 2ptrap

|p| p0

β  

ptrap |p|

β

.

(53)

This distribution has stationary tails: 1−β ρ(p, t) 2ptrap

ptrap |p|

β

for

|p| > p0

τ0 t

1/β

(54)

but a non stationary peak that increases linearly in time: ρ(p, t)

1−β 2pβ0 p1−β trap

t τ0

for

|p| < p0

τ0 t

1/β

.

(55)

This peak is also obtained directly using relation (19) between the height h(t) and the mean number of renewals of eq. (28).

297


4 t = 50τ0

3

ρ(p ( ,t)

t = 20τ0

2 t = 10τ 0 1

t = 5τ0

0 0.0001

0.001

0.01

0.1

p/p / trap

1

Figure 6 6. Time evolution of the momentum distribution ρ(p p, tt)) for subrecoil cooling with a finite mean waiting time. Parameters: β = 0.25, p0 = 1. The tails reach a stationary state. Only a vanishingly narrow part of the peak goes on increasing at long times (note the logarithmic p scale).

Finite mean waiting times are not very favourable for the cooling since only a vanishingly small fraction of atoms goes on accumulating at smaller and smaller velocities (see Fig. 6). Consider now the case of waiting time distributions with infinite mean waiting times (α = 1/β < 1) ensuring. According to eq. (38), one has R(t) −→

t→∞

sin(πα)ptrap 1 α παΓ(α)p0 τ0 t1−α

(56)

(one must write ψ(t) of eq. (11) in the form of eq. (30), thus replacing τ0α by αΓ(α)(p0 /ptrap )ττ0α in eq. (38)). Thus, applying eq. (16) with eq. (8) and eq. (17), the momentum distribution writes sin(πα) ρ(p, t) −→ t→∞ 2πα2 Γ(α)p0 where G(q) = α

0

1

t τ0

α

G

tpβ τ0 pβ0

du uα−1 e−(1−u)q

,

(57)

(58)

298 is a confluent hypergeometric function. Thus ρ(p, t) presents a scaling form and evolves at all times scales, with no stationary state. The tails behave as sin(πα) ρ(p, t) −→ t→∞ 2παΓ(α)

τ0 t

1−α

1 p1−β pβ 0

for |p| > p0

τ0 t

1/β

(59)

and the peak, also obtained directly from eq. (19) and the mean number of renewals (37), behaves as sin(πα) ρ(p, t) −→ t→∞ 2πα2 Γ(α)p0

t τ0

α

for

|p| < p0

τ0 t

1/β

(60)

Infinite mean waiting times are favourable for the cooling (Fig. 7): all atoms accumulate in a narrower and narrower peak in the vicinity of p = 0. The cooling goes on without fundamental limits at long times. The absence of limits is related to the significant weight (cf. !τ " = ∞) of p states with arbitrarily long lifetimes.

3 t = 50τ0

2.5

ρ(p ( ,t)

2 t = 20τ0

1.5

t = 10τ0

1

t = 5τ0

0.5 0 -1

-0.5

0

p/p / trap

0.5

1

Figure 7. Time evolution of the momentum distribution ρ(p, t) for subrecoil cooling with an infinite mean waiting time. Parameters: β = 2, p0 = 1. The full momentum distribution, tails and peak, never reaches a stationary state. All atoms go on accumulating without limit in a narrower and narrower peak as time increases.


4.

299

Bibliographical notes

Fractal time. The theory of fractal time random walks was developed in particular cases in [12] using continuous time random walks and, in the general case, in [16], adding Tauberian techniques to handle Laplace transforms. Fractal times were used in [18] to model turbulent diffusion. Several other applications involving fractal times were presented in [17] and [15]. Subrecoil cooling. The statistical approach to subrecoil cooling was developed in three steps: [5], [2] and [4]. The connection of this approach with renewal theory was stressed in [6]. Renewal processes. The basic theory of renewal processes is presented in a small book [8]. This book does not include infinite mean waiting times but some general expressions for finite mean waiting times are still valid in this case too (see §2.1). Ref. [10], chapter XI, presents renewal theory with a more theoretical viewpoint and includes some results on infinite mean waiting times which were discovered at the beginning of the sixties, in spite of the fact that no application seemed to be known at the time. Renewal processes in a physics context including infinite mean waiting times have been studied systematically in [11] using Laplace transform techniques. The mean number of renewals given by eq. (37) for infinite mean waiting times agrees with eq. (8) in [16] obtained through Tauberian theorems for the fractal time random walk (there is to be a misprint in this reference:< the expression for+ψ(τ ), page 424, line 3, must be replaced by ψ(τ ) ∼ α/ t1+α Γ(1 + α)A(t) ). It also agrees with the renewal theory developed in [11] (eq. (3.6)). The renewal number distribution expressed with Lévy stable laws given by eq. (35) agrees with eq. (5.6) of [10], which is however more complicated. Generalized central limit theorem. The generalized central limit theorem and results on L´ évy stable laws used in §2.3 can be found, e.g. in Appendix B of [7] or in §4.2 of [4]. Details on L´évy stable laws can also be found in §1.2 of [14].

Conclusion This paper has underlined the common statistical core at work in two seemingly opposite problems: a diffusion mechanism (fractal time

300 random walk) and a cooling mechanism (subrecoil laser cooling). This core is made of waiting time distributions with infinite means. Usual theoretical techniques for these problems are based on Tauberian theorems which, through the Laplace transform, imply some loss of physical intuition. Here, we have presented a renewal theory approach which, thanks to the generalized central limit theorem, provides a shortcut to obtain physically relevant quantities. It emphasizes the key contribution of the unusual behaviour of sums of random variables with infinite means. Understanding these stochastic processes with infinite means is not a purely academic game. It has already led to significant improvements of laser cooling strategies [13] and other improvements are under way [9].

Acknowledgments I thank O.E. Barndorff-Nielsen, M. Romeo and M. Shlesinger for discussions.

References [1] A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji. Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping. Phys. Rev. Lett., 61:826–829, 1988. [2] F. Bardou. Ph.d. thesis, 1995. [3] F. Bardou. Cooling gases with L´ évy flights: using the generalized central limit theorem in physics. In O. E. Barndorff-Nielsen, S. E. Graversen, and T. Mikosh, editors, L´ évy processes: theory and applications, Miscillanea no. 11, Aarhus, 1999. MaPhySto. [4] F. Bardou, J.-P. Bouchaud, A. Aspect, and C. Cohen-Tannoudji. Levy ´ Statistics and Laser Cooling. Cambridge University Press, Cambridge, 2002. [5] F. Bardou, J.-P. Bouchaud, O. Emile, A. Aspect, and C. Cohen-Tannoudji. Subrecoil laser cooling and L´ évy flights. Phys. Rev. Lett., 72:203–206, 1994. [6] O. E. Barndorff-Nielsen and F. E. Benth. Laser cooling and stochastics. In M. C. M. de Gunst, C. A. J. Klaassen, and A. W. van der Vaart, editors, State of the Art in Probability and Mathematical Statistics; Festschrift for Willem R. van Zwet, Lecture Notes - Monograph Series, pages 50–71. Inst. Mathematical Statistics, 2000. [7] J.-P. Bouchaud and A. Georges. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep., 195:127–293, 1990. [8] D. R. Cox. Renewal Theory. Methuen (Wiley), London (New York), 1962. [9] F. Bardou et al. Communication at the Workshop on aspects of large quantum systems related to Bose-Einstein condensation (Aarhus, April 2004, and to be published).


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[10] W. Feller. An introduction to probability theory and its Applications. Volume II. I John Wiley and sons, New York, 1971. [11] C. Godr` èche and J. M. Luck. Statistics of the occupation time of renewal processes. J. Stat. Phys., 104:489–524, 2001. [12] E. W. Montroll and H. Scher. Random walks on lattices IV. Continuous-time walks and influence of absorbing boundaries. J. Stat. Phys., 9:101–135, 1973. [13] J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon, and C. Cohen-Tannoudji. Raman cooling of cesium below 3 nk: New approach inspired by Lévy flight statistics. Phys. Rev. Lett., 75:4575–4578, 1995. [14] G. Samorodnitsky and M. S. Taqqu. Stable non-Gaussian random processes. Chapman & Hall/CRC, Boca Raton, 1994. [15] H. Scher, M. F. Shlesinger, and J. T. Bendler. Time-scale invariance in transport and relaxation. Physics Today, pages 26–34, January 1991. [16] M. F. Shlesinger. Asymptotic solutions of continuous-time random walks. J. Stat. Phys., 10:421–434, 1974. [17] M. F. Shlesinger. Fractal time in condensed matter. Ann. Rev. Phys. Chem., 39:269–290, 1988. [18] M. F. Shlesinger, B. J. West, and J. Klafter. Lévy dynamics of enhanced diffusion: Application to turbulence. Phys. Rev. Lett., 58:1100–1103, 1987.


ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA TURBULENCE Xavier Leoncini ´ ¨ LPIIM, Equipe Dynamique des SystEmes complexes, Universit´ é de Provence, Centre Universitaire de Saint J´ érˆ ome, F-13397 Marseilles, France [email protected]

Olivier Agullo ´ LPIIM, Equipe Dynamique des Systèmes complexes, CNRS, Universit´é de Provence, Centre Universitaire de Saint J´ érˆ ome, F-13397 Marseilles, France [email protected]

Sadruddin Benkadda ´ LPIIM, Equipe Dynamique des Systèmes complexes, CNRS, Universit´é de Provence, Centre Universitaire de Saint J´ érˆ ome, F-13397 Marseilles, France [email protected]

George M. Zaslavsky Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA Department of Physics, New York University, 2-4 Washington Place, New York, NY 10003, USA [email protected]

Abstract

Transport properties of passive particles evolving in two dimensional flows are investigated. Flows governed by point vortices and by the Charney-HasegawaMima equation are considered. Transport is found to be anomalous with a non linear evolution of the second moments with time and for all considered cases the characteristic exponent is found to be close to 1.75. The origin of this behavior is traced back to the existence of chaotic jets in these systems.

Keywords: Anomalous transport, plasma turbulence, point vortex, jets 303 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 303–319. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

304

1.

Introduction

In turbulent magnetized plasma, transport problems are often related to confinement, which is one of the last standing issues confronting the realization of magnetically confined controlled fusion devices. Recently, there has been more evidence showing that the transport properties in these systems can be anomalous, in the sense that transport may not be correctly described by Gaussian kinetics. As of today the full understanding of anomalous transport phenomena is far from being complete. There is although a common agreement to link these phenomena to Levy-type processes, moreover the use of fractional derivatives in Fokker-Plank-Kolmogorov type equations captures qualitatively some of the transport properties and is thus a good step towards a proper description of anomalous transport (Zaslavsky 2002). In order to tackle the problem of understanding transport in these system from a dynamical point of view, we chose to proceed gradually, starting from a relatively simple Euler flow governed by a system of point vortices and then increasing a bit the complexity by considering transport properties for a flow governed by the Charney-Hasegawa-Mima equation, which can be considered the minimal system where both vortices and waves exist. Indeed, in a tokamak the presence of a strong magnetic field allows often to reduce the complexity and model the plasma by a system of partial differential equations within a two-dimensional system framework. And many experimental and numerical studies have demonstrated, that in the cases of driven or freely decaying 2D turbulence a number of concentrated vortices develops out of an originally unstructured flow (Tabeling 1998, Carnevale 1991). In many situations, dynamics of this finite-size vortices can be reasonably well approximated by point-vortex models (Zabusky 1982, Fuentes 1996), allowing the reduction of the study of a time dependent field, to a simpler Hamiltonian system of N interacting particles, explaining our choice of a system of point vortices as the starting point for investigating transport properties. In these two-dimensional flows the non Gaussian nature of transport is inherently related to the existence of memory effects and long time correlations. The origin of these memory effects are relatively well understood when dealing with low dimensional systems such as a time periodic flow which belong to the class of 1 − 1/2 degree of freedom Hamiltonians. The dynamics in these systems is typically not ergodic. The phase space is composed of regions known as stochastic seas, with chaotic dynamics, and various islands of quasi-periodic dynamics. The anomalous properties and their multi-fractal nature are then linked to the phenomenon of stickiness observed around islands (Kuznetsov 2000,

Anomalous transport in two-dimensional plasma turbulence

305

Leoncini 2001). However when dealing with more complex systems the loss of time-periodicity complicates the picture. For instance in geophysical flows or two-dimensional plasma turbulence, the islands which were static and well localized in phase space, are replaced by “coherent structures”, which have a life of their own. Hence, tackling the origin of anomalous transports from the chaotic dynamics of individual tracers becomes more subtle. Recently, the existence of a hidden order for the tracers which exhibits the possibility of tracers to travel in each other vicinity for relatively large times was exhibited (Leoncini 2002). This hidden order is related to the presence within the system of chaotic jets. This paper is organized as follows. First we start by basic definitions of the considered systems (point vortex systems and Charney-HasegawaMima equation), numerical settings and typical evolution of the flows are briefly discussed. We then move on to describe the transport properties of passive particles in these flows. For all considered cases transport is anomalous and superdiffusive. Finally we discuss the origin of anomalous transport and trace it back to the existences of chaotic jets within these flows.

2.

Basic definitions

2.1

Point vortices

Systems of point vortices are exact solutions of the two-dimensional Euler equation ∂Ω + [Ω, Ψ] = 0 ∂t ∆Ψ = Ω ,

(1) (2)

where Ω is the vorticity and Ψ is the stream function. The vortices describe the dynamics of the singular distribution of vorticity Ω(z, t) =

N

kα δ (z − zα (t)) ,

(3)

α=1

where z locates a position in the complex plane, zα = xα + iyα is the complex coordinate of the vortex α, and kα is its strength, in an ideal incompressible two-dimensional fluid. This system can be described by a Hamiltonian of N interacting particles (see for instance Lamb 1945), referred to as a system of N point vortices. The system’s evolution writes ∂H ∂H kα z˙α = −2i , z¯˙ α = 2i , (4) ∂ z¯α ∂(kα zα )

306 where the couple (kα zα , z¯α ) are the conjugate variables of the Hamiltonian H. The nature of the interaction depends on the geometry of the domain occupied by fluid. For the case of an unbounded plane, the resulting complex velocity field v(z, t) at position z and time t is given by the sum of the individual vortex contributions: N 1 1 kα , 2πi α=1 z¯ − z¯α (t)

(5)

1 kα kβ ln |zα − zβ | 2π α>β

(6)

v(z, t) = and the Hamiltonian becomes H =−

The translational and rotational invariance of the Hamiltonian H provides for the motion equations (4) three other conserved quantities, besides the energy. Among the different integrals of motion, there are three independent first integrals in involution, consequently the motion of three vortices on the infinite plane is always integrable and chaos arises when N ≥ 4 (Novikov 1975).

2.2

The Charney-Hasegawa-Mima equation

The Charney-Hasegawa-Mima equation can be written in the following form, ∂Ω +[Ω, Φ] = 0 ∂t Ω = Φ − λ∆Φ + gx ,

(7) (8)

where [·, ·] corresponds to the Poisson operator, Ω is a generalized vorticity given by Eq.(8), Φ is a potential and λ and g are parameters. Typically Eqs. (7) and (8) can either describe the evolution of an anisotropic plasma, and are then referred to as the Hasegawa-Mima equation or the evolution of geostrophic flows in which context they are known as the Charney equation. This formal identity has an advantage as the results obtained in this paper for transport properties being agnostic, they should apply in either context. The parameters λ and g may control typically two behavior of the equation. Namely the parameter g is the one which allows wave type solutions to exists, and in plasma turbulence, g −1√corresponds to a characteristic length of the anistropy. The parameter λ is called the hybrid larmor radius and corresponds in this context to screening effects. Typically , if we consider


307

g = 0, point vortex solutions of the equation exists, however the long range interaction between vortices is broken, see Eq.(6), and instead of the logarithm the interaction is governed the modified bessel function K0 .

2.3

Advection equation

For an incompressible fluid, the evolution of a passive particle is given by the advection equation z˙ = v(z, t)

(9)

where z(t) represent the position of the tracer at time t in the complex plane, and v(z, t) is the velocity field. An important feature of this evolution of passive particles is that the evolution equation given Eq.(9) can be rewritten as: z˙ = −i

∂Φ , ∂ z¯

z¯˙ = i

∂Φ , ∂z

(10)

where the potential Φ acts as a time dependant Hamiltonian. This Hamiltonian structure is fundamental as it imposes some constraints on the dynamics of passive tracers, which should be taken into account when carrying a numerical simulation.

2.4

Numerical settings

2.4.1 Charney-Hasegawa-Mima. Simulation of the HasegawaMima equation are computed for different initial conditions and choices of the parameters. The choices made here corresponds largely to make them consistent with the litterature, namely the conditions chosen in ( Annibaldi 2002). The simulations are performed within a square box of size L = 20 and periodic boundary conditions using a pseudo-spectral code. In order to compute the evolution of passive tracers accurately we settled for a somewhat low resolution mesh of 1282 . Fourier transform are computed using a fast Fourier algorithm. For the time evolution, we chose a 4th-order Runge-Kunta integration scheme with typical time step δt = 0.05. In order to avoid numerical instability Eq.(7) could not be kept as is and a dissipation term D as well as a forcing term F were added: ∂t Ω + [Ω, Φ] = D + F .

(11)

For the dissipation we considered a hyperviscous term D = ν(∇Ω)4 , as for the forcing we used a random phase forcing in Fourier space centered around a value k0 for a given δk range.

308 2.4.2 Passive tracers and point vortices. It is important to take special care of the way the dynamics of the tracers and the point vortices are computed in order to characterize the eventual anomalous properties of transport, which if present should find their origin in the existence of “memory effects”, namely long time correlations since the accessible range of speeds is finite. In this perspective, any source of randomness leading to memory loss due to the numerical scheme may then induce a spurious effective diffusive behavior and the Hamiltonian nature of the tracers dynamics imposes necessarly the choice of a simplectic integrator. Moreover in the Hasegawa-Mima situation we needed to compute the speed of particles “exactly” to avoid any source of noise, meaning that we performed an exact back-Fourier transform of the modes describing the evolution of the field. This constraint is numerically expensive, and explains our choice of a relatively low resolution mesh of 1282 . In this setting the evolution of passive tracers may be understood as describing the advection of particles in a flow field generated by 1282 modes interacting through Eq.(7).

2.5

Point Vortex systems

2.5.1 3-vortex systems. A general classification of different types of three vortex motion, as well as studies of special cases were addressed by many authors (Synge 1949; Novikov 1975; Aref 1979; Tavantzis 1988). Among the different possibilities quasiperiodic motion of the vortices is found generically for solutions for which the motion of the vortices is bound within a finite domain. In order to stress the results obtained in (Kuznetsov 1998; Kuznetsov 2000) we consider 3vortex systems in the vicinity of a configuration leading to finite time singular solution. Indeed three vortices can be brought to a single point in finite time by their mutual interaction. This phenomenon, known as point-vortex collapse was studied in (Aref 1979; Tavantzis 1988; Leoncini 2000), the motion is self similar, leading either to the collapse of the three vortices in a finite time, or by time reversal, to an infinite expansion of the triangle formed by the vortices. 2.5.2 4 and 16-vortex systems. Due to the generic chaotic nature of 4-point vortex system, understanding the vortex motion necessitates a different approach than for integrable 3-vortex systems. For identical vortices it is possible to perform a canonical transformation of the vortex coordinates (Aref 1982). For a 4-vortex system, this transformation results in an effective system with 2 degrees of freedom, providing a conceptually easier framework in which a well defined two-dimensional Poincare´ sections can be drawn (Aref 1982; Laforgia 2001). To summa-


309

rize the results, the motion is in general chaotic, except for some special initial conditions, for instance when the vortices are forming a square the motion is periodic and the vortices rotate on a circle, then symmetric deformation (z3 = −z1 and z4 = −z2 ) of the square lead to quasiperiodic motion (periodic motion in a given rotating frame). As we go from the 4 vortex system to 16 vortices the phase space dimension is considerably increased and due to the long range interaction between vortices (see the Hamiltonian 6) the energy does not behave as an extensive variable. Thus, in order to keep some coherence between the four vortex system and the sixteen vortex one, we chose to keep the average area occupied by each vortex approximatively constant. The simulations performed in (Laforgia 2001; Leoncini 2002) indicate that long time vortex pairing exists, in fact the formation of long-lived triplet (a system of 3 bound vortices) is also observed (Leoncini 2002), and thus the relevance of three vortex systems is confirmed. In fact, the formation of triplets or pair of vortices concentrates vorticity in small regions of the plane and in some sense is reminiscent of what is observed in 2D turbulence.

2.6

Charney-Hasegawa-Mima Field

For all vortex systems the velocity Field is completely determined by the position of the vortices and given by Eq.(5), but when dealing with the Charney-Hasegawa-Mima equation, these point vortex solution do not exist. We present here visualization of the fields in a typical configuration. Note also, that when considering the evolution of equation (7) we are faced with a choice of parameters and initial conditions, our approach has been to consider three different cases, with quite different values of the parameters. As mentionned earlier on we chose as a starting base (initial condition, type of forcing , size of the system) similar conditions as those set in (Annibaldi 2002). For the three cases considered, given the initial condition we let the system evolve until it can be considered “sationnary” for the time considered during simulation τf inal = 104 , stationnarity considered being reached by monitoring the evolution of the energy and enstrophy of the system. In order to visualize the field we chose to use levels of the function −∆Φ. The three different cases considered are represented in figures 1, 2 and 3.

2.6.1 Smooth Field. To obtain the “smooth” field depicted in Fig. 1, we carried out simulations with no forcing and low dissipation, the paramaters for this run were λ = 1, g = 0.1, ν = 7 10−6 . Due to this low dissipation the energy may be considered constant for the length of

310

Figure 1. Visualization of the field −∆Φ for the choice of parameters λ = 1, g = 0.1, ν = 7 10−6 , L = 20, N = 1282 with no forcing. The field is “smooth” and appears as being isotropic. A few vortices are present. The black spots marks the position of some passive particles.

the simulation. We can notice that a few distinct vortices are present, during the evolution a merger between two vortices occur, one can also notice an average drift in the y-direction.

2.6.2 Forced Field. To obtain the “forced” field depicted in Fig. 2, we carried out simulations with a strong forcing and dissipation, the paramaters for this run were λ = 4, g = 0.1, ν = 5 10−5 , F = 4, k0 = 6 ± 2, the phases for the random forcing are updated every ∆t = 2 time units. The value of k0 corresponds to physical scales of δx ∼ 2π k0 ≈ 1. With this choice of parameters the system consists of two perturbed vortices, an average drift in the y-direction is also noticeable. 2.6.3 Anisotropic Field. To obtain the “anisotropic ” field depicted in Fig. 3, we carried out simulations with some forcing and a high value for g, the paramaters for this run were λ = 0.125, g = 2, ν = 7.5 10−6 , F = 1.5, k0 = 12 ± 2, ∆t = 2. The value of k0 corresponds to physical scales of δx ∼ 2π k0 ≈ 0.5. In this settings elongated structures as well as a strong drift in the y-direction is present.


311

Figure 2. Visualization of the field −∆Φ for the choice of parameters λ = 4, g = 0.1, ν = 5 10−5 , F = 4, k0 = 6 ± 2 , ∆t = 2. Two big perturbed vortices are present. The black spots marks the position of some passive particles.

Figure 3. Visualization of the field −∆Φ for the choice of parameters λ = 0.125, g = 2, ν = 7.5 10−6 , F = 1.5, k0 = 12 ± 2, ∆t = 2. Elongated structures in the y-direction are present, marking a strong anisotropy. The black spots marks the position of some passive particles.

312

3. 3.1

Transport properties Definitions

Unfortunately the deterministic description of the motion of a passive particle in a chaotic region is impossible. Local instabilities produce exponential divergence of trajectories, thus even an idealized numerical experiment is non-deterministic. The long-time behavior of tracer trajectories is then necessarly studied by using a probabilistic approach. In the absence of long-term correlations, the kinetic description leads to Gaussian statistics. Yet if a phenomenon with associated long time correlations occurs, profound changes in the kinetics can be induced, and often and leads to non-Gaussian statistics and for instance a nondiffusive behavior of the particle displacement variance: !(s − !s")2 " ∼ tµ ,

(12)

where !· · ·" stands for ensemble averaging. For a superdiffusive case the transport exponent µ exceeds the Gaussian value: µ > 1. Within this probablistic approach, the main observables in order to characterize transport properties will be moments of the distributions: Mq (t) = !|s(t) − !s(t)"|q " ,

(13)

where q denotes the moment order. The physical restriction of a finite velocity and of a finite time of our simulations implies that all moments are finite and a power law behavior is expected Mq ∼ Dq tµ(q) .

(14)

with, generally, µ(q) = q/2 as is expected from normal diffusion. The nonlinear dependence of µ(q) is a signature of the multifractality of the transport. Based on the beahvior of µ(q) two types of anomalous diffusion were distinguished, and the notion of weak and strong anomalous diffusion (Castiglione 1999). When the evolution of all of the moments can be described by a single self-similar exponent ν, i.e µ(q) is linear one refers to “weak anomalous diffusion”, whereas the case when µ(q) is nonlinear is named “strong anomalous diffusion”. This distinction is important since in the weak case the PDF must evolve in a self-similar way while a non-constant ν(q) precludes such self-similarity (see the discussions in (Kuznetsov 2000; Leoncini 2001) for details about the non self-similar behavior). Before going on we insist on one last point; what we measure. We chose the arclength s(t) of the path traveled by an individual tracer up to a time t, t

si (t) =

0

vi (t )dt ,

(15)

313

Anomalous transport in two-dimensional plasma turbulence 8 7 6

µ(q)

5 4 3 2 1 0 0

2

4 q

6

8

Figure 4. Large-time behavior for the different moments |s(t)− < s(t) > |q >∼ µ(q) t for the flow driven by 16 point vortices, with µ(2) ≈ 1.77 corresponding to a superdiffusive regime.

where vi (t ) is the absolute speed of the particle i at time t . We recall that in order to consider mixing properties from the dynamical principles it is important to consider the trajectories within the phase space. For the considered case the phase space is “identical” to the physical space where particles evolve (see 10). One advantage of s(t) is that it is independent of the coordinate system and as such we can expect to infer intrinsic properties of the dynamics, moreover problems related to a finite accessible phase space become unrelevant.

3.2

Particles Transport

3.2.1 Point-Vortex systems. In all our simulations, the results show that the transport of tracers in point-vortex systems is strongly anomalous and super-diffusive, hence to avoid redundancy we graphically only present the transport properties of passive tracers obtained for a 16-point vortex flow in Fig. 4. The behavior for the other considered systems is very similar, namely our results show, that for all considered systems µ(q) is well approximated by a piecewise linear function.

314 8

30

7 25

6

µ(q)

5

10

q

log [M (τ)]

20

15

4 3

10

2 5

1 0 2

2.5

3 log10(τ)

3.5

4

0 0

2

4 q

6

8

Figure 5. Left: Moments of distribution of the arclength Mq (τ ) = |s(τ ) − s(τ )|q versus time of tracers evolving in the smooth field for q = 1/2, 1, 3/2, · · · 8. The behavior Mq (τ ) ∼ τ µ(q) is confirmed. Right: Characteristc exponent versus moment order, q vs µ(q), with µ(2) = 1.81. Transport is super-diffusive and almost weakly anomalous.

Even though we only considered few different initial condition for the different vortex system, it is reasonable to assume that the transport properties obtained for such systems are fairly general since all give the same kind anomalous behavior with a transport exponent more or less around µ(2) ± 1.75.

3.2.2 Hasegawa-Mima system. As mentionned in order to study the transport properties of passive tracers in Charney-HasegawaMima flows, we considered three different cases with different choices of parameters and forcing. In these field we followed the evolution of 512 passive particles given by Eq.(9), and measured the arclength si (t) travelled by each tracer i, up to a time τ = 104 . In order to keep a constant accuracy the increments ∆si (t) where recorded for successive diagnostic times. This last feature has also the advantage of allowing better statistics when computing moments. Indeed since we have chosen stationary regimes for the field, we can assume that transport properties are independent from the initial condition of the field and thus use timeinvariance to increase statistics. In all three considered cases the transport is found to be anomalous and superdiffusive, with a characteristic second order exponent all of the same order µ(2) ≈ 1.8. The time behavior of the moments and characteristic exponents is plotted in Fig.5 for reference.

Anomalous transport in two-dimensional plasma turbulence Point vortices Charney-Hasegawa-Mima

Table 1.

3.3

315

4 vortices µ(2) ≈ 1.82 16 vortices µ(2) ≈ 1.77 Smooth Field µ(2) ≈ 1.81 Forced Field µ(2) ≈ 1.73 Anisotropic Field µ(2) ≈ 1.85

Characteristic second moment exponent for the different cases studied.

Universality

A summary of the transport properties obtained for the various system is provided in table 1. The similarity observed for the exponents in these quite different settings of the parameters and regimes of the CharneyHasegawa-Mima equation as well as the one observed in point vortices, raises the possibility of some kind of universal behavior for the transport of passive tracers in these two-dimensional flows.

4.

Stickiness and Jets

In this section, we remind briefly the results presented in (Kuznetsov 1998; Kuznetsov 2000; Laforgia 2001; Leoncini 2001; Leoncini 2002) and discuss the origin of anomalous transport in the considered systems. As mentionned, transport being anomalous, there are long-time correlation and memory effects in the dynamics of tracers. Hence we shall review the origin of these effects and shall start to illustrate the phenomenon of stickiness in the case of three vortex systems for which it is the most obvious. In these systems the structure of the chaotic region is quite complex, with an infinite number of KAM-islands (stratified with regular trajectories) of different shapes and sizes embedded into it. To visualize the phase space and these structures, we construct Poincar´é sections of tracer trajectories (in the co-rotating frame). As already performed in , a plot of Poincare´ section for three vortices and localization of the sticky regions in this plot can be done and are illustrated in Fig.6 . We notice that in this case the sticky regions are located around islands. In their vicinity tracers mimic the quasiperiodic behavior of tracers within the islands, generating long time correlations. In this conetxt, the origin of anomalous transport is fairly well understood when one is able to draw a phase portrait using a Poincar´ é, however, when dealing with a more complex systems, for which the drawing of a phase portrait is not achievable, one has to rely on other techniques. . Namey as we have seen, sticky zones are regions where particles are trapped and therefore are regions where particles remain in each others neighborhood for large times. It becomes therefore natural when

316

Figure 6. Poincare´ Map for a system of point vortices (left). Sticky regions are localized on the Map (right), see Leoncini 2001 for details.

ε

Figure 7.

Tracking of coarse-grained regular jet.


317

dealing with more complex systems to look for places where passive particles remain in each other’s vicinity for large times. We thus look for chaotic jets (Leoncini 2002). These chaotic jets can be understood as moving clusters of particles within a specific domain for which the motion appears as almost regular from a coarse grained perspective. From another point of view, looking for chaotic jets can be understood as a particular case of measurements of space-time complexity (Afraimovich 2003). In order to look for jets we proceed as described in the illustration presented in Fig. 7 which provides an easy and intuitive description of the mechanism used to detect jets. To resume, we consider a reference trajectory r(t) within the phase space. We then associate to this trajectory a corresponding “coarse grained” equivalent, i.e the reunion of the balls ∪B(r(t), ) of radius whose center is the position r(t). Given an -coarse grained trajectory, we analyse the behavior of real trajectories starting from within the ball at a given time and measure the time τ and length s, corresponding to actually how long the trajectory remains and how much it travels before its first escape from the coarse grained trajectory. We then analyse the resulting distributions. This approach has revealed itself very fruitfull when considering the advection of passive tracers in flows governed by point vortices (Leoncini 2002), pinpointing the existence of jets as the origin of anomalous transport in these systems. Typically, Tthe main difficulty in using this diagnostic follows from the fact that data acquisition is not sampled linearly in time nor space, a point which lead in the present case too some difficulties. We recall now the results obtained in (Leoncini 2002). The trapping time distribution of particles within these jet show a power law behavior for the vortex systems, characteristic of stickiness. The characteristic exponent for trapping times observed in vortex systems is typically γ ≈ 2.8, and thus an agreement with the expected γ ≈ µ(2) + 1 relation is observed (see Leoncini 2001; Leoncini 2002 for a discussion on this law), where µ refers to the characteristic transport exponent. In these system the jest are located around the core of the vortices and in the region far away from the vortices. Moreover current investigations for HasegawaMima flows, which will be described in another article, show that jets are also present in these systems. As an illustration we provide in Fig. the localization of a jet in the forced-case of Hasegawa-Mima. In this case the jet is bouncing back and forth between the two structures illustrated in Fig.2, while moving upwards one box size between bounces.

318

Figure 8. Localization of a coarse-grained regular jet in the forced HasegawaMima flow. The jet is bouncing back and forth between the two structures illustrated in Fig.2.

5.

Conclusion

We may conclude from thises analysis that there is some evidence of universal behavior for the transport of passive tracers in two dimensional flows. Indeed in all cases considered the same type of behavior, namely a superdiffusive transport with a characteristic transport exponent µ ≈ 1.75. This phenomenon is explained by the phenomenon of stickiness and the origin of the anomaly is traced back to the existence of coherent non-dispersive jets in these flows. Moreover, given the good agreement found on the γ = µ + 1 law, we may speculate that studying transport properties in these flows can be obtained by directly looking for jets and analysing trapping time. This strategy would then give directly the transport properties as well as the origin of the anomaly, and practically it can reveal itself also less costly numerically.

Acknowledgments The authors would like to thank Dr. L. Kuznetsov for usefull discussions.


319

References V. Afraimovich and G. M. Zaslavsky,Space-Time Complexity in Hamiltonian Dynamics, Chaos 13, 519 (2003) S. V. Annibaldi, G. Manfredi, R. O. Dendy, Non-Gaussian transport in strong plasma turbulence, Phys. Plasmas, 9, 791 (2002) H. Aref, Motion of three vortices, Phys. Fluids 22, 393 (1979) H. Aref and N. Pomphrey, Integrable and chaotic motions of four vortices: I. the case of identical vortices, Proc. R. Soc. Lond. A 380, 359 (1982) G. F. Carnevale, J. C. McWilliams, Y. Pomeau, J. B. Weiss and W. R. Young, Evolution of Vortex Statistics in Two-Dimensional Turbulence, Phys. Rev. Lett. 66, 2735 (1991) P. Castiglione, A. Mazzino, P. Mutatore-Ginanneschi, A. Vulpiani, On Strong anomalous diffusion, Physica D, 134, 75 (1999) O. U. Velasco Fuentes, G. J. F. van Heijst, N. P. M. van Lipzig, Unsteady behaviour of a topography-modulated tripole, J. Fluid Mech. 307, 11 (1996) L. Kuznetsov and G.M. Zaslavsky, Regular and Chaotic advection in the flow field of a three-vortex system, Phys. Rev E 58, 7330 (1998) L. Kuznetsov and G. M. Zaslavsky, Passive particle transport in three-vortex flow. Phys. Rev. E. 61, 3777 (2000) A. Laforgia, X. Leoncini, L. Kuznetsov and G. M. Zaslavsky, Passive tracer dynamics in 4 point-vortex-flow, Eur. Phys. J. B, 20, 427 (2001) H. Lamb, Hydrodynamics, (6th ed. New York, Dover, 1945) X. Leoncini, L. Kuznetsov and G. M. Zaslavsky, Motion of Three Vortices near Collapse, Phys. Fluids 12, 1911 (2000) X. Leoncini, L. Kuznetsov and G. M. Zaslavsky, Chaotic advection near 3-vortex Collapse, Phys. Rev.E, 63, 036224 (2001) X. Leoncini and G. M. Zaslavsky, Jets, Stickiness, and anomalous transport, Phys. Rev.E, 65, 046216 (2002) E. A. Novikov, Dynamics and statistics of a system of vortices, Sov. Phys. JETP 41, 937 (1975) J. L. Synge, On the motion of three vortices, Can. J. Math. 1, 257 (1949) P. Tabeling, A.E. Hansen, J. Paret, Forced and Decaying 2D turbulence: Experimental Study, in “Chaos, Kinetics and Nonlinear Dynamics in Fluids and Plasma”, eds. Sadruddin Benkadda and George Zaslavsky, p. 145, (Springer 1998) J. Tavantzis and L. Ting, The dynamics of three vortices revisited, Phys. Fluids 31, 1392 (1988) N. J. Zabusky, J. C. McWilliams, A modulated point-vortex model for geostrophic, β-plane dynamics, Phys. Fluids 25, 2175 (1982) G. M. Zaslavsky, Chaos, Fractional Kinetics, and Anomalous Transport, Phys. Rep., 371, 641 (2002)


THE ONSET OF SYNCHRONISM IN GLOBALLY L COUPLED ENSEMBLES OF CHAOTIC AND PERIODIC DYNAMICAL UNITS Edward Ott Institute for Research in Electronics and Applied Physics, Department of Physics, and Department of Electrical and Computer Engineering,University of Maryland, College Park, Maryland, 20742 [email protected]

Paul So Department of Physics and Astronomy and the Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia, 22030 [email protected]

Ernest Barreto Department of Physics and Astronomy and the Krasnow Institute for Advanced Study, George Mason University, Fairfax, Virginia, 22030 [email protected]

Thomas Antonsen Institute for Research in Electronics and Applied Physics, Department of Physics, and Department of Electrical and Computer Engineering, University of Maryland, College Park, Maryland, 20742 [email protected]

Abstract

A general stability analysis is given to describe the transition from incoherent to coherent behavior in globally-coupled systems of units whose individual uncoupled dynamics are chaotic and/or periodic.

Keywords: Synchronization, chaos, Kuramoto


322

Introduction Systems that consist of many coupled heterogeneous dynamical units are of great interest in a wide variety of situations. Past work has concentrated on the globally coupled case where the dynamics of the uncoupled units is periodic with a spread in the oscillator frequencies [1]. In that case, for sufficiently low coupling, the individual units oscillate incoherently, and, as the coupling is increased through a critical value, there is a transition to coherent dynamics in which a group of oscillators becomes locked in frequency and phase. Applications include synchrony in chirping crickets [2], flashing fireflies [3], Josephson junction arrays [4], semiconductor laser arrays [5], and cardiac pacemakers cells [6]. Recently, the case in which the individual units are chaotic has been addressed [7, 8, 9]. Here, we discuss a formalism that is capable of treating the onset of synchronism in a general system of globally coupled, heterogeneous, continuous-time dynamical units. No a priori assumption regarding the uncoupled dynamics of the individual units is made. Thus, one can consider chaotic or periodic dynamics of the uncoupled units, including the case where both types of units are present in the same system.

1.

Numerical example

We first discuss a numerical example. We consider an ensemble of coupled Lorenz equations: (1)

(2)

(1)

dxi /dt = σ(xi − xi ) + (k/N ) (2) (1) (2) (1) (3) dxi /dt = ri xi − xi − xi xi (3) (3) (1) (2) dxi /dt = −bxi + xi xi .

N

(1) i=1 xi (t)

(1)

Here, N >> 1 is the number of units in the ensemble. In our analysis, we will take N → ∞. We set σ = 10 and b = 8/3, and draw the parameters ri from a uniform distribution in the interval [28, 52], for which the uncoupled Lorenz equations yield chaotic solutions with no discernable windows. Numerical results for other ensembles, including a periodic ensemble containing a pitchfork bifurcation and a mixed ensemble with both chaotic and periodic units, are reported elsewhere [10]. Figure 1 shows an order parameter x ¯T versus the coupling coefficient k, where x ¯T =

 1 T

t

t+T

N 1

N

i=1

2

xi (t ) (1)

dt

1/2  

characterizes the degree of coherent motion. The time t is chosen large

323

The Onset of Synchronism in Globally Coupled Ensembles

20 15 xT

10 5 0 -6

-5 Figure 1.

k

-4

-3

x ¯T versus k.

enough that the system settles into its time-asymptotic dynamics, while the averaging duration T is chosen large enough that x ¯T becomes essentially independent of T . Note that, if the individual units behave incoherently, the sum is close to zero by the x(1) reflection symmetry of the uncoupled Lorenz equations. We observe a subcritical bifurcation as k decreases through ka = −5.6. Other ensembles investigated include sub- and supercritical Hopf bifurcations. It is our goal to obtain a theory for the critical k values at the onset of coherence. These mark the onset of instability of the incoherent state. We are also interested in the exponential growth rates of these instabilities.

2. 2.1

General Treatment Formulation

We now present our analysis, treating the simplest case. Generalizations are given elsewhere [10]. We consider dynamical systems of the form dxi (t)/dt = G(xi (t), Ωi ) + K(!!x""∗ − !!x(t)""), (1)

(2)

(q)

(2)

where xi = (xi , xi , . . . , xi )T ; G is a q-dimensional vector function; K is a constant q × q coupling matrix; i = 1, 2, · · · , N ; !!x(t)"" is the

324 instantaneous average, !!x(t)"" = lim N −1

N →∞

!xi (t)",

(3)

i

and, for each i, !xi " is the average of xi over an infinite number of initial conditions xi (0) distributed on the attractor of the ith uncoupled system, dxi /dt = G(xi , Ωi ). (4) Ωi is a parameter vector specifying the uncoupled (K = 0) dynamics, and !!x""∗ is the natural measure [11] and i average of the state of the uncoupled system. That is, to compute !!x""∗ , we set K = 0, compute the solutions to Eq. (4), and obtain !!x""∗ from !!x""∗ = lim N −1 N→∞

i

lim τ0−1

τ0 →∞

τ0 0

xi (t)dt .

(5)

In what follows we assume that the Ωi are randomly chosen from a smooth probability density function ρ(Ω). By construction, !!x"" = !!x""∗ is a solution of the N → ∞ globally coupled system (2). We call this solution the “incoherent state” because the coupling term cancels and the individual oscillators do not affect each other. We address the stability of the incoherent state. We envision that, as a system parameter such as the coupling strength varies, the onset of instability of the incoherent state signals the start of coherent, synchronous behavior of the ensemble.

2.2

Stability analysis

To perform the stability analysis, we assume that the system is in the incoherent state, so that for each i at any fixed time t, xi (t) is distributed according to the natural measure. We then perturb the orbits xi (t) → xi (t) + δxi (t), where δxi (t) is an infinitesimal perturbation: dδxi /dt = DG(xi (t), Ωi )δxi − K!!δxi "",

(6)

where

∂ G(xi (t), Ωi ). ∂xi Introducing the fundamental matrix Mi (t) for Eq. (6), DG(xi (t), Ωi )δxi = δxi ·

dMi /dt = DG · Mi ,

(7)

where Mi (0) ≡ I, we can write the solution of Eq. (6) as δxi (t) = −

t −∞

Mi (t)M−1 i (τ )K!!δx""τ dτ,

(8)

325


where !!δx""τ signifies that !!δx"" is evaluated at time τ . Note that, through Eq. (7), Mi depends on the unperturbed orbits xi (t) of the uncoupled nonlinear system (4), which are determined by their initial conditions xi (0) (distributed according to the natural measure). Assuming that !!δx"" evolves exponentially in time (i.e., !!δx"" = ∆est ), Eq. (8) yields ˜ {I + M(s)K}∆ = 0, (9) where s is complex, and GG

˜ M(s) =

t −∞

e

−s(t−τ )

Mi (t)M−1 i (τ )dτ

HH

.

(10)

∗

Thus the dispersion function determining s is ˜ D(s) = det{I + M(s)K} = 0.

(11)

In order for Eqs. (9) and (11) to make sense, the right side of Eq. (10) must be independent of time. We now demonstrate this, and express ˜ M(s) in a more convenient form. Writing the dependence of Mi in Eq. (10) on the initial condition explicitly, we have from the definition of Mi , Mi (t, xi (0))M−1 i (τ, xi (0)) = Mi (t − τ, xi (τ )) = Mi (T, xi (t − T )), where T = t − τ . Using this in Eq. (10) we obtain ˜ M(s) =

GG

∞

−sT

e 0

HH

Mi (T, xi (t − T )dT

∗

.

Our solution requires that this integral converge. Since the growth of Mi with increasing T is dominated by hi , the largest Lyapunov exponent for orbit xi , we require Re(s) > Γ, where Γ = maxxi ,Ωi (hi ). In this case, the integral converges exponentially and uniformly, and the order of the integration and the averaging operations can be interchanged: ˜ M(s) =

∞ 0

e−sT !!Mi (T, xi (t − T ))""∗ dT.

(12)

Here the only dependence on t is through the initial condition xi (t − T ). The average over the initial conditions, which are distributed according to the invariant natural measure, ensures that !!Mi (T, xi (t − T ))""∗ is ˜ is the Laplace transform of !!M""∗ . As shown independent of t. Thus M ˜ below, M(s) can be analytically continued into Re(s) < Γ.

326

2.3

Discussion

˜ M(s) depends only on the solution of the linearized uncoupled system (Eq. (7)). Hence the utility of the dispersion function D(s) given by Eq. (11) is that it determines the linearized dynamics of the globally coupled system in terms of those of the individual uncoupled systems. Consider the kth column of !!M(t)""∗ , denoted [!!M(t)""∗ ]k , which we interpret as follows. Assume that for each of the uncoupled systems i in Eq. (4), we have a cloud of an infinite number of initial conditions distributed according to the natural measure on the uncoupled attractor. Then, at t = 0, we apply an equal infinitesimal displacement δk in the direction k to each orbit in the cloud. The quantity [!!M""∗ ]k δk gives the time evolution of the i-averaged perturbation of the centroid of the clouds as the perturbed orbits evolve back to the attractor and redistribute themselves on the attractor. We now argue that !!M""∗ decays to zero exponentially with increasing time. We consider the general case where the support of the smooth density ρ(Ω) contains open regions of Ω for which Eq. (4) has attracting periodic orbits as well as a positive measure of Ω on which Eq. (4) has chaotic orbits. Numerical experiments on chaotic attractors (including structurally unstable attractors) generally show that they are strongly mixing; i.e., a cloud of many particles rapidly arranges itself on the attractor according to the natural measure. Thus, for each Ωi giving a chaotic attractor, it is reasonable to assume that the average of Mi over initial conditions xi (0), denoted !Mi "∗ , decays exponentially. For a periodic attractor, however, !Mi "∗ does not decay, but periodic orbits exist in open regions of Ω, and when averaged over Ω, there is the possibility that with increasing time, cancellation, causing decay, occurs via the process of “phase mixing”. For this case we appeal to an example. The explicit computation of !Mi "∗ for a simple model limit cycle ensemble is given in Ref. [10]. The result is 1 !Mi "∗ = 2

cos Ωi t − sin Ωi t sin Ωi t cos Ωi t

,

and indeed this oscillates and does not decay to zero. However, if we average over the oscillator distribution ρ(Ω) we obtain ˜ ∗= !!M""

1 2

c(t) −s(t) s(t) c(t)

,

where c(t) = ρ(Ω)cosΩtdΩ and s(t) = ρ(Ω) sin ΩtdΩ. For any analytic ρ(Ω) these integrals decay exponentially with time. Thus, for sufficiently smooth ρ(Ω), there is reason to believe that !!M""∗ decays


327

to zero with increasing time. Conjecturing this decay to be exponential, !!M(t)""∗ < κe−ξt for positive constants κ and ξ, we see that the integral in Eq. (12) converges for Re(s) > −ξ. This conjecture is supported by our numerical results. Thus, using analytic continuation, we can regard Eq. (12) as valid for Re(s) > −ξ.

3.

Numerical implementation

In order to apply Eq. (11) to a given situation, it is necessary to ˜ numerically approximate the matrix M(s). We consider two possible candidate approaches. Approach (i): First approximate the natural measure on each attractor i by a large finite number of orbits initially distributed according to the natural measure. For each initial condition, obtain xi (t) from Eq. (4). Use these solutions in DG and solve Eq. (7). Then average over the natural measure and i to obtain !!M(t)""∗ , and apply the Laplace transform as in Eq. (12). Approach (ii): Since !!M""∗ is the response to an impulse (i.e., the ˜ sudden displacement of each orbit), its Laplace transform M(s) multiplied by exp (st) is the response to the drive I exp (st) added to the right side of Eq. (6). This suggests the following numerical prodedure ˜ for finding M(s). Solve (c,s)

d˜ xi dt

=

(c,s) G(˜ x i , Ωi ) +

σt

∆k ak e

cos ωt sin ωt,

(13)

where s = σ − iω and ak is a unit vector in the direction k. For large t, but ∆k exp (σt) still small in the time interval (0, t), we can regard the average response as approximately linear. Thus, the kth column of ˜ M(s) is −1 −st ˜ [M(s)] [!!x""∗ − !!˜ x""], (14) k ∆k e (c)

(s)

˜i = x ˜ i − i˜ where x xi . Numerically, !!˜ x"" can be approximated using a large finite number of orbits. In Ref. [8], a technique equivalent to this with s taken to be imaginary (s = −iω) was used to obtain marginal stability.

4.

Example

In the example introduced in Section 1, we chose the coupling such ˜ that only the (1, 1) element of M(s) was nonzero. Thus Eq. (11) reduces to ˜ 11 (−iω)k = 0 1+M (15)

ω

328

ω ˜ 11 (−iω) versus ω. The solid black line is Re(M ˜ 11 ), approach (ii); the Figure 2. M ˜ ˜ 11 ), approach solid grey line is Im(M11 ), approach (ii); and the dashed line is Re(M (i).

˜ 11 (−iω) = 0 yields possibly where we have set s = −iω. Solving ImM ∗ multiple roots ω = ω , which, when reinserted into Eq. (15), yield possi˜ 11 (−iω ∗ )]−1 for the critical coupling strengths. ble values k = k∗ = −[M To determine which of these are relevant, we envision that as k is increased or decreased from zero, a critical coupling value is encountered at which the incoherent state first becomes unstable. Hence we are in∗ terested in the roots ωa,b corresponding to the smallest |k∗ | for k∗ both ∗ ∗ negative (k = −|ka |) and positive (k∗ = kb∗ > 0). Growth rates and frequency shifts from ω ∗ can also be simply obtained for k near k∗ by setting k = k∗ + δk, s = −i(ω∗ + δω) + γ and expanding Eq. (11) for small δk, δω, and γ; e.g., γ=−

˜ 11 (−iω)]/∂ω δk ∂Im[M ∗ 2 ˜ 11 (−iω)/∂ω|2 (k ) |∂ M

(16)

˜ 11 /∂ω is evaluated at ω = ω ∗ . where ∂ M ˜ 11 (−iω)] and The black and grey solid lines in Fig. 2 show Re[M ˜ 11 (−iω)] versus ω for the chaotic Lorenz ensemble in Eq. (1). Here, Im[M ˜ 11 (−iω)] we used approach (ii) with ∆x = 2 and N = 20, 000. Im[M ∗ ˜ crosses zero only at ω = 0, where Re[M11 (−iω)] has a prominent peak.

329


0.20 0.15

γ

0.10 0.05 0.00 -0.6

-0.4 Figure 3.

δk

-0.2

0.0

The growth rate γ versus δk.

This gives a critical coupling value of −5.6±0.15 in reasonable agreement with the threshold for coherence observed in Fig. 1. Figure 3 shows the instability growth rate from Eq. (16) versus δk as a solid line, along with values observed from simulations of the full nonlinear system plotted as dots. To obtain the latter data, we first initialize the ensemble in the incoherent state by time evolution with the coupling k set to zero. We then set k = −|k∗ | + δk, plot ln!!x(1) "" versus t, and fit a line to the resulting graph during the exponential growth phase. These data agree well with Eq. (16) for 0 ≥ δk ≥ −0.6 (Fig. 3). Figure 4 shows !!M M11 (t)""∗ versus t obtained by approach(i). !!M M11 (t)""∗ decays as expected for t ≤ 0.7, but then shows apparent divergent behavior. This can be understood on the basis that the individual Mi (t) for each orbit diverge exponentially at their largest Lyapunov exponent. Since !!M M11 (t)""∗ decays, the averaging process must result in a cancellation of the exponential growth components, and this cancellation becomes more and more delicate as time increases. Thus, for any finite N , divergence of the method will always occur at large time. Calculating ˜ 11 (−iω) from the result in Fig. 4 by doing the Laplace transform ReM only over the reliable range 0 ≤ t ≤ 0.7, we obtain the result shown in Fig. 2 as the dashed curve. While there is reasonable agreement with the result from approach (ii) for ω ≥ 0.1 (in Fig. 2), approach (i) fails

330

Figure 4.

M11 (t)∗ from approach (i) versus t, N = 20, 000.

to capture the important sharp increase to the peak at ω = 0 which occurs for for ω ≤ 0.1. This feature corresponds to a time scale 1/ω ∼ 10 which is well past the finite N -induced divergence in Fig. 4. Thus approach (i) yields a value of |k∗ | that is too large (by a factor of order 2). While approach (i) fails in this case, it can be useful in other cases depending on the strength of the divergences that the system exhibits, and particularly in the case of periodic ensembles where Mi does not grow exponentially [10].

5.

Conclusion

We have presented a general formulation for the determination of the stability of the incoherent state of a globally coupled system of continuous time dynamical units. The formalism is valid for both chaotic and periodic dynamics of the individual units. We discuss the analytic ˜ properties of M(s) and its numerical determination. We find that these ˜ are connected: analytic continuation of M(s) to the imaginary axis is necessary for application of the analysis, but in the chaotic case, can ˜ lead to numerical difficulties in determining M(s) (Fig. 4). Our numerical example illustrates the validity of the approach, as well as practical limitation to numerical application.


331

Acknowledgments This work was supported by grants from ONR (physics), NSF (PHYS0098632 and IBN9727739), and NIH (K25MH01963).

References [1] Y. Kuramoto, in International Symposium on Mathematical Problems in Theoretical Physics, edited by H. Araki, Lecture Notes in Physics, Vol. 39 (Springer, Berlin, 1975); Chemical Oscillators, Waves and Turbulence (Springer, Berlin, 1984); A. T. Winfree, The Geometry of Biological Time (Springer, New York, 1980); for a review of work on the Kuramoto model, see S. H. Strogatz, Physica D 143, 1 (2000). [2] T. J. Walker, Science 166, 891 (1969). [3] J. Buck, Q. Rev. Biol. 63, 265 (1988). [4] K. Wiesenfeld, P. Colet and S. H. Strogatz, Phys. Rev. Lett. 76, 404 (1996). [5] D. V. Ramana Reddy, A. Sen and G. L. Johnston, Phys. Rev. Lett. 80, 5109 (1998). [6] C. S. Peskin, Mathematical Aspects of Heart Physiology (Courant Institute of Mathematical Sciences, New York, 1975). [7] A. S. Pikovsky, M. G. Rosenblum and J. Kurths, Europhys. Lett. 34, 165 (1996). [8] H. Sakaguchi, Phys. Rev. E 61, 7212 (2000). [9] D. Topaj, W. -H. Kye and A. Pikovsky, Phys. Rev. Lett. 87, 074101 (2001). [10] E. Ott, P. So, E. Barreto, and T.M. Antonsen, Physica D 173, 226-258 (2002). [11] E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, 1993), Chapter 3.


QUANTUM BREAKING TIME FOR CHAOTIC SYSTEMS WITH PHASE SPACE STRUCTURES A. Iomin Department of Physics, Technion, Haifa, 32000, Israel. [email protected]

G.M. Zaslavsky Courant Institute of Mathematical Sciences, New York University, 251 Mercer Str., New York, NY 10012 and Department of Physics, New York University, 2-4 Washington Place, New York, NY 10003. [email protected]

Abstract The breaking time, known also as the Ehrenfest time, of the quantum– classical crossover scales logarithmically with respect to the Planck constant in chaotic systems. A typical dynamical system is not ergodic and not uniformly hyperbolic. Deviations from the logarithmic scale have been observed for systems with phase space structures, even in the case of strong chaos, when the homogeneous hyperbolicity of phase space does not hold. In this case the breaking time depends on scaling properties of phase space structures. Two examples of chaotic motion with different kind of phase space structures are presented here. The first example is quantum flights studied in the kicked rotor in the presence of the accelerator mode island structure. The second example is a model of periodically kicked harmonic oscillator with dissipation.

Keywords: Semiclassical approximation, quantum–classical breaking time, Ehrenfest time, superdiffusion, chaotic attractor.

1.

Introduction

Classical chaotic dynamics can be characterized by a Lyapunov exponent Λ and infinitely divisible filamentation of the phase flux. Quantized 333 P. Collet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 333–348. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

334 procedure stops the filamentation due to the uncertainty principle and, as a result, breaks the applicability of semiclassical approximation. The corresponding breaking time was found in [1] τ¯h = (1/Λ) ln(II0 /h ¯ ),

(1.1)

where I0 is a characteristic action, indicating a fast (exponential) growth of quantum corrections to the classical dynamics due to chaos. The origin of this time was explained in detail in [2] and gained a wide discussion, see, e.g., [3, 4, 5, 6, 7], and recent results [8, 9]. It should be outlined that it is a breaking time of the exact quantum to classical correspondence between the Hamiltonian equations of motion and the Ehrenfest equations. To stress this phenomenon the breaking time of Eq. (1.1) is named the Ehrenfest time [5]. Expression (1.1) was proved together with evaluation of the remainder, in [7]. Recently, a growing interest to this result reveals with extensive investigations [8, 10] and application in condensed matter physics [11, 12]. The logarithmic scaling in h ¯ for τ¯h corresponds to a fairly good and uniform chaotic mixing (see also a final comment in [13]). A typical dynamical system is not ergodic (or fully chaotic), for example, either typical Hamiltonian dynamics with a complicated phase space structure of islands (see Fig. 1), or a simple dissipative system with a strange or chaotic attractor, as shown in Fig. 2. Two examples of both unitary evolving [14] and dissipative [15] systems will be considered below in the paper. In these cases the quantum breaking time differs from τ¯h . The first system is a quantum kicked rotor (QKR) in the presence of the accelerator mode islands. The classical dynamics of the system is not ergodic due to the presence of infinite islands set in phase space, as shown in Fig. 1, and the Lyapunov exponent is not uniform due to cantori and possible hierarchical structures of islands [16, 17] and their stickiness [18, 19]. This complicates the process of diffusion, transforming the transport from the Gaussian type to the anomalous (fractional) one [20, 17]. Particularly, sticky properties of the island boundaries should impose algebraic laws of the survival probability P (t) ∼ 1/tβ

(1.2)

of a particle to escape after time t from a domain near the islands [20, 19]. We borrow this notation for the exponent from [17] to underline the relation of Eq. (1.2) to the Poicaré recurrences, which asymptotic is PP R (t) ∼ 1/tγ , and γ = 1 + β (see discussion in [17]). The immediate consequence from the scaling property (1.2) is its breaking at some critical time τ ∗ for the case of quantum chaos since a hierarchical dynamical

Quantum breaking timefor chaotic systems withphase space structures

335

chain has no limit t → ∞ and should be abrupt when an island in the ¯ and the quantum effects become important. chain reaches area S ∗ = h This comment, starting from [21], was discussed in detail in [13], where a power law for the Planck constant scaling was suggested τ¯h∗ ∼ 1/h ¯δ

(1.3)

for the breaking time of classical consideration applicability with a value of δ depending on the type of the classical algebraic escape time distribution. There were no quantum simulation in these works and no definite values of δ. Discussion of the algebraic law (1.3) started in [13, 21], was continued in recent publications [22, 23, 24] on the basis of simulations of quantum maps with an explicit evaluation of Eq. (1.2). The answer on the question what is the actual scaling of the breaking time of quantum-classical correspondence with respect to the Planck constant has been suggested in [14]. Using some results of [25, 26] on the strong delocalization effects, and simulation for the quantum kicked rotor (QKR) it was shown in [14] that the result (1.1), being valid for the cases of normal (Gaussian type) diffusion, appears to be an algebraic of the type (1.2) when diffusion becomes anomalous, i.e. superdiffusion, with the second moment of the truncated distribution function as !p2 " ∼ tµ and transport exponent µ > 1. The algebraic behavior for τ¯h∗ with the exponent δ = 1/µ, was predicted by the theory, and confirmed by simulations in [14]. The second example considered in the paper is a quantum analog of a classical chaotic attractor. For this system τ¯h is estimated for the semiclassical applicability in the presence of finite dissipation [15]. The role of dissipation in quantum systems is an object of extensive research, especially due to the different practical needs of the contemporary experimental and theoretical physics. Different aspects related to the quantum dissipative processes can be found in [27, 28, 29, 30, 31]. A special interest in dissipation is related to the decoherence [32] that can prevent quantum computing, since the common opinion is that dissipation should lead to losses of the quantum features of a system [32, 28]. A detailed analysis of related problems can be found in [28], where a quantum counterpart of a dissipative kicked rotor map [33] is considered. It is practically shown in [28] that for small dissipation the time decay of different observables and the destruction of localization take place in the system. A typical situation for the appearance of a classical chaotic attractor includes a fairly strong dissipation and an external pumping [33] (see also the recent rigorous consideration in [34]). This type of chaotic attractor is related to systems that are close to the Hamiltonian ones. (d) The quantization of such systems means that after the time τ¯h quantum

336 corrections will be of the order of 1 and will destroy the (semi)classical behavior of the systems. We will present some results obtained in [15] on (d) the calculation of the the breaking time τ¯h for a dissipative quantum system. The model also corresponds to a kicked harmonic oscillator with dissipation [35] and, with some modification, can be applied to non-linear optics [36]. It should be admitted, also, It is also relevant the kicked Harper model under certain conditions [37, 38, 26]. 0.04

0.6

0.02

0.4

0.2

0

p

p

0

Ŧ0.02 Ŧ0.2

Ŧ0.04 Ŧ0.4 Ŧ0.06 Ŧ0.6

1.74

1.76

1.78

1.82

1.8

1.84

1.86

1.6

1.8

1.7

Figure 1. Section 3.

2.1

2

1.9

2.2

2.3

2.4

q

q

Accelerator mode islands structure for the kicked rotor considered in

The paper is a review of some our results [14, 15, 39] and it is organized as follows. In Sec. 2 we present a rigorous consideration of a quantum nonlinear oscillator to show non-Hamiltonian nature of the equations 5 4 3 2

u

1 0 Ŧ1 Ŧ2 Ŧ3

Ŧ4 Ŧ5 Ŧ10

Ŧ5

0

5

10

v

Figure 2.

A chaotic attractor considered in Section 4.

337


of motion for expectation values. We show that the Liouville theorem on the conservation of the phase space flow violates for observables due to nonlinear quantum effects. In Sec. 3 we show that the breaking time of quantum-classical correspondence for the quantum kicked rotor scales algebraically in the Planck constant with correspondence to the transport exponent of the superdiffusion. Section 4 is devoted to the quantum chaotic attractor. We answer on the question how long the quantum chaotic attractor can be described by classical equations of motion. The obtained results are summarized in Sec. 5.

2.

The Liouville theorem violation

We start with a more general question of the correspondence between time evolution of a classical system and dynamics of its quantum counterpart. The difference begins already with the fact, that quantum mechanics describes a wave packet but not a separate trajectory. In the nonlinear case this difference leads to essential discrepancy between classical and quantum mechanics. To illustrate this statement we present an example, where the Liouville theorem on the conservation of the phase space flow for observables violets due to nonlinear quantum effects [39]. Consider the Hamiltonian of a nonlinear oscillator with the linear frequency ω and the nonlinearity ξ in the form H = ωa† a + ξ(a† a)2 .

(2.1)

The annihilation and creation operators have the commutation rule [a, a† ] = h ¯ . The equation of motion for the annihilation operator a reads ia˙ = (ω + hξ ¯ )a + 2ξa† a2 . (2.2) As a† (t)a(t) = a†(t = 0)a(t = 0) is an integral of motion, the solution directly follows from (2.2) 4

5

a(t) = exp −i(ω + hξ ¯ + 2ξa†0 a0 )t a0 ,

(2.3)

where a†0 = a† (t = 0) and a0 = a(t = 0) . Using the known expressions for normal ordering [40], we rewrite (2.3) in the form: )

4

¯ ˆ exp[−i(ω + hξ a(t) = N ¯ )t] exp (e−2iξht − 1)˜ a†0 a ˜0 /h ¯

5*

a0 ,

(2.4)

ˆ determines the normal ordering of operators: where the operator N )

*

ˆ (a0 )q (a† )p = (a† )p (a0 )q N 0 0

(2.5)

338 and the tilde indicates that quantities a ˜†0 and a ˜0 subjected to the operator ˆ must be regarded as c–numbers. N It is convenient to obtain quantum mechanical observables in the coherent state basis |α0 ". It can be introduced as eigenfunctions of the annihilation operator in the initial moment t = 0 in the form: a0 |α0 " = α0 |α0 ". Therefore we obtain from (2.4) 4

5

¯ )t ¯ α(t) = !α0 |a(t)|α0 " = e−i(ω+hξ exp (e−2iξht − 1)|α0 |2 /h ¯ α0 ,

(2.6)

and the solution for α∗ (t) can be obtained by complex conjugation of (2.6). The time change of the volume in phase space of the observables is determined by the following determinant ∂α ∂α ∂α0 ∂α∗0 J(t) = ∂α∗ ∂α∗ ≡ D(α, α∗ ) − D(α∗ , α) ∂α ∂α∗0 0 4 4 2 2 2 2 = 1 − |α0 | sin hξt ¯ exp − |α0 | sin hξt ¯ , ¯ h ¯ h

(2.7)

where the D-form is D(x, y) = (∂x/∂α0 ) · (∂y/∂α∗0 ) .

(2.8)

The phase volume thus oscillates in time with the quantum period Tq = π/hξ ¯ . Obviously, J(t) equals unity only at times tn = nT Tq (n = 0, 1, 2, . . .). The initial Heisenberg equations of motion for the operators a† (t) and a(t) are Hamiltonian. When these equations are projected by using an arbitrary basis (including the basis of the coherent states), it is not compulsory in the general case for the equations of motion for projections to have the same Hamiltonian structure. However, in the limit h/ ¯ |α0 |2 → 0 the equations for the observables α(t) and α∗ (t) must transform to classical equation of motion, which are Hamiltonian. Hence the expansion in h ¯ can not be regular, and is valid for a limited time. Indeed, in the limit h/ ¯ |α0 |2 → 0 we obtain that J(t) = 1 − 8¯h ¯ |α0 |2 ξ 2 t2 + . . . .

(2.9)

It implies that quantum corrections to classical solutions diverge in time due to nonlinearity. Therefore the classical approach is valid up to time t t0 = 1/hξ ¯ |α0 |.

(2.10)

When the integrable system (2.1) is subject to a perturbation, dynamics of the system becomes complicated, even chaotic. In this case in


339

the semiclassical limit the D-forms in (2.7) can grow exponentially due to the local instability of trajectories. It leads to the logarithmic scale in |α0 |2 /h ¯ , as one can see from (1.1). This nature of D-forms reflects the local instability of trajectories that is transparently seen in the coherent state basis [1, 2, 39], but as it was shown in [3], this result is independent of the choice of the initial basis of wave functions.

3.

Superdiffusive dynamics of quantum kicked rotor

We consider the QKR that corresponds to the standard map in the classical limit p = p + K sin q ,

q = q + p

(3.1)

defined on the cylinder p ∈ (−∞, ∞), q ∈ (−π, π) with a control parameter K and the Lyapunov exponent Λ ∼ ln K for K 1 and for almost all domain excluding areas where K| cos q| < 1. The marginally stable points are defined by the conditions Km = 2πm, p = 2πn, q = ±π/2 with integers (m, n). For 0 < K − Km < ∆K Km a new set of islands appears [18, 41, 43], called tangle islands in [43], as a result of bifurcation. Dynamics inside the islands is known as the accelerator mode [44, 45] and we will call them accelerator mode islands (AMI). Changing of K within the interval ∆K Km influences strongly the topological structure of AMI and consequently the values of the transport exponent µ = µ(K) [42, 17], since the stickiness of trajectories to the island boundaries can be different for different island topologies.

3.1

An analytical estimate of τ¯h

It was found in [17] that for the particular value of K ≡ K ∗ = 6.908745. . ., the stickiness is especially strong. For the K ∗ it appears as a hierarchical set of islands-around-islands with the island sequence 3 − 8 − 8 − 8 − . . .. The first two islands generations are shown in Fig. 1. The island chain satisfies the renormalization conditions S (n+1) = λS S (n) ,

T (n+1) = λT T (n) ,

N (n+1) = λN N (n) ,

(3.2)

where n is a number in the hierarchy sequence, S (n) is an island area, T (n) is the period of the last invariant curve of the corresponding island, N (n) is a number of islands in the chain of the n-th hierarchy level, and λS < 1, λT > 1, λN > 1 are some scaling parameters. The renormalization transform (3.2) can be also extended for Lyapunov exponents Λ in a sticky area of the island’s boundary of the n-th level of the island’s

340 hierarchy [42], Λ(n+1) = λL Λ(n) = λnL Λ(0)

(λL = 1/λT < 1) .

(3.3)

In the absence of the island’s hierarchy, we get just the result (1.1) with Λ = Λ(0) . In the presence of the island’s hierarchy we can introduce particle flux Φ(n) in phase space through the island’s chain of the n-th hierarchical level. It reads Φ(n) = S (n) N (n) /T (n) = Φ(0) (λS λN /λT )n ,

(3.4)

where Φ(0) = S (0) N (0) /T (0) in correspondence to (3.2) and (3.3). For K = K ∗ and the corresponding island’s hierarchy λN = λT [17], Φ(n) = λnS Φ(0) .

(3.5)

The quantum mechanical consideration of the proliferation of islands is meaningful until the smallest island size is of the order of h ¯ . Therefore we get from Eqs. (3.2),(3.4), and (3.5), Smin = h ¯ = S (n0 ) = S (0) λnS0 ,

Φmin = Φ(0) λnS0 = hN ¯ (0) /T (0) ,

(3.6)

and the quantum “cutoff” of the hierarchy appears at ˜ ln λS | , n0 = | ln(¯ h/S (0) )|/| ln λS | ≡ | ln h|/| ¯

(3.7)

˜ = h/I where we introduce a dimensionless semiclassical parameter h ¯ I0 (0) and specify S ≡ I0 . Particularly, for the hierarchy at K = K ∗ we have N (0) = 3 and λN = 8 but it could be many other hierarchies (see more in [46]). After the substitution Λ = Λ(n0 ) we get with Eqs. (3.3) and (3.7), (0) ˜ = (1/h) ˜ 1/µ ln(1/h)/Λ ˜ τ¯h = (1/Λ(n0 )) ) ln(1/h)

(3.8)

µ = | ln λS |/ ln λT .

(3.9)

with The expression of µ through λS , λT was found in [42] for the considered island hierarchy and the expression for τ¯h coincides with the obtained in [25] in a different way. The expression (3.8) is close to Eq. (1.3) up to a logarithmic term and defines δ = 1/µ,

(3.10)

which for K = K ∗ provides µ = 1.25 (see [17]) and δ = 0.8. When there is no island hierarchy, we may put λS → 0 or n0 → 0. This yields

341


transition from the algebraic law (3.8) to the logarithmic one for the breaking time and resolves the paradox discussed in [13]. In addition to this, it was shown in [17] that for the considered islands hierarchy β = 1 + µ = 1 + | ln λS |/ ln λT ,

(3.11)

which gives β = 2.25 for K = K ∗ . We have checked just these values of β and δ by a simulation [14]. 6

0

5.8 Ŧ5

5.6 5.4

Ŧ10

ln W

h

ln P(t)

5.2 5

Ŧ15

4.8 4.6

Ŧ20

4.4

ln W

h

Ŧ25 0

2

4

o

6

8

10

12

Figure 3. Typical evolution of the quantum survival probabilities for N = 25557. Dashed lines correspond to the asymptotic slopes, determined the breaking point τh¯ .

3.2

4.2 7

7.5

8

8.5

9

9.5

ln(1/h)

Figure 4. The quantum break points τh¯ vs dimensionless semiclassical pa˜ The solid line with the slop rameter h. 1/δ ≈ 1.33 ± 0.36 corresponds to least square calculations, and the dashed line is the analytical with δ = 1/µ.

Numerical algorithm for a quantum map

Numerical study of the problem is based on investigation of the quantum survival probability in some domain ∆p ∈ (−π, π) that includes the island hierarchy [47]. The main difficulty appears due to a part of the wave function that belongs to an island interior and “flies” fast along p. A simple way to avoid this type of the “escape” from ∆p is to apply a shift operator ˜ ˜ , Jˆ = exp{−(2π/h)∂/∂ n ˆ } = exp(−2πiq/h)

(3.12)

˜ n = −ih∂/∂q ˜ where q is a dimensionless coordinate, pˆ = hˆ is a dimensionless momentum operator and the wave function Ψt at a discrete time t is considered in the coordinate space. In analogy with [22, 48] we also introduce the absorbing boundary conditions at the edges of the interval

342 ˜ Then ∆n ∈ [−(N + 1)/2, (N − 1)/2] for the momentum eigenvalues hn. the quantum map that keeps the information of the trapping into ∆n part of the wave function is ˆ Ψt Ψt+1 = Pˆ JÛ

(3.13)

˜ n2 /2) exp ( − i(K/h) ˜ cos q) . U = exp ( − ihˆ

(3.14)

with the evolution operator

As usual, for the numerical convenience the dimensionless √ Planck con˜ is taken in a form h ˜ = 2π/(N + g), where g = ( 5 − 1)/2 is the stant h inverse golden mean. The survival probability is defined as P (t) = |Ψt |2

(3.15)

together with definition (3.13). Simulation was performed for 12 values of N from the interval (5 × 103 − 7.5 × 104 ) that corresponds to a good semiclassical approximation for a fairly long time until it fails. A typical behavior of P (t), obtained for K = K ∗ , is shown in Fig. 3. It consists of the crossover from the classical behavior (1.2) with β ≈ 2.25 (the same as in [17]) to some very different dependence. The crossover points τ¯h can be identified in a way shown in Fig. 3 for different ˜ The corresponding result is presented in Fig. 4. It values of N , i.e., h. provides the value of 1/δ ≈ 1.33 ± 0.36 ≈ µ in good agreement with the presented theoretical estimation (see Eq. (3.10)).

4.

How long does quantum chaotic attractor exist? (d)

In this section we will calculate the breaking time τ¯h for a dissipative quantum system. We consider the nonlinear dissipative dynamics of a particle in the presence of a constant magnetic field. The classical counterpart of the model that appeared in [35] is x ¨ + Γx˙ +

2 ωH x

= −k0 T sin(k0 x)

∞

δ(t − nT ),

(4.1)

n=−∞

which corresponds to the cyclotronic motion with the effective cyclotron frequency ωH in the presence of dissipation of the rate Γ, while the perturbation is a periodic set of kicks with the period T and the amplitude , and k0 is a wave number.


343

The quantum problem can be considered in the framework of the following non–Hermitian Hamiltonian: H = (Ω − iΓ/2) a† a − T cos k˜0 (a† + a)

∞

δ(t − nT ).

(4.2)

n=−∞

Here annihilation and creation operators have the same commutation rule [a, a† ] = h ¯ as in (2.1). The complex frequency ωΓ = Ω − iΓ/2 de< +1/2 termines the effective frequency ωH = Ω2 + Γ2 /4 in the presence of a finite width Γ/2 of the levels. The relation between the wave numbers k0 and k˜0 is determined by the linear relation between x and a + a† (see below Eq. 4.4). In the classical limit, the system has chaotic zones, and resonances between the perturbation and the linear oscillator lead to the unlimited pumping of energy [37]. The non-Hermitian nature of H does not change the operator’s algebra, and it produces the following Heisenberg equations ih ¯ a˙ = [a, H] ,

−ih ¯ a˙ † = [H † , a† ] .

(4.3)

It is convenient to transfer to the momentum–coordinate variables. Using the linear transformation , √ † x ˆ = (a + a )/ 2mΩ, pˆ = i mΩ/2(a† − a), (4.4) we obtain from Eqs. (4.4) and (4.3) the following Heisenberg equations pˆ˙ = −Ω2 mˆ x − Γp/ ˆ 2 − k0 T sin(k0 x ˆ) n δ(t − nT ) x ˆ˙ = p/m ˆ − Γx/ ˆ 2.

(4.5)

Let us introduce a coherent state basis |α0 " at the initial time t = 0. Then we define the mean values of the operators at time t, x(t) = !ˆ x(t)" = !α0 |ˆ x(t)|α0 " and p(t) = !ˆ p(t)" = !α0 |ˆ p(t)|α0 ".

(4.6)

This definition also implies that the operators x ˆ(t) and pˆ(t) are normally ordered (see (2.5)), The Heisenberg equations (4.5) averaged over the coherent states |α0 " give the following Ehrenfest equations p˙ = −mΩ2 x − wΓp/2 − k0 T !sin k0 x" x˙ = p/m − Γx/2 ,

n δ(t −

nT ) , (4.7)

where !sin k0 x" ≡ !α0 | sin k0 x ˆ|α0 ". This quantity can be calculated only ¯ , it reads approximately [1, 50] for an arbitrary t. In the first order of h !sin k0 x" ≡ !sin(k0 x ˆ)" = sin(k0 x) − hk ¯ 02 sin(k0 x)D(x, x) ,

(4.8)

344 where D(x, x) = (∂x)/(∂α0 ) · (∂x)(∂α∗0 ), and x ≡ x(t). The differentiation over α0 is replaced by the differentiation over the initial momentum and coordinate. From Eqs. (4.4) and (4.6), we obtain 4

5

D(x, x) = (1/2mΩ) · (∂x/∂x0 )2 + mΩ (∂x/∂p0 )2 /2 ,

(4.9)

where x0 = x(t = 0) and p0 = p(t = 0). Following [35], we integrate the Ehrenfest equations (4.7) over the period T . Between two consequent 2 kicks, the equation of motion is x ¨ + Γx˙ + ωH x = 0. At a kick in the moment tn = nT , the shift conditions are x(tn + 0) = x(tn − 0),

p(tn + 0) = p(tn − 0) − k0 T !sin k0 xn ",
and p =< pˆ > /kf . This infiabsorption of a photon, x = kf < x nite set of nonlinear ordinary differential equations possesses an infinite number of the integrals of motion, the total energy integral ∞

W =

∞

α 2 √ δ p − n + 1 un cos x − zn , 2 2 n=0

n=0

(7)

Hamiltonian chaos and fractals in cavity quantum electrodynamics

355

the Bloch-like integral for each n Rn2 = u2n + vn2 + zn2 , and the global integral

∞

(8)

Rn = 1.

(9)

n=0

The infinite-dimensional nonlinear dynamical system (5) is a quantum generalization of the five-dimensional semiclassical set of the equations of motion for the same problem that was derived in [12] with a classical field. The later one was shown [12, 14] to be chaotic with positive values of the maximal Lyapunov exponent in some ranges of the system’s control parameters. 1

z

(a)

0

-1

0

20

40

60

80

τ 100

Figure 1. Periodic Rabi oscillations in a resonant coherent quantum field with < n >= 10 and p0 = 50.

In general, a pure quantized field state is an infinite superposition of the photon-number states |f " =

∞

cn |n" ,

(10)

n=0

where |cn |2 is the probability for observing n photons. The most classical of single-mode quantum states is a coherent state |α" = e−|α|

2 /2

∞ αn n=0

n!

|n" ≡

∞

cn (α) |n" ,

(11)

n=0

whose photon probabilities follow a Poisson distribution with the mean number of photons < n >= |α|2 and the root mean square spread ∆n =

356 |α|. In Fig. 1 the population inversion z(τ ) =

∞

zn (τ )

(12)

n=0

for an initially excited atom interacting in resonance with the field that is initially in a coherent state (11) with < n >= 10 is shown. In order to understand peculiarities of the Rabi oscillations in Fig. 1, look at Fig. 3a demonstrating the scheme of gedanken experiments. An atom starts at x = 0 and moves to the right with the initial momentum p0 = 50. It reaches the first node of the standing wave, where the coupling coefficient with the field mode is zero at time moment τ1 = π/2αp0 31.4 in dimensionless units (we choose α = 0.001 in our computer simulations). One can see in the figure the respective slowing down of the Rabi oscillations at the moments of time τs = (1 + 2s)π/2αp0 when the atom transverses the s-th node. The pronounced peaks of revivals of the Rabi oscillations occur at the moments τr = rπ/αp0 , when the atom transverses the r-th antinode of the standing wave where its coupling with the field is maximal. In spite of a complicated character of the Rabi oscillations, shown in Fig. 1, they are regular because at exact resonance, δ = 0, all the values of the real-valued amplitudes un are conserved during the evolution. With initially excited atom we have un (0) = vn (0) = 0 for each n. It immediately follows from Eqs.(5) that the atom moves with a constant velocity through the cavity. It means that x is a linear function of time, and the Hamilton-Schrodinger ¨ equations (5) reduce to the periodically modulated linear Bloch-like equations with periodic solutions. The exact solution for the atomic population inversion can be found for any initial pure field state √ ∞ 2 n+1 zn (0) cos sin αp0 τ . (13) z(τ ) = αp0 n=0

It is a periodic function with the period to be equal to π/αp0 and the maxima at τr = rπ/αp0 (r = 0, 1, 2, ...). Out off resonance, δ = 0, the quantum-classical hybrid may demonstrate chaos. To diagnose chaos it is instructive to compute the maximal Lyapunov exponent λ whose values depend on initial conditions, on the detuning δ, the mean number of photons in the mode, and on the recoil frequency α. It has been computed in [15] with Fock field states that in the range α ∼ 10−4 ÷ 10−2 and δ ∼ −2 ÷ 2 the maximal Lyapunov exponent may be positive and one may expect chaotic atomic motion in the respective ranges of α and δ.


357

A feasible scheme for detecting manifestation of chaos with hot twolevel Rydberg atoms moving in a high-Q microwave cavity has been proposed in [14]. The same idea could be realized with cold usual atoms in a high-Q microcavity. Consider a 2D-geometry of a gedanken experiment with a monokinetic atomic beam propagating almost perpendicularly to the cavity axis x. In a reference frame moving with a constant velocity in the perpendicular direction, there remains only the transverse atomic motion along the axis x. One measures atomic population inversion after passing the interaction zone. Before injecting atoms in the cavity, it is necessary to prepare all the atoms in the same electronic state, say, in the excited state, with the help of a π-pulse of the laser radiation. It may be done with only a finite accuracy, say, equal to ∆zin for the initial population inversion zin . The values of the population inversion zout are measured with detectors at a fixed time moment. If we would work with the values of the control parameters corresponding to the regular atomfield dynamics, we would expect to have a regular curve zout –zin . In the chaotic regime, the atomic inversion at the output can be predicted (within a certain confidence interval ∆z) for a time not exceeding the so-called predictability horizon τp

1 ∆z ln , λ ∆zin

(14)

which depends weakly on ∆zin and ∆z. Since the maximal confidence interval lies in the range |∆z| 1 and λ may reach the values of the order of 1.5 [15], the predictability horizon in accordance with the formula (14) can be very short: with λ =0.5 the predictability horizon τp may be of the order of 10 in units reciprocal of the vacuum Rabi frequency Ω0 that corresponds to tp 10−7 s with the realistic value of Ω0 108 rad· s−1 . In the regular regime, the inevitable errors in preparing ∆zin produce the output errors ∆zout of the same order. In the chaotic regime, the initial uncertainty increases exponentially resulting in a complete uncertainty of the detected population inversion in a reasonable time. It is demonstrated in Fig. 2, where we plot the dependence of the values of z(τ ) = zout at τ = 100 (Fig. 2a) and τ = 200 (Fig. 2b) on the values of z(0) = zin in the chaotic regime with an initially Fock field at δ = 0.4 and λ 0.05. Simulation shows that an initial error ∆zin = 10−4 in preparing the atomic electronic state leads to complete uncertainty ∆zout 2 in a rather short time. To see the difference, it is desirable to carry out a control experiment at the exact resonance (δ = 0) when the atomic motion is fully regular with any initial values. The dependence zout –zin with δ = 0 is demonstrated in the inset in Fig. 2b with all the other control parameters and initial values being the same.

358 zout

1

0

(a) -1 -1

zin

0

1

1

zout

1

0

0

0

1

(b) -1 -1

0

zin

1

Figure 2. Dependence of the output values of the atomic population inversion zout on its initial values zin with δ = 0.4 (a) at τ = 100 and (b) at τ = 200 with the inset showing this dependence at the exact atom-field resonance, δ = 0. The field is initially in the Fock state with n = 10.

4.

Dynamical atomic fractals

In this section, we treat the atom-photon interaction in a high-Q cavity as a chaotic scattering problem. Let us consider the scheme of scattering of atoms by the standing wave shown in Fig. 3a. Atoms, one by one, are placed at the point x = 0 with different initial values of the momentum p0 along the cavity axis. For simplicity, we suppose that they have no momentum in the other directions (1D-geometry). We compute the time the atoms need to reach one of the detectors placed at the cavity mirrors. The dependence of this exit time T on the initial atomic momentum p0 is studied under the other initial conditions and parameters being the same. To avoid complications that are not essential to the main theme of this section, we consider the cavity with only two standing-wave lengths. Following to the scheme in Fig. 3a, let us consider scattering of atoms by the standing wave field initially prepared in the coherent state (11). Before injecting into a cavity, atoms supposed to be prepared in the


359

(a) atoms

standing wave x

0 detectors

(b) 1.5 3

1

1

x/π 0.5

0 2 -0.5 0

50

100 150 200 250 300 350 400 450

τ

Figure 3. (a) Schematic diagram showing scattering of atoms at the standing wave and (b) sample atomic trajectories.

excited state |2" with z(0) = 1 and the following initial conditions: un (0) = vn (0) = 0, ∀n, < n >n zn (0) = e− . n!

(15)

Fig. 4 demonstrates the respective exit-time function T (p0 ) with δ = 0.1, α = 0.001, and < n >= 10. A coherent state is not a sharply defined state, as a Fock one, but a superposition of an infinite number of Fock states. We used a truncated basis of 1000 Fock states for the cavity mode in our simulation. Resolution of one of the unresolved structures in Fig. 4a is shown in Fig. 4b. Further magnification of the function in Fig. 4b, which is shown in Fig. 4c, reveals again a self-similar structure with smooth and singular zones. The exit time T , corresponding to both smooth and unresolved p0 intervals, increases in average with increasing the magnification factor. It follows that there exist atoms never reaching the detectors in spite of the fact that they have no obvious energy restrictions to leave the

360 (a) 1000

T 800

600

400

200

0

5

10

15

20

25

p0 30

(b) 1000

T 800

600

400

200

0 25

25.2

25.4

25.2516

25.2518

p0 25.6

(c) 1000

T 800

600

400

200 25.2514

Figure 4.

p0

25.252

Coherent atomic fractal with different resolutions.

cavity. Tiny interplay between chaotic external and internal dynamics prevents these atoms from leaving the cavity. The similar phenomenon in Hamiltonian systems is known as dynamical trapping [17]. Different kinds of atomic trajectories are shown in Fig. 3b. A trajectory with the number m transverses the central node of the standing-wave, before being detected, m times and is called m-th trajectory. There are also special separatrix-like mS-trajectories following which atoms in infinite time reach the stationary points xs = ±πs (s = 0, 1, 2, . . . ), ps = 0, transversing m times the central node. These points are the anti-


361

nodes of the standing wave where the force acting on atoms is zero. A detuned atom can asymptotically reach one of the stationary points after transversing the central node m times. The trajectory with number 1, showing in Fig. 3b, is close to a separatrix-like 1S-trajectory. The smooth p0 intervals in the first-order structure in Fig. 4a correspond to atoms transversing once the central node and reaching the right detector. The unresolved singular points in the first-order structure with T = ∞ at the border between the smooth and unresolved p0 intervals are generated by the 1S-trajectories. Analogously, the smooth p0 intervals in the second-order structure in Fig. 4b correspond to the 2-nd order trajectories with singular points between them corresponding to the 2Strajectories and so on. There are two different mechanisms of generation of infinite exit times, namely, dynamical trapping with infinite oscillations (m = ∞) in a cavity and the separatrix-like motion (m = ∞). The set of all initial momenta generating the separatrix-like trajectories is a countable fractal. Each point in the set can be specified as a vector in a Hilbert space with m integer nonzero components. One is able to prescribe to any unresolved interval of a m-th order structure a set with m integers, where the first integer is a number of a second-order structure to which trajectory under consideration belongs in the first-order structure, the second integer is a number of a third-order structure in the second-order structure mentioned above, and so on. Unlike the separatrix fractal, the set of all initial atomic momenta leading to dynamically trapped atoms with m = ∞ seems to be uncountable. We collect an exit time statistics with 6 · 105 events by counting atoms with initial momenta in the range 9 ≤ p0 ≤ 30 reaching the detectors. The plot of the respective histogram of exit times, shown in Fig. 5, demonstrates a few local maxima. The first maximum around T 150 ÷ 160 corresponds, mainly, to atoms which, being initially directed to the right with comparatively low momenta p0 10, turn back before reaching the central node and are registered by the left detector. For such atoms T1 π/2αp0 157 at αp0 = 0.01. However, the atoms, to be injected initially with the momentum p0 20 ÷ 25 and registered by the right detector, may contribute to the first maximum as well because, after transversing the central node, they can be accelerated and gain the values of the momentum p up to 50. The second local maximum around T 200 corresponds, mainly, to atoms with p0 20 ÷ 30 which transverse the central node and are registered by the right detector for the time T2 ∼ 3π/2αp0 . The other local maxima of the PDF are not so pronounced as the first ones, they are formed by the atoms transvers-

362 ing the central node a few times. The PDF in Fig. 5 demonstrates an exponential decay at the tail up to the exit times T = 700. log10p

5 4 3 2 1 0 -1 100

Figure 5.

300

500

T 700

Exit-time distribution in the coherent field.

It should be noted that with a Fock field and initial atomic momenta in the range 8 ≤ p0 ≤ 40 we have found an algebraic decay with the characteristic exponent γ −3.72 at the PDF’s tail (≤ 300T ≤ 4000).

5.

Conclusion

We have studied in the strong-coupling regime the fundamental interaction between a two-level atom with recoil and a quantized radiation field in a single-mode cavity in the mixed quantum-classical formalism modeling quantum evolution of the electronic-field purely quantum system to be disturbed by translational atomic motion. We have shown that even in the absence of any other interaction with environment the Hamiltion-Schr¨ odinger dynamical system provides the emergence of classical Hamiltonian dynamical chaos with fractal-like structures from cavity quantum electrodynamics.

Acknowledgments I am grateful to Leonid Kon’kov and Michael Uleysky for preparing the figures. This work was supported by the Russian Foundation for Basic Research under Grant No. 02–02–17796, by the Program “Mathematical Methods in Nonlinear Dynamics” of the Russian Academy of Sciences, and by the Program for Basic Research of the Far Eastern Division of the Russian Academy of Sciences.

References [1] G.M. Zaslavsky, Phys. Rep. 80 (1981) 157. [2] F. Haake, Quantum Signatures of Chaos, Springer, Berlin (1991).


363

[3] H. Walther, Phys. Rep. 219 (1992) 263. [4] J. M. Raimond, M. Brune, S. Haroche, Rev. Mod. Phys. 73 (2001) 565. [5] P.I. Belobrov, G.M. Zaslavskii, G.Kh. Tartakovskii, Sov. Phys. JETP 44 (1976) 945. [6] R.M. Dicke, Phys. Rev. 93 (1954) 493. [7] S.V. Prants, L.E. Kon’kov, Phys. Lett. A 225 (1997) 33. [8] S.V. Prants, L.E. Kon’kov, JETP 88 (1999) 406. [9] S.V. Prants, L.E. Kon’kov, I.L. Kirilyuk, Phys. Rev. E 60 (1999) 335. [10] S.V. Prants, L.E. Kon’kov, Phys. Rev. E 61 (2000) 3632. [11] V.I. Ioussoupov, L.E. Kon’kov, S.V. Prants, Physica D 155 (2001) 311. [12] S.V. Prants, V.Yu. Sirotkin, Phys. Rev. A 64 (2001) 033412. [13] E.T. Jaynes, F.W. Cummings, Proc. IEEE 51 (1963) 89. [14] S.V. Prants, JETP Letters 75 (2002) 651. [15] M. Uleysky, L. Kon’kov, S. Prants, Comm. Nonlin. Sci. Numeric. Comm. 8 (2003) 329. [16] V.Yu. Argonov, S.V. Prants, JETP 96 (2003) 832. [17] G.M. Zaslavsky, Physics of Chaos in Hamiltonian Systems, Academic Press, Oxford, 1998.


INERT AND REACTING TRANSPORT M. Cencini Dipartimento di Fisica, Universit` a di Roma ”la Sapienza” and Center for Statistical Mechanics and Complexity INFM UdR Roma1 Piazzale Aldo Moro 5, I-00185 Roma, Italy [email protected]

D. Vergni Istituto Applicazioni del Calcolo, CNR Viale del Policlinico 137, I-00161 Roma, Italy [email protected]

A. Vulpiani Dipartimento di Fisica, Universit` a di Roma ”la Sapienza” and Center for Statistical Mechanics and Complexity INFM UdR Roma1 Piazzale Aldo Moro 5, I-00185 Roma, Italy [email protected]

Abstract

Some aspects of passive transport in fluid flows are reviewed. Two classes of problems are considered: inert substances advected by fluid flows, and substances that chemically (or biologically) react while are advected. Concerning the former issue, we discuss it in the Lagrangian framework. In particular, we address the problem of anomalous diffusion in the asymptotic single particle motion, and we introduce a scale dependent description of the particles pair separation evolution. The Lagrangian description is very useful to characterize the non asymptotic properties of transport that often are interesting ones. For the problem of reacting transport we study the dependence of the front speed on the flow characteristics, considering the case of reaction that are slow or fast with respect to the typical time scales of the advection. Moreover, we comment about the role of Lagrangian chaos on the propagation properties.

Keywords: Transport and reaction in fluids.

365 P. Co C llet et al. (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 365–399. © 2005 Kluwer Academic Publishers. Printed in the Netherlands.

366

1.

Introduction

The dynamics of fields (e.g. the concentration of some species) advected by a velocity field is a problem of considerable practical and theoretical interest in many disciplines as astrophysics, geophysics, chemical engineering and disordered media [1, 2]. Though typically the advected field backreacts on the flow modifying the velocity field, often the advected field can be considered as passive, i.e. transported without dynamical effects on the velocity field. This is the case we shall consider in the following. Under this simplifying assumption, in a given velocity field the most general equation describing the evolution of the concentrations of N species, θi (x, t), can be written as ∂t θi + u · ∇θi = Di ∆θi +

1 fi (θ1 , · · · , θN ) , τi

(1)

where on the l.h.s. the second term accounts for the transport by an incompressible velocity field; on the r.h.s the first term represents molecular diffusion (Di is the diffusion constant for the i-th specie), and the second one takes into account possible chemical or biological processes (of characteristic time scale τi ) taking place among the different substances. In this contribution we shall consider separately the problem of inert transport (i.e. fi (θ1 , · · · , θN ) = 0), and that of reacting transport. As far as inert transport is concerned, one usually considers a unique field θ that evolves according to the advection-diffusion equation ∂t θ + u · ∇θ = D∆θ ,

(2)

being D the molecular diffusivity; eventually a source term, which represents the injection of scalar fluctuations, may be added in the r.h.s.. Given the field u the main goal is to understand the dynamical and statistical properties of the field θ. Remarkably, in the last few years, much progress has been reached in this direction and we have now a satisfactory understanding of the statistics of passive fields in terms of the motion of advected passive particles. The interested reader may consult the recent review [3] where an exhaustive discussion on passive fields in turbulent flows can be found. The problem of transport can be recast in terms of particles motion indeed Eq. (2) is nothing but the Fokker-Planck equation of the stochastic process describing the motion of test particles: √ ˙ x(t) = u(x(t), t) + 2Dη(t) (3)

367

Inert and reacting transport

where u(x(t), t) is the Eulerian velocity field at the particle position x(t), and η is a Gaussian white noise with zero mean and !ηi (t)ηηj (t )" = δij δ(t − t ). After the seminal works of Arnold and Aref it is now well recognized that particle motion can be highly non trivial even in simple laminar velocity fields due to the so-called Lagrangian chaos [4]. Therefore already the single particle motion presents very interesting features. For instance, the dispersion properties are greatly enhanced by the combined effects of the molecular diffusivity and the advection by the velocity field [1, 4]. Indeed at very large times and scales (with respect to the typical time and length scales of u), the test particle undergoes a Brownian process with an enhanced diffusion coefficient [5], i.e. E E !(xi (t) − xi (0))2 " 2 Dii t where eddy diffusion coefficient Dii > D contains the effect of the velocity field. In terms of the field θ this means that the coarse-grained concentration !θ" (where the average is over a volume of linear dimension larger than the typical velocity length scale) obeys the Fick equation: E 2 ∂t !θ" = Dij ∂xi xj !θ"

i, j = 1, . . . , d ,

(4)

where d is the space dimension. To compute DE given the velocity field there are now well established techniques (see e.g. [5, 6]). Moreover, it is also well known that under certain conditions anomalous diffusion may take place, i.e. !(x(t) − x(0))2 " ∼ t2ν with ν = 1/2 [2]. In the following we discuss in details the necessary conditions to observe anomalous diffusion. In particular, we consider the case of incompressible velocity fields where either standard diffusion (ν = 1/2) or superdiffusion (ν > 1/2) may appear [5]. Though interesting and relevant to many problems, the single particles motion is essentially determined by the large scale properties of the velocity field, which often hide much more interesting phenomena taking place at small scales. Moreover, the standard or anomalous diffusive properties are asymptotic features that in realistic situations may not be attained. In this sense it is more interesting to study the relative motion of two particles. This is indeed characterized by a variety of behaviors in dependence on the statistics of the velocity field at different length scales. For instance, at very small scales (where the velocity field is smooth) particles separate exponentially, i.e. !ln |x2 (t) − x1 (t)|" λt + const, being λ the Lyapunov exponent. On the other hand, in the inertial range of fully developed turbulent flows particle separates as !|x2 (t) − x1 (t)|2 " ∼ t3 , i.e. the Richardson dispersion law holds, in spite of the diffusive single particle behavior at large scales. Since, as shown in the above example, the relative dispersion properties depend

368 on the behavior of the velocity field at different scales, we introduce a scale dependent description of particle pairs separation in term of the Finite Size Lyapunov Exponent [7]. This kind of description is also very useful to account for the non asymptotic properties of dispersion and, e.g., to describe transport in finite systems where finite size effects are present. The problem of reacting transport is much more difficult, because the presence of the production term fi (θ1 , . . . , θN ) in the transport equation (1) makes it to be non linear, and new phenomena appear. Here we consider the simplest non trivial case of Eq. (1), i.e.: a unique scalar field θ(x, t) evolving according to the advection-diffusion-reaction equation 1 ∂t θ + u · ∇θ = D∆θ + f (θ) . τ

(5)

Where θ represents the fractional concentration of the reacting substance with the following glossary: θ = 0 indicates fresh material which has still to react, 0 < θ < 1 means coexistence of fresh material and products, and θ = 1 means that the reaction is over [8]; equivalently one may think of θ as the concentration of a biological organism which is transported by the flow and grows according to the dynamics f (θ) [9]. Although very generic form can be considered for f (θ) (see [8, 9]) here we mainly discuss the case of f (θ) which are convex functions (f (θ) < 0) with f (0) = f (1) = 0 and f (0) = 1. A typical example is f (θ) = θ(1 − θ). This class of production terms belong to the so-called Fisher-Kolmogorov-PetrovskyPiscounoff (FKPP) type [10, 11]. With this choice θ = 0, 1 are the unstable and stable steady states of the dynamics, respectively. Once at the initial time a small portion of the system is such that θ = 0 the reaction starts and a front connecting the unstable and stable states propagates. In absence of stirring, u = 0, it is known that Eq. (5), for FKPP nonlinearity, generates afront propagating, e.g., from left to right with asymptotic √ speed v0 = 2 D/τ , and the thickness of the reaction region is ξ = 8 Dτ [10, 11]. It is worth recalling that the problem of front propagation has been extensively studied in many different fields [12, 13] such as chemical reaction fronts [8], flames propagation in gases [9] and population dynamics of biological communities [12, 13]. In many of these systems the reaction takes place in moving media, i.e. fluids, so that it is important to understand how the presence of a flow modifies the propagation properties. As a generic feature, in presence of a non-zero velocity field the front propagates with an average speed vf greater than v0 [14, 15, 16]. How-


369

ever, if f (θ) is not convex, under certain circumstances, the flow may stop (“quench”) the reaction [17]. The front velocity vf is the result of the interplay among the flow characteristics (i.e. intensity U and length-scale L), the diffusivity D and the production time scale τ . Here our major concern will be to discuss the dependence of vf on such quantities. For instance, introducing the Damk¨ öhler number Da = L/(U τ ) (the ratio of advective to reactive time scales) and the P´ éclet number P e = U L/D (the ratio of diffusive to advective time scales), one can seek for an expression of the front speed as an adimensional function vf /v0 = φ(Da, P e) ≥ 1. In particular, we study the case of cellular flows. We will see that a crucial role in determining φ(Da, P e) is played by the renormalization of the diffusion coefficient and chemical time scale induced by the advection [14]. Moreover, we consider an important limit case, i.e., the so called geometrical optics limit, which is realized for (D, τ ) → 0 maintaining D/τ constant [18]. In this limit one has a non zero bare front speed, v0 , while the front thickness ξ goes to zero, i.e., the front is sharp. Physically speaking, this limit corresponds to situations in which ξ is very small compared with the other length scales of the problem. Also in this case we provide a simple prediction for the front speed, which turns out to be expressible as an adimensional function vf /v0 = ψ(U/v0 ). Other interesting questions concern the modification of the front geometry as a consequence of advection. In particular, one may ask if the presence of Lagrangian chaos has a role in front dynamics. We shall briefly discuss this problem in the framework of the geometrical optics limit. The material is organized as follows. In Section 2 we discuss single particle and two-particles motion in laminar and turbulent flows. Emphasis is put on the conditions for anomalous diffusion and on the nonasymptotic properties of particles pairs separation. Section 3 is devoted to the study of front propagation in fluid flows. After a brief discussion on some general results which do not depend on the specific properties of the velocity field we shall analyze in details the case of cellular flows in different regimes.

2.

Transport of inert substances

As stated in the introduction the dynamical and statistical properties of advected passive fields are tightly related to those of test particles. Therefore, from the study of particle motion one can predict many aspects of the dynamics of advected scalar fields.

370 In particular, the small scale features of the scalar field can be understood studying the relative motion of test particles, as the following discussion will clarify. Consider for instance the transport equation for θ ∂t θ + u · ∇θ = D∆θ + Φ , (6) where we added an external source of tracer fluctuations, Φ, which acts at a given length scale LΦ . The link with particle trajectories is evident by solving (6) with the method of characteristics:

t θ(x, t) = −∞ ds Φ(x(s; t), √ s) ˙ t) = u(x(s; t), s) + 2D η(s) , x(s;

x(t; t) = x ;

(7)

the second equation is nothing but Eq. (3) where we explicitly fixed the final position to be x. By using (7) one can then connect the statistics of particle trajectories to the correlation functions of the scalar field. For instance, let us consider the simultaneous two-point correlations !θ(x1 , t)θ(x2 , t)" =

t −∞

ds1

t

−∞

ds2 !Φ(x1 (s1 ; t), s1 )Φ(x2 (s2 ; t), s2 )" ,

(8) with x1 (t; t) = x1 and x2 (t; t) = x2 . With a convenient choice of the correlation function of the forcing, e.g. !Φ(x1 , t1 )Φ(x2 , t2 )" = χ(|x1 − x2 |)δ(t1 − t2 ), and exploiting space homogeneity, Eq. (8) can be further simplified to the form C2 (R) = !θ(x, t)θ(x + R, t)" =

t −∞

ds

dr χ(r) p(r, s|R, t) .

(9)

where p(r, s|R, t) is the probability density function for a pair to be at a separation r at time s, under the condition to have a separation R at time t. It is now clear that the knowledge of p(r, s|R, t) allows to predict the behavior of C2 and so of the scalar spectrum. Moreover, assuming that χ(r) drops to zero for r > Lf and χ(0) = χ0 , (9) may be approximated as C2 (R) ≈ χ0 T (R; Lf ), where T (R; Lf ) is the average time the particles took to reach a separation O(Lf ) starting from a separation R (while going backward in time). This calls also for studying relative dispersion in terms of the time T (R1 ; R2 ) needed to reach a separation R2 being started from R1 . As we shall see, this is at the core of our approach to relative dispersion. Similar reasonings can be extended to correlations functions involving more than two points. This leads to consider the relative motion of more than two particles, which may be highly non trivial [19]. A detailed discussion of these aspects is beyond our aims, therefore we limit our

371


discussion to single and two particles properties. The interested reader may consult [3] for a discussion on the more than two particles aspects and their consequences on the scalar field properties.

2.1

Standard and anomalous diffusion

Investigating the diffusive properties of single particle motion allows to predict the characteristics of the macroscopic motion of concentration fields (cfr. Eq. (4)). In this framework it is important to identify the conditions which may lead to anomalous diffusion that brings as a consequence the failure of the Fickian description of transport. Under these circumstances Eq. (4) does not hold anymore. From Eq. (3) it is easy to obtain the following relation [20]: 2

!(xi (t) − xi (0)) " =

0

t

dt1

0

t

dt2 !vi (x(t1 )) vi (x(t2 ))" 2 t

t 0

dτ Cii (τ ) , (10)

where Cij (τ ) = ! vi (x(τ )) vj (x(0)) "

(11)

˙ is the correlation function of the Lagrangian velocity, v = x. From Eq. (10) it is not difficult to understand that anomalous diffusion can occur only when one, or both, of the following conditions are violated i) finite variance of the velocity: !v 2 " < ∞ ii) fast enough decay of the auto-correlation function of Lagrangian velocities: 0t dτ Cii (τ ) < ∞.

If both !v 2 " < ∞ and 0t dτ Cii (τ ) < ∞ then one has standard diffusion and the effective diffusion coefficients are ∞ 1 E Dii = lim !(xi (t) − xi (0))2 " = dτ Cii (τ ) . (12) t→∞ 2 t 0 Let us now examine two examples in which the above conditions are violated and anomalous diffusion takes place. It is worth remarking that here we use the term anomalous diffusion to indicate a non standard diffusion in the asymptotic regime. Sometimes in the literature the term anomalous is used also for long (but non asymptotic) transient behaviors. Violation of i) can be obtained in the so-called L´ évy flight model [21]. The simplest instance is the discrete (in time) one dimensional case, where the particle position x(t + 1) at the time t + 1 is obtained from x(t) as follows: x(t + 1) = x(t) + U (t), (13)

372 and U (t)’s are independent variables identically distributed according to a α–Lévy–stable distribution, Pα (U ), i.e.:

dU eikU Pα (U ) ∝ e−c|k|

α

and

Pα (U ) ∼ U −(1+α)

for

| U | 1

(14) with 0 < α ≤ 2. An easy computation gives !x(t) " = Cq t if q < α, and !x(t)q " = ∞ if q ≥ α. Though !x2 " = ∞ for any α < 2, one can consider the Levy ´ flight as a sort of anomalous diffusion in the sense that xtypical ∼ t1/α t1/2 . However, in spite of the relevance of the α–Lévy–stable distribution in probability theory, the Levy ´ flight model, in our opinion, has a rather weak importance for physical systems. This is due to the very unrealistic property of infinite variance. Physically more interesting is the Lévy walk model [22], that is still described by (13) but now U (t) is a random variable with finite variance but non trivial time correlations, so that ii) is violated. Let us assume that U (t) can assume the values ±u0 and maintains its value for a duration T which is a random variable with probability density ψ(T ). The origin of the possible anomaly is transferred to the correlation function of the Lagrangian velocity: the idea is that one has to generate a correlation such that Cii (τ ) ∼ τ −β with β < 1. By taking ψ(T ) ∼ T −(α+1) standard diffusion is realized for α > 2, while anomalous (super) diffusion takes place for α < 2: q

!x(t)2 " ∼ t2ν

q/α

 

ν=

1/2 α>2 (3 − α)/2 1 < α < 2  1 α < 1.

(15)

Besides the above simplified models, more interesting is the understanding of the anomalous diffusion in incompressible velocity fields or deterministic maps. In this direction Avellaneda, Majda and Vergassola [23, 24] obtained a very important and general result about the asymptotic diffusion in an incompressible velocity field u(x). If the molecular diffusivity D is non zero and the infrared contribution to the velocity field are weak enough, namely

ˆ !| u(k) |2 " 0 and ν >

1 ; 2

(19)

strong anomalous diffusion when !| x(t) − x(0) |q " ∼ tq ν(q)

ν(q) = const ν(2) >

1 2

(20)

374 and ν(q) is a non-decreasing function of q. In terms of the probability P (∆x, t) of observing a displacement ∆x = x(t) − x(0) at time t, weak anomalous diffusion amounts to the scaling property: P (∆x, t) = t−ν F (∆x t−ν ) (21) where the function F is not necessarily the Gaussian one. On the contrary strong anomalous diffusion is not compatible with the scaling (21). In the case of weak anomalous diffusion, it is natural to conjecture F (z) ∝ e−c|z| , α

(22)

where in general α is not determined by ν. However, an argument á la Flory due to Fisher [28] suggests that 

P (∆x, t) ∼ t−ν exp −c

|∆x| tν

1 1−ν

  ,

(23)

1 i.e., α = 1−ν . Remarkably the random shear flow examined in the previous section is in agreement with the Fisher’s prediction, as shown by Bouchaud at al. [29] indeed for ζ = 0 i.e. ν = 3/4 one has F (a) ∼ 4 e−c|a| for | a | 1. While for the properties of dispersion the detailed functional dependence of P (∆x, t) is not particularly important, it has a non trivial role in determining the propagation properties in reactive systems [30].

2.3

Strong anomalous diffusion in chaotic flows

If (16) holds then anomalous diffusion may appear only for D = 0 and very strong Lagrangian velocity correlations. The latter condition can be realized, e.g., in time periodic velocity fields in which the Lagrangian phase space has a complicated self-similar structure of island and cantori [31]. In such a case superdiffusion is essentially due to the almost trapping of the ballistic trajectories, for arbitrarily long time, close to the cantori that are organized in complicated self-similar structures. In this framework an interesting example is the Lagrangian motion in velocity field given by a simple model that mimics the Rayleigh–Bénard convection [32], and is described by the stream function:

2π 2π (x + B sin ωt) sin y ψ(x, y, t) = ψ0 sin L L

,

(24)

where the velocity is given by u = (∂ ∂y ψ, −∂ ∂x ψ), ψ0 = U L/2π (being L the periodicity of the cell, here we use L = 2π) and U the velocity

375

E

D / \ 0


2

Z L / \0

E Figure 1. The turbulent diffusivity D11 /ψ0 vs the frequency ωL2 /ψ0 for different values of the molecular diffusivity D/ψ0 . D/ψ0 = 3 × 10−3 (dotted curve); D/ψ0 = 1 × 10−3 (broken curve); D/ψ0 = 5 × 10−4 (full curve).

intensity. The even oscillatory instability is accounted for by the term B sin ωt, representing the lateral oscillation of the rolls [32]. At fixed B, the control parameter for particle diffusion is ≡ ωL2 /ψ0 , i.e., the ratio between the lateral roll oscillation frequency (ω) and the characteristic circulation frequency (ψ0 /L2 ) inside the cell. Different regimes take place for different values of . For instance, at ∼ 1 the synchronization between the circulation in the cells and their global oscillation is a very efficient way of jumping from cell to cell. This mechanism, similar to stochastic resonance, makes the effective diffusivity as a function of the frequency ω very structured [27] (see Fig. 1). Moreover, in the limit of vanishing molecular diffusivity, anomalous superdiffusion takes place in a narrow window of ω values around the peaks, i.e. !(x(t) − x(0))2 " ∝ t2ν(2)

with

ν(2) > 1/2 .

(25)

as reported in Fig. 2 (left). The presence of genuine anomalous diffusion E is confirmed by the fact that effective diffusivity diverges as D11 ∼ D−β with β > 0 (see Fig. 2 (right)), as suggested in [5]. The remarkable property of the flow (24) is that moments of the particle displacement display a strong anomalous behavior (20), indeed Fig. 3 (left) shows that q ν(q)’s is a non trivial function of q. In particular, the curve q ν(q) vs q displays a nonlinear behavior. A closer inspection shows that two linear regions are present: the first one up to q ∼ 2, the second elsewhere.

376 The two linear regions are associated to two different mechanisms in the diffusion process. For small q’s, i.e. for the core of the probability distribution function P (∆x, t), only one exponent (ν1 ≡ ν(q) 0.65 for q < 2) fully characterizes the diffusion process. This means that the typical, i.e. non rare, events obey a (weak) anomalous diffusion process.

5

10

10

4

10

D 11

E

2

Chaotic dynamics and transport in classical and quantum systems

Chaotic Dynamics And Transport In Classical And Quantum Systems

Chaotic Dynamics and Transport in Classical and Quantum Systems