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Non-equilibrium Statistical Physics with Application to Disordered Systems

Manuel Osvaldo Cáceres

Non-equilibrium Statistical Physics with Application to Disordered Systems

123

Manuel Osvaldo Cáceres Centro Atómico Bariloche and Instituto Balseiro Comisión Nacional de Energía Atómica and Universidad Nacional de Cuyo, and Comisión Nacional de Investigaciones Científicas y Técnicas San Carlos de Bariloche Rio 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 2017 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the 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 information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective 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 book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer International Publishing AG The 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 be worth our efforts. I am not referring to the superficial aspects of beauty such as those which only concern external qualities or appearance; not because I hold them in contempt, which would be far from my intention, but because those bear no relation to science. I rather mean that deeper beauty which comes from the harmonious order of all parts to which only pure intelligence is susceptible.” HENRI POINCARÉ

Dedicated to the memory of Nico 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 by Reverté S.A. As compared with that Spanish edition, the first eight chapters have continued with the same topic, while the contents have been revised and expanded. In particular, advanced exercises with their solutions have been incorporated at the end of each chapter, Appendix I is new and introduces an approach to quantum open systems, and Chap. 9 is new and has been included, in the English edition, with the purpose of relating the stochastic approach with the important study of the relaxation from 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 critical revision of the English translation.

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Foreword

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

Innovations The physical-mathematical motivation is the main aspect throughout this text. Academic issues regarding probability theory and stochastic processes are presented, as well as new pedagogical aspects in the presentation of the nonequilibrium statistics theory, stochastic differential equations, and disorder. Possible representations for stochastic processes are detailed, and a functional theory is presented for solving linear differential equations with arbitrary noises. In Chap. 4, I talk about the irreversibility problem in particular, and, in this context, we discuss the FokkerPlanck dynamic; the relaxation theory of nonstationary time-periodic Markovian systems is also presented. In Chap. 6, I introduce a presentation of transport phenomena in finite and infinite lattices. In Chap. 7, the anomalous diffusion theme is generally presented. In Chap. 8, bases are given in order to establish the existing relationship between the microscopic aspects of linear response theory and the calculation of the diffusion coefficient in amorphous systems. In Chap. 9, a review on fluctuations around metastable and unstable points is given, and Kramers’

ix

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Foreword

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

Applications Different applications and exercises are almost homogeneously found throughout the whole text. In Chap. 2, we introduce, as an application of the theory of random variables, the theory of fluctuations around thermodynamic equilibrium, originally developed by Einstein and later in grater detail by Callen and Landau. In Chap. 3, several physical applications are given, from the stochastic processes’ theory to the study of the relaxation in the solid-state area, and also the study of stochastic differential equations and its relationship with the Fokker-Planck equation by means of Stratonovich stochastic differential equations. In Chap. 4, we present general aspects regarding the concept of irreversibility by Onsager and the theory of temporal fluctuation (first fluctuation-dissipation theorem). Several applications of the fluctuation-dissipation theorem to simple, mechanical, electrical, and magnetic systems are also presented in this chapter. In Chap. 5, we will talk about general aspects of the linear response theory developed by Green and Callen, and we will also talk about the (second) fluctuation-dissipation theorem using a magnetic system to introduce an intuitive presentation of it. Other fundamental theorems regarding the linear response theory are also deduced, and some applications in solid state are presented. In Chap. 6, the theory of diffusive transport in ordered media is presented. Emphasis is particularly laid on the analysis of discrete and continuous time Markovian random walks and, in general, on master equations with applications on the study of finite systems with special (absorbing, reflective, and periodic) boundary conditions; finally, we briefly present the statistics problem of the first passage random time through a given boundary. In Chap. 7, we present two alternative and complementary techniques to confront the problem of diffusion in (amorphous) disordered media. The first is based on the effective medium approximation, while the second is based on the non-Markovian random walk theory. Emphasis is laid on the calculation of the diffusion coefficient in disordered media, 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-diffusion problem and the analysis of diffusion with inner states are presented. Chapter 8 deals with certain quantum aspects of the transport and irreversibility problem. We particularly discuss in detail the formulation of Kubo regarding the analysis of the linear response from a microscopic point of view (third fluctuation-dissipation theorem) and the calculation of the electric conductivity (Green-Kubo formula). Also, we broadly discuss the formula of Scher and Lax for the calculation (within the classical limit) of the electric conductivity in (nonmetallic) disordered materials. Some examples and applications in a Lorentz’ gas are presented. Finally, we discuss the relationship between anomalous diffusion and certain characteristics of fractal geometry. In Chap. 9, a review on fluctuations around metastable and unstable points

Foreword

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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 and generalized 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 stochastic processes and its applications in physics, engineering, chemistry, and biology. In this case, Chaps. 1 and 3 constitute the core of a course on random variables, stochastic processes and their relationship with stochastic differential equations. Chapter 2 serves as a presentation to the theory of Einstein that deals with fluctuations around thermodynamic equilibrium, while Chaps. 4 and 5 end the course of nonequilibrium statistics with the analysis of irreversibility in the context of Fokker-Plank equation and the linear response theory. Also Chap. 9 can be included in this introduction study of stochastic process, with the aim of applying the theory of the first passage time 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 diffusion in disordered, or amorphous, media and its relationship with the calculation of transport coefficients in the context of the linear response theory, which can be studied independently of Chaps. 2, 3, and 4. In this course, Chaps. 6 and 7 give a detailed presentation of the anomalous diffusion problem. Chapter 8 is focused on the microscopic presentation of Kubo’s formula for the calculation of electric conductivity. Appendix G.1 particularly presents a alternative demonstration of Kubo’s formula (or third theorem), which, from a pedagogical point of view, is easier than that originally introduced by Kubo. This is because, in this new presentation, we use elementary concepts of the time-dependent perturbation theory of quantum mechanics instead of the algebra of superoperators (Liouville-Neumann operator). Appendix I presents a review on quantum open systems with an application to the quantum random walk model. In general, there are different options regarding exercises throughout the whole text. In particular, we have exercises labeled as “optional,” which should be skipped at first reading. On the other hand, sections and chapters indicated with an asterisk are more advanced topics which should later require a second reading. There are also advanced exercises with their solutions presented at the end of each chapter. Appendices from A to H are written with the aim of presenting for the sake of completeness certain physical-mathematical aspects regarding some topics discussed throughout the text. Finally, those sections labeled as “excursus” are specialized comments for those readers who wish to know more about a certain topic. The history of science shows that the interest toward noise (fluctuations) has varied according to its perception. During the nineteenth century, noise was considered as “annoying,” not only in theoretical physics but also in experimental

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Foreword

physics. In the beginning of the twentieth century, the study of fluctuations surrounding equilibrium and its symmetries gave origin to the linear response theory, which includes the majestic and ground-breaking works done by Onsager (fluctuation-dissipation), while in the last decades of that same century, noise became essential for understanding self-organized structures out of equilibrium (synergetic). Simultaneously, in the last three decades, disorder (spatial noise) has also occupied a fundamental role in the comprehension of anomalous transport problem. In the last 40 years, nonequilibrium statistics has made huge progress in the complex understanding of fluctuations and mesoscopic phenomena induced by noise. Nowadays, fluctuation concepts besides equilibrium, stochastic dynamics, noise-induced phase transition, stochastic resonance, chaotic regime, anomalous transport, disorder, and fractal geometry, among others, are being used more and more in basic subjects of exact sciences. It is thus necessary to introduce these basic elements to students in order to prepare them for bigger transformations that they may surely deal with when facing a unified statistical theory of nonequilibrium. This text is expected to give a general idea in order to pave the way for readers toward understanding nonequilibrium statistics and its applications to anomalous transport (e.g., localization). San Carlos de Bariloche, Rio Negro, Argentina 2017

Manuel O. Cáceres

Acknowledgments

I am pleased to express my gratitude to students, colleagues, and friends, who in one way or another have collaborated in the preparation of this book. Many of their names appear in the references I have used throughout this text. I would like to thank the director of the Instituto Balseiro for logistical support to realize this effort, and in particular the staff of English Academic Corps and the Department of Graphics. My thanks also to the National Atomic Energy Commission, and the Centro Atómico Bariloche and Instituto Balseiro, National University of Cuyo, which for over more than 30 years provided the essential habitat for training and scientific research.

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List of Notations and Symbols

List of Notations rv si F-P FPTD MFPT ME pdf mv sp sde Prob. sirv GWN RW DOS CTRW

random variable statistical independent Fokker-Planck first passage time distribution mean first passage time master equation probability distribution function mean value stochastic process stochastic differential equation probability statistical independent random variable Gaussian white noise random walk density of states continuous time random walk

List of Symbols N Z O .1/ Re C Im  .x/

natural numbers integer numbers of order one real value complex value imaginary value step function xv

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hh   ii h   i ım;n ı .n/ .x/ W .t/ dW .t/ ı .x/

List of Notations and Symbols

cumulant mean value Kronecker symbol n-th derivative of the delta function Wiener process differential of a Wiener process Dirac delta function

Contents

1

Elements of Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Introduction to Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Axiomatic Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Conditional Probability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Bayes’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Frequency Scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Properties of the Probability Density PX ./ . . . . . . . . . . . . . . 1.4 Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 The Simplest Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Examples in Which G.k/ Cannot Be Expanded in Taylor Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Characteristic Function in a Toroidal Lattice . . . . . . . . . . . . 1.4.4 Function of Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . 1.5 Cumulant Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Transformation of Random Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Fluctuations Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Many Random Variables (Correlations) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Statistical Independence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Marginal Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.3 Conditional Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Multidimensional Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10.1 Cumulant Diagrams (Several Variables). . . . . . . . . . . . . . . . . . . 1.11 Terwiel’s Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12 Gaussian Distribution (Several Variables) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.12.1 Gaussian Distribution with Zero Odd Moments . . . . . . . . . . 1.12.2 Novikov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.13 Transformation of Densities in n-Dimensions. . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 5 5 6 8 8 10 11 14 16 18 21 22 24 26 29 30 30 31 32 34 35 37 38 40 41 43 xvii

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Contents

1.14

Random Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14.1 Continuous Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14.2 Discrete Time Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.1 Circular Probability Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.2 Trapezoidal Characteristic Function . . . . . . . . . . . . . . . . . . . . . . . 1.15.3 Using Novikov’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.4 Gaussian Operational Approximation . . . . . . . . . . . . . . . . . . . . . 1.15.5 Cumulant Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.6 Addition of rv with Different Supports. . . . . . . . . . . . . . . . . . . . 1.15.7 Phase Diffusion (Periodic Oscillations) . . . . . . . . . . . . . . . . . . . 1.15.8 Random Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.15.9 Properties of the Characteristic Function . . . . . . . . . . . . . . . . . . 1.15.10 Infinite Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

46 47 48 51 51 51 52 54 54 55 56 57 58 59 60

Fluctuations Close to Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . 2.1 Spatial Correlations (Einstein’s Distribution) . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Gaussian Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Minimum Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Thermodynamic Potential ˆ . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Fluctuations in Terms of P; V; T; S . . . . . . . . . . . . . . . 2.3 Fluctuations in Mechanical Variables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

61 61 62 68 71 71 73

2.3.1 Fluctuations in a Tight Rope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temporal Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Time-Dependent Correlation in a Stochastic Toy Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Energy of a String Deformation . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

74 78 79

Elements of Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Time-Dependent Random Variable . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 The Characteristic Functional (Ensemble Representation). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Kolmogorov’s Hierarchy (Multidimensional Representation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Overview of the Multidimensional Representation . . . . . . . 3.1.5 Kolmogorov’s Hierarchy from the Ensemble Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Conditional Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Markov’s Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 The Chapman-Kolmogorov Equation . . . . . . . . . . . . . . . . . . . . . 3.4 Stationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 83

2.4 2.5

3

79 80 81

84 88 91 93 93 94 96 98

Contents

2-Periodic Nonstationary Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brownian Motion (Wiener Process) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Increment of the Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Increments of an Arbitrary Stochastic Process . . . . . . . . . . . . . . . . . . . . . . 3.8 Convergence Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.2 Markov Theorem (Ergodicity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8.3 Continuity of the Realizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Gaussian White Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9.1 Functional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Non-singular Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 The Singular Case (White Correlation) . . . . . . . . . . . . . . . . . . 3.11 Spectral Density of Fluctuations (Nonstationary sp) . . . . . . . . . . . . . . . 3.12 Markovian and Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12.1 The Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Einstein Relation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Generalized Ornstein-Uhlenbeck Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15 Phase Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.15.1 Dielectric Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.16 Stochastic Realizations (Eigenfunction Expansions) . . . . . . . . . . . . . . . . 3.17 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17.1 Langevin Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17.2 Wiener’s Integrals in the Stratonovich Calculus . . . . . . . . . . 3.17.3 Stratonovich’s Stochastic Differential Equations . . . . . . . . . 3.18 The Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.19 The Multidimensional Fokker-Planck Equation . . . . . . . . . . . . . . . . . . . . 3.19.1 Spherical Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.1 Realization of a Campbell Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.2 Gaussian White Noise Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.3 On the Realizations of a Continuous Markov Process . . . . 3.20.4 On the Chapman-Kolmogorov Necessary Condition . . . . . 3.20.5 Conditional Probability and Bayes’ Rule. . . . . . . . . . . . . . . . . . 3.20.6 Second-Order Markov Process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.7 Scaling Law from the Wiener Process . . . . . . . . . . . . . . . . . . . . . 3.20.8 Spectrum of the Wiener Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.9 Time Ordered Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.10 On the Cumulants of Integrated Processes . . . . . . . . . . . . . . . . 3.20.11 Dynamic Versus Static Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.12 On the van Kampen Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.13 Random Rectifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.20.14 Regular Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 3.6

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100 101 103 104 105 106 108 109 110 111 113 113 113 115 118 123 125 127 130 131 133 137 137 139 141 146 155 158 160 160 161 161 162 164 165 165 166 167 168 170 170 172 174 176

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Irreversibility and the Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Onsager’s Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Entropy Production in the Linear Approximation. . . . . . . . . . . . . . . . . . . . 4.2.1 Mechanocaloric Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Onsager’s Relations in an Electrical Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Ornstein–Uhlenbeck Multidimensional Process . . . . . . . . . . . . . . . . . . . . . 4.4.1 The First Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . 4.5 Canonical Distribution in Classical Statistics . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Stationary Fokker–Planck Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 The Inverse Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Probability Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 The One-Dimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Multidimensional Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Kramers’ Equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Generalized Onsager’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.5 Comments on the Calculation of the Nonequilibrium Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 Nonstationary Fokker–Planck Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Eigenvalue Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.2 The Kolmogorov Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.3 Evolution Over a Period of Time . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.4 Periodic Detailed Balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.5 Strong Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Microscopic Reversibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.2 Regression Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 Detailed Balance in the Master Equation . . . . . . . . . . . . . . . . . . 4.9.4 Steady-State Solution of F-P (Case Jst D 0; D D ı D). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.5 Inhomogeneous Diffusion Around Equilibrium . . . . . . . . . . 4.9.6 Chain of N Rotators (The Stationary F-P distribution) . . . 4.9.7 Asymptotic Solution of the F-P Dynamics for Long Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.8 2-Periodic Nonstationary Markov Processes . . . . . . . . . . . . 4.9.9 Time-Ordered Exponential Operator . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Irreversibility and Linear Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Wiener-Khinchin’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linear Response, Susceptibility. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 The Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Relaxation Against a Discontinuity at t D 0 . . . . . . . . . . . . . . 5.2.3 Power Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

179 180 183 184 187 190 191 195 197 199 200 203 203 205 210 212 213 214 214 216 217 221 223 227 227 228 229 230 231 232 235 236 237 238 241 241 244 246 249 252

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5.3

Dissipation and Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Brownian Particle in a Harmonic Potential. . . . . . . . . . . . . . . . 5.3.2 Brownian Particle in the Presence of a Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 On the Fluctuation-Dissipation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Theorem II: The Green-Callen’s Formula . . . . . . . . . . . . . . . . . 5.4.2 Nyquist’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 Spectrum of the Dichotomic Process . . . . . . . . . . . . . . . . . . . . . . 5.5.2 On the Rice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Ergodicity in Mean of the Mean-Value . . . . . . . . . . . . . . . . . . . . 5.5.4 Ergodicity in Mean of Other “Statistical Quantities” . . . . . 5.5.5 More on the Fluctuation-Dissipation Theorem. . . . . . . . . . . . 5.5.6 The Half-Fourier Transform of Stationary Correlations . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

Introduction to Diffusive Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Properties of T1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Moments of a Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Realizations of a Fractal Random Walk . . . . . . . . . . . . . . . . . . 6.3 Master Equation (Diffusion in the Lattice) . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Formal Solution (Green’s Function) . . . . . . . . . . . . . . . . . . . . . . . 6.3.2 Transition to First Neighbors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Solution of the Homogeneous Problem in One Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Density of States, Localization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Models of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Stationary Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Short Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Long Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Boundary Conditions in the Master Equation . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 The Equivalent Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 Limbo Absorbent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.4 Reflecting State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.5 Boundary Conditions (Method of Images) . . . . . . . . . . . . . . . . 6.5.6 Method of Images in Finite Systems . . . . . . . . . . . . . . . . . . . . . 6.5.7 Method of Images for Non-diffusive Processes . . . . . . . . . . 6.6 Random First Passage Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Survival Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 On the Markovian Chain Solution . . . . . . . . . . . . . . . . . . . . . . . . .

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6.7.2 6.7.3 6.7.4

Dichotomic Markovian Chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kramers-Moyal and van Kampen  Expansions . . . . . . . . . Enlarged Master Equation (Stochastic Liouville Equation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.5 Enlarged Markovian Chain (Noisy Map) . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

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327 328 330 332 333

Diffusion in Disordered Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Disorder in the Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Effective Medium Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 The Problem of an Impurity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Calculation of the Green Function with an Impurity. . . . . . 7.2.3 Effective Medium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Short Time Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.5 The Long Time Limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Anomalous Diffusion and the CTRW Approach . . . . . . . . . . . . . . . . . . . . . 7.3.1 Relationship Between the CTRW and the Generalized ME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Return to the Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3 Relationship Between Waiting-Time and Disorder . . . . . . 7.3.4 Superdiffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Diffusion with Internal States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1 The Ordered Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 The Disordered Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3 Non-factorized Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Power-Law Jump and Waiting-Time from an Entropic Viewpoint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Telegrapher’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.3 RW with Internal States for Modeling Superionic Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4 Alternative n-Steps Representation of the CTRW . . . . . . . . 7.5.5 Distinct Visited Sites: Discrete and Continuous Time Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.6 Tauberian Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

335 335 337 338 340 341 344 344 350

Nonequilibrium Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Fluctuations and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Transport and Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Transport and Kubo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Theorem III (Kubo). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Kubo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Application to the Electrical Conductivity . . . . . . . . . . . . . . . . 8.4 Conductivity in the Classical Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Conductivity Using an Exponential Relaxation Model . . .

387 387 389 390 392 392 395 396 397

354 356 361 365 369 369 371 372 375 375 378 378 380 382 384 385

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8.5

Scher and Lax Formula for the Electric Conductivity . . . . . . . . . . . . . . . 8.5.1 Susceptibility of a Lorentz Gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2 Fick’s Law (Static Limit). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3 Stratified Diffusion (the Comb Lattice) . . . . . . . . . . . . . . . . . . . 8.5.4 Diffusion-Advection and the CTRW Approach . . . . . . . . . . . 8.6 Anomalous Diffusive Transport (Concluded) . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 The CTRW Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 The Self-Consistent Technique (EMA) . . . . . . . . . . . . . . . . . . . . 8.6.3 Fractional Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7 Transport and Mean-Value Over the Disorder . . . . . . . . . . . . . . . . . . . . . . . 8.8 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.1 Quantum Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8.2 Classical Diffusion with Weak Disorder . . . . . . . . . . . . . . . . . . 8.8.3 Fractal Waiting-Time in a Persistent RW . . . . . . . . . . . . . . . . . . 8.8.4 Abel’s Waiting-Time Probability Distribution . . . . . . . . . . . . 8.8.5 Nonhomogeneous Diffusion-Like Equation. . . . . . . . . . . . . . . 8.8.6 Diffusion with Fractional Derivative . . . . . . . . . . . . . . . . . . . . . . 8.8.7 Anomalous Diffusion-Advection Equation. . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

399 402 407 410 411 413 413 415 417 419 420 420 422 422 423 425 426 427 427

Metastable and Unstable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Metastable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Decay Rates in the Small Noise Approximation . . . . . . . . . . 9.1.2 The Kramers Slow Diffusion Approach . . . . . . . . . . . . . . . . . . . 9.1.3 Kramers’ Activation Rates and the Mean First Passage Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.4 Variational Treatment for Estimating the Relaxation Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.5 Genesis of the First Passage Time in Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Unstable States. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1 Relaxation in the Small Noise Approximation . . . . . . . . . . . . 9.2.2 The First Passage Time Approach . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.3 Suzuki’s Scaling-Time in the Linear Theory . . . . . . . . . . . . . . 9.2.4 Anomalous Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.5 Stochastic Paths Perturbation Approach for Nonlinear Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.6 Genesis of Extended Systems: Relaxation from Unstable States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Additional Exercises with Their Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Dynkin’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The Backward Equation and Boundary Conditions. . . . . . . 9.3.3 Linear Stability Analysis and Generalized Migration Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

429 430 430 433 435 438 439 444 444 446 446 449 450 457 463 463 464 465 470

xxiv

Contents

A Thermodynamic Variables in Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . A.1 Boltzmann’s Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1.1 Systems in Thermal Contact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 First and Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . .

473 473 474 476

B Relaxation to the Stationary State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 B.1 Temporal Evolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 479 B.2 Lyapunov Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 482 C The Green Function of the Problem of an Impurity . . . . . . . . . . . . . . . . . . . . . 485 C.1 Anisotropic and Asymmetric Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485 C.2 Anisotropic and Symmetrical Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487 D The Waiting-Time Function

.t/ of the CTRW . . . . . . . . . . . . . . . . . . . . . . . . . . 489

E Non-Markovian Effects Against Irreversibility. . . . . . . . . . . . . . . . . . . . . . . . . . . 493 E.1 ˆ.t/ and the Generalized Differential Calculus . . . . . . . . . . . . . . . . . . . . . . 496 F The Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.1 Properties of the Density Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.2 The von Neumann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3 Information Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F.3.1 Quantum Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

499 500 501 505 505

G Kubo’s Formula for the Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 G.1 Alternative Derivation of Kubo’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 510 H Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 H.1 Self-Similar Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 H.2 Statistically Self-Similar Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 I

Quantum Open Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.1 The Schrödinger-Langevin Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2 The Quantum Master-Like Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.2.1 Approximations to Get a Quantum ME . . . . . . . . . . . . . . . . . . . I.3 Dissipative Quantum Random Walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I.3.1 The Tight-Binding Quantum Open Model . . . . . . . . . . . . . . . . I.3.2 Quantum Decoherence, Dissipation, and Disordered Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

529 529 532 535 536 537 540

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