Non-equilibrium Statistical Physics withApplication to Disordered Systems

Manuel Osvaldo Cáceres

Non-equilibrium StatisticalPhysics with Applicationto Disordered Systems

123

Manuel Osvaldo CáceresCentro Atómico Bariloche and Instituto

BalseiroComisión Nacional de Energía Atómica

and Universidad Nacional de Cuyo,and Comisión Nacional de InvestigacionesCientíficas y Técnicas

San Carlos de BarilocheRio Negro, Argentina

Original Spanish edition published by Reverté, Barcelona, 2002

ISBN 978-3-319-51552-6 ISBN 978-3-319-51553-3 (eBook)DOI 10.1007/978-3-319-51553-3

Library of Congress Control Number: 2017933961

© Springer International Publishing AG 2017This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,broadcasting, reproduction on microfilms or in any other physical way, and transmission or informationstorage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodologynow known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in this publicationdoes not imply, even in the absence of a specific statement, that such names are exempt from the relevantprotective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in this bookare believed to be true and accurate at the date of publication. Neither the publisher nor the authors orthe editors give a warranty, express or implied, with respect to the material contained herein or for anyerrors or omissions that may have been made. The publisher remains neutral with regard to jurisdictionalclaims in published maps and institutional affiliations.

Printed on acid-free paper

This Springer imprint is published by Springer NatureThe registered company is Springer International Publishing AGThe registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

. . . for my parents and my family. . .

“As if it were some practical purpose, geometricians always talk about squaring, extending,adding, when in fact science is grown for the sole purpose of knowing.”PLATON (República, Libro VII, 527)“Scientists study Nature not because of its usefulness but for the joy they find in its beauty.If Nature were not beautiful it would not merit our studying it and life itself would not beworth our efforts. I am not referring to the superficial aspects of beauty such as those whichonly concern external qualities or appearance; not because I hold them in contempt, whichwould be far from my intention, but because those bear no relation to science. I rather meanthat deeper beauty which comes from the harmonious order of all parts to which only pureintelligence is susceptible.”HENRI POINCARÉ

Dedicated to the memory ofNico G. van Kampen

whose influence on my permanent work runs deeper than I can know.

Preface to the First English Edition

The first edition of this book appeared 13 years ago in Spanish, published byReverté S.A. As compared with that Spanish edition, the first eight chapters havecontinued with the same topic, while the contents have been revised and expanded.In particular, advanced exercises with their solutions have been incorporated at theend of each chapter, Appendix I is new and introduces an approach to quantum opensystems, and Chap. 9 is new and has been included, in the English edition, with thepurpose of relating the stochastic approach with the important study of the relaxationfrom steady states. This topic brings the opportunity to develop, with some detail,the theory of first passage time in physical problems.

I should like to use this occasion to thank Prof. Dr. V. Grunfeld for the criticalrevision of the English translation.

vii

Foreword

This text is the result of several courses in nonequilibrium statistics, stochasticprocesses, stochastic differential equations, anomalous diffusion, and disorder,which I have been giving during the last 25 years at Instituto Balseiro, CentroAtómico Bariloche (Argentina). This book is aimed at university students ofphysics, chemistry, mathematics, science in general, and engineering. Readers areexpected to have a prior knowledge of mathematics and elements of physics froma fourth-year university course. However, less well-known concepts of physics andmathematics are developed not only in sections and special exercises throughout thewhole text but also in appendices. Some concepts of quantum mechanics, especiallythose which first-year students are still not acquainted with, are briefly presented inAppendices F, G, and I and in guided exercises, according to their needs.

Innovations

The physical-mathematical motivation is the main aspect throughout this text. Aca-demic issues regarding probability theory and stochastic processes are presented, aswell as new pedagogical aspects in the presentation of the nonequilibrium statisticstheory, stochastic differential equations, and disorder. Possible representations forstochastic processes are detailed, and a functional theory is presented for solvinglinear differential equations with arbitrary noises. In Chap. 4, I talk about theirreversibility problem in particular, and, in this context, we discuss the Fokker-Planck dynamic; the relaxation theory of nonstationary time-periodic Markoviansystems is also presented. In Chap. 6, I introduce a presentation of transportphenomena in finite and infinite lattices. In Chap. 7, the anomalous diffusiontheme is generally presented. In Chap. 8, bases are given in order to establish theexisting relationship between the microscopic aspects of linear response theory andthe calculation of the diffusion coefficient in amorphous systems. In Chap. 9, areview on fluctuations around metastable and unstable points is given, and Kramers’

ix

x Foreword

activation rate and Suzuki’s scaling time are presented. A generalized scaling theoryto study the lifetime from arbitrary nonlinear unstable points is presented.

Applications

Different applications and exercises are almost homogeneously found throughoutthe whole text. In Chap. 2, we introduce, as an application of the theory of randomvariables, the theory of fluctuations around thermodynamic equilibrium, originallydeveloped by Einstein and later in grater detail by Callen and Landau. In Chap. 3,several physical applications are given, from the stochastic processes’ theory tothe study of the relaxation in the solid-state area, and also the study of stochasticdifferential equations and its relationship with the Fokker-Planck equation by meansof Stratonovich stochastic differential equations. In Chap. 4, we present generalaspects regarding the concept of irreversibility by Onsager and the theory oftemporal fluctuation (first fluctuation-dissipation theorem). Several applications ofthe fluctuation-dissipation theorem to simple, mechanical, electrical, and magneticsystems are also presented in this chapter. In Chap. 5, we will talk about generalaspects of the linear response theory developed by Green and Callen, and wewill also talk about the (second) fluctuation-dissipation theorem using a magneticsystem to introduce an intuitive presentation of it. Other fundamental theoremsregarding the linear response theory are also deduced, and some applications insolid state are presented. In Chap. 6, the theory of diffusive transport in orderedmedia is presented. Emphasis is particularly laid on the analysis of discrete andcontinuous time Markovian random walks and, in general, on master equations withapplications on the study of finite systems with special (absorbing, reflective, andperiodic) boundary conditions; finally, we briefly present the statistics problem ofthe first passage random time through a given boundary. In Chap. 7, we presenttwo alternative and complementary techniques to confront the problem of diffusionin (amorphous) disordered media. The first is based on the effective mediumapproximation, while the second is based on the non-Markovian random walktheory. Emphasis is laid on the calculation of the diffusion coefficient in disorderedmedia, the displacement variance analysis as a function of time and its scaling laws(universal or not), which depend on the kind of disorder. Finally, the super-diffusionproblem and the analysis of diffusion with inner states are presented. Chapter 8deals with certain quantum aspects of the transport and irreversibility problem.We particularly discuss in detail the formulation of Kubo regarding the analysis ofthe linear response from a microscopic point of view (third fluctuation-dissipationtheorem) and the calculation of the electric conductivity (Green-Kubo formula).Also, we broadly discuss the formula of Scher and Lax for the calculation (withinthe classical limit) of the electric conductivity in (nonmetallic) disordered materials.Some examples and applications in a Lorentz’ gas are presented. Finally, we discussthe relationship between anomalous diffusion and certain characteristics of fractalgeometry. In Chap. 9, a review on fluctuations around metastable and unstable points

Foreword xi

is given. Emphasis is placed on establishing the connection between Kramers’activation time and the theory of the first passage time in stochastic process.Suzuki’s scaling theory—for the lifetime from an unstable point—is presented andgeneralized to study lifetime in critical points, as well as for non-Markovian process.

How This Course Is Designed

This textbook might be useful for the introduction to the study of stochasticprocesses and its applications in physics, engineering, chemistry, and biology. In thiscase, Chaps. 1 and 3 constitute the core of a course on random variables, stochasticprocesses and their relationship with stochastic differential equations. Chapter 2serves as a presentation to the theory of Einstein that deals with fluctuations aroundthermodynamic equilibrium, while Chaps. 4 and 5 end the course of nonequilibriumstatistics with the analysis of irreversibility in the context of Fokker-Plank equationand the linear response theory. Also Chap. 9 can be included in this introductionstudy of stochastic process, with the aim of applying the theory of the first passagetime in order to tackle the study of the lifetime at metastable and unstable points.

We can also design a course of introduction to the study of anomalous diffusionin disordered, or amorphous, media and its relationship with the calculation oftransport coefficients in the context of the linear response theory, which can bestudied independently of Chaps. 2, 3, and 4. In this course, Chaps. 6 and 7 givea detailed presentation of the anomalous diffusion problem. Chapter 8 is focused onthe microscopic presentation of Kubo’s formula for the calculation of electric con-ductivity. Appendix G.1 particularly presents a alternative demonstration of Kubo’sformula (or third theorem), which, from a pedagogical point of view, is easier thanthat originally introduced by Kubo. This is because, in this new presentation, weuse elementary concepts of the time-dependent perturbation theory of quantummechanics instead of the algebra of superoperators (Liouville-Neumann operator).Appendix I presents a review on quantum open systems with an application to thequantum random walk model.

In general, there are different options regarding exercises throughout the wholetext. In particular, we have exercises labeled as “optional,” which should be skippedat first reading. On the other hand, sections and chapters indicated with an asteriskare more advanced topics which should later require a second reading. Thereare also advanced exercises with their solutions presented at the end of eachchapter. Appendices from A to H are written with the aim of presenting for thesake of completeness certain physical-mathematical aspects regarding some topicsdiscussed throughout the text. Finally, those sections labeled as “excursus” arespecialized comments for those readers who wish to know more about a certaintopic.

The history of science shows that the interest toward noise (fluctuations) hasvaried according to its perception. During the nineteenth century, noise wasconsidered as “annoying,” not only in theoretical physics but also in experimental

xii Foreword

physics. In the beginning of the twentieth century, the study of fluctuationssurrounding equilibrium and its symmetries gave origin to the linear responsetheory, which includes the majestic and ground-breaking works done by Onsager(fluctuation-dissipation), while in the last decades of that same century, noisebecame essential for understanding self-organized structures out of equilibrium(synergetic). Simultaneously, in the last three decades, disorder (spatial noise) hasalso occupied a fundamental role in the comprehension of anomalous transportproblem.

In the last 40 years, nonequilibrium statistics has made huge progress in thecomplex understanding of fluctuations and mesoscopic phenomena induced bynoise. Nowadays, fluctuation concepts besides equilibrium, stochastic dynamics,noise-induced phase transition, stochastic resonance, chaotic regime, anomaloustransport, disorder, and fractal geometry, among others, are being used more andmore in basic subjects of exact sciences. It is thus necessary to introduce these basicelements to students in order to prepare them for bigger transformations that theymay surely deal with when facing a unified statistical theory of nonequilibrium. Thistext is expected to give a general idea in order to pave the way for readers towardunderstanding nonequilibrium statistics and its applications to anomalous transport(e.g., localization).

San Carlos de Bariloche, Rio Negro, Argentina Manuel O. Cáceres2017

Acknowledgments

I am pleased to express my gratitude to students, colleagues, and friends, who inone way or another have collaborated in the preparation of this book. Many of theirnames appear in the references I have used throughout this text. I would like to thankthe director of the Instituto Balseiro for logistical support to realize this effort, and inparticular the staff of English Academic Corps and the Department of Graphics. Mythanks also to the National Atomic Energy Commission, and the Centro AtómicoBariloche and Instituto Balseiro, National University of Cuyo, which for over morethan 30 years provided the essential habitat for training and scientific research.

xiii

List of Notations and Symbols

List of Notations

rv random variablesi statistical independentF-P Fokker-PlanckFPTD first passage time distributionMFPT mean first passage timeME master equationpdf probability distribution functionmv mean valuesp stochastic processsde stochastic differential equationProb. probabilitysirv statistical independent random variableGWN Gaussian white noiseRW random walkDOS density of statesCTRW continuous time random walk

List of Symbols

N natural numbersZ integer numbersO .1/ of order oneRe real valueC complex valueIm imaginary value� .x/ step function

xv

xvi List of Notations and Symbols

hh� � � ii cumulanth� � � i mean valueım;n Kronecker symbolı.n/ .x/ n-th derivative of the delta functionW .t/ Wiener processdW .t/ differential of a Wiener processı .x/ Dirac delta function

Contents

1 Elements of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction to Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Axiomatic Scheme� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Conditional Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2.2 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.4 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Frequency Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.1 Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.3.2 Properties of the Probability Density PX.�/ . . . . . . . . . . . . . . 10

1.4 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 The Simplest Random Walk� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.4.2 Examples in Which G.k/ Cannot Be

Expanded in Taylor Series� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.4.3 Characteristic Function in a Toroidal Lattice� . . . . . . . . . . . . 181.4.4 Function of Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . 21

1.5 Cumulant Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.6 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.7 Transformation of Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.8 Fluctuations Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.9 Many Random Variables (Correlations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.9.1 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.9.2 Marginal Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311.9.3 Conditional Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

1.10 Multidimensional Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.10.1 Cumulant Diagrams (Several Variables). . . . . . . . . . . . . . . . . . . 35

1.11 Terwiel’s Cumulants� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.12 Gaussian Distribution (Several Variables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

1.12.1 Gaussian Distribution with Zero Odd Moments . . . . . . . . . . 401.12.2 Novikov’s Theorem� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

1.13 Transformation of Densities in n-Dimensions. . . . . . . . . . . . . . . . . . . . . . . . 43

xvii

xviii Contents

1.14 Random Perturbation Theory� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.14.1 Continuous Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.14.2 Discrete Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

1.15 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.15.1 Circular Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 511.15.2 Trapezoidal Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . 511.15.3 Using Novikov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.15.4 Gaussian Operational Approximation . . . . . . . . . . . . . . . . . . . . . 541.15.5 Cumulant Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 541.15.6 Addition of rv with Different Supports. . . . . . . . . . . . . . . . . . . . 551.15.7 Phase Diffusion (Periodic Oscillations) . . . . . . . . . . . . . . . . . . . 561.15.8 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.15.9 Properties of the Characteristic Function . . . . . . . . . . . . . . . . . . 581.15.10 Infinite Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2 Fluctuations Close to Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . 612.1 Spatial Correlations (Einstein’s Distribution) . . . . . . . . . . . . . . . . . . . . . . . . 61

2.1.1 Gaussian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.2 Minimum Work� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.2.1 The Thermodynamic Potential ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . 712.2.2 Fluctuations in Terms of �P; �V; �T; �S . . . . . . . . . . . . . . . 71

2.3 Fluctuations in Mechanical Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.3.1 Fluctuations in a Tight Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742.4 Temporal Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.5 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.5.1 Time-Dependent Correlation in a StochasticToy Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.5.2 Energy of a String Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 80References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 Elements of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.1.1 Time-Dependent Random Variable . . . . . . . . . . . . . . . . . . . . . . . . 833.1.2 The Characteristic Functional

(Ensemble Representation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.1.3 Kolmogorov’s Hierarchy (Multidimensional

Representation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.1.4 Overview of the Multidimensional Representation . . . . . . . 913.1.5 Kolmogorov’s Hierarchy from the Ensemble

Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.3 Markov’s Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

3.3.1 The Chapman-Kolmogorov Equation . . . . . . . . . . . . . . . . . . . . . 963.4 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

Contents xix

3.5 2�-Periodic Nonstationary Processes� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.6 Brownian Motion (Wiener Process) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.6.1 Increment of the Wiener Process� . . . . . . . . . . . . . . . . . . . . . . . . . 1033.7 Increments of an Arbitrary Stochastic Process� . . . . . . . . . . . . . . . . . . . . . . 1043.8 Convergence Criteria� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1063.8.2 Markov Theorem (Ergodicity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083.8.3 Continuity of the Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.9 Gaussian White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.9.1 Functional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.10 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.10.1 Non-singular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1133.10.2 The Singular Case (White Correlation)� . . . . . . . . . . . . . . . . . . 113

3.11 Spectral Density of Fluctuations (Nonstationary sp)� . . . . . . . . . . . . . . . 1153.12 Markovian and Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

3.12.1 The Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . 1233.13 Einstein Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1253.14 Generalized Ornstein-Uhlenbeck Process� . . . . . . . . . . . . . . . . . . . . . . . . . . . 1273.15 Phase Diffusion� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

3.15.1 Dielectric Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1313.16 Stochastic Realizations (Eigenfunction Expansions) . . . . . . . . . . . . . . . . 1333.17 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

3.17.1 Langevin Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1373.17.2 Wiener’s Integrals in the Stratonovich Calculus . . . . . . . . . . 1393.17.3 Stratonovich’s Stochastic Differential Equations . . . . . . . . . 141

3.18 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1463.19 The Multidimensional Fokker-Planck Equation� . . . . . . . . . . . . . . . . . . . . 155

3.19.1 Spherical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583.20 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

3.20.1 Realization of a Campbell Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 1603.20.2 Gaussian White Noise Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1613.20.3 On the Realizations of a Continuous Markov Process . . . . 1613.20.4 On the Chapman-Kolmogorov Necessary Condition . . . . . 1623.20.5 Conditional Probability and Bayes’ Rule. . . . . . . . . . . . . . . . . . 1643.20.6 Second-Order Markov Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1653.20.7 Scaling Law from the Wiener Process . . . . . . . . . . . . . . . . . . . . . 1653.20.8 Spectrum of the Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 1663.20.9 Time Ordered Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1673.20.10 On the Cumulants of Integrated Processes . . . . . . . . . . . . . . . . 1683.20.11 Dynamic Versus Static Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.20.12 On the van Kampen Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1703.20.13 Random Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1723.20.14 Regular Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

xx Contents

4 Irreversibility and the Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . 1794.1 Onsager’s Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1804.2 Entropy Production in the Linear Approximation. . . . . . . . . . . . . . . . . . . . 183

4.2.1 Mechanocaloric Effect� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1844.3 Onsager’s Relations in an Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 1874.4 Ornstein–Uhlenbeck Multidimensional Process . . . . . . . . . . . . . . . . . . . . . 190

4.4.1 The First Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . 1914.5 Canonical Distribution in Classical Statistics . . . . . . . . . . . . . . . . . . . . . . . . 1954.6 Stationary Fokker–Planck Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

4.6.1 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1994.6.2 Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

4.7 Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.7.1 The One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2034.7.2 Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2054.7.3 Kramers’ Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2104.7.4 Generalized Onsager’s Theorem� . . . . . . . . . . . . . . . . . . . . . . . . . 2124.7.5 Comments on the Calculation of the

Nonequilibrium Potential� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2134.8 Nonstationary Fokker–Planck Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

4.8.1 Eigenvalue Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2144.8.2 The Kolmogorov Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2164.8.3 Evolution Over a Period of Time . . . . . . . . . . . . . . . . . . . . . . . . . . 2174.8.4 Periodic Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2214.8.5 Strong Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

4.9 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.9.1 Microscopic Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2274.9.2 Regression Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2284.9.3 Detailed Balance in the Master Equation . . . . . . . . . . . . . . . . . . 2294.9.4 Steady-State Solution of F-P (Case Jst

� D 0;

D�� D ı��D) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2304.9.5 Inhomogeneous Diffusion Around Equilibrium . . . . . . . . . . 2314.9.6 Chain of N Rotators (The Stationary F-P distribution) . . . 2324.9.7 Asymptotic Solution of the F-P Dynamics for

Long Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2354.9.8 2�-Periodic Nonstationary Markov Processes . . . . . . . . . . . . 2364.9.9 Time-Ordered Exponential Operator . . . . . . . . . . . . . . . . . . . . . . 237

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

5 Irreversibility and Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.1 Wiener-Khinchin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.2 Linear Response, Susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

5.2.1 The Kramers-Kronig Relations� . . . . . . . . . . . . . . . . . . . . . . . . . . . 2465.2.2 Relaxation Against a Discontinuity at t D 0 . . . . . . . . . . . . . . 2495.2.3 Power Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Contents xxi

5.3 Dissipation and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2535.3.1 Brownian Particle in a Harmonic Potential . . . . . . . . . . . . . . . . 2535.3.2 Brownian Particle in the Presence of a

Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2555.4 On the Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258

5.4.1 Theorem II: The Green-Callen’s Formula . . . . . . . . . . . . . . . . . 2595.4.2 Nyquist’s Formula� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

5.5 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2635.5.1 Spectrum of the Dichotomic Process . . . . . . . . . . . . . . . . . . . . . . 2635.5.2 On the Rice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2645.5.3 Ergodicity in Mean of the Mean-Value . . . . . . . . . . . . . . . . . . . . 2655.5.4 Ergodicity in Mean of Other “Statistical Quantities” . . . . . 2675.5.5 More on the Fluctuation-Dissipation Theorem. . . . . . . . . . . . 2685.5.6 The Half-Fourier Transform of Stationary Correlations . . 271

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6 Introduction to Diffusive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2736.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

6.1.1 Properties of T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2746.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

6.2.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2796.2.2 Moments of a Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2806.2.3 Realizations of a Fractal Random Walk� . . . . . . . . . . . . . . . . . . 284

6.3 Master Equation (Diffusion in the Lattice) . . . . . . . . . . . . . . . . . . . . . . . . . . . 2876.3.1 Formal Solution (Green’s Function) . . . . . . . . . . . . . . . . . . . . . . . 2916.3.2 Transition to First Neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2936.3.3 Solution of the Homogeneous Problem

in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2946.3.4 Density of States, Localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296

6.4 Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2976.4.1 Stationary Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3006.4.2 Short Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3026.4.3 Long Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303

6.5 Boundary Conditions in the Master Equation . . . . . . . . . . . . . . . . . . . . . . . . 3056.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3056.5.2 The Equivalent Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3066.5.3 Limbo Absorbent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3086.5.4 Reflecting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3096.5.5 Boundary Conditions (Method of Images) . . . . . . . . . . . . . . . . 3116.5.6 Method of Images in Finite Systems� . . . . . . . . . . . . . . . . . . . . . 3126.5.7 Method of Images for Non-diffusive Processes� . . . . . . . . . . 319

6.6 Random First Passage Times� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.6.1 Survival Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

6.7 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3266.7.1 On the Markovian Chain Solution . . . . . . . . . . . . . . . . . . . . . . . . . 326

xxii Contents

6.7.2 Dichotomic Markovian Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3276.7.3 Kramers-Moyal and van Kampen � Expansions . . . . . . . . . 3286.7.4 Enlarged Master Equation (Stochastic

Liouville Equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3306.7.5 Enlarged Markovian Chain (Noisy Map) . . . . . . . . . . . . . . . . . . 332

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

7 Diffusion in Disordered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3357.1 Disorder in the Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3357.2 Effective Medium Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

7.2.1 The Problem of an Impurity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3387.2.2 Calculation of the Green Function with an Impurity. . . . . . 3407.2.3 Effective Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3417.2.4 Short Time Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3447.2.5 The Long Time Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

7.3 Anomalous Diffusion and the CTRW Approach . . . . . . . . . . . . . . . . . . . . . 3507.3.1 Relationship Between the CTRW and the

Generalized ME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3547.3.2 Return to the Origin� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3567.3.3 Relationship Between Waiting-Time and Disorder� . . . . . . 3617.3.4 Superdiffusion� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365

7.4 Diffusion with Internal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.4.1 The Ordered Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.4.2 The Disordered Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3717.4.3 Non-factorized Case� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

7.5 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3757.5.1 Power-Law Jump and Waiting-Time from an

Entropic Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3757.5.2 Telegrapher’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3787.5.3 RW with Internal States for Modeling

Superionic Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3787.5.4 Alternative n-Steps Representation of the CTRW .. . . . . . . 3807.5.5 Distinct Visited Sites: Discrete and

Continuous Time Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3827.5.6 Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

8 Nonequilibrium Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3878.1 Fluctuations and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3878.2 Transport and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3898.3 Transport and Kubo’s Formula� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 390

8.3.1 Theorem III (Kubo). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3928.3.2 Kubo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3928.3.3 Application to the Electrical Conductivity . . . . . . . . . . . . . . . . 395

8.4 Conductivity in the Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3968.4.1 Conductivity Using an Exponential Relaxation Model . . . 397

Contents xxiii

8.5 Scher and Lax Formula for the Electric Conductivity . . . . . . . . . . . . . . . 3998.5.1 Susceptibility of a Lorentz Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4028.5.2 Fick’s Law (Static Limit) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4078.5.3 Stratified Diffusion (the Comb Lattice) . . . . . . . . . . . . . . . . . . . 4108.5.4 Diffusion-Advection and the CTRW Approach . . . . . . . . . . . 411

8.6 Anomalous Diffusive Transport (Concluded) . . . . . . . . . . . . . . . . . . . . . . . . 4138.6.1 The CTRW Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4138.6.2 The Self-Consistent Technique (EMA) . . . . . . . . . . . . . . . . . . . . 4158.6.3 Fractional Derivatives� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417

8.7 Transport and Mean-Value Over the Disorder . . . . . . . . . . . . . . . . . . . . . . . 4198.8 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

8.8.1 Quantum Notation� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4208.8.2 Classical Diffusion with Weak Disorder . . . . . . . . . . . . . . . . . . 4228.8.3 Fractal Waiting-Time in a Persistent RW. . . . . . . . . . . . . . . . . . 4228.8.4 Abel’s Waiting-Time Probability Distribution . . . . . . . . . . . . 4238.8.5 Nonhomogeneous Diffusion-Like Equation. . . . . . . . . . . . . . . 4258.8.6 Diffusion with Fractional Derivative . . . . . . . . . . . . . . . . . . . . . . 4268.8.7 Anomalous Diffusion-Advection Equation. . . . . . . . . . . . . . . . 427

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

9 Metastable and Unstable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4299.1 Metastable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

9.1.1 Decay Rates in the Small Noise Approximation . . . . . . . . . . 4309.1.2 The Kramers Slow Diffusion Approach . . . . . . . . . . . . . . . . . . . 4339.1.3 Kramers’ Activation Rates and the Mean First

Passage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4359.1.4 Variational Treatment for Estimating the

Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4389.1.5 Genesis of the First Passage Time in Higher

Dimensions� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4399.2 Unstable States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444

9.2.1 Relaxation in the Small Noise Approximation . . . . . . . . . . . . 4449.2.2 The First Passage Time Approach . . . . . . . . . . . . . . . . . . . . . . . . . 4469.2.3 Suzuki’s Scaling-Time in the Linear Theory . . . . . . . . . . . . . . 4469.2.4 Anomalous Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4499.2.5 Stochastic Paths Perturbation Approach for

Nonlinear Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4509.2.6 Genesis of Extended Systems: Relaxation

from Unstable States� . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4579.3 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 463

9.3.1 Dynkin’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4639.3.2 The Backward Equation and Boundary Conditions. . . . . . . 4649.3.3 Linear Stability Analysis and Generalized

Migration Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470

xxiv Contents

A Thermodynamic Variables in Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . 473A.1 Boltzmann’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

A.1.1 Systems in Thermal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474A.2 First and Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 476

B Relaxation to the Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479B.1 Temporal Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479B.2 Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482

C The Green Function of the Problem of an Impurity . . . . . . . . . . . . . . . . . . . . . 485C.1 Anisotropic and Asymmetric Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485C.2 Anisotropic and Symmetrical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487

D The Waiting-Time Function .t/ of the CTRW . . . . . . . . . . . . . . . . . . . . . . . . . . 489

E Non-Markovian Effects Against Irreversibility. . . . . . . . . . . . . . . . . . . . . . . . . . . 493E.1 ˆ.t/ and the Generalized Differential Calculus . . . . . . . . . . . . . . . . . . . . . . 496

F The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499F.1 Properties of the Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500F.2 The von Neumann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501F.3 Information Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

F.3.1 Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505

G Kubo’s Formula for the Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507G.1 Alternative Derivation of Kubo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 510

H Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515H.1 Self-Similar Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515H.2 Statistically Self-Similar Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519

I Quantum Open Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529I.1 The Schrödinger-Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529I.2 The Quantum Master-Like Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532

I.2.1 Approximations to Get a Quantum ME . . . . . . . . . . . . . . . . . . . 535I.3 Dissipative Quantum Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 536

I.3.1 The Tight-Binding Quantum Open Model . . . . . . . . . . . . . . . . 537I.3.2 Quantum Decoherence, Dissipation, and

Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545