Contemporary Problems in Statistical Physics

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

51HJTL Society for Industrial and Applied Mathematics Philadelphia 1994

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

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

94-11368

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

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is a registered trademark.

Contents

Contributors

xi

Preface

xiii

Introduction

xv

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

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

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

CONTENTS 25 26 27 28 29 30 32 35

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

CONTENTS

vii

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

91 93 96

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

103

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

147

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

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

viii

4.4.

4.5.

4.6.

4.7. 4.8. 4.9.

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

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

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

CONTENTS 5.8. 5.9. 5.10. 5.11.

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

ix 228 232 239 242 244 245 248

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Contributors

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

XI

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Preface

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

xiv

PREFACE

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

Introduction

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

xvi

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

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

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

INTRODUCTION

xvii

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

xviii

INTRODUCTION

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

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

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

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

Chapter 1

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

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

Introduction

A chemical reaction in solution, such as

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

2

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

3

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

4

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

Models of molecular motion

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

5

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

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

6

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

7

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

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

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

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

8

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

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

where illustrated in Fig. 1.2. 1.3.

Typical VACFs within this famiily are

First-passage times for diffusions

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

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

5.0.

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

10

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

11

backward equation

with boundary conditions

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

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

The survival probability can be obtained from w since

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

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

and performing the deconvolution for the case n = 3 gives

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

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13

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

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

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

14

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

for coalescing Brownian motion and

for annihilating Brownian motion, where

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

15

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

for CBM and

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

16

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

17

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

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

First-passage times for integrated diffusions

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

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

where p is the transition density of the unconstrained singular diffusion y(^). a is any point within the sphere, and w(yo.y
18

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

passage time u and the state of the particle y = (x, v) at the instant of first passage. The interior integral is over the set S-, the product of the surface of the sphere, and the set of all inwardly directed velocities. The equation is useful because in certain cases the function p is already known. Taking Laplace transforms gives

However, in general there is no simple way to separate out the transformed density w. 1.4.1. Return times and impact velocities. If yo is on the surface of the sphere, with nonzero initial velocity, the first passage density w becomes the density of the first return. Only one case has been solved explicitly [53]. Using the Lebedev-Kontorovich transform pair [31], McKean was able to obtain the joint distribution of the hitting time t and impact velocity v for first return to the origin in the one-dimensional case in which v(t) — W(t\ i.e., the velocity process is simple Brownian motion. By scaling, it can be shown that the time taken to return when the initial velocity is VQ has the same distribution as VQ times the return time in the case VQ = —I. Similarly, the impact speed is multiplied by a factor VQ . Taking the initial velocity to be —1 the joint density becomes

from which the marginal density of t can be obtained by integrating out v over (0,oo). The first return time is also called the half-winding time. In the velocity-position plane it represents the time taken for the process to wind through 180°. Asymptotically, the half-winding density decreases as £~5/4 for large t. The slow rate of decrease has the surprising consequence that although the mode of the density is approximately 1, there is a 95% chance that no return will occur until t — 100, 000. The density and distribution function of the half-winding time are evaluated numerically in Fig. 1.4. 1.4.2. Hitting times from return times. Return densities can be used to obtain first-passage densities to arbitrary levels. A probability flux argument enables the joint density of the time to hit a sphere and the associated impact velocity to be expressed in terms of first-return densities starting from the surface. Starting from an arbitrary exterior point yo the joint density w(yo,y,t) is given by

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

19

FlG. 1.4. The density and distribution function of the half-win din (j time of integrated Brownian motion.

where nx is the inward normal on the surface of the sphere at the point x and S+ is the product of the sphere surface and the set of outward-directed velocity vectors [4]. Since both y and y' have x coordinates that arc on the surface of the sphere, the term w(y'.y.u) inside the integral is therefore a return density. The equation can be explained as follows. The first-passage density can be thought of as the rate at which particles cross the barrier for the first time. They will cross the barrier at a particular place x and with a particular velocity v. The first term on the right of (1.39) is the bulk rate at which the barrier is crossed. Because the diffusion is singular, particles flow across the barrier rather than diffuse. The bulk rate is therefore the concentration at the barrier multiplied by the component of velocity normal to the surface. The bulk rate includes particles crossing for the first time as wrell as those that have crossed before. The second term on the right of (1.39) corresponds to particles which have already crossed the barrier, have left the sphere at some time t — u. and subsequently, at time t. are returning to the barrier for the first time. The equation is therefore a simple accounting equation, which is possible for integrated processes because boundary crossings occur only finitely often. A similar argument shows that the following expression is identically zero:

20

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSIS

where y E S+. Combining (1.39) and (1.40) and integrating over y using the divergence theorem, w(yo,t) is therefore given by

In other words, the hitting-time density w(yo, t) can be expressed as a weighted integral of the return-time density. Again, in the simplest case with v(t) = W(t) some analytical progress is possible. Goldman [34] has shown that the density w(t) of the time to hit level x > 0 starting from XQ = 0, VQ = 0 has asymptotic form

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

where

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

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

21

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

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

22 1.5.

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS Computer simulation methods

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

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

23

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

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

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

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

24

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

25

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

26

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

1.6.

Independent pairs

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

27

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

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

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

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

where

28

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

where N represents the number of ion pairs present and

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

29

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

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

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

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

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

The independent reaction times approximation

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

30

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

31

of the initial separation gtj ,

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

32

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

33

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

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

34

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

MICROSCOPIC NONHOMOGENEOUS SYSTEMS

35

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

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

36

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

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

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

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

REFERENCES

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[43] N. J. B. Green, M. J. Pilling, S. M. Pimblott, and P. Clifford, Stochastic modeling of fast kinetics in a radiation track, J. Phys. Chem.. 94 (1990). pp. 251 258. [44] N. J. B. Green and S. M. Pimblott, Asymptotic analysis of diffusion-influenced kinetics with a potential. J. Phys. Chem., 93 (1989), pp. 5462 5467. [45] , Stochastic modelling of partially diffusion-controlled reactions in spurs kinetics, J. Phys. Chem., 96 (1992), pp. 9338-9348. [46] J. M. Harrison, Brownian Motion and Stochastic Flow Systems. Wiley. New York. 1985. [47] K. M. Hong and J. Noolandi, Solution of the Smoluchovski equation with Coulomb potential. II. Application to fluorescence quenching, J. Chem. Phys.. 68 (1978). pp. 5172 5176. [48] S. Karlin and H. M. Taylor, A First Course in Stochastic Processes, 2nd ed.. Academic, New York, 1975. [49] , A Second Course in Stochastic Processes, Academic, New York. 1981. [50] A. Knpperrnann, Diffusion kinetics in radiation chemistry: An assessment, in Physical Mechanisms in Radiation Biology, USAEC 721001, 1975. pp. 155 176. [51] A. A. Lushnikov, Binary reaction 1 +1 —» 0 in one dimension, Phys. Lett. A. 120 (1987). pp. 135 137. [52] R. M. Lyndcn-Bell. D. J. C. Hutchinson, and M. J. Doyle, Translational molecular motion and cages in computer molecular liquids, Molec. Phys., 58 (1986). pp. 307 315. [53] A. P. McKean. A winding problem for a resonator driven by white noise. J. Math. Kyoto Univ.. 2-2 (1963), pp. 227-235. [54] D. A. McQuarrie, Stochastic approach to chemical kinetics, J. Appl. Prob., 4 (1967), pp. 413-478. [55] G. N. Milshtcin. Approximate integration of stochastic differential equations, Theory Prob. Appl., 19 (1974), pp. 557-562. [56] . A method of second-order accuracy integration of stochastic differential equations, Theory Prob. Appl., 28 (1978), pp. 396-401. [57] L. Monchick, J. L. Magec, and A. H. Samuel, Theory of radiation chemistry IV. Chemical reactions in the general track composed of N particles, J. Chem. Phys., 26 (1957), pp. 935-941. [58] H. Mori, Transport, collective motion and, Brownian motion. Prog. Theor. Phys., 33 (1965). pp. 423 455. [59] P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II, McGrawHill. New York. 1953. [60] A. Mozurnder, Mean recombination time of diffusion-controlled geminate reaction, J. Chem. Phys., 76 (1982), pp. 5107-5111. [61] A. Mozumder and J. L. Magee, Theory of radiation chemistry VII. Structure and reaction in low LET tracks, J. Chem. Phys., 45 (1966), pp. 3332-3341. [62] A. Mozumder, S. M. Pimblott, P. Clifford, and N. J. B. Green, Electron-ion geminate escape probability in anisotropic media, Chem. Phys. Lett., 142 (1987). pp. 385-388. [63] R. M. Noyes, Effects of diffusion rates on chemical kinetics, Prog. React. Kinet., 1 (1961), pp. 129-160. [64] E. Pardoux and D. Talay, Discretization and simulation of stochastic differential equations, Acta Applic. Math., 3 (1985), pp. 23-47. [65] J. B. Pedersen, The reactivity dependence of the recombination probability. J. Chem. Phys., 72 (1980), pp. 3904-3908.

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

Chapter 2

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

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

Introduction

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

41

42

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

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

FLUCTUATIONS IN NONLINEAR SYSTEMS

43

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

and measurement of $(0;) simply involves recording f ( t ) over a sufficiently long time 2t0 and calculating a Fourier transform. Since $(cj) is a physical observable it is advantageous to express the characteristics of a noise-driven system in terms of it. The shape of the power spectrum depends on the source of the noise. For example, if the noise results from coupling to a thermal bath then 4>(o;) is determined by the density of states of the elementary excitations of the bath, the coupling constants between the bath and the system of interest, and the temperature. The shape of $(u) is used to differentiate, very roughly, between two types of noise: white and colored. White noise is characterized by a totally flat spectrum <£(u;) = const., in analogy with white radiation where all spectral components are of the same intensity. Noise whose spectrum deviates from a constant value is called "colored" (an additional reason for using optical terms in the context of noise is that "normal" incoherent radiation is itself noisy; the electric and magnetic fields fluctuate about their zero-average values). Strictly speaking, any physical noise is colored: <E>(o;) must necessarily vanish as uj —-» oo, since otherwise the time correlation function 0(t) would diverge as t —> 0, i.e., the mean-square value of the noise would be infinite. However, if all the characteristic eigenfrequencies and reciprocal relaxation times of a noise-driven system are small compared to the frequencies over which <£(u;) changes, the effects of the color of the noise are expected to be minor and the noise can be assumed to be white. A macroscopic Brownian particle in a liquid is a typical example where the latter approximation holds; the characteristic reciprocal duration of the collisions with solvent molecules that give rise to the noisy forces acting on the Brownian particle is of order 1013 sec~] and exceeds by many orders of magnitude the typical frequencies of mechanical motion of the particle. Fluctuations in systems driven by white noise have been investigated for nearly 90 years (see [1], [2], [6], [9]-[15], and references therein), and the principal aspects of the problem are generally fairly well understood (although there are still problems to be solved, both numerical and qualitative, as discussed subsequently). A "massive attack" on the effects of the color of noise started only about a decade ago. Although several topical reviews on the subject have already appeared [15] [17], a number of important new results have been obtained since. Some of these results and their underlying concepts are outlined herein. We primarily consider the effects of weak noise, i.e., the situation where the characteristic noise intensity D given by the maximum value of the power spectrum of the noise,

44

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

is small (in all states of the system if the noise is state-dependent), so that the root-mean-square fluctuational deviations of a noise-driven system from its attractor (or attractors, if several of them coexist) are small compared to the distances between the attractors and to other characteristic length scales of the phase space of the system. Our aim is to demonstrate mainly the qualitative features of the pattern of fluctuations; in this spirit we shall avoid rigorous mathematical proofs and limit ourselves to physical ideas and to "physical rigor." We note that in many papers the terminology "colored noise" is reserved for exponentially correlated noise,

The analysis below is not limited to this particular type of colored noise. A function that characterizes much of the behavior of a fluctuating system is the statistical distribution, i.e., the stationary probability density (the probability density achieved as t —> oo). The shape of this distribution depends substantially on whether dissipation and fluctuations in the system are both due to its coupling to a thermal bath (i.e., whether the noise is of thermal origin), or whether the driving noise is of nonthermal origin. In the first case there is a relation between the noise driving the system and the dissipative forces that extract energy from the system (cf. §2.2). As a consequence, the shape of the stationary distribution of the system (which in this case is an equilibrium distribution in the thermodynamic sense) for sufficiently weak coupling is Gibbsian regardless of the shape of the power spectrum $(0;). If the power spectrum is not flat, i.e., if the noise is colored, the relaxation toward the equilibrium state occurs via an equation of motion that is not time-local (contains explicit memory terms), that is, the relaxation is "retarded." The fluctuation-dissipation relation can sometimes be described phenomenologically, as in the case of Brownian motion [1] (cf. [18] and [19] and references therein for a discussion of fluctuation-dissipation relations in systems driven by colored noise). In some cases the noise and dissipation can be calculated from a microscopic model of the dynamical system of interest, a heat bath, and their coupling. The latter approach was first applied by Bogoliubov [20] to the problem of a linear oscillator coupled to a phonon bath; the corresponding quantum analysis of the dynamics of a linear oscillator coupled to a heat bath was first given by Schwinger [21] (see also Senitzky [22]). If the noise driving a system is of nonthermal origin, the statistical distribution of the system in the stationary state (if indeed one exists) is nonGibbsian. However, for sufficiently weak noise and nonbifurcational parameter values, the distribution still has a maximum (maxima) at the attractor(s) and is Gaussian near the maximum (maxima), just as in the case of thermal equilibrium. The dependence of the parameters of the Gaussian distribution(s) on the shape of the power spectrum of the driving noise is discussed briefly in §2.2. The spectral densities of the fluctuations of systems driven by thermal noise and bv nonthermal colored noise are also considered there.

FLUCTUATIONS IN NONLINEAR SYSTEMS

45

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

Spectral density of fluctuations and statistical distribution near a stable state

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

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

46

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

with

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

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

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

FLUCTUATIONS IN NONLINEAR SYSTEMS

47

where

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

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

we can write

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

where

and

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

48

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

is given by (2.18) with

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

FLU C T U ATI ONS IN NON LI NEAR SYSTEMS

49

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

If $(u;) were constant (white noise), the derivative Q'(t) would be singular as t —> 0 since the integral over a; of a;/(a2 + a;2) diverges. If, however, <£(a;) decays to zero with increasing u;, as is the case for any physical noise, this derivative necessarily vanishes since the integral of cj$(o;)/(a2 + cu2) converges and <£>(o;) is an even function of uj (and also the stationary correlation function Q(t] is an even function of t } . 2.2.2. Statistical distribution near the maximum. The expressions obtained above in the case of weak noise lead in a straightforward way to an expression of the parameters of the statistical distribution p(x] of the system near a maximum in terms of the spectral densities $nm(u). It follows from (2.10) that the components (3nm of the matrix j3 formed from the means of the products of the displacement components are given by

By the ergodic hypothesis that permits the interchange of time averages and ensemble averages, one can interpret the matrix elements (3nm as second-order moments of the statistical distribution p(x):

Since the driving noise is zero-mean Gaussian, the higher moments of p(x) can also be expressed in terms of the !3nm. Indeed, according to (2.12),

and therefore all the odd moments of p(x] vanish while the relationships among the even moments are similar to those of the noise. For example,

50

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

(cf. (2.3)). This implies that the statistical distribution p(x) is Gaussian near the maximum, i.e., of the form [24]

The expression for the distribution takes an even simpler form for a onedimensional system. It follows from (2.29) that in this case

with x = x\ (an alternative derivation of (2.30) based on the path-integral formulation is presented in §2.3). An important feature of the statistical distribution arises from the fact that the spectral density of the noise 3>(u) and hence /3 scale as D. As a result, it is evident in (2.29) and (2.30) that the width of the Gaussian distribution is proportional to the noise intensity D, i.e., the distribution becomes increasingly narrower with decreasing noise. The distribution also becomes narrower as the maximum of the power spectrum of the noise moves away from the range of large susceptibility of the system, i.e., the range where the Green functions le-

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

where

FLUCTUATIONS IN NONLINEAR SYSTEMS

51

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

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

where

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

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

52

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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53

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

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

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

Large fluctuations: Method of the optimal path

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

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55

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

We shall call x a "spatial" variable. If the system is monostable, i.e., if the potential U(x] has only one minimum at x = xst [Uf(xst] = 0], its intrinsic motion is characterized by the relaxation time tr = l/U"(xst). The spatial scale of the most probable fluctuations of the system is given by the root-mean-square displacement Ax — (i1/2 about the stable position xst. According to (2.25), Ax is proportional to the square root of the noise intensity D (cf. the last paragraph in §2.2.2). On the other hand, the decay of correlations of the noise is characterized by the correlation time tc given by the reciprocal width of the narrowest peak (or dip) of the power spectrum <£(u;) of the noise defined in (2.2). (In general, colored noise may be characterized by several times equal to the reciprocal positions (excluding the peak at LJ = 0 if present) and widths of all the peaks of $(0;), but it is obviously the smallest of the reciprocal widths that gives the time over which a value of f ( i ] is forgotten.) "Large" fluctuations are those that cause the system to move away from st x by a distance that substantially exceeds Ax. Apart from the spatial scale, there is also a time scale associated with large fluctuations. It is obvious that the sojourn in the vicinity of a remote point x and the return to xst takes a time of order

Large noise "outbursts" that cause the system to stray far away from the stable state xst are rare. If the farthest state x reached as a consequence of such an outburst is sufficiently far from xst, then the time T(x) between these large noise "outbursts" substantially exceeds t0. It is easy to imagine the character of such large fluctuations. During a time t ~ T(x] (but t ^> t0) the system fluctuates mostly near the stable state. Then the system makes an "excursion" to x of characteristic duration t0. The successive excursions to x are thus statistically independent of one another, since the previous excursion has been forgotten by the time the next excursion occurs. Of course, in addition to the excursions to a given x, other excursions to remote points may be taking place. The duration of each of them is of order t 0 , since this is the only time that characterizes the correlation of fluctuations or the deterministic evolution of the system (whichever takes more time). The intervals between excursions to any extreme value that is sufficiently far from the stable state are also statistically independent of one another. It is evident from the "physical" notion of the probability p(x}dx as the relative length of time spent in a small vicinity dx around the point x (ergodic hypothesis) that p ( x ) / p ( x s t ) ~ t0/T(x). It is also clear from the above picture that T(x) might be called a mean first-passage

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

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

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57

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

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

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

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

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

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

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

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

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59

vanish as t —> ±00, and for which the integral j dt f(t)F(—id/dt)f(t] exists. The smoothness condition on f(t) imposes no physical restriction in practice since noise generated by real physical sources does not have singularities. The results considered below refer mostly to the case where $~1(o;) is a finite series in cj; in particular, for exponentially correlated noise as in (2.6), <&~l(u} = (1 + u2t^}/D. This corresponds to f(t] being a component of a Markov process with the total number of components equal to the degree of the polynomial $~1(u;) divided by two [45]. Since the power spectrum $(u;) is positive for all u;, the argument of the exponential in (2.46) is negative for all /(*)• The crucial point for the mathematical formulation of the method of the optimal path in systems driven by weak Gaussian noise is that if a system such as that described by (2.38) has been moved from a stable state in the phase space to another point by an external force, it must have been subjected to finite forcing applied over some time. Although different realizations of the force can result in the same final destination of the system at time t, the point is that the trajectories leading there are essentially deterministic, not random, and that they are independent of the noise intensity D. Therefore, if D is sufficiently small, the probabilities of all of these realizations are seen from (2.46) to be exponentially small. Moreover, it is also seen from (2.46) that the probabilities of different realizations differ exponentially from each other. Thus one would expect there to exist a realization f(t) which is much more probable than the others, and it is just this realization that is associated with the optimal path fopt(t) of the noise. The optimal path xopt(t) of the system is obtained from the equation of motion (e.g., from (2.38)) with f(t) = fopt(t)It follows from (2.40), (2.41), and (2.46) and from the above arguments that the statistical distribution p(x) is exponentially small for points x that are removed from the stable state xst by a distance that substantially exceeds the root-mean-square displacement Ax ~ ((x~Xs*)2}1'2 oc D 1 / 2 . Since we have limited ourselves to seeking p(x) to logarithmic accuracy in the noise intensity it is convenient to write it as

Here, the argument R(x) of the exponential is just the value of the integral (l/2)fdtf(t)F(-id/dt)f(t) in (2.46) for the optimal path fopt(t). The factor C(x)p(xst) (the coefficient p ( x s i ) is written explicitly simply for convenience) is a smooth function of x on the scale Ax oc D1/2 that allows for the normalization (the denominator in (2.41)) and for the fact that not only the optimal path but a "tube" of paths surrounding it contribute to a steepest-descent evaluation of the integral in the numerator in (2.41). The width of this tube is determined by the form of the probability density functional p of (2.46) and also by the form of the equation of motion of the system and the value of x. It is obvious from (2.46), however, that p[/(t)j drops sharply for those /(t) that differ from fopt(t] by more than ~ D 1 / 2 , and therefore this width is proportional to D1'2

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as well. The analysis of the prefactor C(x) in (2.47) for some limiting cases is briefly discussed below. To calculate the argument R(x) of the exponential in (2.47), we note that it is the motion of the system from the stable state xst to a given point x that we are monitoring. Therefore, the integral in the argument of the exponential in (2.46) should be minimized subject to the inter-relationship (2.38) between the optimal path fopt(t) and the associated path xopt(t). Therefore, R(x), fopt(t}, and xopt(t] are given by the solution of the following variational problem [45]:

with the boundary conditions

and the relation (2.38). The first term in the functional 3ft in (2.48) is the argument of the exponential in (2.46) (without the coefficient —1/D). The second term allows for the inter-relationship between /(£) and x(t); X(t) is an undetermined Lagrange coefficient. With this term, the functional 3ft should be minimized independently with respect to both f(t] and x(t). The boundary conditions (2.49) correspond to the physical picture of the motion described above. Prior to arrival at a remote point x at the instant t — 0 as a result of a large fluctuation, the system spends a long time that substantially exceeds t0 (of the order of the mean first-passage time T(x)) near the stable state xst . Here the noise is weak, so that x — xst oc \f(t)\ oc D1/2. The boundary conditions on x(t] and f ( t ) for t —> — oo express this fact to within an error proportional to Z)1/2 (reflecting the uncertainty in the position of the system near xst and of the noise near zero). This error can be neglected since all contributions to R(x] that vanish when D —->• 0 are ignored. The conditions (2.49) for t > 0 follow from the fact that the calculation of the statistical distribution p(x) to logarithmic accuracy does not require us to follow the further evolution of the system once it has reached the -given point x; therefore, the driving noise f ( t ) can be allowed to decay back toward zero for t > 0 "on its own" independent of x(t), and therefore one can set X(t) = 0 (this boundary condition has also been taken into account in (2.48) by setting the upper limit in the second integral in 3ft equal to zero). We note that X(t) can be discontinuous for t = 0. At the same time, f ( t ) itself and also several derivatives of f ( t ) are continuous (except for the case of white noise; see below). Because of this continuity, there arises a "postaction" : the decay of f ( i ] for

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

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

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

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with

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

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

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

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

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

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

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

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

where

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

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R(x] in (2.63) is the well-known result for systems driven by white noise and can be obtained directly from (2.62). It leads to a Boltzmann distribution p(x) in (2.47) [24] with an "effective temperature" of D/2F(0) = (1/2)$(0). This is in agreement with the observation (cf. Fig. 2.3(a)) that the effective intensity of the "white" noise is proportional to $(0) for \tc <^.tr. The second term in R(x) is the correction due to the color of the noise. We stress that this correction, although small compared to the main term, can change the statistical distribution p(x) exponentially strongly, since it can substantially exceed the characteristic noise intensity D. Another point to note is that this correction term may be positive or negative depending on the shape of the power spectrum of the noise. In other words, the color of the noise may either "squeeze" or "extend" the statistical distribution. The former situation occurs when Ic > 0 since R(x] then exceeds its white-noise value. The latter occurs if tc < 0. An example of a power spectrum <3>(o;) that leads to a negative finite-correlation-time-induced correction to R(x] is considered in §2.4. If the noise is exponentially correlated (cf. (2.6)), the correlation time tc in (2.64) is precisely the time tc that parametrizes the correlation function of the noise, and the correction to R(x] is then positive. Equation (2.47) with (2.63) in this case coincides with the result of others (see [49] and [50] and also [17], [51]-[53], and references therein). These results have been obtained by several methods. In particular, the approaches based on the derivation of "effective Fokker Planck equations" for the statistical distribution by various approximation procedures have been reviewed in detail by Lindenberg et al. [17]. The approach of [49] and [50] exploits the fact that the exponentially correlated noise /(t) itself, and the composite system [x(t), /(£)] consisting of the dynamical system of interest and the noise, can be viewed as Markov processes, with the equation for x(t) being of the form (2.38) and that for f ( t ) being of the form

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

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

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

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

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

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

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

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

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

form

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

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

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

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

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

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

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

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

and

so that finally [45]

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

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

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

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

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

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

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

If the motion in this region lasts for a time ot
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To summarize: it follows from the results of the present section that the dependence of the tails of the statistical distribution of a system driven by Gaussian noise on the noise intensity is of the activation type. The problem of calculating the activation energy for reaching a given point can be reduced, via the method of the optimal path, to a variational problem which in turn in many cases translates to a boundary-value problem involving a set of ordinary differential equations. The form of the equations is determined by the shape of the power spectrum of the noise. They can be solved analytically in several limiting cases, including those of large and small characteristic correlation times of the noise. In general it is straightforward to investigate the equations numerically. 2.4.

Fluctuations induced by quasi-monochromatic noise

The general formulation presented above clearly demonstrates that small fluctuations about stable states of noise-driven systems as well as the statistical distribution of large fluctuations depend crucially on the shape of the power spectrum of the noise. In many cases the noise turns out to be "truly colored": its power spectrum is peaked at a certain frequency u;0, and the halfwidth F of the peak is much smaller than u;0,

An example of such noise is "normal" (incoherent) nearly monochromatic light. Such light possesses a specific "color," and it would therefore be reasonable to use the terminology "colored" for any noise with a similar peaked power spectrum. However, to avoid confusion, we have used and will continue to use the standard terminology; to distinguish this sort of noise from other forms of colored noise we will therefore call it quasi-monochromatic noise (QMN) [62]. QMN is generated by a variety of noisy systems capable of singling out a frequency and can be viewed as the result of filtering broad-band noise through a highly selective system. Examples of such systems include various electromagnetic or acoustic high-Q cavities: their eigenvibrations excited at random by an external noise produce QMN [5], [6]. Another well-known example is that of local and quasi-local (resonant) vibrations of impurities in crystals [63] (see [64] for recent work); such vibrations are mainly characterized by a single frequency and are coupled to a broad band of other modes of the crystal. Their thermal fluctuations are a typical example of QMN. The spectral densities of fluctuations of local and resonant vibrations have been thoroughly investigated both theoretically and experimentally (see [33], [63], and [65] for reviews). Yet other examples of QMN are provided by random vibrations of fragments of macromolecules (in particular, enzymes [66]), and of engineering structures. The simplest type of QMN (which will be the specific noise that we refer to as QMN below) is that produced by a harmonic oscillator of frequency UJQ

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and friction coefficient F driven by white noise (cf. (2.21)):

The power spectrum of QMN (i.e., the spectral density of the fluctuations of the oscillator) is of the form

(cf. (2.18) and (2.23)). The characteristic noise intensity D =
Noise with the power spectrum (2.89) with parameters F and u;0 that are unrelated beyond the constraint (2.87) has sometimes been called "harmonic noise" [66] . Such noise is a simple example of colored noise with two correlation times given by F-1 and u;"1. Although strong effects related to the presence of two times are expected to arise (and indeed do arise) when u0 ^> F, some features already become apparent for cj2 > 2F2 when the peak of the power spectrum <£(u;) in (2.89) is positioned at a finite frequency, in contrast to that of the exponentially correlated noise. Changes in the structure of 3>(u;) related to the shift of its peak to finite frequency are expected to affect the fluctuations of the system noticeably when the position of the peak is of the order of the reciprocal relaxation time t~l of the system (cf. Fig. 2.3). However, some effects arise even for large tr. In particular, it can be seen from (2.64) that the parameter tc in the expression for the correction to the activation energy ^?(.r) when the correlation time of the noise is finite is of the form [45]

It follows from (2.63) and (2.91) that, depending on the ratio u; 2 /F 2 . the correction to /?(./') is either positive (if u;2 < 4F 2 ) or negative (if ^2 > 4F 2 ) and therefore the distribution p(x] becomes either squeezed or extended relative to the distribution associated with white noise. The most interesting and nontrivial effects associated with a noise that is characterized by two correlation times rather than a single one might be expected to occur when these times are substantially different, as is the case for quasi-monochromatic noise with F
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This is the case that is analyzed further in the present section [45], [62]. The analysis is presented in the context of the simplest model of a noise-driven system, as described in (2.38). 2.4.1. Double-adiabatic approximation for a QMN-driven system. To gain insight into the characteristic features of the fluctuations of the dynamical variable x ( t ) in the parameter range (2.92) we note that, when (2.87) is fulfilled, the QMN f ( i ) consists mostly of nearly periodic random vibrations of frequency UJQ. In fact, according to (2.88), f ( t ) is of the form

The complex amplitudes f±(f] of the random vibrations vary smoothly in time; their correlation time is equal to F"1,

and the nonresonant addition 6f(t) is small on the average (it has been omitted when estimating (|/± 2 )). The rapidly oscillating random force f ( t ) gives rise to fast oscillations of the noise-driven system, i.e., to rapidly oscillating terms x±(t)exp(±iu0t) in the coordinate x(t) in (2.38) in addition to a smooth contribution xc(i). Because of the inequality u0tr
Equation (2.38) is the only equation for the variables x±(t) and xc(t), and therefore two of these three must be specified separately in some fashion. To do this we note that if the amplitudes x±(t) and xc(t) were time-independent, then the force U'(x] would be a series in exp(±zu;0£). The coefficients of this series would be expressed in terms of x±,xc and they would obviously be of order t~l « U"(xst}. The same expansion is physically meaningful if

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x±,xc vary smoothly with t compared to exp(±2CJ0i)- When substituting the corresponding expansion into (2.38) it is convenient to set the x-dependent coefficients of the rapidly oscillating terms exp(±zu;0t) on the left-hand side of the equation equal to the corresponding coefficients in f ( t ) on the right-hand side. Then to lowest order in (uj0tr)~l and F/u;0, i.e., upon neglect of U'() and x± compared with uj0x±, one finds

To write the equation for xc(t) it is convenient to introduce an auxiliary threevariable-dependent potential V(xc,x+,X-),

which is thus the value of U(x) averaged over the period 27r/u;0 for constant xc,x±. Then

i.e., if one neglects 6f(t) the function xc(t) is smooth. It is easy to see that the rapidly oscillating corrections to xcj e.g., those oscillating as exp(±2zu 0 t) and arising from the nonlinearity of U(x) (which has not been assumed small), are of order (uj0tr)~l . Since the amplitudes f ± ( t ) and thus x±(t] practically do not vary over the time ~ tr
Equations (2.96) and (2.99) show that, with varying amplitudes f± of the rapidly oscillating noise components, the amplitude of the forced oscillations of the system varies as well; the latter variation causes the change of the effective potential V(xc,x+,X-) that determines the motion of the center xc of the vibrations and, because the relaxation time of the system is short compared to the decrement/increment time of /-t, the center xc occupies the minimum of the potential V for given x + , x__. A naive picture of the optimal fluctuation that brings the system to a given point x constructed with the help of (2.95), (2.96), and (2.99) is as follows. The amplitude f+(t)\ of the noise increases from a root-mean-square value of order Dl/2/uj0 with an increment of order F, and the amplitude x+(t)\ increases accordingly. At the same time, the center xc of the vibrations shifts (initially xc = xsi within an accuracy of D1/2/^). Finally the vibrating coordinate x(t] reaches, for the first time, a given value x. This obviously happens at the turning point of the vibrations, i.e., when the deviation of x ( t ) from the instantaneous position xc(t) of the center of the vibrations is maximal. If the time is set so that this event occurs at the time origin t — 0, then

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Although the subsequent behavior of the system once it has reached the given x is not of interest in the present context, we note that, with an overwhelming probability, the amplitude \f+(i}\ decays with a decrement F for t > 0, and the coordinate x(t) follows this decay adiabatically and returns to the stable position. This picture makes it straightforward to obtain the expression for the activation energy R(x) for reaching a given point x. Indeed, (2.99) and (2.100) give the values of x c (0) and x±(0) in terms of x, and the value of x+(0) gives the minimal (optimal) value of the amplitude |/+(0)| of the noise necessary to bring the system to a given x. The statistical distribution of the squared amplitudes |/+ 2 is exponential: it is of the form of the energy distribution of an underdamped oscillator in thermal equilibrium at temperature D,

(the energy of the oscillator is (1/2)/ 2 + (l/2)u; 2 / 2 = 2u; 2 |/ + | 2 ). We note that the functions /+ and /_ here are related to / and / via (2.93) without 6f together with the relation / = iLJ0^l^i'fl,exp(ii'LJ0t}. The term 8f contains the small contributions that would otherwise be included in /+ and /_ (and would give rise to small rapidly oscillating terms in f ± ) . Omission of 8f simply places these contributions back in f±. Allowing for the inter-relation (2.90) between the maximum value D of the power spectrum of the noise and the effective temperature D, and for (2.96), one obtains

where x + (0) is related to x via (2.99) and (2.100). A consistent derivation of (2.102) based on the set of variational equations (2.52), (2.53) is given by Dykman [45]. One remarkable feature of the statistical distribution of a QMN-driven system is immediately seen from (2.99), (2.100), and (2.102) for systems with a symmetric potential U(x) (no matter how nonlinear). If U(x] = U(—x) and x = 0 is the stable equilibrium position, it is seen from (2.99) that xc = 0 for all x± and thus x+(0) = x_(0) = x/2, so that

Thus, according to (2.103) the distribution is independent of the shape of the potential provided the latter is symmetrical. This invariance is a consequence of the fact that it is the amplitude of the rapidly oscillating noise that is fluctuating initially. The system follows these fluctuations adiabatically, and for high UJQ the amplitude of the forced oscillations is nearly independent of the shape of U(x]. The independence of the activation energy R(x] of the curvature U"(0) of the potential U(x] for x — 0 and also of its nonlinearity was tested in analog electronic experiments [62]. The data obtained for symmetric potentials are

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shown in Fig. 2.6. The results were compared with (2.103) and also with the exact expression (2.30) for the Gaussian statistical distribution of a system with the parabolic potential U(x) = ax2/2: the "steepness" parameter (3 in (2.30) when the power spectrum of the noise is of the form (2.89) takes the form

(to lowest order in F/a and in a/uJ0 the value of 3 [ as given in (2.104) coincides with 2(dR(x)/dx2)/D as given in (2.103)). The agreement of the data measured for the harmonic potential with the expression (2.104) was excellent within the experimental uncertainty of ±2 ( /t. Furthermore, the data are nearly independent of the curvature U"(0) = a for a broad range of values of a and of the anharmonicity of the potential U ( . r ) .

FIG. 2.6. The activation energy H(x) for reaching a point ,r by a system (2.38) driven by quasi-monochromatic noise 'in the case of a symmetric parabolic potential U(x) — (l/2)x 2 . as obtained from, an analog experiment [62]. The parameters of tJie QMN are: u0 = 9.81. F = 0.021. and the characteristic intensity D = 160 (note that the effective noise intensity FD/a;2 % 0.035 -C 1). The experimental data for U(x) = (l/2);c 2 + (l/4),r 4 . and for U(x) = (l/6).r 2 were found to be coincident with those for U(x) — (l/2),r 2 within the experimental error. The smooth curve r<-presc-nts the exact theory (2.30) with (2.104).

2.4.2. Quasi-singularity of the activation energy: Breakdown of the adiabatic approximation. For asymmetric potentials U(.r) the activation energy R(x) as given in (2.99). (2.100). and (2.102) is asymmetric. The- specific feature that follows from these expressions is that /?(.r) may be singular for some x — x if the values of the coordinate that lie beyond .? (with respect to xst] are adiabaticallv inaccessible. The onset of such "forbidden" regions

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can be understood on the basis of our previous description of the motion: with increasing amplitude |/+ of the noise, the amplitude x+ of the forced vibrations increases as well, and as a result the limiting point x of (2.100) reached in the course of the vibrations also shifts. The dependence of the latter shift on the variation of x + , allowing for the fact that the position of the center of the forced vibrations xc itself depends on x+ , is of the form

Here V" stands for a second derivative of V(xc,x+,x^} and the subscripts c, + , — stand for differentiation with respect to x c ,x + ,x_, respectively. We have taken into account that the dependence of xc on x± in the adiabatic approximation is given by (2.99) and that V+ = V_ when x+ = X-. It is seen from (2.105) that for x± — x± such that

a further increase of the amplitude of the vibrations does not drive the system into the region beyond the point

This obviously happens because the dependence of the adiabatic position of the center of vibrations, x" d (x + ,x_), on the amplitude x+ for x± = x± becomes too fast and with increasing amplitude the system as a whole actually moves away from x. It follows from (2.102) and (2.105)-(2.107) that the activation energy R(x] has a square-root singularity for x = x in the adiabatic approximation [45], R(x) = R(x] — [C(x — x)]1/2, where the value of C can easily be expressed in terms of the derivatives of the potential V for x± — x^ . Penetration into the "forbidden" region occurs via fluctuations for which the adiabatic approximation (2.99) does not hold. For high-frequency noise the nonadiabaticity comes primarily from contributions to the variables xc and x± which, although still slow compared to the rapidly oscillating factors exp(±zu;o£), vary over time scales that are not slow compared to the relaxation time of the system, i.e., xc ~ x± ~ t~l. The onset of these terms is related to the onset of a large (compared to its value of O((r/a; 0 )|/+|) in the adiabatically accessible range) nonresonant term 8f(t) in the driving noise (2.93) which varies over the same characteristic time, \6f/6f\ ~ t~l. To estimate the steepness of the activation energy R(x) in the adiabatically inaccessible range we note that, according to (2.46) and (2.89), the magnitude of the logarithm of the probability of a "smooth" (compared to exp(±iu;0£)) fluctuation 6f(t) of duration ~ tr is of order u2(8f}2tr/T2D. Since in the range of x in question the only time scale for slow variables is tr , the deviations x — x, xc — x^d(x+1 £-), x± — x± caused by the force 8f(t) are all of order 8ftr, and the activation energy R(x) must be of the form R(x) ~ cu2(x — x)2/trr2,

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79

i.e., the value of R(x) changes by an order of magnitude compared with the value (2.102) in an extremely narrow interval x — x ~ (F^) 1 / 2 . A consistent theory that describes the smearing out of the square-root singularity of R(x) arising from the adiabatic approximation has been given by Dykman et al. [62]. It is based on the solution of the boundary-value problem (2.49), (2.52), and (2.53) for a QMN-driven system. The result that makes it possible to follow the change of R(x) from the comparatively smooth function far from the threshold point x to a steep function beyond x is of the form

Here all the derivatives of the potential F(.x c ,x - + , x _ ) are calculated for x± = x±(Q),xc = x° d [x + (0),x_(0)]; the function 6/(0) is the value of the smooth component of the noise at the instant t = 0 when the system reaches the given x. Far from the threshold x =• x in the adiabatically accessible range of x where the force <5/(0) is small [O(F)j, the expression (2.108) for R(x] coincides with (2.102), and the function R(x) is smooth, \d\uR(x)/dx\ ~ 1. In the vicinity of x the terms containing the force 6/(0) become substantial and R(x] becomes steep. Far in the adiabatically inaccessible range \d\nR(x}/dx ~ (Ftj,)" 1 / 2 3> 1, in complete agreement with the qualitative estimate given above. We stress that it is the argument of the exponential of the statistical distribution that becomes very steep. Therefore, the distribution p(x] itself is expected to vanish extremely sharply in the adiabatically inaccessible range. In contrast to the change of the character of the distribution at the inflection point of the potential when the correlation time of the noise is large (cf. §2.3.4), the threshold point x given by (2.106) and (2.107) is not immediately associated with a singular point of the potential of the system ('although it is determined entirely by the potential and does not depend on the parameters of the noise). In this sense the appearance of a singularity in the distribution is "hidden," i.e., there is no a priori reason to expect any unusual behavior at x — x. The onset of an extremely sharp behavior of the logarithm of the statistical distribution in QMN-driven systems was observed in an analog electronic experiment [62]. The potential of the system was of the quartic bistable form (2.84). The singular points x as given by (2.106) and (2.107) are

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x = ±(5/3) 1 / 2 . If. for example, the system is placed initially in the vicinity of the stable state x — — 1 and the quasi-stationary distribution over the range of fluctuations about this state is investigated, then only the point x = — (5/3)1/2 plays a role. The results for R(x) obtained in [62] for this case are shown in Fig. 2.7. It is seen from Fig. 2.7 that R(x) is indeed extremely steep beyond the point x = — (5/3)1/2, although it is of course not vertical, as would be predicted within the adiabatic approximation. The nonadiabatic theory (2.108) is in good agreement with the experiment (the data were obtained for u0 — 9.81,F = 0.021 and the theory does not contain any adjustable parameters). Another most peculiar feature of the quasi-stationary distribution plotted in Fig. 2.7 is that it spreads beyond the point x — 0, i.e., over the range of attraction of another stable state. We shall address this feature in §2.6 in the context of the transition probabilities between stable states, in particular those for QMN-driven systems. It follows from the results of the present section that for a narrow-band high-frequency driving noise the shape of the statistical distribution of the system is not determined by the fluctuations at frequencies lower than the reciprocal relaxation time of the system (cf. Fig. 2.3). Instead, the distribution is determined by the high-frequency fluctuations. The shape of the tails of the distribution is qualitatively different from that for white-noise-driven systems, and the logarithm of the distribution (and not only the distribution itself) can be extremely steep.

FIG. 2.7. The activation energy R(x) for reaching a point x starting from an equilibrium position xst = — 1 in the QMN-driven system with potential U(x) = — (l/2)x2 + (l/4)x 4 as obtained from an analog experiment [62]. The eigenfrequency and bandwidth of the QMN are the same as in Fig. 2.6, and the noise intensity is D = 189. The experimental data (jagged line) are compared with the double-adiabatic theory (full curve, singular at x = — \/5/3) and the expression (2.108)(the nonsingular full curve).

FLUCTUATIONS IN NONLINEAR SYSTEMS 2.5.

81

Pre-history problem

The above analysis of large fluctuations in a system driven by Gaussian noise was based on the concept of the optimal path. It was assumed on physical grounds, and then demonstrated by making use of the path-integral method, that among the various paths along which the system can arrive at a given point x there is an optimal path that corresponds to the most probable fluctuation among very infrequent ones. This path starts in the vicinity of the stable equilibrium point x st , and the duration of the motion far from the range of small fluctuations about xst is given by the time t0 of (2.39), namely, by the larger of the relaxation time tr of the system and the noise correlation time tc. Although the concept is physically and mathematically clear, the optimal paths of both the system and the noise have so far only been described as the solutions of the variational problem (2.48). In the present section (see also [67]) we introduce physically observable characteristics related to the optimal path of a system that further elucidate the approach. Such characteristics can be expressed in terms of the "prehistory" probability density p h ( x , t ; X f , t f } that the system fluctuating about a given stable state xst was at a point x at time t < tf, given that at time tf it is at x-f. The calculation of this probability density constitutes a prehistory problem. In the general case of a multistable system p h ( x , t ; x f , t f ] is not a standard two-time conditional probability; rather, it is the ratio of the three-time transition probability density w ( x f j t f j x , t : x i , t l ) (the probability density of the transitions X{ —» x —» Xf) to the standard two-time transition probability density w ( x f , t f , X i , t i ) , with the initial value of the coordinate x% lying in the vicinity of xst and the initial instant ti having been assumed to be such that t — tt and tf — tL substantially exceed the relaxation time of the system and all correlation times of the noise but, at the same time, are small compared to the reciprocal probability W~~l of the escape from the given stable state (see §2.6). As a consequence, both ti and xz have ostensibly dropped out from ph ( x . t , \ X f , t f ) ,

From the definition of the pre-history probability density it obviously follows that

Under quasi-stationary conditions only the instant tf when the system has been observed at Xf is singled out, and therefore

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The concept of the pre-history probability density appeals to the fact that the visits of the system to a given point Xf are infrequent and that the corresponding noise outbursts leading to these visits are therefore mutually independent and uncorrelated. The probability density of crossing a given point at a given instant before arrival at a final destination Xf is then precisely the quantity that characterizes the distribution of the paths arriving at this final point. Since (by definition) the optimal path xopt(t; Xf) for reaching Xf at the instant t = 0 is the most probable of these paths, the function Ph(x, t; £/, tf) for a given t — tf should have a sharp maximum if x lies on this path, i.e., if By investigating the pre-history probability density experimentally one can therefore not only visualize the optimal paths themselves, but one can also directly test the general concepts of the optimal path and the optimal fluctuation presented in §2.3. Such an investigation should establish the range of parameters within which the concepts are applicable to a given system and for a given type of noise. 2.5.1. General expression for the pre-history probability density. The pre-history probability density p^ depends on the stable state that had been occupied initially and from which the system arrives at x and x/. Persisting with our goal of simplicity, in what follows we give the theoretical formulation for the case of a monostable system; the generalization to the case of multistable systems is straightforward (the experimental data discussed below refer to a bistable system). We note that for a monostable system one can set ti = —oo in (2.109), and Ph(x, t', #/, tf) can then be expressed in terms of a two-time probability density: it is given by the ratio pz(x, t, ; x/, tf)/p(xf, tf). The other simplification made below is that the pre-history problem will be formulated in the context of the simplest type of equation of motion of a noise-driven nonlinear system, namely, (2.38). As always, we will assume the driving noise to be Gaussian. It is convenient to express the two- and threetime transition probability densities of (2.109) in the form of a path integral similar to that used in (2.41) and based on the fact [36] that each path of the noise f ( t ) generates a certain trajectory x(t) of the dynamical system. The weighting factor in the integral over the paths of the noise is the probability density functional T>[f(t)] that determines the probability density of a given realization f ( t ) . The appropriate expression follows from (2.109) and is

Here, just as in (2.41), the ^-functions and the lower limits of the integrals show that the paths f ( t r ) that contribute to the transition probability densities are those that generate system paths that start at a point Xi a long time before arrival at x/, pass through Xf at the instant £/, and, in the case of the threetime transition probability density w ( x f , t f ' , x , t ; X i , — o o ) , also pass through the

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83

point x at the instant t. The explicit form of the probability density functional p[/(t)] in the case of colored Gaussian noise with the power spectrum $(u;) is given in (2.46). The realizations of the force f ( i ] in (2.112) that bring the system to given point(s) at given instant(s) are "deterministic," i.e., although of fluctuational origin, they do not depend on the noise intensity D. However, their probabilities do, and therefore for sufficiently small D both numerator and denominator in (2.112) are exponentially small (the denominator was already considered in §2.3). It is thus reasonable to calculate them to logarithmic accuracy and to write the pre-history probability density as

where C(x, x/; t — tf) is a smooth function and where

The two terms in /o(x, xj; t) come from the two transition probability densities in (2.109). They are determined by the values of the argument of the exponential in the probability density functional (2.46) for the optimal paths of the noise that result in the appropriate trajectories of the system. The inequalities in (2.114) express how weak the noise intensity D must be for the concept of the optimal path to be applicable. The further analysis is completely analogous to that in §2.3 and shows that the variational problem for the optimal path that arrives at the point xj at the (running) time t' = 0 via the point x at the instant t' = t is of the form

with the boundary conditions

In a sense, the functional 3ft in (2.115) coincides exactly with that in (2.48) and, as in (2.48), the minimum in (2.115) should be taken with respect to f ( t f ) and x ( t f ) independently since their inter-relation as given by the equation of motion (2.38) has been taken into account by introducing the undetermined

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coefficient X(tf). The difference between the present variational problem and that considered in §2.3 for the optimal path arriving at x/ directly lies in the boundary conditions: in the present case the system is to pass a given point x at time t before arriving at Xf. Obviously, the value of p(xf,xf,Q) as given by (2.115) and (2.116) coincides exactly with the activation energy (2.48) for reaching the point xy,

which is not surprising since R(xf) in (2.48) determines the logarithm of the transition probability density w(xf,0;xst, —oo). Since p(xf, x/; 0) is determined by the optimal path xopt(t'; x/) that arrives at Xf without the additional constraint of passing a given point at a given previous time, one has the inequality p(x,xf,i) > p>(x/,xy;0). It is obvious from (2.115) and (2.116) that the equality holds only when x coincides with x0pt(t'iXf} (for a given t]. Thus the maximum of the probability density P h ( x , t ; x f , $ ) is indeed achieved on the optimal path,

According to (2.114) and (2.115), ph decreases exponentially away from the optimal path, and for weak noise its shape is Gaussian near the maximum, i.e.,

when x — xopt(t]Xf}\ is sufficiently small (in particular, as compared to xst — Xf\). The dispersion of Ph(x, t; x/, 0) has been denoted by Da(t;Xf). In view of the normalization (2.110), equations (2.113), (2.114), and (2.119) give not only the argument of the exponential but also the prefactor in the pre-history probability density,

(We have taken into account that the prefactor C in (2.113) and (2.114) is smooth on the scale Ax ~ (Der)1/2 and have therefore set it equal to its value at x = xopt(t-Xf).) The analysis of the pre-history transition probability has thus been reduced to the calculation of two functions: the optimal path x o p t ( t ; x f ) , and the dispersion parameter <j(t;xf}. The calculation of these functions for various types of power spectra of the noise can be done numerically. (The variational equations are obviously of the form (2.52) and (2.53), but the boundary conditions for the problem of the optimal path itself and the pre-history problem as a whole are different.) In some cases these functions can be calculated analytically.

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85

2.5.2. Pre-history probability density for systems driven by white noise. Simple analytical results can be obtained if the driving noise is white, whence the operator F in (2.115) is equal to 1. The optimal path in this case is given by (2.62), while the expression for cr(t;x/) can be shown to be of the form [67]

with

Experimental data [67] on the pre-history probability density are shown in Figs. 2.8 and 2.9. They were obtained via analog electronic simulation of Brownian motion as described by (2.38) with the potential (2.84). The system was initially placed in a state within the range of attraction of the stable equilibrium position xsi = -1. The paths that arrived at a given point Xf were stored, and it was their distribution that gave, by definition, the prehistory probability density. It is seen clearly in Fig. 2.8 that the function P h ( x , t \ x f , t f ) obtained in this way exhibits a very sharp ridge. This ridge provides the visualization of the optimal path. The shape of the optimal path Xapt(t\Xf], and tne characteristic width a(x;xf] as defined by assuming the function pn to be Gaussian near the maximum, are shown in Fig. 2.9. It is obvious that the function a(x; x/) is strongly nonmonotonic: the pre-history probability density is very sharp for x near the final point Xf and its peak is

FIG. 2.8. The pre-history probability density Ph(x, t; x/,0) for the white-noisedriven system with the bistable quartic potential (2.84) [67]. The final position is X = —0.3 7 and the characteristic noise intensity is D = 0.07.

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FIG. 2.9. The dispersion parameter a(x; x/) of the pre-history probability density (cf. Fig. 2.8) as a function of x measured in the analog experiment and compared with the theory (2.122) (curves) [67] for (a) xf = -0.30, D = 0.07 (circles); (b) Xf = -0.55, D = 0.0265 (triangles)- (c) xf = -0.75, D = 0.0085 (pluses). Inset: the optimal path (curve) as given by (2.62) is compared with a path along the ridge (data points) of the experimental distribution PH(X, t; x/, 0) for x/ = —0.30, D = 0.07.

comparatively narrow for x close to the equilibrium position xst = — 1, while for intermediate x it is much flatter and the width cr(x;xf) is correspondingly larger. The surprising behavior of the width <j(x; Xf) of the ridge of the pre-history probability density observed in [67] can be explained on the basis of (2.121) and (2.122). In particular, it immediately follows from these equations that for x close to x/, cr(x]Xf) = (xf — x)/U'(xf) is small, i.e., the bunch of paths that have arrived at x/ is sharply squeezed in the immediate vicinity of Xf. This is also evident from qualitative arguments: for Xf remote from stationary points the velocity Jt/'^)! of the motion along the optimal path is of order unity, and the motion from x to Xf for x close to Xf takes a short time. Therefore, the diffusional smearing (which gives rise to smearing of the bunch of paths) should be small. It is also evident that for x close to the stable state xst the width of the peak of p^ should coincide with that of the stationary statistical distribution of the system and that in fact p h ( x , t ; x f , t f ) —> p(x) for t —» —oo, where p(x) is the stationary distribution considered in §§2.2 and 2.3 (the distribution over x does not depend on whether in the far future (tf — t—> oo) there might be an outburst of noise that drives the system to Xf). Both of these features are seen very clearly in Fig. 2.8. The dome-like shape of a(x-Xf) is related to the particular shape of the potential (2.84); we note, however, that such a shape will always arise for bistable potentials provided

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87

Xf is sufficiently close to the saddle point xs [(U'(xs) = 0 and U"(xs) < 0)] because it follows from (2.121) and (2.122) [67] that a(x;x/) is proportional to (xf — £.s)~2 for small Xf — xs and both x — xs and x — xst ~^> Xf — xs , i.e.. for x between xst and xj the dispersionCT(X,XJ)does become large. It is evident from Fig. 2.9 that the expression (2.121) for a(x;x/) explains the experimental data, at least qualitatively. The discrepancy between theory and experiment is most likely related to the fact that the noise intensities investigated experimentally were not sufficiently small for the theory to be applicable; in particular, the width of the peak of ph is seen from Fig. 2.8 to be comparable to the distance between the singular points of the potential U(x] of the system, while in the theory the width of the peak has been assumed to be much smaller. Another important point to be addressed in the future is the interpretation of the data when Xf approaches a saddle point where a(x]Xf] diverges arid the present theory becomes inapplicable. It follows from the results of the present section that the formulation of the pre-history problem has made it possible to visualize optimal paths in noisedriven systems and to investigate their statistical distribution. Through this approach it has been possible to provide direct experimental verification of the fundamental concept of the optimal path. 2.6.

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

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

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

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

FLUCTUATIONS IN NONLINEAR SYSTEMS

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

89

are. the stable equilibrium

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

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

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

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

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

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

The

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

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where

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

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

FLUCTUATIONS IN NONLINEAR SYSTEMS

93

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

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

94

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

maximum of the spectral peak, the halfwidth F of the peak, and the relaxation time tr of the system satisfy the inequality (2.92), T
FIG. 2.12. Two samples of the trajectory x(t) of a symmetrical bistable system driven by quasi-monochromatic noise exhibiting an example of occasional large fluctuations from each of the attractors that result in passage across the saddle point (the potential is of the form (2.84), so that xs = 0), as observed in an analog experiment [62]. The eigenfrequency and halfwidth of the QMN are the same as in Fig. 2.6, and the noise intensity is D = 192.

FLUCTUATIONS IN NONLINEAR SYSTEMS

95

This observation is in complete agreement with the inequality (2.127): the activation energy -R(O) for the mean first-passage time to the point xs — 0 is less than the activation energy R% of fluctuational transitions (see below), and therefore the transition probability is exponentially smaller than the reciprocal mean first-passage time to the saddle point. The value of JR(0) is given immediately by (2.99), (2.100), and (2.102) [45],

The double-adiabatic approximation explained in §2.4 does not hold for the entire optimal path resulting in a transition. One of the scenarios for escape is that, for some value x+ = x+ , the "relaxation time" of the center of vibrations xc diverges,

Having reached the corresponding value of x®d, the center of vibrations goes over to the other branch of the solution of the equation Vc (x c ,x + , x_) — 0 and approaches the saddle point as the amplitude x+ of the forced vibrations (and thus the noise amplitude) falls to zero (the details are given in [45]). It follows from this picture that it is not the vibrating coordinate itself but the center of vibrations that should reach the saddle point for the transition to occur, and also that the transition probability is given by the probability of the realization of the noise amplitude resulting in the amplitude of the forced vibrations given by (2.133), so that, as explained when (2.102) was derived, the activation energy of the transition is of the form

In the particular case that the potential is of the form (2.84), one obtains (see [45] and [76])

which indeed exceeds the activation energy (2.132) of the mean first-passage time. The value RI given in (2.135) fits the activation energy of the transition obtained experimentally [62] very well. (The data for R(x] have been discussed in §2.4; we note that the range where the distribution about state i is quasistationary extends to the values of x for which R(X,XI) = Rt, i.e., far beyond the range of attraction to the state z; in the particular case of the potential (2.84) and Xi = — 1 the boundary of quasi-stationarity is x = v/2/3; cf. also experimental data in Fig. 2.7.) In conclusion, the color of noise not only drastically changes the value of the activation energy of the transition probability, but it also changes the entire pattern of the transition. The transition probability differs exponentially strongly from the mean first-passage time to the saddle point (the top of the

96

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

potential barrier)—the system can recross the saddle point many times without completing a transition. The method of the optimal path makes it possible to reduce the problem of the calculation of the activation energy of a transition to a variational problem with intuitively clear boundary conditions. 2.7.

Conclusion

It follows from the above discussion that the present status of the investigation of colored-noise-driven dynamics is very promising. Several new theoretical and experimental results have been obtained within the last few years, including the prediction and observation of the features of large fluctuations in systems driven by quasi-monochromatic noise, and the visualization of optimal paths. Although some features of the dynamics are now well established qualitatively and the appropriate mathematical techniques have been developed, such as those used here to find optimal paths for large fluctuations and fluctuational transitions, there are still both qualitative and quantitative problems to be addressed. For example, if a bistable system is driven by noise, what is the shape of the far tails of the stationary distribution? This distribution is shaped by large noise outbursts that bring the system to a given point in the phase space from one rather than the other attractor with an overwhelming probability. A simple-minded picture, then, is that the full phase space is separated into the "strips" attributed to fluctuational arrivals from either attractor; this picture is similar to that of deterministic motion in the absence of noise where the phase space is separated into the ranges from which a system goes to one or the other attractor and, as in this latter picture, one can investigate the question of the structure of the separating manifold (fluctuational separatrix). However, the topology of the phase space with respect to fluctuational arrivals may be more complicated, and this simple-minded picture may not be generally applicable. The outstanding problem of greatest importance is that of large fluctuations in systems driven by non-Gaussian noise. Since for Gaussian noise the probability of large noise outbursts decreases extremely rapidly with increasing magnitude of the outburst, even small deviations of the probability density functional from a Gaussian form can result in drastic changes in the transition probabilities and other quantities associated with large fluctuations in noisedriven systems. In many cases of physical interest the method of the optimal path will still be applicable, but more detailed inspection is necessary in each case. There also exist a number of less "global" problems that are nevertheless interesting and important. These include, for example, a systematic algorithm for the numerical solution of the variational problem for optimal paths, the calculation of the prefactors in the expressions for the tails of the statistical distribution and for the transition probability, the shape of the spectral density of fluctuations in nonlinear (especially underdamped) noise-driven systems.

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Chapter 3 Percolation Shlomo Havlin and Armin Bunde

Abstract Percolation has been found to be a useful model for characterizing disordered systems in nature. In its simplest form, one considers a lattice where each site is occupied randomly with probability p or empty with probability 1 — p. At the critical concentration pc, for the first time an "infinite" spanning cluster of occupied neighbor sites is formed. Percolation can be viewed as a system that exibits a purely geometrical critical phenomena. Near pc, many geometrical properties behave as power laws of p~ pc, in analogy with thermal critical phenomena. The critical exponents characterizing these power laws are known exactly only for a limited number of cases. Their values are usually estimated from numerical simulations and series expansion methods. There exists an upper critical dimension, d = dc = 6. above which mean field-type approximations can be applied and the exponents are known exactly. Percolation clusters and their substructures are well characterized by fractal geometry. This chapter reviews the structural properties as well as the dynamical properties of the percolation system. Also, variations of the percolation model such as invasion percolation, directed percolation, and correlated percolation are discussed. Application of the percolation model to glasses, surfaces, and other fields is also reviewed. 3.1. Introduction The percolation model suggested in 1957 by Broadbent and Hammersley [1] has been found useful for characterizing many realizations of disordered systems in nature. Examples are porous media, polymer gels, fragmentation and fractures, dispersed ionic conductors, ionic glasses, galaxies, forest fires, and epidemics. In this review we present a brief introduction to percolation as well as a description of recent developments in this field. For recent books and reviews on percolation and its various applications, see [2] [15].

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For simplicity, we begin by describing the site percolation model on a square lattice. Each site on the lattice is chosen to be occupied randomly with a probability p or empty with a probability I — p . Two occupied sites belong to the same cluster if they are connected by a path of nearest neighbor-occupied sites. When p increases, the average size of the clusters increases. For the infinite system, there exists a critical concentration j9c, below which only finite clusters exist and above which an infinite cluster is formed (see Fig. 3.1). The average size of the finite clusters diverges when approaching pc from both below and above. The bond percolation is defined similarly, by randomly occupying bonds on the lattice with probability p. Here, nearest neighbor-occupied bonds form connected clusters. The value of pc depends on the dimension and the type of the lattice as well as on the type of percolation (e.g., site or bond). Several examples for values of pc are given in Table 3.1.

FIG. 3.1. Square lattice of 20 x 20. Sites are randomly occupied with probability p(p = 0.2,0.59,0.8). Sites belonging to finite clusters are marked by full circles and sites on the infinite cluster are marked by open circles (after [9]).

TABLE 3.1 Percolation thresholds for several two- and three-dimensional lattices and the Cayley tree. a Exact [3], [16]; bNumerical method [17]; cMonte Carlo [2]; dSeries expansion [18]; e Monte Carlo [19]; f Series expansion [20]. Lattice Triangular Square Honeycomb Diamond Face centered cubic Body centered cubic Simple cubic (1st nn) Simple cubic (2nd nn) Simple cubic (3rd nn) Hypercubic (d = 4) Cayley tree

Percolation threshold Site Bond 1/2° 2sin(7r/18)a b l/2 a 0.5927460 C 0.6962 1 -2sin(7r/18) a C 0.43 0.388C C 0.198 0.119C 0.245C 0.1803d e 0.31161 0.2488d 0.137/ 0.097/ 0.197C 0.160C l/(z-l) l/(z-l)

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As seen in Table 3.1 only a few critical concentrations are known exactly. The others are known only from approximate methods such as Monte Carlo simulations or series expansions. In general, although percolation is a very simple problem to formulate and to understand, only a few results for the percolation model can be obtained exactly. Several special cases, such as percolation in d = I or percolation on a Cayley tree, can be treated exactly [2], [3]. For d = 1, it is easy to see that pc = 1, since all sites must be occupied in order to obtain an infinite cluster connecting both sides of the infinite system. Let us mention several other exact results. The existence of a critical concentration, 0 < pc(d) < 1, for d > 2 has been proven by Broadbent and Hammersley [1], [21]. A related theorem on the existence of an infinite cluster was proven by Grimmett [5] and Stirzaker [22]. The probability ifj(p] that there exists an infinite cluster satisfies

It was also shown [5] that a unique infinite cluster exists above pc(d] for all d. Equation (3.1) does not say what happens at p = pc(d). It was proven only for d = 2 that at p — p c (2) an infinite cluster does not exist [4]. For d > 3 there is no proof, but it is believed that no such cluster exists at p = pc(d}. Kesten [23] has shown that for d —> oo, pc(d) ~ l/2oL We have described site arid bond percolation, where either sites or bonds of a given lattice have been chosen randomly. If sites are occupied with probability p and bonds are occupied with probability q, we speak of site-bond percolation. Two occupied sites belong to the same cluster if they are connected by a path of nearest neighbor-occupied sites and occupied bonds between them. For q — 1, site-bond percolation reduces to site percolation; for p = 1 it reduces to bond percolation. In general, both parameters characterize the state of the system. Accordingly, a critical line in p-q space separates both phases, which for p = 1 and q = 1 gets the values of the critical bond and site concentrations, respectively. Site-bond percolations can be relevant for gelation, forest fires, and epidemic processes in dilute media [24]-[27]. The most natural example of percolation is perhaps the continuum percolation model, where the positions of the two components of a random mixture are not restricted to the discrete sites of a regular lattice. As a simple example, consider a sheet of conductive material with circular holes punched randomly in it (Fig. 3.2), or a cube with spherical pores—the "Swiss Cheese" model. An interesting example of interface percolation is the model of dispersed ionic conductors [28]. In its continuum "Swiss Cheese" version [29], a cube represents a bad ionic conductor and the spherical pores represent the dispersed insulating phase. Each pore of radius r is surrounded by a highly conducting spherical shell of width A. In this problem two critical concentrations occur: at p'c the highly conducting interface percolates, and at p'('. all conducting paths are destroyed.

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FIG. 3.2. Continuum percolation: Swiss cheese model. Another example of percolation is the percolation surface. Here, one asks what the critical concentration, pc, is in d = 3 in which for the first time a simply connected (no holes) surface spans the whole system. This problem is of interest since it is related to lattice gauge theory [30]. Using duality arguments Kertesz and Herrmann [31] showed that such a percolation surface has the same correlation exponent as the usual d = 3 bond percolation. Finally, it should be mentiond that Flory [32] and Stockmayer [33] already studied a problem related to percolation in 1941-1943. They studied a polymerization process in which small molecules form very large molecules when chemical bonds are added randomly between the small molecules. This process is the mechanism for gelation, such as that occurring when boiling an egg. Their theory is known today as percolation on a Cayley tree. The question of whether the gelation is described by the percolation theory, i.e., if bonds are added randomly, is still controversial (see de Gennes [34] and Kolb and Axelos [35]). 3.2. Critical phenomena The percolation problem can be viewed as a purly geometrical critical phenomenon. Geometrical properties behave near pc as powers of p—pc in analogy to thermal critical phenomena [36] where thermodynamical quantities behave as powers of T — Tc, where Tc is the critical temperature. The exponents characterizing these powers are called critical exponents. A geometrical quantity having this property is POO, the probability that a site on the lattice belongs to the incipient infinite cluster. Below pc, Px = 0, whereas above pc, Px increases with p as The mean diameter of the finite clusters below and above pc is characterized by the correlation length £, which is defined as the average distance between two sites belonging to the same finite cluster. When p —* pc, £ diverges as

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The mean number of sites (mass) of a finite cluster scales as

The exponents v and 7 are the same above and below pc. The exponents (3, z/, and 7 describe the critical behavior of the percolation transition, and are called the critical exponents. The exponents are universal and depend neither on the structural details of the lattice (e.g., square or triangular) nor on the type of percolation (site, bond, or continuum), but only on the dimension d of the lattice. This universality property is a general feature of phase transitions, where the order parameter vanishes continuously at the critical point (second-order phase transition). For example, the magnetization in all three-dimensional magnetic materials is described by the same exponent /?, independent of the crystalline structure or the type of short-range interactions between the spins. The exponents /?, ^, and 7 are not the only critical exponents characterizing the percolation transition. The size distribution of percolation clusters, for example, is described by other exponents, a, r, and a. However, as we will show later, there exist relations between these exponents, and all of them can be obtained from the knowledge of just two of them. In Table 3.2, the values of the critical exponents (3, v^ and 7 in percolation are listed for two, three, and six dimensions.

TABLE 3.2 Exact values and best estimates for the critical exponents in percolation. a Exact [37], [38]; b Numerical simulations [19]; cExact [39], [40].

Percolation Order parameter POO/ (3 Correlation length £: v Mean cluster size S: 7

d=2 5/36° 4/3a 43/18°

d=3 0.417 ±0.0036 0.875 ± 0.0086 1.795 ±0.005 6

d> 6 lc 1/2C lc

Let us define the distribution ns to be the number of clusters of size s per lattice site. From this it follows that sns is the probability that a given site in the percolation system belongs to a cluster of size s. Thus

When approaching criticality, p —•> pc, we assume that ns ~ s~ T /(z), where f ( z __> Q) — const,/(z > 1) —> 0. This assumption is motivated z ^ (p^pc^S(j^ by the exact solution of percolation on the Cayley tree; see, e.g., [3]. Using the above assumption for ns one obtains [2], [3], [39], [40] for the mean mass of a cluster,

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The exponents a, (3, v, 7, r, and a are related by the four relations

The exponents considered here describe the geometrical properties of the percolation transition. The physical properties associated with the transition also show power-law behavior near pc and are characterized by critical exponents. Examples are the conductivity in a random resistor or random superconducting network, or the spreading velocity of an epidemic disease near the critical infection probability. It is possible that these "dynamical" exponents cannot generally be related to the geometric exponents. The dynamical properties of percolation will be discussed in §3.4. (See also [41].) At the end of this section we note that all quantities described above are defined in the thermodynamic limit of large systems. In a finite system, POO, for example, is not strictly zero below pc. An approach to finite-size effects will be discussed in the following section. 3.3. Fractal properties In this section we present studies on the geometrical properties of percolation. The fractal concept pioneered by Mandelbrot [42] has been found very useful for this purpose. 3.3.1. The fractal dimension df. As first found by Stanley [43], the structure of percolation clusters can be well described by the fractal concept. We begin by considering the infinite cluster at the critical concentration pc. A representative example of the infinite cluster is shown in Fig. 3.3. As seen in the figure, the infinite cluster contains holes of all sizes, similar to the Sierpinski gasket. The cluster is self-similar on all length scales (larger than the unit size and smaller than the lattice size), and can be regarded as a fractal. The fractal dimension df describes how, on the average, the mass M of the cluster, within a sphere of radius r, scales with r, In random fractals M(r) represents an average over many different cluster configurations or, equivalently, over many different centers of spheres on the same infinite cluster. Below and above pc, the mean size of the finite clusters in the system is described by the correlation length £. At pc, £ diverges and holes occur in the infinite cluster on all length scales. Above pc, £ also represents the linear size of the holes in the infinite cluster. Since £ is finite above pc, the infinite cluster can be self-similar only on length scales smaller than £. We can interpret £(p) as a typical length up to which the cluster is self-similar and can be regarded as a fractal. For length scales larger than £, the structure

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FIG. 3.3. Percolation cluster at the critical concentration. pc.

is not self-similar and can be regarded as homogeneous. The crossover from the fractal behavior at small length scales to a homogeneous behavior at large length scales is best illustrated by a lattice composed of Sierpinski gasket unit cells of size £ (see Fig. 3.4). If our length scale is smaller than £, we see a fractal structure. On length scales larger than £, we see a homogeneous system which is composed of many unit cells of size £. Mathematically, this can be summarized as

FIG. 3.4. Lattice composed of Sierpinski gasket cells of size £.

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It is possible to relate the fractal dimension df of the infinite percolation cluster to the critical exponents (3 and v. The probability that an arbitrary site within a circle of radius r smaller than £ belongs to the infinite cluster is the ratio between the number of sites on the infinite cluster and the total number of sites,

This equation is valid for r — a£, where a is an arbitrary constant smaller than 1. Substituting r = a£ into (3.7) yields

Both sides are powers of p — pc. Substituting (3.2) and (3.3) into (3.8) yields [39], [43]

Thus the fractal dimension of the infinite cluster at pc is not an independent exponent, but depends on (3 and v. Since (3 and v are universal exponents, df is also universal. Numerical results [2] for the dependence of M(L) on the system's linear size L for site percolation on a triangular lattice are shown in Fig. 3.5. At large values of L, the curve in the double logarithmic plot approaches a straight line with slope df = 91/48, in agreement with (3.9).

FIG. 3.5. The size of the largest cluster at the critical concentration pc = 1/2 of the triangular lattice as a function of the linear size L of the lattice. For large L, the slope of this log-log plot is very close to the predicted value df = 91/48 (after [2]).

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It can be shown that (3.9) also represents the fractal dimension of the finite clusters below pc, as long as their linear size is smaller than £. Below pc there exist clusters with a linear size larger than £. These clusters are called lattice animals and their fractal dimension is smaller than df [44], [45]. For example, the fractal dimension of lattice animals in d = 2 is df = 1.56 [46], [47] and in d=3, df =2 [48]. 3.3.2. The graph dimension d^. The fractal dimension is not sufficient to fully characterize a percolation cluster. This becomes evident, for example, when comparing pictures of diffusion-limited aggregates (DLA) (see, e.g., [3], [12], [25], [49]-[51]) and percolation clusters. The geometrical structure of the two clusters is very different. The percolation cluster has loops on all length scales, while the DLA has almost no loops (see Fig. 3.6). In d = 3, both structures retain their characteristic differences, but their fractal dimensions are nearly the same: df = 2.5.

FIG. 3.6. Typical DLA cluster.

For additional characterization of a fractal let us consider the shortest path between two sites on the cluster (see Fig. 3.7). We denote the length of this path, which is called the "chemical distance" [52] [61]. by L The graph dimension d(. which is also called the "cherrncar or the "topological" dimension, describes how the cluster mass A/, within the chemical distance t from a given site, scales with t,

While the fractal dimension df characterizes how the mass of the cluster scales with the ''Euclidean" distance r (see (3.5)), the graph dimension df characterizes how the mass scales with the chemical distance (.

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FIG. 3.7. Shortest path between two sites A and B on a percolation cluster.

The simplest useful way to calculate df is to choose an arbitrary site on the cluster and to determine the number M(r) of all sites within a distance r from this site. To calculate di, an arbitrary site is chosen on the cluster and one determines the number M(i] of all sites which are connected to this site by a shortest path of length smaller than or equal to t. As for M(r), averages have to be performed on M(t] over many random realizations. In regular homogeneous lattices, both di and df coincide with the Euclidean space dimension d. For linear fractals such as self-avoiding walks, the fractal dimension is df = (d + 2)/3 (1 < d < 4) [62], [63] while di = 1. The graph dimension for lattice animals was found to be di = 1.33 (d = 2) and di = 1.47 (d = 3) [47]. Combining (3.5) and (3.10) one obtains an expression relating the chemical distance t and the Euclidean distance r between two sites, Equation (3.11) can be written as I ~ r d ™«, where dm(n = \jv has the meaning of the fractal dimension of the minimum path between two sites [56], [61]. The graph dimension d# (or v} is useful for distinguishing between different fractal structures which may have the same fractal dimensions. In d — 3, for example, DLA clusters and percolation clusters have approximately the same fractal dimension, df = 2.5, but very different v values: v = I for DLA [60] but v ^ 0.75 for percolation [56], [59], [61], [64]. While the fractal dimension of percolation df has been related to the known critical exponents, (3.9), no similar relation is known for the graph dimension di. The values of di or v are known only from approximate methods, mainly numerical simulations (see Table 3.3). An epsilon expansion approach up to order e was derived by Cardy and Grassberger [57]. The concept of the chemical distance also plays an important role in the description of spreading phenomena, such as epidemics and forest fires, which propagate along the shortest path from the seed. The velocity with which the fire front or the epidemic propagates is related to the exponent v (see, e.g., [3], [41], [64], [65]). The relations between M, r, and t are fully characterized by the probability densities 0(M r), (M t), and <j>(r t) [59]. The quantity 0(M r) is the probability of finding M cluster sites within a circle of radius r. Accordingly,

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4>(M 1} is the probability of finding M cluster sites within i chemical shells, and 0(r £} is the probability that two cluster sites separated by chemical distance I are within a Euclidean distance r of each other. The mean quantities M(r), M(f), and r(i) given in (3.5), (3.10), and (3.11) represent the first moments of these distributions, e.g., M(r) = / M'cf)(M' \ r)dM'. Assuming that percolation clusters are characterized by a single length scale, it is expected that the probability density 0(M r) scale as [41], [59]

Similar scaling forms have been found for 4>(M i] and (r For (f)(r 1} a more detailed functional form was suggested, in analogy with the distribution of the end-to-end distances -), the of self-avoiding walks (see, e.g., [62]). In a recent work [66], probability density that for a given r, the chemical distance is was calculated numerically for d — 2. The data suggests that with g' = g — df. This relation between g and g' can be derived by assuming that both 4>'(i r) and 4>(r (,} have the same functional form and are related by 1} t)both are normalized: r) with respect to I. Fig. 3.8 shows numerical with respect to r and results supporting the scaling form (3.13a).

FIG. 3.8. Plot of P(R | £)R as a function of Rjtv, calculated for d = 2 percolation clusters of size t = 4000, at criticality. The solid curve is a best fit to equation (3.13a).

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3.3.3. Fractal substructures. Next we show that df and df are not the only exponents characterizing the structure of a percolation cluster at pc. A percolation cluster is composed of several fractal substructures, which are described by other exponents. To identify these substructures let us apply a voltage drop between two sites at opposite edges of a metallic percolation cluster. The backbone of the cluster consists of those sites (or bonds) which carry the electric current. The dangling ends are those parts of the cluster which carry no current, i.e., those parts that are connected to the backbone by a single site. The red bonds (or singly connected bonds) [43], [67] are those bonds that carry the total current, i.e., when a red bond is cut, the current stops. The blobs, finally, are those parts of the backbone that remain after the red bonds have been removed (see Fig. 3.9).

FIG. 3.9. Schematic plot of the backbone of a percolation cluster. The backbone consists of blobs and red bonds represented by solid lines, and the dashed lines represent the dangling ends.

Other substructures of the percolation cluster are the external perimeter (which is also called the hull], the skeleton, and the elastic backbone. The hull consists of those sites of the cluster which are adjacent to empty sites and are connected with infinity via empty sites. In contrast, the total perimeter also includes the holes in the cluster. The external perimeter is an important model for random fractal interfaces. The skeleton is defined as the union of all shortest paths from a given site to all sites at a chemical distance t [69]. The elastic backbone is the union of all shortest paths between two sites [70]. The fractal dimension dp of the backbone is smaller than the fractal dimension df of the cluster (see Table 3.3). This reflects the fact that the mass of the cluster is dominated by the dangling ends. The value of the fractal dimension of the backbone is known only from numerical simulations [71]. Note also that the graph dimension di of the backbone is smaller than that of percolation. In contrast, v is the same for both backbone and percolation cluster, indicating the universal nature of v. This can be understood since every two sites on a percolation cluster are located on the corresponding backbone. Thus the

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relation between r and t, represented by z>, is the same for both the percolation cluster and its backbone. The fractal dimensions of the red bonds dred and the hull dh are known from exact analytical arguments. It has been proven by Coniglio [67], [68] that the mean number of red bonds varies with p as and the fractal dimension of the red bonds is therefore
where i/> = 1 — o. For d = 2, ip = 55/91 and for d — 3, i\) = 0.54. It was found numerically [77] that for d = 3 more than 99.8% of the occupied sites are in the external perimeter, supporting [76]. In Table 3.3 the values of the fractal dimension df and the graph dimension df of the percolation cluster and its fractal substructures are summarized. The values for six dimensions are known rigorously from exact considerations. TABLE 3.3 Fractal dimensions of percolation and of the substructures composing percolation clusters. Exact results and best estimates are presented: a Exact [37], [38]; b Numerical simulations [19]; cNumerical simulations [59], [61]; ^-Numerical simulations [71], [72]; eNumerical simulations [64]; /'Exact [67], [68]; 9Exact [73], [74]; h Numerical simulations [76].

Fractal dimensions df df CLB ^red dh

Space dimension d —2 d=3

91/48° 1.6765 ±0.0006 e 1.647±0.004 e 3/4/ 7/49

b

2.524 ± 0.008 1.885 ±0.015C 1.74±0.04 d 1.143±0.01b 2.548 ± 0.014^

d >6

4 2 2

2 4

116 3.4.

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS Transport properties

In this section we review the dynamical properties of percolation systems. For this purpose we assign to each site or bond a physical property such as conductivity or elasticity. We show that due to the fractal nature of percolation near pc, the physical laws of dynamics are modified and one obtains anomalous laws. For example, we discuss transport properties of percolation systems by assuming that occupied sites are conductors, empty sites are insulators, and electric current can flow only between nearest neighbor conductor sites. The specific conductivity is no longer constant but depends anomalously on the size of the system. Diffusion is discussed by assigning finite transition rates between neighboring occupied sites. It was found that diffusion on percolation does not follow Fick's law, in which the mean square displacement is proportional to time. Vibrational properties will be discussed in the context of bond percolation by assuming that occupied bonds can be represented by springs with a finite spring constant, while empty bonds are disconnected. If all bonds in the lattice are occupied, one has normal phonons. When the system is diluted randomly, localized modes occur for large frequencies, which have been introduced and called fractons by Alexander and Orbach [78], and their density of states shows anomalous frequency behavior. As discussed above, the fact that the percolation transition is a critical phenomenon implies that physical quantities related to the transition can be characterized by power laws of \p — pc . The same applies to the dynamical properties. The first empirical evidence for power-law behavior was given by Last and Thouless [79] in 1971 when studying a Id diluted conducting material. Their data suggest that above pc = 0.6, in the critical regime, the conductivity a behaves as

with IJL greater than 1. 3.4.1.

Transport on fractal substrates.

(a) Total resistance. In homogeneous conductors where the density of the conducting material is constant the total resistance p scales as

In fractal conductors, the density is proportional to Ldf~d and approaches zero for L —•> oo. This is a consequence of the fact that in fractal structures holes of all sizes up to the size of the system exist. If we increase L. we increase the size of the (nonconducting) holes as well, and by this we decrease the conductivity a. Due to self-similarity, a is decreased on all length scales.

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leading to the power-law dependence

which defines the exponent jl. In the following we shall see that for percolation clusters, jj, is related to the exponent p, defined in (3.16) and the correlation exponent v by jl = ^jv. As a consequence of (3.17), the total resistance behaves as

where now £ = 2 — d + ji is greater than the value 2 — d for homogeneous conductors (see (3.17)). It is instructive to calculate ( for the Sierpinski gasket [80]. The method is very simple and can be used in a straightforward way to obtain the resistance exponents of other finitely ramified deterministic fractals, such as the Mandelbrot-Given fractal [81]. For infinitely ramified fractals, see [82]. Consider a voltage difference between the top of the Sierpinski gasket and the two edge points in the bottom line (see Fig. 3.10) and compare the endto-end resistances of a gasket of length L and a smaller gasket of length L/2. According to Kirchhoff's law, the end-to-end resistance between (top to bottom) of the large system, p(L), is the resistance of one small resistor, p(L/2), plus the resistance of one small resistor in parallel with two small resistors, i.e.,

Using (3.18) we obtain [80] (Sierpinski gasket).

FIG. 3.10. Sierpinski gasket with a voltage difference V between the top site and the two sites at the bottom corners. The arrows describe the direction of the current. By symmetry, the total current along the horizontal direction is zero.

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FIG. 3.11. Resettling of transit time for traversing the gasket. The walker enters the gasket at the top vertex and (a) takes a time T to exit through the lower O vertices. (b) The resettled gasket, T —> Tf and A and B are transit times from the internal (decimated] vertices to the lower O-vertices. For the Mandelbrot-Given fractal one finds in an analogous way [81]

(b) Diffusion. To illustrate anomalous diffusion, consider a random walker on the Sierpinski gasket. Due to holes, bottlenecks, and dangling ends in the fractal, the motion of a random walker is slowed down. Since holes, bottlenecks, and dangling ends occur on all length scales, the motion of the walker is slowed down on all length scales. Fick's law for the mean square displacement, (r2(t}) oc Dt, is no longer valid. Instead, the mean square displacement is described by a more general power law,

where the new exponent dw, called the "diffusion exponent" or "the fractal dimension of the walk," is always greater than 2. To demonstrate the anomalous diffusion on fractals and to calculate dw for the Sierpinski gasket, we use a simple renormalization scheme. Consider, for example, the mean transit time T needed to traverse a gasket unit from one of its vertices to one of the remaining vertices O (Fig. 3.1 la). One can then calculate the corresponding time T' for exiting a rescaled fractal unit by a factor of two (Fig. 3.lib). This is done by exploiting the Markov property of the random walk on the fractal. Thus T' equals the time T to exit the first gasket unit, plus A1 the mean transit time needed to leave the rescaled unit from then on. Using the same reasoning for the times A and B (the mean exit times starting from the decimated internal vertices), one gets [83]

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The solution is T' = 5T (and A = 4T, B = 3T), which is the rescaling of time for a diffusion process on the gasket, upon the rescaling of length by a factor of two. It follows that dw = In 5/ In 2 = 2.322. This renormalization scheme is easily generalized for the d-dimensional Sierpinski gasket, yielding for the analogous transit times,

with the solution

and

(c) Relation between diffusion and conductivity. The resistance exponent ( and the exponent dw can be related by the Einstein equation

which relates the dc conductivity u of the system to the diffusion constant D = Iimt^oo(r 2 (t)}/2dt of the random walk. In (3.24), e and n denote charge and density of the mobile particles, respectively. Simple scaling arguments can now be used to relate dw to C and /}. Since n is proportional to the density of the substrate, n ~ L d f ~ d , the right-hand side of (3.24) is proportional to Ldf-dt2/d™-1. The left-hand side of (3.24) is proportional to L~^. Since the time a random walker takes to travel a distance L scales as Ld"- , we find L~^ ~ Ldf-d+2-du, ^ f rom which the "Einstein relation" [78]

follows. Since df = In 3/ In 2 for the Sierpinski gasket and df — In 8/ In 3 for the Mandelbrot- Given fractal [81], we obtain (Sierpinski gasket), Mandelbrot-Given fractal). For random fractals, the determination of dw and C is not that easy to find in general. In percolation, these exponents for 2 < d < 6 cannot be calculated exactly and only numerical estimates exist. 3.4.2. Transport on percolation clusters. In the previous section we discussed anomalous transport in fractal structures, which are self-similar on all length scales. Here we discuss transport in percolation systems above pc where fractal features appear only at length scales below the correlation length £. For r > £, the infinite cluster can be regarded as homogeneous. Since the correlation length is the only relevant length scale we expect that, similar to

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

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

where

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Thus, the bounds for £ are

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

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

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

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

where the exponent u is related to dw by

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

FIG. 3.12. Plot of rP(r, t) as a function ofr/tl'dw for different values of r and t. The solid line represents the theory of equation (3.47). For r/{r2)1/2
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The form of (P(r, t)} is relevant for the theory of self-avoiding walks (SAW) on percolation. SAWs in percolation clusters are important model systems for polymers in random media. From the form of (P(r, t)), equation (3.47), one can derive a Flory-type theory for the end-to-end exponent v of self-avoiding walks in percolation, giving [41]

where the index B is associated to the backbone of the percolation cluster. Equation (3.48b) was also supported by other scaling arguments [100], [101] 3.4.3. Fluctuations in diffusion. Consider a random walk starting from a given site on a percolation cluster. We are interested in the fluctuations in the probability density P(r, t) of finding the random walker after t time steps on a site a distance r from its starting point. For large clusters, there will be many sites at distance r (see Fig. 3.3), and the probability of being on different sites may be very different. We will show, using an analytical approach, that the histogram of these probabilities is very broad and has rnultifractal features [92], [93 . On the other hand, one can also consider the probability P(i, t) of finding the walker on sites at a fixed chemical distance t from the starting point. We will show that this histogram is relatively narrow and can be represented by a single exponent. In the general case, the qih moment (P9(r, t)} can be written as

where the sum is over all Nr sites located a distance r from the origin (Nr may include many configurations or a single configuration with a very large number of cluster sites). The sum in (3.49) can be written as sums over different t values (Nm values of tm),

According to the definition (3.13a), Nm/Nr = c/)'(l r) is the probability that two sites separated by a distance r are at chemical distance t. Thus, the moments in r space and i space are generally related by

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

We assume now that P(£,t) has narrow distribution, i.e., (P? (.£,£)} = (P(t,t))*, where (P(i,t)) is of the form [41]

Here, dw characterizes how the mean chemical distance traveled by the walker scales with time, L(t) ~ tl/d™. Substituting 0(r £) from (3.13a) and (3.52) into (3.51) we obtain [92], [93]

From the form of (P(r, £)), (3.47a), and (3.53) we obtain for q > 0

with

Since r(q) is nonlinear, the average moments cannot be described by a finite number of exponents, showing the multifractal nature of P(r, £). Next we calculate the histogram -/V(log P) giving the number of sites with values P between InP and InP + d\nP [P = P ( r , t ) ] . We use the identity

and compare (3.51) with (3.56). By changing the variables in (3.51) from i to P and using (3.52) we obtain

with

and

From the moments one can deduce easily the relative width A = ((P2) — (P) 2 )/(P) 2 of the histogram A^(logP) in i- and r-space. In ^-space, A is constant (similar to A = 0 in linear fractals), while in r-space A increases exponentially with r. Numerical results for A^(lnP) in d = 2 are shown in

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Fig. 3.13. The theoretical results for the histogram (dashed line) (see (3.57)), are in good agreement with the numerical results. Similar to the case of random walks on linear fractals, the large fluctuations in r-space are in marked contrast to the very narrow fluctuations in l-spa.ce, which a posteriori justifies the assumption (Pi(l,t)) ~ (P(^,t))(r i} and (Pq(£, t } ) . Although (f)(r \ I) is rather narrow in £, the resulting histogram of P(r,t) is logarithmically broad since i scales logarithmically with P through (3.52). In a recent work [99] it was shown that the multifractality of P(r, t) described above is restricted for q moments, qm-m < q < qm&x- The values of qm[n and gmax depend on the values of r and t,

FIG. 3.13. Representative plot of the histogram N(lnP} versus \lnP\ for fixed r and t (full line) and for fixed t and t (dotted line), with t = 1000, r — 30, and i = 80. The dashed line represents the theoretical result (3.57) (after [92]).

3.5.

Fractons

In the previous section we considered transport properties on percolation networks. Next we consider the elastic properties on percolation networks of (harmonic) springs. For simplicity we will mostly assume scalar force constants and consider only briefly the case of vectorial force constants where different components of displacements of neighboring masses are also coupled. The scalar force constants are either kA = 1 (with probability p) or ks — 0 (with

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

probability 1 — p). We begin with the elastic modulus of the percolation network. 3.5.1. Elasticity. Consider a small constant stress field applied at opposite faces of the elastic network. Due to the stress field, the network is expanded; the relative expansion AZ//L is proportional to the strength of the field. The proportionality constant is the elastic coefficient e, which characterizes the response of the network to the external stress field. The inverse elastic coefficient is the elastic modulus K [103]. The behavior of K is similar to the behavior of the dc conductivity. Below pc, the network is disconnected and the elastic modulus is zero. Above pc, opposite faces become increasingly connected by springs if p is increased, the network becomes stiff er and the macroscopic elastic modulus K, increases, For scalar force constants, the elasticity exponent p,e is identical to the conductivity exponent // [104]. For vectorial force constants Webman and Kantor [105] showed that p,e is considerably larger than //. Similar to the conductivity exponent, the elasticity exponent n& cannot be derived exactly and only upper and lower bounds can be derived [6]. The bounds yield 3.67 < p,e < 4.17 in d = 2 and 3.625 < \ie < 3.795 in d = 3. These values are considerably larger than those for scalar spring constants, fjLe =• 1-3 in d = 2 and /ie — 2.0 in d = 3. Experimentally, elastic modula between 3 and 3.6 have been observed in gels [106]-[108] showing the importance of vectorial force constants in real materials. Next we consider the vibrations of the network. Similar to diffusion, it is convenient to discuss the vibrations of the infinite cluster and the vibrations of the whole system separately. 3.5.2. Vibrational excitations and the phonon density of states. Consider N = Ldf particles located at the sites of a fractal embedded in a (i-dimensional hypercubic lattice, where neighbor particles are coupled by strings. Denoting the matrix of spring constants between nearest neighbor particles i and j by k°j , the equation of motion reads

where uf is the displacement of the ith atom along the a-coordinate. For simplicity we assume that the coupling matrix k** can be considered as a scalar quantity, kf- = kij8ap. Then different components of the displacements decouple, and we obtain the same equation,

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for all components uf = Ui. This equation is identical to the diffusion equation when Ui(t] is replaced by P(i,t) and the second time derivative is replaced by a first time derivative. In the diffusion equation, the matrix klj = kjz describes the jump probabilities between sites i and j and obeys the same symmetries as the matrix of the spring constants. The solution of (62) can be obtained following standard classical mechanics: The ansatz Ui(i] = Aiexp(—iut] leads to a homogeneous system of equations for the N unknowns AI^ from which the N real eigenvalues u ; Q > 0 , a : = l , 2 , . . . , . / V , and the corresponding eigenvectors ( A " , . . . , Ajy) can be determined. It is convenient to choose an orthonormal set of eigenvectors (^". • . • , lAjv)- Then the general solution of (62) becomes Ui(t] — Re < 5Ia=1 cQ^f exp(—zo; Q t) >, where the complex constants ca have to be determined from the initial conditions. If at t = 0 only the fco-th atom is displaced, i.e., 1^(0) = iifc 0 (0)6ifc 0 , we have

The solution of the corresponding diffusion equation can be found accordingly

where ea — CJQ. According to the initial condition P(z,0) = 6 z fc 0 , denotes the probability of being at the origin of the walk. We obtain the average probability (P(r, t)} that the walker is at time t at a site separated by a distance r from the starting point by (a) averaging over all sites Z + /CQ, which are at distance r from /CQ, and (b) choosing all sites of the fractal as starting points ko and averaging over all of them:

where

and the inner sum here is over all JVr sites i, which are at distance r from ko. For r — 0 we have simply ?/>(0, a) — I/ AT (since the eigenvectors are normalized) and thus

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CONTEMPORARY PISTICAL PHYSICS

In the limit of N —> oo the sum over a can be transformed into an integral over 6 by introducing the energy density of states n(e),

Accordingly, we can find n(e) by the inverse Laplace transform of (P(0, £)}. It is easy to verify that (P(0,£)) ~ t ~ d f / d w (see (3.48)), implies ra(e) ~ e <*/M«-i. From n(e) we obtain the vibrational density of states z(u}. Since ea = uia and, by definition, n(e)dt = z(uj}du, we have [78]

The exponent ds = 2df/dw has been termed fracton dimension [78] or spectral dimension [109], and replaces the Euclidean dimension d in the expression for the phonon density of states. For percolation clusters, ds is close to 4/3 for all dimensions [78]. The vibrational excitations in fractals are called fractons [78] . In contrast to regular phonons, fractons are strongly localized in space. From the above treatment it is easy to verify that

where here z(uj] is normalized to unity. The inverse Laplace transform of (P(r, £)}, (3.47), can be performed by the method of steepest descent, yielding

with

and

Using (3.47b) one obtains d^ — 1, i.e., the fractons are localized vibrations and decay by a simple exponential, with a localization length A (a;). Equation (3.71) is general and also includes the case of regular lattices. In this case, dw — 2 and c(dw) — i, and we recover the well-known result that harmonic vibrations in regular lattices are infinitely extended. For a different derivation of (3.71b) we refer to [41] and [112]-[114].

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3.5.3. Vibrations of the infinite cluster. In (3.68) we showed that the fracton density of states is proportional to the Laplace transform of the probability of return to the origin in the corresponding diffusion problem. At the critical concentration we obtained (see (3.69)) z(uj] ~ ujds~l, where [78]

is the spectral dimension of the percolation cluster. Above pc, the infinite cluster is self-similar on length scales below £ and homogeneous on length scales above £. Accordingly, the time t% ~ ^dw a random walker needs to explore the fractal labyrinth is the relevant time scale in diffusion, and e^ = l/t^ is the relevant frequency scale. Since the frequency scales for diffusion and vibrations are related by e = u;2, the characteristic frequency LJ^ for vibrations scales as

Below u>£, corresponding to large time and length scales, z(ui) shows normal phonon behavior, z(uj) ~ ujd~l, while above u;^, z(ui) shows anomalous fracton behavior, z(uj) ~ uds~l. Both frequency regimes can be bridged by the scaling ansatz [115]

where we require n(x] ~ x° for x ^> 1 and n(x) ~ x ~d* for x
The different behavior of phonons and fractons can be observed directly by looking at the vibrational amplitudes in the different frequency regimes. Figure 3.12 shows the vibrational amplitudes in the infinite percolation cluster above pc for two different frequencies: (a) uj < LJ^: phonon regime, and (b) u) ~^> uj^. fracton regime. One can clearly see the different features in the two regimes. In the phonon regime (a), even close to u;^, large regions of the cluster vibrate, i.e., the phonons represent extended vibrations. In the fracton regime (b), only small portions of the cluster vibrate, i.e., the vibrations are localized. 3.5.4. Vibrations in the percolation system. Similar to diffusion, the vibrational spectrum is characterized by different exponents ds and d's on the infinite cluster and in the percolation system, respectively. To see this, we first consider z(uj] at the critical concentration and utilize the relation between z(uj} and the probability of return to the origin, (P(0,t)). As in §3.4, we first consider the restricted ensemble of clusters of fixed size s. The mean probability of being at the origin on these clusters can be written as

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

and the average of (P5(0, £)} over all clusters gives (P(0, £)},

The sum can be performed as in §3.4, equations (3.33)^(3.35). For large times (P(0, t)) approaches a constant. The leading time-dependent correction is

which immediately yields

Note that d's is obtained from ds by exchanging the fractal dimension df describing the infinite percolation cluster, with the space dimension d that describes the percolation system as a whole. Following the scaling approach from §3.4, one can easily extend the discussion to p > pc. Since u;^ is the only characteristic frequency scale, we expect that

where m(x) ~ const for x 3> 1 and m(x) ~ xd~d's for x
The existence of both frequency regimes as well as the validity of the scaling assumptions (3.75) and (3.80) has been confirmed by numerical calculations [116], [117]. It has been found that the scaling functions n and ra are smooth and monotonic functions of the scaling variable x. By employing the analogy between the diffusion equation and the vibrational equation the more general case of a nonzero second force constant ks can also be treated [118]. We mention here only the behavior close to PC, where diffusion is characterized by the time scales tfx — (/A//B) and t'x — (fA/fBYdw^^+s^Below £ x , diffusion is anomalous and characterized by d'w; above £ x , diffusion is normal. The crossover times correspond to the vibrational crossover frequencies u/x = (^x)' 1 / 2 and u'x = (t'x)~1/2- In close analogy to the above treatment, one obtains z(uj) ~ ujd~l for u; u>'x. Now, however, the phonon and the fracton regime do not fit smoothly to each other but are separated by a frequency gap. In the frequency gap between o/x and o/x, there occurs a pronounced maximum of z((jj}, which is reminiscent of the Van Hove singularity in an ordered lattice consisting only of B springs. Thus, in contrast to the crossover discussed above, a broad maximum in z(u] separates the phonon regime from the fracton regime.

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To some extent, this situation is similar to the situation with vectorial force constants, where, according to Feng [119], a second crossover length lc occurs, which depends on the relative strength of the bond-stretching arid bond-bending force constants. For £ 1CJ the density of states shows a crossover from an effective spectral dimension ds = 0.8 at intermediate frequencies to ds — 2d//dw at larger frequencies. Numerical simulations [120] confirm this behavior. Support of the fracton concept comes from neutron and light scattering experiments in several amorphous structures and in aerogels [121]-[127]. Recently, by combining neutron and Raman spectroscopies, two crossovers in the vibrational density of states of aerogels have been observed by Vacher et al. [124], where z(uj] changes from a;2 at very low frequencies to cj°-3 at intermediate and a;1-2 at large frequencies. This has been interpreted as the first experimental indication of the crossovers from a phonon regime to a bending and a stretching regime predicted in [119]. Since the behavior of z(u) at small frequencies determines the behavior of the specific heat C at low temperatures, we also expect the corresponding crossover in the specific heat, as a function of temperature. Small frequencies correspond to low temperatures; large frequencies correspond to high temperatures. Accordingly, below a crossover temperature T^ determined by u;^, we expect "phonon behavior" C ~ T d , while above T^ we expect ''fracton behavior" C ~ Td* [125]. For vectorial force constants, a third crossover should eventually occur. A crossover behavior of the specific heat at low temperatures has indeed been observed in a large number of amorphous systems, but the results so far do not unambiguously support the fracton interpretation. 3.6.

Other types of percolation

In this section we consider modifications of the percolation problem which are useful for describing several physical, chemical, and biological processes, such as the directed percolation problem and the invasion of water into oil in porous media, which is relevant for the process of recovering oil from porous rocks. 3.6.1. Directed percolation. Consider bond percolation on a square lattice at concentration p, and assign an arrow to each bond such that vertical bonds point in the positive x direction and horizontal bonds point in the positive y direction (see Fig. 3.14). To illustrate directed percolation, let us now assume that each bond is a conducting diode where electric current can flow only along the directions of the arrows. There exists a critical concentration pc that separates an insulating phase from a conducting phase. The critical concentration is larger than in ordinary percolation (see Fig. 3.14): pc = 0.479 for the triangular lattice an pc ^ 0.644701 for the square lattice [128].

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

FIG. 3.14. Directed percolation on a square lattice below the percolation threshold. Although a path exists between the top and bottom lines, a current cannot go through. Hence the percolation threshold in directed percolation is larger than pc in ordinary percolation.

The structure of the directed percolation clusters is strongly anisotropic and there exist two characteristic correlation lengths £_L and £y, perpendicular and parallel to the main direction, here the x-y direction. This anisotropy represents the feature that the clusters at pc are self-affine rather than selfsimilar objects. The critical behavior of both correlation lengths is described by

and

with i/± ^ z/ij, and both are different from the exponent v. Using scaling theory the following relation can be derived:

which is a generalization of (3.4c). Other relations in (3.4c) which do not include v, such as r — 2 + /?/(/? + 7), are also valid for directed percolation. Substituting the mean field exponents 7 = (3 = v\\ — 2v_\_ = 1 in (3.84) yields the upper critical dimension dc = 5 for directed percolation [129]. The best estimates for the critical exponents have been achieved using 35 terms in the series expansion method [130]-[131] yielding for d = 1,v\\ — 1.7334 ± 0.001, Z/L - 1.0972 ± 0.0006, and 7 = 2.2772 ± 0.0003, suggesting the rational values i/|| = 26/15, z/± = 79/72,7 = 41/18, and /3 = 199/720. Note that the values of 7, /?, and other static exponents characterizing directed percolation differ from ordinary percolation. For directed percolation in d = 3, the known values for the critical exponents are: v\\ = 1.27, z/j. = 0.735,/? = 0.6. and 7^1.58 [128], [132].

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Recently, progress has been made in relating directed percolation to selforganized criticality [134] [135], and in calculating the spectrum of its transfer matrix [136] and its fractal dimension [137]. It also has been shown that in directed percolation, hyperscaling is violated [138]. In a recent work [139], it was argued that anomalous interface roughening in porous media is due to pinning and might be related to directed percolation. Measurements of interfaces formed when a wet front propagates in paper yield an anomalous roughening exponent a = 0.63 ± 0.04, which can be explained by the width w of a directed percolation path in d — 2. The width can be identified by w ~ £ L ~ £^ x ^ ~ £?, where a = 0.63 for d = 2 and a = 0.5 for J

>-L

-,||

»S|| 7

It was shown recently [141] that directed polymers on percolation clusters are also related to the directed percolation problem. Above pc the scaling behavior of the transverse fluctuations of directed polymers in d = 1 + 1 dimensions, defined by 6x ~ K, is governed by the standard exponent £ — 2/3, whereas at criticality (p = p c ), £ = v±/v\\ = 0.63. 3.6.2. Invasion percolation. Consider the flow of water into a porous rock filled with oil. The porous medium can be considered as a network of pores which are connected by narrow throats. When the water invades the rock, the oil is displaced by the water since water and oil are both incompressible immiscible fluids. The flow rate is kept constant and very low so that viscous forces can be neglected compared to capillary forces. The invasion percolation was introduced as a model to describe this dynamical process [142]-[144]. In this model, we consider a regular lattice of size L x L, which here represents the oil (the "defender"). Water (the "invader") is initially placed along one edge of the lattice. To describe the different resistances of the throats to the invasion of the water, we assign to each site of the lattice a random number between 0 and 1. The invading water follows the path of least resistance. At each time step that perimeter site of the invader which has the lowest random number is occupied by the water and the oil is displaced (see Fig. 3.15). Due to iricompressibility and immiscibility properties of the fluids, oil surrounded completely by water cannot be replaced, and therefore oil can be trapped in some regions of the porous medium. Thus, if by the Monte Carlo

FIG. 3.15. Invasion percolation on a 5 x 5 square lattice (a) at the initial time and (b) after 15 time steps.

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

process a closed loop is generated by the invader, it can no longer enter this region. The fractal dimension of the clusters of water was found numerically to be df = 1.82 [144], [145]. Experiments on air slowly invading a d — 2 network of ducts with glycerol performed by Lenormand and Zarcone [146] yielded df = 1.8, in good agreement with the numerical results. This value of df suggests that the invasion percolation model is in a different universality class from regular percolation. Support for this conclusion comes from recent numerical results for the graph dimension, yielding di = 1.40±0.06 (or v = l/dmin — 0.77) [147]. This value for di is significantly smaller than that of percolation (see Table 3.3). On the other hand, if the defender is compressible, closed regimes can continue to grow and the fractal dimension of the invading fluid is larger [144], df = 1.89 (d = 2) and df = 2.5 (d = 3), in close agreement with percolation. 3.6.3. Correlated percolation. In studies of the percolation model and its variants, spatial disorder has usually been assumed to be uncorrelated. However, the nature of disorder in real systems is seldom uncorrelated. For example, the permeability of rock formations is known not to vary randomly in space, but to be consistently high over extended regions of space and low over others; thus it is strongly correlated in space. The standard lattice percolation model assumes independent occupation of sites, characterized by the probability p. To introduce correlations, the lattice system can be characterized by an infinite set of random variables #(r), which receive the value 1 if r is an occupied site and 0 if r is an empty site. When 9(r) are uncorrelated one obtains the standard percolation model. It is common to define the correlation function g(r - r') = (0(r)0(r')) - (#(r)}2 = {#( r )#( r/ )) ~ p2. It was shown by Harris [148] that short-range correlations will not be relevant when v > 2/d. Indeed, this criterion is fulfilled for percolation in all dimensions and thus short-range correlations added to the ordinary percolation problem will not change its universality class. The possibility that the elements in a percolation problem experience a long-range spatial correlation has been of longstanding interest [148]-[150]. When the occupancy of different sites are long-range correlated, the critical behavior of percolation may change. An example of algebraically long-range correlations is the Ising percolation system [151] in d = 2, where the critical behavior is different from ordinary percolation. For long-range correlations such as g(R) ~ R-(2~x^ the universality class criterion can be obtained in analogy to Harris [148]. The correlated percolation can be introduced directly in the occupation probabilities, p(r). It can be shown that the above-defined correlation function g(R) ~ (p(r)p(r-\-R)) ~ .R~( 2 ~ A ). It is necessary to estimate the flue1 /*?

tuations of p(r), 8p$ = ((p(r) — p)2)t fluctuations can be evaluated as

, within a regime of linear size £. These

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Thus the correlations will be irrelevant only when or for A < 2 — 2/zA For A > 2 — 2/v the correlations are expected to modify the universality class [149], [150], and the new correlation exponent is expected to be The study of such long-range correlations has been handicapped by the inability to perform computer simulations. In a recent work [152] an algorithm for generating long-range power-law correlations in the site-occupancy variables of the percolation model was developed. In this method one replaces the random occupancy variables of ordinary site percolation by longrange correlated variables {p(r)}, represented by the power-law form The effects of such correlations on the structural and dynamical properties of percolation were studied. The percolation threshold pc is found to decrease continuously with the strength A of the correlation (see Fig. 3.16). For A —> 2, p c (A) extrapolates to pc — 1/2, which can be understood from (84) since for A —» 2, the asymmetry between vacant and occupied clusters breaks. There are numerical indications (see Fig. 3.16) that the fractal dimensions of the backbone ds and the red bonds dft change continuously and rather dramatically with A even though the fractal dimension of the cluster as a whole does not seem to change. It is also found that the conductivity exponent ((A) is different from random percolation and varies continuously with A (see Fig. 3.17). To study the correlation length exponent v, we first carry out a Monte Carlo renormalization group (MCRG) calculation, in the spirit of Reynolds, Stanley, and Klein [153], but with variables that are correlated instead of random. To this end, we must find the fixed point p*(L) of the renormalization group (RG) transformation for several values of the lattice size L. From the fixed point we can find the exponent v as a function of A by using the scaling relation

FIG. 3.16. The phase diagram pc(\] •

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FIG. 3.17. Variation of exponents with correlation strength A. (a) Fractal dimension of the backbone dsB- (b) Conductance exponent £. Note the trend towards and compactness as

We find agreement of z/(A) with the predictions of Weinrib and Halperin [149], [150] for A < 1.0; specifically, we obtain no change in the exponent from the random percolation value for A < 0.5; for 0.5 < A < 1.0, our value agrees with the prediction ^(A) = 2/(d — A), but for A > 1.0, our values are consistently lower than this prediction. Finally, we compare our results for long-range correlated percolation with the structural properties of Ising clusters at criticality. It is known that the Ising model at criticality possesses correlations of a power-law form,

which corresponds to the correlation strength X = d — 77 — 1.75 in two dimensions. Consequently, it is interesting to compare our clusters at A = 1.75 to the "bare" critical Ising clusters (not Ising droplets [2]). The percolation threshold for Ising clusters is known to be pc = 0.5 and we find pc = 0.529 ± 0.005 for A = 1.75. Since the two systems are identical up to two-point correlations, the difference between our result and the Ising model could be due to the presence

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139

of higher-order correlations in the latter. We also note a trend in the variation with A of the values of structural exponents for our correlated percolation clusters that is similar to that in the variation with q of the same exponents for droplets in the g-state Potts models. Recently, a new model [154], [156], [157] for ionic transport in glasses has been developed by which peculiar anomalies (mixed alkali effect, anomalous increase of the conductivity with the concentration of mobile ions; for a review, see, e.g., [158]) can be explained by the emergence of correlated percolation structures, which facilitate the migration of particular ions in glass. In the model, the sites have a memory. If only one type of mobile ion (A ions) is present in the glass, one can distinguish between two types of sites, A and C. The C sites are the natural interstitial sites for the mobile ions in the glassy network. If a C site is occupied for more than r time steps by an A ion, its local environment is modified and the C site converts to an A site, which is more adapted to the A ion. Accordingly, an A ion is more likely to jump to a neighboring A site than to a neighboring C site. If an A site is not occupied by an A ion for more than r time steps, it loses its memory and converts back to a C site. The mismatch between A ions and C sites leads to a correlated clustering of the A sites that form fluctuating pathways for the mobile ions and tend to freeze in at lower temperatures. An element of percolation is introduced which determines the ion transport and leads to the drastic increase of the ionic conductivity with increasing concentrations of the mobile ions. If two types of ions, A ions and B ions, can move in the glass, they modify the local network structure according to their particular type. Therefore, one must distinguish among A, B, arid C sites. A ions prefer to jump to nearest neighbor A sites, while B ions prefer to jump to nearest neighbor B sites. Due to this memory effect, both types of ions create their own correlated, fluctuating pathways and have the tendency to stay on them. If A ions are progressively substituted by B ions, the long-range mobility of the A ions decreases since more and more pathways consisting of A sites become blocked by pathways consisting of B sites, whereas the diffusion coefficient of the B ions increases. There is an intersection of the diffusion curves of both types of ions at some ratio of concentration where both A and B ions have the same mobility. As a consequence, the total ionic conductivity goes through a deep minimum when A ions are progressively substituted by B ions. The minimum is close to the intersection of the diffusion curves and becomes more pronounced if the temperature is lowered. This effect indeed occurs in glasses, where it is known as "mixed alkali effect." It occurs in all vitreous alkali conductors of the general formula xX 2 O(l-x)Y 2 O-(SiO 2 or B 2 O 3 or GeO 2 ) where X 2 O arid Y 2 O are different networkmodifying alkali oxides, and in related sulphide systems. Mixed "alkali" effects are also seen in glasses containing protons. Ag+ and T1+ ions, and there is a corresponding effect in mixed F~/C1~ glasses where the anions are the

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[147] S. Havlin, A. Bunde, and J.E. Kiefer, The minimal path in invasion percolation, unpublished. [148] A. B. Harris, Effect of random defects on the critical behavior of Ising model, J. Phys. C, 7 (1974), p. 1671. [149] A. Weinrib and B. I. Halperin, Critical phenomena in systems with long-rangecorrelated quenched disorder, Phys. Rev. B, 27 (1983), p. 413. [150] A. Weinrib, Long-range correlated percolation, Phys. Rev. B, 29, 1984, p. 387. [151] A. Coniglio, Fractal structure of Ising and Potts clusters: Exact results, Phys. Rev. Lett., 62 (1989), p. 3054. [152] S. Prakash, S. Havlin, M. Schwartz, and H. E. Stanley, Structural and dynamical properties of long-range correlated percolation, Phys. Rev. A, 46 (1992), p. R1724. [153] P. J. Reynolds, H. E. Stanley, and W. Klein, Large-cell Monte Carlo renormalization group for percolation, Phys. Rev. B, 21 (1980), p. 1223. [154] P. Maafi, A. Bunde, and M. D. Ingram, Ion transport anomalies in glasses, Phys. Rev. Lett., 68 (1992), p. 3064. [155] A. Bunde, P. Maafi, and M. D. Ingram, "Diffusion Limited Percolation": A model for anomalous transport in glasses, Ber. Bunsenges. Phys. Chem., 95 (1991), p. 977. [156] A. Bunde, M. D. Ingram, P. Maafi, and K. L. Ngai, Diffusion with memory: A model for mixed alkali effects in vitreous ionic conductors, J. Phys. A, 24 (1991), p. L881. [157] , Mixed alkali effects in ionic conductors: A new model and computer simulations, J. Non-Cryst. Solids, 131-133 (1991), p. 1109. [158] M. D. Ingram, Ionic conductivity in glasses, Phys. Chem. Glasses, 28 (1987), p. 215.

Chapter 4 Aspects of Trapping in Transport Processes Frank den Hollander and George H. Weiss

Abstract The trapping problem is one in which particles move about randomly in a space containing randomly located traps, which may or may not themselves be mobile. Typically the space is Rd or T,d in d dimensions, the particles and traps are spheres or points, and the motion is Brownian motion or random walk. In the most commonly studied models the particles (as distinguished from the traps) are annihilated on coming into contact with a trap. The function of greatest interest in such models is the so-called survival probability, (S(t)}, which is the probability that a particle remains untrapped for at least a time t averaged over all trap configurations. Other quantities that have been studied are the mean-squared displacement conditional on survival and the mean time to trapping. A complete solution of the trapping problem is known only in very special cases. However, there have been many investigations of this general problem in the literature of chemistry, physics, and mathematics. These have produced a variety of rigorous, asymptotic, approximate, heuristic, and simulation results. In this article we review a number of these approaches and discuss presently open questions. 4.1.

Introduction

4.1.1. Reaction kinetics. In freshman chemistry one learns how to write down equations describing the kinetics of a given reaction, typified by

The interpretation of this reaction scheme in physical terms is that, when an A molecule and a B molecule react, the A is annihilated while the B remains intact. If [A] and [B] denote the concentrations of species A and B, respectively, then the kinetic equation for the disappearance of species A in this reaction is where k is known as the rate constant. There are a number of assumptions implicit in a transition from the reaction in (4.1) to the kinetic equation in (4.2). The most important of

147

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these is that the rate of disappearance of species A in (4.2) depends only on the macroscopic concentrations of A's and 5's, whereas the reaction scheme in (4.1) refers to a microscopic process in which individual ^4's interact with individual J5's. From this observation one expects that local fluctuations in the reactant concentrations may have to be taken into account to obtain a correct description of kinetic behavior. Indeed, when a single A and B react to annihilate the A, they necessarily deplete the concentration of ^4's in the region surrounding them. Moreover, annihilation of A's is faster in regions with many £Ts than it is in regions in which £?'s are comparatively rare. This implies that many-body effects should be incorporated into the analysis. The development of a satisfactory microscopic theory for the reaction noted in (4.1) requires consideration of at least two additional physical effects. The first, completely absent in the kinetic equation in (4.2), is that molecules are mobile and that therefore local variations in concentration occur even in the absence of reactions. The second, not fully accounted for in (4.2), is that the way in which two molecules interact depends on their distance from one another. Two related models are generally used to describe particle motion. The first is that of a diffusion process or Brownian motion, and the second is that of a random walk. These are known to be intimately related [34], [35]. A simple definition of Brownian motion in Rd (Euclidean space in d dimensions) is that at time t a particle, initially at the origin of coordinates, is found in an infinitesimal volume element (r, r-\-dr) = (xi,x\+dxi;x2i x^+dx^] . . . ; x^, X&+ dxd) with probability p(r; i)dx\dx2 • • • dxj-, where the function p(r; t} is the Gaussian

In this equation D is a constant referred to as the diffusion constant and r = ||r|| is the radial distance. The probability density in (4.3), which in physical terms is proportional to the concentration of particles, can be found as the solution to the diffusion equation

subject to the initial condition p(r; 0) = <$(r) ? where <5(r) is the Dirac delta function centered at the point r — 0 [62] . A simple definition of a random walk, for those not familiar with this terminology, is that the position of a particle can be regarded as a sum of random variables, i.e., Yn = X\ -f X<2.-\- • • • + Xn, where the Xn, referred to as the steps of the random walk, can either be scalar or vector quantities [97], [117]. In the trapping problems treated in this review the Xi will be regarded as independent random variables with a common distribution. The lattice random walk description is appropriate for atomic motion in solids, in which crystalline structure is most naturally described in terms of a lattice rather than a continuum. Typically the lattice

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will be Zd (the integers in d dimensions) and so the Xz will take on values in Z d . One of the problems of considerable interest in modern statistical physics is that of the transport of matter in disordered, as opposed to translationally invariant, media. Such problems arise, for example, when one tries to force water at high pressure through a rock in order to force out any oil that may be hidden in the interstices, or in groundwater transport [3], [69]. In addition, there arc a large number of other physical situations that suggest the ubiquitous occurrence of disordered media in nature (cf. [50]. [58], and [78]). When the property of translational invariance does not hold, one generally can no longer write down a diffusion equation and is perforce required to devise new mathematical methods to describe the transport process. These are most naturally formulated as problems in the theory of random walks. 4.1.2. Smoluchowski's model. A Polish physicist, Marian von Smoluchowski, was the first to attempt to reconcile the macroscopic model in (4.2) with a microscopic model like (4.1) by taking local density fluctuations into account [96]. He proposed that the dependence on [A] and [B] in (4.2) should be retained, since it so successfully describes much of chemical kinetics, but that the rate constant k should be taken to depend on time, a relation which we write as k = k ( t ) . His analysis was directed at finding the actual form of k(t) by relating it to a microscopic model of the actual reaction. For this purpose he proposed a model which is grossly oversimplified from the physical point of view yet leads to results in agreement with a number of experiments [81]. Smoluchowski's model is one which assumes that the B species is sufficiently rare so that one can picture the system as consisting of a single B surrounded by a swarm of A's. The B is taken to be an immobile sphere of radius R. The A's are taken to be point particles which do not react with one another and which diffuse freely outside the B. The initial distribution of the A's is Poisson, which means that the probability that an A is found in an infinitesimal volume dV is equal to XdV where A is the density. To more precisely characterize the state of a single particle, we let p(r. t TO) denote the probability density for the location of an A at a point r at time t, given that it is at TO at t — 0. Then free diffusion of the A's is equivalent to the assumption that p(r,t|ro) satisfies (4.4) subject to the initial condition p(r,0|ro) = <5(r — TO). In order to find the form taken by k(t) in (4.2) one first solves (4.4) for a fixed TO and then integrates over all TO with the appropriate concentration factor or Poisson density. As a model of the reaction process Smoluchowski assumed that whenever an A comes into contact with the surface of the B it is instantaneously eliminated from the system, the B remaining unchanged by the reaction. This qualitative description is translated into a mathematical formula through the assumption that the surface of the B is an absorbing boundary, i.e., the analysis requires one to solve (4.4) with the boundary condition p(r,t|ro) = 0 for all t > 0 and r — B. The reaction rate

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is then identified with the flux into the sphere, i.e., the number of A's that impinge on the surface of the B per unit time, a quantity which equals

with tJd the area of the surface of the ball of unit radius in Rd. A straightforward calculation using this formula shows that k(t) in d = 3 takes the form

This approaches a constant in the limit t —» oo, which means that at sufficiently long times (4.2) provides a useful description of the kinetics. In contrast, in d = I one has k(t}—\(D/Tit}^ so that a classical description of reaction kinetics is not possible because k(t) tends to zero. 4.1.3. Extensions of Smoluchowski's model. Aside from the unphysical behavior of k(t) at t = 0, the Smoluchowski model has a number of features that are unrealistic from the physical point of view. Many investigators have attempted to rectify these shortcomings by introducing more sophisticated models. As an example, to go a step beyond the Smoluchowski model one can study a concentration of immobile, randomly distributed, B spheres all having absorbing surfaces. Such a model incorporates some of the many-body effects ignored by Smoluchowski. To calculate the reaction rate, k(t), one attempts to solve the diffusion equation, but this time subject to an absorbing boundary condition on each B. In this picture the reaction rate is identified with the total flux into all of the 5's. Most investigators have concentrated their efforts on calculating the flux of particles initially uniformly distributed throughout space (see §4.6.4). Although most results are based on a model with static .B's, another extension allows the .A's and B's to move with different diffusion constants characteristic of the two species. This generalization, even when only a single B is present, leads to extremely difficult mathematical problems and has been relatively little explored. In addition to the flux, other physically significant quantities have been studied, although not in as great detail. An example of these is the meansquared displacement of a particle at time t conditioned on survival. For an ordinary isotropic diffusion process (such as Brownian motion) in a Euclidean trap-free space a standard result is that the mean-squared displacement is strictly proportional to time, namely, IdDt in d dimensions. However, if the space contains traps, then one may anticipate that the mean-squared displacement of a particle surviving for a time t should be smaller than in a trap-free space, since undersampling the space is one way in which the particle can avoid hitting the traps. It is of direct interest to know whether the meansquared displacement is still asymptotically proportional to £, with perhaps a smaller coefficient, as in the trap-free case, or whether the functional form of the ^-dependence changes.

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The characterization of the kinetic behavior of a diffusing particle in a system containing a fixed configuration of trapping spheres is a particular formulation of what has become known in the physics literature as the trapping problem. The resulting mathematical problem is, as such, quite intractable from the point of view of obtaining a solution in closed form, except for a special class of one-dimensional models (see §4.2). However, the variety of questions raised, the mathematical techniques used to study them, and the results so far obtained have transformed the field into a rich area of research. Some of these problems will be reviewed in §4.2-4.6. As a second formulation of the trapping problem, one allows traps to not necessarily be discrete, but rather to be smeared out over R d . This formulation is translated into the problem of solving diffusion equations having the general form

in which £ is a diffusion operator (which may incorporate any relevant force field acting on the particles), and p is a space-dependent nonnegative reaction rate [119], [2]. The term reaction means that if C = 0 (so that no motion is possible), then p(ro,t|ro) = exp[—p(ro)t], which says that particles disappear from the system by first-order kinetics. It may be noted that a choice of the function p(r] as a linear combination of delta functions allows one to formulate the above-described trapping problem in terms of equations of the form (4.7) [99]. We shall not describe any of the analytical approaches to models formulated in terms of (4.7), but rather restrict our remarks to the many ramifications of the trapping model as described above. 4.1.4. Rosenstock's model. Two closely related physical phenomena serving to motivate the study of the trapping problem come from the fields of metallurgy and solid state physics. These contexts suggest a slightly simpler mathematical problem than that of trying to extend Smoluchowski's analysis of diffusion-controlled reactions, since the microscopic structure of a solid is most naturally described in terms of a lattice and the diffusion process can therefore be replaced by a random walk on this lattice. Beeler and Delaney [10], and later Beeler [9], appear to have been the first to discuss such models in a study of the trapping of defects in a crystal lattice, with a view to modelling the process of defect annealing. Rosenstock had earlier discussed the lattice trapping model without specific applications [87] and later proposed that it might be a possible model for luminescent emission from an organic solid with traps [88]. The essential physics in Rosenstock's work is one in which sites can be one of two types, either a host or a trap, the specific category being randomly and independently assigned to each site. A photon is absorbed at a host site and excites the molecule occupying that site. After some time the excitation is transferred randomly to a nearest-neighbor site. If this site is a host, then the excitation is either emitted as luminescence with some probability or else

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it is transferred to a nearest-neighbor site again. If the site is a trap, then the excitation is absorbed and the possibility of luminescence vanishes. The trapping problem is related to another question that is of mathematical interest. Consider the motion of a random walk Yn = X\ + X<2 + • • • + Xn on Z rf . Assume, as did Beeler and Rosenstock, that the probability for any given site on the lattice to be a trap is equal to c > 0, independently of the state of the other sites. We shall use the physicist's notation "(Z)" to denote the expected value of a random variable Z. In this notation the probability that a random walker will survive for n steps or more, S(n), can be expressed exactly as where Rn is the (random) number of distinct sites visited by an n-step random walk, i.e., Rn = \{Yo,Y\, . . . , Yn}\. The average is taken with respect to all n-step random walks on the trap-free lattice. The average over the random positions of the traps is implicit. The random variable, Rn, is referred to in the mathematical literature as the cardinality or range of the random walk. Equation (4.8) can be understood as follows: If the random walker has survived for n steps, then each of the Rn distinct sites visited cannot have been a trapping site. This event has a probability equal to (1 — c)Rn when averaged over the configuration of traps. Thus, the study of the survival probability in the simplest version of the trapping problem is equivalent to the analysis of the cardinality of a random walk. While the random walk itself is a Markov process, the cardinality is not. This considerably complicates any analysis. Only in d — 1, and when the random walk is restricted to step only between nearest-neighbor lattice points, does the cardinality coincide with the span, defined as the sum of the maximum extensions of the random walk in both directions [28], [116]. This allows one to calculate the distribution of Rn explicitly (see §4.2). A related quantity, which has been analyzed by a number of investigators in the physics literature, is the mean trapping time, (n), obtained from (4.8)

by

[113], [90], [91], [47]. In both sets of studies mentioned above the most extensively analyzed functional is the survival probability as a function of time. A motivation for analyzing properties of other quantities, exemplified by the mean-squared displacement of untrapped walkers, comes from the currently intensively studied area of the transport of matter in disordered or amorphous media [43], [22]. Of course, it is possible to imagine many kinds of disorder, depending on the physical source of the disorder. Hence, the trapping model should be regarded as just a single example of such media, namely one corresponding to a uniformly random distribution of the traps.

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4.1.5. Other reaction schemes. We finally comment that there are a vast number of generalizations of the plain vanilla trapping problem as defined by the reaction scheme in (4.1). There is currently an enormous research effort in different branches of physics and mathematics aimed at trying to understand a variety of features of the kinetic behavior of chemical reactions in low dimensions. Thus, for example, it is interesting to consider in which particles annihilate in pairs when they collide. This too can be categorized as a trapping problem, namely one in which all of the particles are regarded as being both traps and diffusing species. The kinetic behavior of such reactions in confined geometries (i.e., in spaces with low dimension) differs from the predictions of the classical theory of chemical kinetics [77], [111], [107], [108], [19]-[21]. We shall not discuss these more complicated problems here, noting that even the simplest version of the underlying physical problem gives rise to a rich variety of mathematical problems, many of which remain open for future research. 4.2. Trapping in one dimension: A solvable example Before we embark on an exposition of some of the literature on asymptotics, approximations, and simulations of the survival probability S(n) in (4.8) and of the mean trapping time (n) in (4.9), let us first consider a simple example in which it is possible to obtain some of the interesting results in closed form. Consider a symmetric random walk on the integer lattice Z, constrained to make jumps between nearest-neighbor points only. That is, the position of the random walk at step n is Yn = X\ -f X% + • • • + Xn, with YQ = 0 and with the Xi being independent, identically distributed, random variables given by Pr{Xi = 1} = Pr{Xz = —1} = T^. If such a random walk takes place in the presence of traps, then its survival depends only on the positions of the two traps on either side of the origin that are closest to the initial origin. The locations of these traps will be denoted by —£2 and i\, respectively, where £1,^2 > 0. Since the random walk cannot jump over a trap, the positions of traps further away from the origin are irrelevant. This property reduces the solution of the trapping problem to one on a finite interval, thereby allowing us to calculate a number of results in closed form. In §§4.2.1-4.2.5 we shall sketch two such results, which will be used in §4.3 to test different approximations. 4.2.1. The mean trapping time (n). Suppose, as we did in §4.1.4, that traps are distributed randomly and independently, each site being a trap with probability c > 0. In this case i\ and t^ are random variables characterized by the joint distribution Now suppose that i\ and i^ are given and that we start the random walk at site x, where — £2 < x < i\. What is the average number of steps taken

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before the random walker hits one of the traps? Let us denote this average by (n x) . This function satisfies the recursion relation

which must be solved subject to the boundary conditions

Equation (4.12) is obtained by noting that the average number of steps must increase by one at each step, and that the process is regenerated from either of the points x ± 1, each with probability 1/2. Since the endpoints of the interval are trapping points, it follows that trapping at either of these points is immediate, which explains (4.13). The solution to (4.12) subject to (4.13) is easy, namely In our formulation of the trapping problem the walk starts at x = 0, so that

Having this result we can compute the trapping time by averaging over all values of t\ and £2 using the distribution in (4.11). In this way we find

This answer is exact for every c > 0, and has the advantage of being rather simple in form [72]. 4.2.2. The survival probability 5(n). A calculation of the survival probability of a random walker initially at x is slightly more complicated. The simplest starting point for the calculation is to find the probability, pn(y\x], that a random walker, initially at x, arrives at site y at step n, in the presence of traps at —^2 and t\. This function satisfies the recursion relation

whose solution is to be found subject to the initial condition

and the boundary conditions

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Once pn(y\x] is known, the probability that a random walker initially at x survives until at least step n, S(n\x}, is given by

Hence the problem of calculating S(n x) reduces to that of solving the system of equations in (4.17)- (4.19). To find this solution, put L = l\ +£% and expand pn(y\x) in a finite Fourier series that incorporates the boundary conditions in (4.19). This prescription amounts to writing

which clearly vanishes when the random walker is at either of the two traps, y = i\ or y = —I?. On substituting (4.21) into (4.17) we find that the amjTl(x} satisfy

or

We compute am$(x} = (2/L} sin[irm(x -\- t^/L] by matching (4.21) with the initial condition (4.18), leading to

The resulting expression for the survival probability in (4.20) is

If we set x — 0 in this expression, and average over t\ and t^ using the joint distribution in (4.11) (remembering that L = ^ 1 + ^ 2 ) ? then we find for the survival probability in the one-dimensional random trap problem

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4.2.3. Large-n asymptotics for S(n). It appears to be impossible to reduce the sum in (4.26) to a simpler looking form. However, it is possible to find an asymptotic expression for S(n) valid in the limit n —» oo. To do so, we observe that, since the cosine term is maximal when k — 0, we need only retain this term in the sum over fc, finding

where the symbol " ~ " means that the ratio between the two sides goes to one as n —> oo. Since the function cos(w/L) increases to 1 as L —> oo, it is plausible to suppose that for large n and an exponential scale the sum can be approximated by the term in (4.27) in which the decrease of (1 — c}L~l with increasing L roughly balances the increase of cosn(7T/L). In other words, we want to find the location and value of the maximal term in the sum in (4.27). (There is also a contribution from the prefactor (l/L)2c2 cot2[-7r/2L] which can, however, be neglected in calculations of the maximum, carried out to lowest order in 1/L on an exponential scale.) For this purpose let A = — ln(l — c). Then, since

the maximal term in the expansion in (4.27) is at L, which is the solution to

It follows that L —>• oo in the limit n —> oo so that we may replace tan(?r/L) by TT/L. Hence the maximizing value of L is given by

After the appropriate substitutions are made, the maximal summand can be written as a n exp(—6 n ) where

Without a more precise knowledge of the corrections to 6 n , as well as an estimate of the contributions of the remaining terms in the series in (4.27), one cannot state that an provides a good estimate of the prefactor of the survival probability. In any case the above argument is at least a demonstration that

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It is not difficult to establish, as we have already suggested, that in fact one has the asymptotic estimate

In other words, on an exponential scale the sum in (4.27) is indeed dominated by the term at Lmax alone. The computation given above shows that the contribution from terms at L with L/L max outside of any neighborhood of 1 is negligible, because these terms have an exponent strictly larger than 6n, while those inside only contribute to the prefactor an. The reader is invited to write out the mathematical estimates that rigorously prove the validity of (4.33) [5], [114]. Equation (4.33) is a useful result showing that S(n) decays at long times as a stretched exponential, the exponent being proportional to ns with a coefficient that depends in a nontrivial way on the density, c, of traps. In addition, (4.30) can be interpreted as saying that the long-time survival probability is mainly determined by large trap-free intervals of length (7r 2 n/A)3, which depends not only on the trap concentration, but also on the step number. Another one-dimensional model, one with so-called non-Markovian trapping conditions, has been analyzed in which the time-dependent behavior of the survival probability differs from that in (4.33) [118]. In this model trapping occurs with probability Oj at the jth encounter of a random walker with a trapping point. If the average number of encounters ((j) — Y^j J^j] required to produce a trapping event is finite, then one finds the same asymptotic ndependence of the survival probability as in (4.33). However, when Oj follows an asymptotic decay law proportional to j~' a+1 ', where 0 < a < I (with the result that (j) = oo), it is found that S(n] decays as n~a for large n. In such a model it is clearly the large number of walker-trap encounters required to produce a trapping event that dominates the behavior of the survival probability at long times, rather than the trap-free intervals, as is the case when (j} is finite. 4.2.4. Small-n behavior of S(n}. The trapping problem in one dimension is also sufficiently simple that one can estimate the behavior of the survival probability at short times. In this regime changes in the survival probability will be due mainly to particles found in the smaller intertrap intervals, in contrast to the long-time survival probability which principally reflects particles initially located in the larger intertrap intervals. A derivation of the desired result that presents minimal mathematical difficulties is based on a diffusion model rather than the discrete random walk considered in §§4.2.2 and 4.2.3. As in our earlier exposition, we first examine the survival probability within a fixed interval of length L terminated by traps at x — 0 and x = L, later performing the average with respect to L. Let p(x,t\xo] be the probability density for the position of the diffusing particle at time t, given that it is

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initially located at XQ. The survival probability for such a particle, S(t\XQ), is related to p(x, t\x$) by

which is the analog of (4.20). We shall be interested in the survival probability for a particle that is initially uniformly distributed throughout the interval [0, L], which is the same as fixing the initial position of the particle at x — 0 and shifting the trap interval uniformly. This probability will be denoted by , which is found from S(t\xo] by one further integration

Thus, the crucial function required for the evaluation of SL(£) is the probability density p(x,t\xo}. This is found as the solution to the diffusion equation (4.4) in one dimension subject to the boundary conditions p(0, t\xo) = p(L,t\xo) = 0. Standard techniques for the solution of partial differential equations (cf. [13]) suffice to produce the result, similar to that in (4.24),

After performing the integrations indicated in (4.34) and (4.35) one finds

which is analogous to, but simpler than, (4.25). Note that at t — 0 the sum is equal to 1 [1]. We shall be interested in finding the form of this function in the short-time limit, i.e., Dt/L? —» 0. There are a number of ways to do this. For example, one can take a Poisson transformation of the series in (4.37) [41] or use the more physical method of images to produce an equivalent series whose short-time behavior is easily determined. Here we shall folllow a more heuristic approach. AND CALCULATE Let us define a dimensionless time parameter the small-r behavior of

The derivative of this function is the series

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Because r is a small parameter we can approximate the sum by an integral

Hence, taking account of the initial condition SL(O) = 1,

The average of this function with respect to the distribution of L is found in terms of the physical time t as

The form of the probability density for L in the integral takes into account the fact that L is a sum of two negative exponential random variables, each with mean A^ 1 , namely the average distances to the two traps closest to the origin. Note that the result in (4.42) is a stretched exponential with an exponent equal to 1/2. The power 1/2 is larger than the power 1/3 that appears in (4.33), which characterizes the long-time behavior. This indicates that there is a crossover region between short- and long-time behavior. 4.2.5. Preview of extensions. Our analysis to this point was developed for a lattice model in which each point is independently designated as a trap with probability c, or for a diffusion model with a constant density A of traps. The advantage of working in one dimension is that, because of the availability of an exact expression for the survival probability in a fixed interval terminated by two traps, one can also consider models that include correlations between the locations of neighboring traps. For example, if the probability that two adjacent traps are separated by a distance rn is taken to be

then an analysis similar to that given above shows that the asymptotic form of the survival probability is

in which the power of n is 1/2 instead of the 1/3 appearing in (4.33) 114]. Indeed, one expects the decay of the survival probability to be faster than for an

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uncorrelated placement of traps, since (4.43) implies trap-free regions smaller on average than those for the standard model (provided that both trap placements are comparable in the sense that the densities of traps are equal). In §§4.4 and 4.5 we shall discuss extensions of (4.16) and (4.33) to higher dimensions. An example of such a generalization, for the symmetric nearestneighbor random walk on Zd, is the result

with A = — ln(l — c) and K^ a positive constant depending on the dimension [30]. The proof of this extension requires the use of probabilistic techniques much more sophisticated than those based on an explicit representation of 5(n), as in §§4.2.2-4.2.4. 4.3.

What do approximations, heuristics, and numerics tell us about

5(n)? There are no exact results available for S(ri) that are valid for all n, except in d = I where the simplified geometry allows considerable progress to be made, as explained in §4.2 (recall (4.26)). Because of this, a considerable part of our information about survival probabilities in higher dimensions has been gained either from asymptotic approximations or from simulations. We review some of these in §§4.3.1-4.3.6. 4.3.1. The Rosenstock approximation. The first approximation was suggested by Rosenstock [89], who proposed the replacement of the exact form given in (4.8) by the approximation

All that is required in this approximation is the average of the number of distinct sites visited in a random walk of n steps, (Rn). The form of (4.46) is independent of the dimension and of the transition probabilities associated with the random walk, although, of course, (Rn) will itself depend on these parameters. Fortunately, the asymptotic behavior of (Rn} in the limit n —> oo is not too hard to calculate for all kinds of random walks [70], [72]. For convenience, we sketch a derivation in Appendix A. Here we simply note the results of that analysis. Let us suppose that the random walk is defined by the single-step transition probability p(j), where j = (ji, J2, • • • , j d ) £ ^• That is, p(j) is the probability that the random walker makes a jump j in a single step. We state results for random walks which are symmetric and have finite second moments, i.e.,

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In d = 1 we have the large-n behavior (see Appendix A)

The combination of this estimate with the approximation in (4.46) implies that S(n) is asymptotically proportional to exp(—Kn*} with K = X(j(~)^ where A = — ln(l — c). However, we have seen from the exact analysis of the onedimensional nearest-neighbor case in §4.2.3 that the correct asymptotic form of the survival probability is in fact equal to exp(—K'n^} (recall (4.33)) with 2 K' — (3/2)(?rA)3, which proves that the Rosenstock approximation gives the wrong result in the limit of large n. In d > 2 we have (see Appendix A):

where the Green's function p(0; z) is defined by the multiple integral

in which p(6) is the characteristic function or Fourier series associated with the single-step transition probabilities, i.e.,

It is relatively simple to show that the integral in (4.50) converges at z — 1 when d > 3 but not in lower dimensions [72], [97]. By substituting (4.49) into (4.46) and recalling the exact Donsker-Varadhan result given in (4.45), we see that the Rosenstock approximation incorrectly predicts the large-n behavior of S(n) in any number of dimensions. Should one then discard the Rosenstock approximation altogether? The answer to this question appears to be: No! The reason behind this is that there is evidence, based mainly on the results of simulation studies [123], that the Rosenstock approximation provides a useful approximation to S(n) at small and moderately long times provided that the trap concentration is not too large (which is generally the case in physical applications). This statement requires qualification because the definition of the asymptotic regime will depend both on the dimension and on the trap concentration, and to a lesser degree on the detailed form of the single-step transition probability (on the understanding that (4.47) is satisfied). What is meant by the large-n limit of S(n], i.e., how large does n have to be in order for the behavior in (4.45) to show up? No information on this point is available from the derivation given by Donsker and

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Varadhan [30]. Whatever is known related to this problem is the result of a number of simulations and other approximate analyses (e.g., [36], [44]). Such data indicate that the Donsker-Varadhan regime is accurate only at values of n in which S(ri) is of the order of 10~10 or less for trap concentrations c > 0.01. Such small survival probabilities are hardly useful for applications, although it is also true that in solid state physics physical concentrations equivalent to trap densities of the order of 10~5 are the ones of major interest. No one, as far as we know, has simulated trapping problems at such low concentrations, and indeed it would be very difficult to get reliable results, since sampling variances in this regime are necessarily very large. Because the Donsker- Varadhan result is valid only in a very remote regime, accurate approximations to the survival probability at shorter times are indispensable for physical applications. Parenthetically, we note that Jensen's inequality for convex functions implies that the Rosenstock approximation is a rigorous lower bound for S(n). That is (cf. [35]),

The original formulation by Rosenstock [87] and Beeler and Delaney [10] in terms of lattice random walks is appropriate in the setting of solid state physics. However, it should be noted that the lowest-order Rosenstock approximation can be shown to coincide with results obtainable from the continuum Smoluchowski model [101]. Corrections to the approximate values, i.e., higher-order terms, will depend on whether one is interested in the continuum or discrete version of the model. 4.3.2. The truncated cumulant approximation. In their extension of the Rosenstock approximation, Zumofen and Blumen [122], [123] begin by observing that the exact expression for S(n) given in (4.8) can be regarded as a moment generating function. That is, one can expand S(n) into the series

By the same token one can write a second series representation in terms of the cumulants of Rn, written {Kj(n)},j = 1, 2, 3, ... [98]. To define the cumulants we define the associated cumulant generating function

The survival probability is expressed in terms of

by

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

163

Equations (4.53)-(4.54) allow us to define the jth cumulant, Kj(n), in terms of the mean (Rn) and the central moments {/t^)c = ((Rn — (Rn)}1) of order i < j. The first four relations are

so that, for example, the second cumulant is just the variance of jRn. Zumofen and Blumen point out that the Rosenstock approximation is just a truncation of the cumulant expansion in (4.54) at j = I . They suggest, as an extension of this idea, that one can truncate the expansion at any higher value of j, which should yield more refined estimates of the survival probability (in a sense which they do not specify more precisely). When A and n are small, such an approach is especially appealing because the series in (4.54) presumably converges very rapidly. The fly in the ointment implied by this strategy for successive approximations to S(n) is that it is not easy to compute higher moments of Rn (except in the case of one dimension). Torney [106], for example, has studied the variance of the range, (R^)c = (R%) ~ (Rn)2i deriving approximations in closed form to this quantity by combining analytical and computational techniques. The earliest rigorous results on the properties of Rn are due to Jain and Orey [51] and Jain and Pruitt [52], [53]. What they have shown is that as n —+ oo, under the assumption of (4.47),

where the cj are constants calculable in terms of the single-step transition probabilities that characterize the random walk. However, it is not known how large n must be before the asymptotic regime in (4.57) is reached, nor is much known about cumulants of order j > 3. 4.3.3. Systematic corrections. There have been a number of numerical investigations whose object is to determine the accuracy of approximations derived by truncating the cumulant expansion in (4.54). For a summary of some of these see the review by Blumen, Klafter, and Zumofen [16]. To assess this accuracy it is necessary to generate the asymptotic form of the cumulants given in (4.56). It is possible to find a systematic expansion of (Rn} beyond the leading-order term given in (4.49) (see Appendix A). As an example, in d = 3, for large n, and subject to (4.47) one has

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

where a = l/p(0; 1) and b = (2^/7ri)cricr 2 cr3^(0; 1) (recall (4.50) and see (B2)). A similar result holds in any number of dimensions (cf. [63], [46]). An analytical calculation of corrections to the results for (R'fyc in (4.57) requires a much greater level of mathematical effort, and to our knowledge has not been carried out to date. As an example of the accuracy that can be obtained using the cumulant expansion truncated at various orders, we present in Fig. 4.1 simulation data obtained by Zumofen and Blumen [123], together with the cumulant predictions. The symbol $n denotes S(n), while p stands for the trap density c. The notation $jjn refers to a truncation of the cumulant expansion at term j. It is evident from the figure that, for the range of c and n considered, the cumulant expansion truncated at second order gives excellent agreement with the simulated data, except in d = 1 where none of the approximations are accurate except at very small step numbers. The data plotted for d = 2 and 3 suggest that the approximation becomes more accurate as the trap concentration decreases or as the number of dimensions increases. A more precise discussion of the validity of the truncated cumulant approximation will be given in §4.4.1. It would be virtually impossible to simulate the trapping process down to the more physically realistic trapping densities of the order of 10~°, because of restrictions on computer time. However, as stated, the simulation results do indicate that the accuracy of the truncated cumulant expansion improves as c decreases. Parenthetically, we note that truncation of the cumulant expansion can lead to estimates of S(n] that exceed 1 when n is sufficiently large. This, of course, is an artifact of the approximation. This is easily seen in the case of truncation at the second order in d = 2 and d = 3 (see (4.49) and (4.57)), which suggests that the method of approximation can at best be useful when the trap concentration is sufficiently low and n is only moderately large. A slightly different approach has been suggested by Weiss [113], based on the theorem of Jain and Pruitt [52] that the distribution of Rn is asymptotically Gaussian. The Gaussian form allows us to evaluate the average in (4.8) explicitly. The result in this way agrees with that given by the cumulant expansion truncated at the second order. 4.3.4. Heuristic derivation of the asymptotic form of the DonskerVaradhan tail. As observed already in (4.45), Donsker and Varadhan [30] have derived a result equivalent to the statement that for large n

in which K& is a positive constant depending on d and on the random walk (subject to the conditions in (4.47)). The derivation involves some rather sophisticated mathematical analysis based on the so-called theory of large

165

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(b)

(a)

(c)

FIG. 4.1. The survival probabilities in The variable p is the concentration, 3>n are the results of simulation of the survival probability and $.;-.n is the jth curnulant approximation to the survival probability.

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

deviations [112]. It is not readily summarized, but we shall give a brief sketch of the main ideas in §4.5.4. At about the same time that the rigorous result was presented in the mathematical literature a number of heuristic derivations appeared in the physics literature. These were aimed at finding (4.59) and various generalizations [8], [40], [55], [56]. One can remark, following Kayser and Hubbard [55], that the noninteger exponent in (4.59) precludes the use of a technique based on perturbation theory. A further remark is that (4.59) gives only the lowest-order term on an exponential scale. A rigorous estimate of the prefactor of the exponential term is not known at this time, but there is a calculation by Lubensky [64] , based on nonrigorous field-theoretic methods, which addresses this point. According to Lubensky's results the prefactor can be significant in determining the survival probability for moderately large n. The simplest approach to a heuristic derivation of (4.59) is found in the work of Grassberger and Procaccia [40]. They start from the sensible observation that the survival probability at large step numbers must be dominated by random walks in regions in which there are few traps. The probability that a given volume F containing V lattice points is trap-free is equal to (1 — c)v ' . Since we are interested in long times and large volumes, we can replace the lattice by a continuum, allowing us, in turn, to replace the random walk by a Brownian motion. Let us assume that F is bounded by a surface consisting only of trapping sites. Grassberger and Procaccia choose the bounding surface to be a sphere, but this choice is not crucial [8]. One can then calculate the probability that a diffusing particle inside the sphere with an absorbing surface will survive until time t. Since the sphere is a bounded geometric figure, the survival probability at long times necessarily decreases exponentially in t. But what is crucial for the argument is the coefficient of t that appears in the exponent. This is easily found for the case of a sphere, but a more heuristic scaling argument will furnish its dependence on V for more general shapes, as we next show. A simple way of deriving this coefficient is to note that the survival probability of a particle initially at the origin satisfies the diffusion equation in (4.4) with absorbing boundary conditions on the surface of F. We shall not write a detailed solution to this equation, since this is hard to do in general. Rather we rewrite the equation in terms of dimensionless parameters. The scaling can be done quite straightforwardly provided that F has a characteristic length indicative of its volume V. For example, the characteristic length for a sphere is just the radius .R, while for an ellipse with axes RI, R^i • • • 5 Rd a reasonable choice would be (R\R<2 • • • Rd)^ • Let us denote this length by Rc. There is only one further parameter in the problem, and that is the diffusion constant D, which has the dimension (length)2/time. Hence a dimensionless time variable that is proportional to the time is

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

167

But the volume V in d dimensions is proportional to R£, which allows us to redefine the time variable as

Hence the joint probability for there to be a trap-free region of volume V around 0 and for the diffusing particle to stay for at least a time t within this volume is given by

where A = — ln(l — c) and A is a constant that depends on the shape of F. On maximizing the function S(V, t] with respect to V, we arrive at an inequality for the survival probability that can be written

where Kj is a constant. The reason for the inequality is our assumption that the trap-free region is nicely shaped and that there are traps everywhere along its boundary. It is not at all obvious that such a situation will necessarily yield the dominant effect in determining the survival (see §§4.5.1 and 4.5.2). The term on the right-hand side of (4.63) agrees in form with the rigorous asymptotic result of Donsker and Varadhan insofar as the dependence on t and A is concerned. As noted earlier, there should be a prefactor multiplying the exponential, but neither the rigorous nor the heuristic theories provide a value for this multiplier. The point emphasized by Grassberger and Procaccia, which is of greatest interest in the context of applications, is that the survival probability cannot fall off as a negative exponential, which might be a first naive guess. However, as d increases, the simple exponential gets to be an increasingly more accurate approximation to the true result. A variation of the above heuristic argument can be used to suggest a form for the long-time behavior of the mean-squared displacement, {r 2 (t))*, of a diffusing particle conditioned to remain untrapped at time t (the subscripted asterisk indicates that the moment is to be calculated conditional on survival) . The result is the scaling behavior [40]

K again denoting a constant. In d = I the exponent is 2/3, which is in agreement with a more detailed calculation along the lines of the analysis in §4.2 [114] (recall (4.30)). As the dimension increases, (4.64) predicts an increasing localization of untrapped random walks. A rigorous proof of (4.64) is known only in d = I and 2 (see §4.5.2). The Grassberger Procaccia argument, as we have seen, leads to a lower bound for the survival probability. A complementary and lengthier argument was given by Kayser and Hubbard [55] to establish an upper bound which has

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

the same functional form as (4.63), except that the constant K^ appearing in the exponent differs from that found by Grassberger and Procaccia. The d combination of these results serves to show that the quantity [\i\.S(t)]/td+2 is bounded above and below by a constant. This, in itself, is still weaker than the rigorous result of Donsker and Varadhan, but at least provides a heuristic reasoning that might be more accessible to the physicist. In a later paper, Kayser and Hubbard [56] extend their calculation to allow for imperfect trapping. The result of that calculation suggests that the asymptotic ^-dependence of \nS(t] is the same as for perfect trapping. 4.3.5. Numerics: Exact enumeration techniques. From the mathematician's point of view, the form of the asymptotic survival probability derived by Donsker and Varadhan is a perfectly rigorous result. However, from a more practical point of view it is meaningless without some accompanying estimate of how large the time must be to validate its accuracy. This problem was studied by Havlin et al. [44] by using the so-called method of exact enumeration [43]. This technique is one that essentially solves a variety of lattice random walk problems exactly (within the accuracy of the computer arithmetic) at the expense of requiring large amounts of computer memory. The basic idea can be illustrated by means of a very simple example. Suppose that one has a random walk on Z starting at a point jo between two traps at 0 and at N. Let us further suppose that the probability that a random walker at site j moves to j + 1 or to j — 1 is equal to 1/2, so that jumps longer than a single lattice spacing are forbidden. Let us ask for the probability that the random walker survives for n or more steps before being trapped. This is precisely the lattice model discussed in §4.2, but here we outline the numerical, as opposed to the exact, solution. In the exact enumeration technique one assigns one memory register for every site of the lattice, initially putting a 1 in the register corresponding to JQ and a 0 in all of the other registers. The calculation then proceeds by recursion. The content of the register corresponding to site j after n steps will be denoted by Mn(j). A trapping site will always have a 0 in its register. This remains unchanged throughout the numerical computation. First, suppose that the initial position JQ is not adjacent to a trap, which imposes the requirement jo 7^ 1, TV — 1. Then three registers are modified in the course of a single step, namely those associated with JQ and JQ ± 1 , while the remaining registers retain their original value of 0. The contents of the registers at step 1 are

Next, suppose that JQ is adjacent to a trap, e.g., JQ = 1. The contents of the registers are then changed to

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

169

That is. a contribution of 1/2 from site 1 is swallowed by the trap at 0. The recursion is similar at jo = N — 1. Summarizing, at step n we perform a recursion that is equivalent to the random walk process, i.e.,

Finally, the survival probability at step n is just

The reader will notice that no approximations have been made in this prescription and any errors in the resulting approximation are attributable solely to the finite precision of the calculation. The combination of (4.67) and (4.68) is trivial to implement on a computer. While the general method of exact enumeration as just outlined makes no approximation, it does require a step that is not exact when applied to the trapping problem. Namely, the initial step in the calculation is to assign the positions of the traps randomly, using the probability c to decide whether or not a given site is a trap (recall (4.11)). One then fills each of the registers for nontrapping sites by a 1, and those for traps by a 0. and solves for the survival probability by the recursion procedure suggested in (4.67) and (4.68). However, to derive a general solution of the trapping problem it is necessary to lay down a number of configurations and average the results of the exact enumeration calculation with respect to the members of this set. The crucial point required for the success of the ensuing calculation is, of course, that one should not need an astronomical number of configurations to find the appropriate exponent for the survival probability. In the simulations reported in Havlin et al. [44] for c > 0.5. no more than 100 200 configurations were used to analyze the trapping problem in d = 2 for a lattice size of 1.000 x 1.000. A similar number of replications was used for the case d = 3 with a lattice size of 100 x 100 x 100. There is no available analysis to predict how many configurations are needed to obtain reasonably accurate results for a given value of r. but the problem is solved in a practical sense through an examination of successive convergents. A second numerical method has also been applied in Havlin et al. 44] that is useful only for values of c close to 1. In that method the survival probability is calculated exactly for small clusters of nontrapping sites. A cluster is defined to be a connected group of k nontrapping sites surrounded by / traps and containing the origin. Because c is taken to be large, one need only generate relatively small clusters. The probability of occurrence of a given k. /-cluster is equal to (1 — c}kcl. Any calculation of the survival probability requires taking account of the specific form of the cluster. We may subsume this factor into a single index ?'. which counts all k. /-clusters. Let S[, i(n] denote 1 the probability

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

that a random walker initially uniformly distributed on such a cluster survives until step n. The average survival probability can then be expressed as

When c is sufficiently close to 1 only a small number of easily enumerated clusters need be taken into account. Some typical results obtained by this method for — \nS(n) in d = 3 are shown in Fig. 4.2, plotted as a function of the parameter p — [— ln(l — c)]s for concentrations 10~3(+), 10~4(A), and 10~5(o). The convergence to linearity is evident. The values of c and n leading to this behavior correspond to the regime p > 10, or to values of S(n) of the order of 10~13 or less. Rosenstock and Straley [91] have computed the first few terms of (4.69) exactly.

FIG. 4.2 plots of - in s(n) as a function of

4.3.6. The trapping problem on a fractal. One of the popular approaches to the study of transport in disordered media is based on models of a specific class of media known as fractals [43], [22]. The concept of a fractal is an old one [42], but has recently been popularized in the writings of Mandelbrot [66], [67]. A fractal, roughly speaking, is a geometric object whose properties are unchanged at any level of magnification. One of the most striking examples of a fractal surface, suggested by Mandelbrot, is that of a coastline. Its shape often appears to be quite irregular in an atlas. However, if one were to examine it on successively finer length scales, then it would be found to exhibit the same kinds of irregularities as on the scale shown in the atlas.

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

171

There are two types of fractals: deterministic and random. A widely studied example of a deterministic fractal is the so-called Sierpinski gasket, generated as shown in Fig. 4.3. One of the striking features of a fractal is that its dimension, properly defined, need not be an integer even though the fractal structure itself is embedded in a space of integral dimension. As an example of the way in which the properties of a random walk on a fractal differ from those on a uniform medium, consider the mean-squared displacement, (r 2 (n)}, of a symmetric nearest-neighbor random walk on Z rf , which is easily shown to be equal to

In contrast, for large n the mean-squared displacement of a random walk on a fractal typically behaves like

where C is a constant and dw is an exponent generally greater than 2. Now, one of the characterizations of so-called anomalous diffusion, which commonly occurs in disordered media, is that (r2(n}} has the same behavior as in (4.71) with the diffusion exponent dw different from 2. One reason, therefore, for the wide use of fractals as approximate models of disordered media is precisely this similarity. Another is that the fractal structure allows the application of a number of techniques based on scaling and decimation, which are generally intractable for disordered media. Two of the dimensions that characterize static properties of a fractal are the fractal (or Hausdorff) dimension df and the spectral dimension ds. The fractal dimension is related to the density of sites. It is defined by counting the

FIG. 4.3. A schematic drawing of the 3-level Sierpinski gasket. Higher levels are obtained in an iterative manner.

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number of sites N(R) inside a sphere of radius R (in the appropriate number of dimensions) , and writing

Notice that df = d'mZd. The fractal dimension is known exactly for a number of deterministic fractals, and is generally determined from simulated data for random fractals. For the Sierpinski gasket in d dimensions [66],

so that df = 1.585 for the two-dimensional Sierpinski gasket shown in Fig. 4.3. The spectral dimension ds, on the other hand, can be expressed as a combination of dw (see equation (4.71)) and df by the relation

and for the Sierpinski gasket in d dimensions it is found to be

Hence ds — 1.365 for the two-dimensional Sierpinski gasket shown in Fig. 4.3 [79]. Blumen, Klafter, and Zumofen [15], [16] have carried out a number of studies of trapping on Sierpinski gaskets and other fractals, approximating the survival probability by the technique of truncated cumulant expansions as described in §4.3.2. For the Sierpinski gasket it is known that (Rn) and (R^}c have the large-n behavior

where a and b are constants [4], [79]. For random fractals the simulated data were fitted to

where a, 6, a, and (3 are constants [124]. Results of such simulations are shown in Fig. 4.4, the notation being the same as in Fig. 4.1. The same qualitative conclusions can be drawn as for the trapping problem on Zd, namely that the accuracy of an approximation based on truncation after the second cumulant generally suffices for many physical applications. Furthermore, the accuracy improves as either the number of dimensions is increased or the trap concentration is decreased.

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

173

FIG. 4.4. Survival probabilities for trapping on a fractal, together with successive cumulant approximations. <3>n is the survival probability as calculated by simulation and &j,n ^ the jth cumulant approximation to .

4.4

.

Some results for (n)

Having elaborated on the quantity S(n) in §4.3, we proceed with a discussion of the mean trapping time (n) in §§4.4.1-4.4.2. 4.4.1. Low trap density asymptotics. An obvious characterization of kinetic behavior in trapping processes is the mean trapping time (n). In §4.3 we saw that there are good estimates of S(n) for both large and small n. But in order to compute

we must also have some estimate of the behavior of S(n) in the intermediate-n regime. When n is fixed, the cumulant expansion in (4.53)-(4.55) converges rapidly provided that A is sufficiently small, which corresponds to a low trap density. This is because 1 < Rn < n -f 1, which implies that the jth cumulant, KJ(H], is at most of order n j . Consequently, the series in (4.54) can be expected to converge rapidly, at least when n -C A^ 1 .

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

Fortunately, however, the convergence is much better. Indeed, (4.49) and (4.57) show that, for random walks satisfying (4.47), as n —>

with c^ and bj being positive constants depending on d and on the random walk. Thus, a comparison of the first two cumulants shows that for all n < A"1

In fact, for small A the regime in which this inequality holds is

In other words, as long as n satisfies (4.81) we may truncate the expansion in (4.54) after the second term, thereby obtaining the approximation

in accordance with the observations made in §§4.3.2 and 4.3.3. The important observation now is that for small enough A the term \KI(TI) is very large when n reaches the end of the regime indicated by (4.81). This implies that the survival probability is already very small. Hence there is good reason to expect that (n), as defined in (4.78), should be well approximated by restricting n to the range specified in (4.81) and by substituting (4.79) and (4.82) into the sum in (4.78). In this way one should find the first few terms in an expansion of (n) for small A, as will be explained in more detail below. A more complete analysis requires that we also have some information about the cumulants Kj(n) with j > 3 in order to assess the accuracy of the approximation in (4.82), i.e., to ensure that the higher cumulants do not spoil the expansion of (n) up to the terms computed. Very little is known about the behavior of the higher cumulants, except for a few rough order estimates. However, these estimates turn out to be enough in d > 3. as we shall next demonstrate.

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

175

Jain and Pruitt [53] proved that equivalently,

This estimate allows us to produce the order estimate

To see why this is true, we first note that (4.79) gives

Together with (4.83) this yields the case j = 4 in (4.84). Newt, since 1 < Rn < n + 1, we have \(R^)C < nj~*(R%)c for j > 4, allowing us to assert that ftj(n) = O [n J ~ 4 K4(n)]. This estimate, combined with KI(TI) ~ a^n (from (4.79)) and with (4.83) and (4.85), gives (4.84) for j > 4. Finally, the Cauchy-Schwarz inequality implies that

which proves (4.84) for j = 3. On substituting (4.84) into (4.54) and (4.55) we finally can write, as a rigorous approximation,

from which we see that (4.82) is indeed the correct approximation, as long as we keep only the leading-order term in ^(n). Thus, we have indeed convinced ourselves that a summation of the second cumulant approximation for S(n) over the regime indicated in (4.81) gives the first few terms in the expansion of (n) for small A in d > 3. Presumably this will also be true in d = 2, but unfortunately there are no estimates available in this case to verify (4.84). To turn the above argument into a precise mathematical statement requires some work. Furthermore, in order to find a correct expansion we must also include terms in the asymptotic expansion of Ki(n) beyond the leading order given in (4.79). But these can be readily found through the formalism in Appendix A [63]. To summarize what is known, we present a list of results obtained by den Hollander [47]. These should be understood to be valid in the

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

limit c —> 0 :

In this equation we have used a number of constants which are defined in terms of a = o-i(72 • • • (Jd (cf. (4.47)) and p(0; 1) (cf. (4.50)) as follows:

w = a more complicated function of the random walk

The computation for d = 3 is written out in Appendix B. The expansions in (4.88) are exact, but only valid for sufficiently small c since nothing is known about the contributions from higher-order terms. A comparison of successive terms shows that for d > 3 the expansions are accurate up to c < 0.05, and for d — 2 up to c < 0.001. The three-dimensional result is adequate for most applications in solid-state physics. 4.4.2. A rigorous inequality. To end this section on a somewhat happier note, we mention an upper and a lower bound on (n} valid for all trap concentrations and for arbitrary random walks in any dimension. The inequality is [48]

The proof is based on a combinatorial inequality known as the FKG-inequality. For random walks that satisfy (4.47) the lower bound produces the leading term in (4.88) for d > 2. It also produces the next term in d > 3. but with coefficients slightly different from the correct ones. The upper bound is finite if and only if d > 3 and. of course, coincides with the leading term in (4.88).

ASPPECTS OF TRAPPING IN TRANSPORT PROCESSES 4.5.

177

A rigorous look at survival at long times

The elucidation of the large-n behavior of S(n) poses a delicate mathematical problem, as it is governed by the rare event that the walk avoid the traps for a long period of time. In §4.2 we had a brief look at the relatively easy case of a symmetric nearest-neighbor random walk in d — I. More generally now, we must examine two possible scenarios that determine the survival. I. The origin (which is the initial position of the random walk) lies in a large trap- free region and the walk tends to stay within this region ( "localization" ) . II. The origin lies in an average trap- free region and the walker sneaks around the traps ( "delocalization" ) . In scenario I there is an untypical fluctuation in the location of traps, but the random walker moves about freely in the trap- free region, i.e., its motion at short to moderate times is essentially that of a typical random walker in the absence of obstacles. In scenario II, on the other hand, there is a typical fluctuation in the location of traps, but the random walk is required to safely tunnel through the field of traps, i.e., its motion exhibits untypical behavior as compared to a free random walk. The questions suggested by these considerations relate to which of these scenarios is the more significant one in producing the long-time behavior of the survival probability, and how to tackle the problem of finding this out from a mathematical point of view. We discuss these points in §§4.5.1-4.5.3. 4.5.1. Large deviations. The answer to these questions was given in a seminal paper by Donsker and Varadhan [30]. The formalism they used and developed for their analysis is termed the theory of large deviations, a combination of probabilistic and variational techniques. This theory was tailored to produce a precise description of untypical events for functionals of Markov processes, exemplified by the range Rn of interest in the context of the trapping problem (recall (4.8)). They found that if the step distribution of the random walk satisfies the conditions in (4.47), then scenario I is the dominant effect. We shall now describe some of this work. For simplicity we consider only the isotropic case in which o\ — a\ = • • • — cr\ and we abbreviate The key result in the Donsker- Varadhan paper is the stretched exponential decay (recall (4.45))

he function a^(A) is identified astified as

Here the infimum runs over the class Ud of nonempty bounded open subsets of Rd with negligible boundary, u)(U] denotes the (Lebesgue) volume of [/, and

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

v(U} is the principal Dirichlet eigenvalue of — ^A in [/, i.e.,

The infimum in (4.93) is taken over the class \£(t/) of smooth functions with compact support in U that satisfy the normalization condition The variational problems posed in (4.92) and (4.93) can be solved and the flA identified as

with MA = uj(Bd) and v& — v(Bd), Bj being the ball of unit radius in Rd. The infimum in (4.92) is attained at

Notice that uj^ — 7T2/r(l-|-|). Also notice that both v^ and the corresponding minimizer in (4.93) can be computed in terms of Bessel functions, e.g., v d — 5^| _, 5 where £p is the smallest zero of the Bessel function of the first 2

kind of order p [I]. How should these results be interpreted? The rough idea is that, for any set of the form V = Un^- (U 6 U oo the largest contribution to S(n) comes from the set C7, where the sum of the two exponents is minimal, which is the solution to the variational problem posed in (4.92). Equation (4.95) says that the large trap-free region mentioned in scenario I is the ball of radius r(i(X)n'd+2 . A striking feature of the analysis is that the function a^(A) depends on d, a 2 , and A in a nontrivial way. Equation (4.91) is a result that is only correct to leading order. Correction d terms could be any multiplicative factor decaying like exp[o(nd+2)]. Part of the difficulty in trying to get more precise results comes from the fact that (4.95) identifies the best scenario only roughly as the walk filling a ball of radius rd(A)nrf+2. Any excursion of the walk outside this ball which covers an area with a linear size o(n^+2) in at least one of the d directions does not show up in the scaling limit, nor does any such area inside the ball that is missed by the walk. 4.5.2. Localization. A finer analysis of the walk associated with the best scenario for long-time survival was carried out by Schmock [93] in d = 1, and by

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Bolthausen [17] and Sznitman [103] in d = 2. Let Pn denote the law of the nstep walk on the free lattice and Pn the law of the n-step walk in the presence of traps conditioned on survival until step n. Then Pn can be expressed in terms of Pn as

Bolthausen proved that in d = 2 and with Pn-probability tending to 1 in the limit n —> oo

for every e > 0 and some x £ (1 — e)B(0, r^(A)n4). That is, the walk precisely covers a ball of radius equal to r^(A)n4 [1 -f o(l)] centered at a point x that is itself located somewhere in the ball of the same radius centered at the origin (which is the initial position of the random walker). The latter simply says that the spherical trap-free region may be shifted away from the origin, as long as it still contains the origin. (This property is related to the fact that the variational formula in (4.92) is invariant under translation.) Equation (4.97) is a strong result, as it excludes any excursions outside the ball no matter how small. It is therefore referred to as the confinement property. Sznitman strengthened the result in (4.97) by giving the probability law for the center x. Namely, he proved

In this result (Zt)t>0 is a mixture of Brownian motions starting from the origin and conditioned on never leaving 5(x,r^(A)), where the center x is random according to the density function ip(x)/ $ 'ip(x}dx with 0(x) being the minimizer of (4.93). The same result as (4.98) had been found earlier in d — 1 by Schmock [93], with the exponents 1/4 and 1/2 both replaced by 1/3. It is believed that (4.97) and (4.98) are also true in d > 3, but there is as yet no proof of this, and thus the confinement property in d > 3 remains an open question. 4.5.3. Drift. All of §§4.5.1-4.5.2 relates to zero-mean finite-variance random walks. Quite a different situation occurs when the random walk has a drift. In that case the existence of large trap-free regions in scenario I ("localization" ) does not play as significant a role as when no drift is present because the random walk has a tendency to drift out of them often. Therefore, one must expect scenario II ("delocalization") to be the dominant one. However, it turns out that the behavior is more subtle, namely, a crossover between the two scenarios occurs at a certain critical value of the drift. A one-dimensional

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system was first analyzed by Movaghar, Pohlmann, and Wiirtz [73]. Later a full analysis for an arbitrary number of dimensions was given by Eisele and Lang [31]. We shall now describe some of the results obtained in the latter investigation. The setting is continuous space and time, and the traps are modelled as spheres of radius p, their centers being located at the points of a Poisson process on Rd with density A. Overlapping of traps is allowed. Diffusive transport is modelled in terms of Brownian motion subject to a drift h = (/i, 0 , . . . ,0). In terms of the standard Brownian motion X(t) in Rd the resulting process, X/i(£), is

A particle hitting the surface of a sphere is assumed to be trapped instantly. Let S(t) be the probability of survival until time t. The main result of Eisele and Lang is that S(t) decays exponentially at long times, i.e.,

but that the rate constant b d ( X , h , p } displays two regimes, depending on whether h is above or below a critical value, namely

in which

and hc = hc(d,X,h,p) is a critical drift. The values of bj and hc are the solutions to some variational formulas in the spirit of (4.92) and (4.93) but having a more complex form. In d = I the exact values are found to be

both independent of p. Eisele and Lang were able to find bounds on these quantities in d > 2 and to prove that

The two regimes in (4.102) reflect a transition between localized and delocalized behavior of the Brownian motion, i.e., a crossover from scenario I to scenario II. Indeed, for h < hc the drift is apparently small enough so that the Brownian particle prefers to be localized rather than throwing itself

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at the mercy of the traps. It pays exp(—^h 2 t] to stay localized until time t, and conditional on this localization the survival decays as a Donsker-Varadhan exponential exp(/c^t^+2)j; hence the rate constant is \h2. On the other hand, for h > hc the drift is large enough to make delocalized motion the preferred behavior, and in this case the rate constant is the result of a more complex interplay between the Brownian motion and the traps, in the competition between diffusion and absorption. For large values of h the Brownian particle will move roughly along a straight line with speed h. Its absorption rate will then be h\pd~luj(i~\, as in (4.104), because pd~lUd-i is the cross-section of a trap. Sznitman [104] refined (4.100) for h < hc by proving that

with ad(A) the same function as appears in (4.94) (with a2 — 1). This is equivalent to the statement that conditional on localized behavior the survival is indeed similar to that in the zero-drift case. In d — I the rate constant in (4.105) depends on h. In d > 2, on the other hand, it is independent of /i, but the statement is restricted to h < h* where h* is some value strictly below hc. An open problem is whether or not the same result holds up to hc. It is possible that h* represents a second critical value, defining an intermediate regime for h G (h*Jic] with yet different behavior. There also seems to be no result yet in the spirit of (4.97) and (4.98). Finally, the following remark: As mentioned earlier, all of the simulations indicate that the Donsker-Varadhan tail sets in only at extremely small values of S'(n), which suggests that from the applied point of view the results in (4.91) and (4.100) are not very useful. On the other hand, the theory described in this section reveals an underlying structure of competing effects which, aside from its own inherent interest, has become a focal point in the research on a number of trapping problems. 4.6.

Extensions and generalizations

In §§4.6.1 4.6.6 we give a brief overview of some extensions of the plain vanilla trap problem discussed to this point. Although much remains to be done in the analysis of even the simplest versions of this problem, applications of the theory suggest a number of extensions that have only recently begun to be explored. The degree of the mathematical difficulties that arise in analyzing these extensions is as great, if not greater, than that encountered for the models introduced so far. 4.6.1. Trap distributions other than Poisson. If the distribution of traps is not Poisson (i.e., there are correlations between the positions of the traps), then we can no longer express the survival probability in the succinct

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form given in (4.8), because the survival is no longer solely dependent on the range of the random walk, but depends also on the shape of the set of sites visited. This generally presents only minor problems in d — 1, because of the grossly simplified topological structure (cf., e.g., the result cited in (4.44)), but in d > 2 it raises serious complications. Kayser and Hubbard [56] argued heuristically that if the traps are modelled as spheres in Rd whose centers are Poisson distributed subject to the constraint that no overlaps are allowed (which is equivalent to hard-core interactions in physical terminology), then the Donsker-Varadhan tail in (4.91) should still be valid. However, they argued that the Poisson density parameter A should be replaced by the so-called pressure, £>(A), associated with the trap distribution. The most general rigorous results bearing on this problem have been obtained by Sznitman [105]. He proved that the Kayser-Hubbard suggestion is indeed correct, and in fact even holds quite generally, namely, when the trap distribution is a Gibbs point process with a pair interaction potential V : Rd —> (-co, oo] that is translationally invariant, symmetric, bounded below, stable, and of finite range. The pressure is defined by the statistical mechanical relation [25]

with A the so-called activity parameter which controls the trap density. The Poisson case corresponds to V = 0 and p(X) = A. A particularly interesting aspect of Sznitman's result is that it holds independently of whether the Gibbs measure is unique or not, and whether or not it breaks the translational invariance. Apparently the survival depends on large fluctuations in the trap field on a volume scale rather than small fluctuations on a surface scale (which are the ones responsible for a phase transition). Other examples of non-Poisson trap distributions are considered in [23] and [75]. Here the traps are organized as segments of static polymer chains, with a random shape, size, and location. For instance, the chains can be self-avoiding walks of a random length N with starting points having a Poisson distribution with density cp. In that case cp is the polymer density and (N}cp is the trap density. The results obtained so far are heuristic and still preliminary. One particularly interesting aspect of these investigations is that if the distribution of N is bounded or has a sufficiently rapidly decaying tail (e.g., a Gaussian), then S(n) has the same Donsker-Varadhan tail as in (4.91), but with A replaced by Ap, which is the activity corresponding to CP (i.e., e~Xp = 1 — CP). In other words, the polymer density controls the decay, not the trap density. Some rigorous support for this fact comes from Sznitman [105], who shows that the Donsker-Varadhan tail is the same, including the constant in the exponent, when all trapping sites in Zd are replaced by some finite set U C Z d , which is

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fixed but otherwise arbitrary (or a nonpolar finite set U C Rd in the case of Brownian motion). 4.6.2. Moving traps. So far we have discussed properties of systems in which the traps are randomly distributed but immobile, while the particles are allowed to perform some type of random motion. Our motivation for doing so was to study generalizations of the original Smoluchowski model for reaction kinetics, as described in §4.1.2. It is clearly unrealistic to apply the Smoluchowski model for reactions in solution, where it is to be expected that not just one but all reactants are able to diffuse. However, the development of a theory for even grossly simplified versions of the moving species problem poses some extremely difficult mathematical questions. These have so far largely defied rigorous analysis. In §§4.6.2.1 to 4.6.2.3 we shall discuss some aspects of these problems. Before talking about what is known, we first describe some of the motivating applications behind a study of this area. 4.6.2.1. Segregation of reactants in confined geometries. Recall that the original Smoluchowski model refers to the reaction A -f B —> B in (4.1), and its properties were studied only for three-dimensional systems [96], [81]. There are, however, a number of effects that appear in various types of reacting systems in dimensions less than three or in restricted geometries. These have provoked a considerable amount of research effort in recent years, stimulated by the seminal analysis first given by Ovchinnikov and Zeldovich [77]. One of the most notable implications of setting restrictions on the geometry is that the laws describing reaction kinetics are no longer the classical ones, exemplified by (4.2). This deviation occurs because the basic assumption of perfectly mixed reactants is violated. Some representative papers making this point are: Ovchinnikov and Zeldovich [77]; Toussaint and Wilczek [111]; Kang and Redner [54]; Kopelrnan [59], [60]; Kuzovkov and Kotomin [61]; ben-Avraham, Burschka, and Doering [6]; and Bramson and Lebowitz [19] [21]. Perhaps one of the most startling effects, peculiar to the annihilation reaction A+B —> 0 and probably to some other reactions as well, is that an initially uniform mixture of A's and B's in d = 1 will eventually separate into disjoint islands containing only pure constituent reactants. This effect is somewhat weaker in d — 2 and 3, and disappears altogether in d > 4. The case d — 4 is critical and in d > 5 the classical reaction rate equation in (4.2) can be used [21]. Of course, an initially separated mixture of A's and £?'s in one dimension will remain separated. A simple model can be devised to illustrate how such segregation comes about. Consider the reaction A + B —> B. Suppose that there is a single immobile trap of species B and an initial Poisson distribution of particles of species A allowed to diffuse independently of one another. One can then ask about how the average distance (L(t)} between the B particle and the closest untrapped A changes as a function of time. In d — 1 both the As and the B can be idealized in terms of point particles. A straightforward calculation

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[115] shows that

in the limit t —> oo where C here and later denotes a constant. That is, the presence of the B tends to repel the ^4's through progressive depletion. This cumulative separation is a prototype of the tendency towards segregation in more complicated reactions. The repulsion also exists in d = 2, but can be shown to be much weaker [45], namely (L(t)) ~ (ln£)2. In this calculation the B is represented as a circle, while the A's are points. In d > 3 the average distance of mobile particles from an immobile trapping sphere tends asymptotically towards a constant. One therefore expects, following Smoluchowski, that in d > 3 one can safely use classical reaction rate equations after some transient period. But in lower dimensions (which also covers some fractal media that are characterized by noninteger dimensionality) one expects quite different rate equations to apply. Note that here the critical dimension is d — 2, as opposed to d = 4, which is the critical dimension for the reaction A + B -*0. The results cited in the last paragraph were derived on the assumption that the B is immobile and the A's are mobile. One can perform an exact calculation in d = 1 for the opposite case in which the A's are immobile and the B is mobile [7]. In this case it is readily shown that

In d — 2 one finds that (L(t)) ~ (7hit but, in contrast to the result in (4.108), this result is based only on simulation. It is obvious that the outstanding question is how these results are changed when both ^4's and 5's are allowed to diffuse. A solution to the problem in closed form has not been found, and indeed sets almost insuperable mathematical problems. The only result available at the time of writing this article is a simulation study by Schoonover et al. [94] of a one-dimensional system containing a single B and a set of v4's, both mobile. The result of that investigation suggests that (4.108) can be generalized to

where 1/4 < a < 1/2. Good agreement with the simulated data is obtained by choosing a to be the constant

in which DT and DW are the diffusion constants for the trap (— B) and for the walkers (= vl's), respectively. Equation (4.109) is not, however, firmly enough established to exclude a result like

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4.6.2.2. Perturbation techniques for low trap density. A number of papers have appeared that treat the possibility of diffusion by both species in a dilute system for the A + B —> B reaction. The most ambitious of these, by Szabo, Zwanzig, and Agmon [100], considers the problem of calculating the survival probability of a single diffusing particle in a field of N diffusing traps found initially in a segment of length L on R. (One expects that, because of the severely restricted motion in low dimensions, the formulation which leads to the greatest deviation from classical results should occur in d = 1.) In such a system one can write a backward equation for the survival probability S^L^I^o? ^i 5^25 • • • 5^;v) conditioned on the particle being initially at XQ and the traps being initially at xt, i — 1, 2, . . . , N. This equation is just the adjoint to the forward diffusion equation, namely,

The difficulty is that this equation must be solved subject to the boundary conditions SN,L,(t\xo, £1, ^2 5 • • • 5 XN] — 0 whenever XQ = xt, i = 1 , 2 , . . . , J V . Thus, the diffusion equation is separable, but the boundary conditions are not. It is possible to transform to relative coordinates, yi — X{ — XQ. in which case (4.111) is transformed into an equation with mixed spatial derivatives and the boundary conditions become slightly simpler. In neither formulation is a solution in closed form known, except for N = 1 and 2. Hence, as in so many other physical problems, one wants, at a minimum, to develop a perturbation scheme which allows the calculation of a series of successive approximations. which presumably furnish more accurate results as more terms are retained in the expansion. We describe such a perturbation scheme below. The original statement of the physical problem requires that we consider survival in the presence of an infinite number of traps that are initially Poisson distributed with a constant density A. So, as a last step in a calculation based on a finite number of traps, one must take the thermoclynamic limit, i.e., both L and N tend to infinity subject to the constraint

Let SV,L(£) denote the survival probability averaged over all initial positions of the traps, where these positions are distributed uniformly over the specified interval of length L. Szabo, Zwanzig, and Agmon then proceed by expanding SW,L(t) in the series

The finite series on the right-hand side of this equation is written by analogy with expansions used in the theory of imperfect gases [25]. The coefficient

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an(t) is found by requiring the expansion to be exact for 1,2, . . . , n traps, respectively. Exact expressions for the an(t) can be found in terms of integrals with respect to the relative coordinates yi(= XQ — Xi). The first three of these are found to be (ao(t) = 1)

where the range of the t/'s can be taken as (—00,00) and where •SW(£12/1,2/2, ••• ? 2A/v) is just the survival probability at time t of a particle in the presence of N traps expressed in terms of coordinates relative to the diffusing particle (i.e., the solution of (4.11) with L — oo in relative coordinates). Higher-order terms than those shown in (4.114) are readily generated. However, it is by no means trivial to calculate 5W(£|2/i,2/2 5 • • • il/N) beyond N = 2, and even the solution of (4.111) in the case N = 2 has a quite complicated form. In the thermodynamic limit, (4.113) provides what is essentially a perturbation expansion of S\ (t) in powers of the trap concentration, where we write S\(t) to denote the right-hand side of (4.113), i.e., the survival probability in the limit given by (4.112). A device that often improves the convergence properties of a perturbation series is that of exponentiating the original series, i.e., writing the perturbation series in the exponential form

This is equivalent to an expansion in terms of cumulants, in which the bi(t) can be expressed in terms of the dj(t) with j < i. The first three of these functions are:

The lowest-order approximation that follows from (4.115) is

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which is equivalent to the original Smoluchowski approximation since it is calculated in terms of a single trap and a single particle. To convert from absolute to relative coordinates for the evaluation of (4.117) we set the origin at one of the particles, say at the trapping site. The survival probability S\(t\y] can then be found as the solution of the diffusion equation in (4.4) with diffusion constant DT + DW- The equation is to be solved subject to the boundary condition 5i(t|0) = 0. The result of such a calculation is

On substituting this into (4.117) one finds that the lowest-order approximation to the survival probability is S\(t] ~ exp — I\\/(DT + Dw}t/ir\ (compare with (4.42)). The steps beyond this lowest-order approximation, although apparently quite straightforward, are actually impossible to carry out in a convenient mathematical form. In fact, only the second-order approximation, involving two traps, can be found as a somewhat complicated series. Some notion of the complicated nature of the computation of S2(t\yi, 2/2) can be gleaned from (4.111). If one defines fj, — D\\r /(Dw + DT}, then the equation for 52(£|2/1,2/2) is readily shown to be

The sign of the mixing term is positive when the two particles are initially on the same side of the trap and negative when they are on opposite sides. Equation (4.119) is to be solved subject to the boundary conditions, written in terms of relative coordinates as ^(tjO,^) = S2(t\yi,Q} ~ 0. By transforming to new spatial variables one can reduce (4.119) to a two-dimensional diffusion equation, the boundary being replaced by an absorbing wedge of angle 20 = cos"1 fj,. Carslaw and Jaeger [24] have shown that the solution to such an equation can be written in terms of a rather complicated infinite series. The series in (4.115) truncated after n = 2, together with the definition of bi(t] and 62(^) in (4.116), then provides the second-order approximation to the average survival time. Szabo, Zwanzig, and Agmon [100] solved the resulting equations numerically, finding close agreement between the case of equal diffusion constants (/i = 1/2) and the case of static traps, at all but very long times. 4.6.2.3. Large-t asymptotics of survival for moving traps. The work of Berezhkovskii, Makhnovskii, and Suris [11], [12] has mainly focussed on deriving a Donsker-Varadhan-type tail for the survival probability in the moving trap case, using the heuristic approach suggested by Balagurov and Vaks [8] and Grassberger and Procaccia [40], which is described in §4.3.4. The

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results of the analysis suggest a much richer variety of asymptotic behavior than is found in the immobile trap case because the large trap-free regions that played such a significant role there are now transient. We briefly mention some ideas in the derivation and describe the predicted results. The principal idea is that there are now three types of terms which must be considered in calculating the bounds on the survival probability. The first and second correspond to terms that also appear in the heuristic derivation in the case of immobile traps (see §4.3.4). These are the probability that a d- dimensional ball of radius R has no traps inside at t = 0 and the probability that the particle remains within the ball for a time t. The third, which is specific to the case of moving traps, is the probability that the original ball remains trap- free for a time t, i.e., no traps diffuse into it from the outside. This probability decays exponentially with time. The results of the analysis predict that at very long times the survival probability will not decay to zero with the characteristic Donsker-Varadhan exponent, but rather with an exponent proportional to t. The authors further suggest, based on their somewhat heuristic analysis, that there may be a range of times, before the final exponential decay sets in, during which the decay of the survival probability appears to follow a Donsker-Varadhan law. Because of the heuristic nature of the argument it would be extremely useful if there were some verification of these results, at least in terms of simulations. 4.6.3. Kinetics in the presence of decaying traps. Den Hollander and Shuler [49] consider a variation of the trapping problem in which initially randomly distributed traps on a lattice are allowed to decay independently according to a given law characterizing the trap lifetimes. Let c(k) be the probability that a given site is a trap at time k. That is, c(0) is the probability that a site is initially a trap, and c(fc)/c(0) is the probability that the trap survives until time k, where c(k) is a decreasing function of k. Let A& be the indicator random variable for the event that the random walker reaches a new site at step &(Ao = 1). Then A& is related to the number of distinct sites visited in k steps, R^, by A& = R^ — Rk-i- In this notation it turns out that one can replace the representation of the survival probability in (4.8) by

Nondecaying traps correspond to the choice c(k) = c. A first question that can be asked about this model is whether a random walker will eventually be trapped or not. Put in mathematical terms, one wants the condition under which limn-^ S(ri) = 0. It is clear that if the traps disappear instantly, then there will be a positive probability that the walker is never trapped. It is less clear whether the same is also true when the decay of the traps is not instantaneous. Den Hollander and Shuler prove that

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a necessary and sufficient condition for trapping to be certain is that

This is a convenient characterization since the asymptotic form of {A^} can be computed in terms of generating functions, as is demonstrated in Appendix A. The proof of (4.121) does not suffice to find the large- n behavior of S(n}. However, bounds for this probability have been derived for the case c(k) ~ Ak~^ for large k. where A and 7 are positive constants, and 7 must be small enough to satisfy the condition in (4.121). For example, for the random walk with mean zero and variance 0 < cr2 < oo in d = 1, and for any A and 0 < 7 < 1/2, it can be shown that there exist constants K\ and K^, independent of a, A. and 7, such that for sufficiently large n

It is tempting to conclude that one has Donsker Varadhan-type kinetics in which the exponent 1/3 is simply changed to (1 — 27) /3. One can further argue on heuristic grounds that in d > 2 the exponent d/(d + 2) appearing in (4.91) in this model becomes

This suggested form of the exponent shows a crossover at 7 = 2/d. However, there is as yet no proof in support of (4.123), either mathematical or based on simulation studies. In Bolthausen and den Hollander [18] a continuous version of the decaying trap model is studied (i.e.. Brownian motion among decaying trapping spheres). The Donsker-Varadhan tail with exponent -^—^(0 < 7 < |) is established rigorously and the constant in the exponent is identified in terms of a variational formula similar to (4.92). 4.6.4. Further trapping models suggested by applications to chemical reactions. There are a large number of further trapping models that have been analyzed in the literature of both chemistry and physics, a few of which we mention briefly. As mentioned in §4.1, the original trapping problem arose as a very idealized version of a variety of processes. One weakness of most mathematical formulations of the trapping model is that they are phrased in terms of points and lattices, whereas physical entities are characterized by sizes and shapes. It is clearly impossible to take complicated shapes into account, either of traps or diffusing particles, in a mathematical theory that attempts to describe the kinetic behavior of real physical systems. However, there is a

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considerable literature available in which the authors try to analyze diffusion in the presence of a collection of identical spherical traps (representative papers in this area include Felderhof and Deutch [32]; Bixon and Zwanzig [14]; Felderhof, Deutch, and Titulaer [33]; Cukier [26], [27]). The problem is formulated in terms of a three-dimensional diffusion equation, in which the concentration or probability density vanishes at each one of the surfaces of the spheres that make up the system. A complete and rigorous solution to such an equation appears beyond reach, even when the trapping spheres are situated in a regular configuration. On the other hand, one can make a certain amount of progress in tackling such problems through the use of a variety of approximate techniques. As an example of how one goes about translating the physical problem into mathematical terms we consider a model analyzed by Muthukumar and Cukier [74]. These authors were concerned with determining the reaction rate for a three-dimensional system in which diffusing particles are continuously injected into a system containing N traps, so that the rate of injection is balanced by the rate of trapping. This is a simplification of the more general problem, in the sense that time no longer plays a role because equilibrium is reached between injection and trapping. One is interested in the limit N —* oo defined by the requirement that the trapping centers be Poisson-distributed in space with some finite density. In the resulting formulation of the model the questions that have been addressed refer to how the diffusing particles are distributed in the presence of the traps, and to how one can determine the steady-state reaction rate for such a system. Let CQ be the uniform concentration of diffusing particles in the absence of any trap and let c(r) be the equilibrium solution to the diffusion equation DV2c — 0 when there is a single spherical trap of radius R located at the origin. The diffusion equation is to be solved subject to c(R) = 0. In such a system the solution, which must be spherically symmetric, is trivially found to be

and the total flux into the trap, which can be identified with the reaction rate, is just that found from (4.5) and (4.6), namely k = 47tDRco. The problems that interest the physicist relate to how the presence of more than a single trapping center affects the trapping rate, and the choice of algorithm defining an effective diffusion constant. More generally, suppose that there is a field superimposed on the system, which we denote by 0(r). This may account for external boundary effects or other sources of reactant particles. In the presence of such a field the equation to be solved for the concentration is

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whose solution can be written in terms of the Green's function G(r, r') = l/(47rD|r-r'|) as

Let RJ be the position of any point on the surface of the jth trap. This can be written relative to the center of mass of the trap, R^, as Rj — R^ + Fj(Qj), in which rj(fjj) is a vector from the center of mass to the surface of the trap and 17 j is the angular orientation of this vector. In addition to these variables it is also necessary, in certain applications, to take into account the spatial dependence of the reactive part of the trap surface. This will be inserted in the form of a function which we denote by <jj(fij). The modification to (4.125) that includes all of these features is

which is exact for a given configuration of the N traps. It is generally quite difficult to solve (4.127) for a specific set of traps, but one is usually interested in the average concentration, (c(r)}, taken with respect to the distribution of the N trapping centers. On the assumption that the reaction can be represented as being first order, we can replace the average of (4.127) by an effective-medium equation for (c(r)}, namely

where ]^(r) is a function that contains all the many-body effects that depend both on the location, size, and shape of the sinks and on the reaction strengths on the trapping surfaces. Equation (4.128) is, of course, easily solvable in terms of the Fourier transforms of (r) and XX r )- Denoting these by 0(cj) and 51 (u;), and noting that because of the spherical symmetry these can only be functions of uj = |u; , we can rewrite (4.128) in Fourier space as

allowing us to solve for (c(u;)} and then to represent (c(r)} in terms of an inverse Fourier integral. The concealed difficulty in this approach lies in finding the function XX r ) m terms of the parameters describing the underlying physical system. Successive approximations to this function are given in the paper by Muthukumar and Cukier [74]. A related, but slightly more general, approach to the solution in the time domain is found in the paper by Bixon and Zwanzig [14], with later modifications by Kirkpatrick [57] based on diagrammatic techniques widely used in statistical physics. 4.6.5. Reversible trapping. We very briefly mention a number of further extensions of the simplest models of trapping. Until now the discussion was based on the assumption that trapping is irreversible. A variety of applications

192

CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

in the theory of diffusion-controlled reactions, metallurgy, and solid-state physics suggest the importance of determining effects of reversible reactions on the kinetic behavior of the system. Richards [82]-[85], Richards and Myers [86], and Lyo [65] have analyzed some aspects of this type of model with stationary traps. A number of arguments can be adduced to show that at short times and in d = 3 the survival probability of the mobile species is

where a and b are constants which can be expressed in terms of the parameters of the underlying physical model that defines the geometric shape of the traps as well as the reaction rate. A number of studies have appeared which compare several approximate theoretical predictions with the results of simulations (cf., for example, [120] and [121]). A general conclusion following from these simulations, which are necessarily restricted to short times, is that most of the approximate methods are useful only when there is a small volume fraction of traps. However, it should be emphasized that this is the most interesting regime for physical applications. 4.6.6. The variational approach. An approach that a number of investigators have followed is that of deriving bounds on the reaction rate using a variational analysis (cf., for example, [80], [29], [109], [92], and [38]). They study a steady-state problem in which the disappearance of particles due to reaction is balanced by the injection of particles into the system. They divide the infinite space into two parts, the void space T in which diffusion by a mobile particle is possible, and the nonvoid space Rd\T of trapping particles. The two spaces are separated by a surface. In such a system c(r) is the solution to where c(r) = 0 when r G M d \F, and (c) is the analog of a bulk concentration. That is, (c) is found by forming the ensemble average (c(r)/p(r)} where /r(r) — 1 for r e F and /r( r ) = 0 otherwise. The average is to be interpreted as first integrating over a finite volume V and then passing to the limit V —> oo, i.e.,

One performs the average subject to the constraint that the trap density, trap size distribution, and correlation function between trapping centers all remain fixed. The quantity to be calculated in this formulation is the reaction rate, k. This is equivalent to setting a self-consistency requirement on the solution to (4.131), as we show below. Because the resulting boundary value problem is quite complicated, finding a solution in closed form is clearly out of the question. However, a solution can be found in terms of a variational formulation.

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

193

To somewhat simplify the analysis one works in terms of the normalized function u>(r) = Dc(r)//c(c), which replaces (4.131) by the slightly simpler

where w = 0 on Rd\T (the boundary condition). The trapping rate is then given in terms of the indicator function for F as (recall (4.132))

which, with the help of (4.133), can be rewritten in terms of the gradient of w as

[38]. A variational principle now follows from the fact that the quadratic functional in (4.135) is minimized by precisely the function w(r] that satisfies (4.133) together with the boundary condition. Hence one has

for any u(r) that satisfies (4.133) but not necessarily the boundary condition. Rubinstein and Torquato [92] have, in a similar way, derived an upper bound for the reaction rate in order to complement the result in (4.136). Inequalities for the reaction rates in a number of problems of physical interest have been derived in this way by specializing the class of trial functions. As an example, Miller and Torquato [68] use this approach to discuss the effect of polydispersivity of trap sizes on trapping rates. A review of some of the work utilizing the variational approach is contained in a review by Torquato [110]. 4.7.

Afterword

We have seen, in the course of this article, that the source of the mathematical trapping problem is to be found in a number of chemical and physical theories that bear on kinetic phenomena. The general research area is still very much an open one, with a rich selection of problems awaiting further analysis and possible resolution. Acknowledgments

We are indebted to Drs. Gert Zumofen and Alex Blumen for good copies of figures that have appeared in their papers on the cumulant approximation (Figs. 4.1 and 4.4). We are grateful to Dr. Attila Szabo for a careful reading of this manuscript and for pointing out the intimate relation between the Smoluchowski model and a number of kinetic properties which have been calculated by more laborious means.

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

4.8. Appendix A 4.8.1. Derivation of the large-n asymptotics of (Rn) in d dimensions. Our object in this appendix is to find a generating function for (Rn] and then, by applying a Tauberian theorem, to infer its large-n behavior (recall (4.48), (4.49), and (4.58)). We consider a random walk on Zrf with p(j) the probability that at any step the displacement is j. Associated with p(j) is the characteristic function or Fourier series p(0) defined by p(0) — £)jp(J) exp[^(j • 0)} as in (4.51). Let Pn(j) be the probability that the random walker is at site j at step n. This set of probabilities satisfies the recursion relation

where the sum extends over all sites of the lattice. Let the initial position of the random walker be at 0 and define the Fourier series pn(0) in terms of the

(A2) On multiplying (Al) by exp[z(j • 0)] and summing on j we find the trivial recursion relation

The inverse of the series in (A2) then gives us an explicit relation for p n (j), which may be written as the d-fold integral

Since p(0) appears in the integrand as a power it is useful to also introduce a generating function with respect to the step number n, namely

When we multiply (A4) by zn and sum on n, we get

Let the generating function of the (Rn) be denoted by R(z). To find this function we need the probability that the random walker reaches j for the first time at step n. This probability will be denoted by /n(j) and can be related to the pn(j) through the convolution equation

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES

195

The sum accounts for the possibility that the random walker reaches j for the first time at step 0 < k < n and returns to j again after an additional steps (/o(j) = 0). Let /(j; z] denote the generating function of the /n(j) with respect to n. Equation (A7) allows us to express /(j; z) in terms of the known function p(j; z] as

In order to relate the generating function R(z) to the p(j;z) we observe that the probability, A&, that the random walker visits a new site at step k > I is

and that the expected number of distinct sites visited after n steps is

Next note that since XljPn(j) = 1 it follows that

The final step in the calculation is to combine (A8)-(A11) to conclude that

While this last expression for R(z] is exact, it is of little use in deriving exact expressions for the (Rn}- However, it can be utilized to find a large-n approximation by appealing to a Tauberian theorem for power series [35]. The exact form needed for our purposes is the following: Let L(x] be a slowly varying function, i.e., linix^oo L(cx]/L(x] — I for every c > 0. Let U(y] be a series

with an > 0 having the singular behavior

Then the Tauberian theorem allows us to conclude that

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CONTEMPORARY PROBLEMS IN STATISTICAL PHYSICS

Moreover, if it is also known that the a& are monotone in A:, then an asymptotic estimate of an can be found by differentiating this formula, i.e.,

Let us see how this formalism works in the case d = 1, based on (A6) and (A12). The function R(z) is clearly singular at z = 1 because of the factor (1 — z)2 that appears in it. However, there is a further contribution to the singular behavior due to the divergence of p(Q;z) in the limit z —> 1. To determine this contribution we first observe that j5(0) = 1 because Y^JP(J} — 1Hence there is a potential divergence of the integral in (A6) because the denominator vanishes when z = 1 and 0 = 0. We shall assume, in the simplest form of the calculation, that ^ j p ( j ) = 0 and 0 < ^2jj2p(j) = cr2 < oo. One can then expand p(0) in the neighborhood of 9 = 0 to lowest order as

If p(6] = 1 has no other root than 9 = 0 (which is the case unless the random walk moves on a sublattice of Z [39]), then the singularity of p(Q; z) at z — 1 is found by approximating the denominator of the integral representation of ) in (A6) inserting the quadratic function in (A17). Thus, we have

which is valid as z —> 1. In going from the first integral to the second we have replaced the limits ±?r by ±00, since the integral is nonsingular outside any neighborhood of 0 = 0. A multiplicative factor of z has also been omitted, since we are only interested in values of z near 1. Substitution of (A18) into (A12) yields

This, together with the Tauberian theorem mentioned earlier and the monotonicity of (Rn) apparent from (A10), implies the asymptotic result in (4.48). The asymptotic results in (4.49) for d > 2 can be derived by essentially the same technique. Such results depend critically on the assumption that p(j) has a zero mean and a finite variance (recall (4.47)). Only partial results along these lines can be found when the random walk has an infinite variance [37]. It appears to be difficult to derive asymptotic results for higher moments of Rn in a similiar manner (cf. [106]).

ASPECTS OF TRAPPING IN TRANSPORT PROCESSES 4.9.

197

Appendix B

4.9.1. Derivation of the small-c asymptotics for (n) in d = 3 (equation (4.88)). In this derivation we assume that p(j) satisfies the conditions set out in (4.47) and, in addition, has a finite fourth moment. One can show, from the d = 3 analog of (A18), that the function p(0; z] for z —>• 1 behaves as [63]

where u = p(0; 1) and v — l/(2^7rcr). This expansion, together with (A12), implies that the average number of distinct sites visited after n steps has the large- n expansion

Moreover, Jain and Pruitt [52] have proved that

An approximate expansion of the survival probability in terms of cumulants, truncated at the second-order term, is equivalent to writing

in which

Let n(x) be the function inverse to x(n). Then the Euler-Maclaurin formula can be invoked to provide an estimate of (n) as

After A = — ln(l — c) is expanded as a series in c, (B5) implies, together with (B2) and (B3), that the lowest-order terms in the expansion of n(x) for large x are

Substitution of this expansion into (B6) yields the lowest-order terms of the expansion in (4.88), after taking into account the integrals J^° xe~xdx — 1 and

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[118] G. H. Weiss and S. Havlin (1985), Non-Markovian reaction sites and trapping, J. Chem. Phys., 83, pp. 5670-5672. [119] G. Wilemski and M. Fixman (1974), General theory of diffusion-controlled reactions, J. Chem. Phys., 58, pp. 4009-4019. [120] L. H. Zheng and Y. C. Chiew (1989), Computer simulation of steady-state diffusion-controlled reaction rates in dispersions of static sinks: Effects of sink sizes, J. Chem. Phys., 93, pp. 2658-2660. [121] (1990), Computer simulation of diffusion-controlled reactions in dispersions of spherical sinks, J. Chem. Phys., 90, pp. 322-327. [122] G. Zumofen and A. Blumen (1981), Survival probabilities in three-dimensional random walks, Chem. Phys. Lett., 83, pp. 372-375. [123] (1982), Random-walk studies of excitation trapping in crystals, Chem. Phys. Lett., 88, pp. 63 67. [124] G. Zumofen, A. Blumen, and J. Klafter (1984), Scaling behaviour for excitation trapping on fractals, J. Phys. A., 17, pp. L479-L485. [125] (1991), Transient A + B —>• 0 reaction on fractals: Stochastic and deterministic aspects, J. Stat. Phys., 65, pp. 1015-1023.

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Chapter 5

Stochastic Resonance: From the Ice Ages to the Monkey's Ear Frank Moss

Abstract The process whereby external or internal noise operates on a nonlinear, multistable system, modulated by a weak periodic function in order to induce or facilitate switching events among the stable states, has been called stochastic resonance (SR). We may associate the switching events with the information flow through the system. For every frequency of the modulation, the information flow is optimum for a specific noise intensity, that is, for a specific Kramers transition rate, hence the term resonance. Two physical quantities which characterize the response of such a system have been the objects of a flurry of recent experimental and theoretical activity: the Fourier transform, or autocorrelation function of the appropriate state variable, and the probability density of the residence or escape times. The former have been used to obtain the power spectra and hence the signal-to-noise ratios of the response, while the latter directly reflect the rates and symmetry properties of the system. Theoretical challenges arise, because the calculation of these quantities pose specific problems for theorists characteristic of nonstationary Fokker- Planck systems. Applications and potential applications span the range from theories on the onset of the ice ages through nonlinear signal processing to noise-assisted information flow in the sensory neurons of living systems. In this paper, we will review the recent activity, include a discussion on the historical foundations of SR, outline several current and potential applications, and speculate on the connection of SR to modern neuroscience. 5.1.

Introduction

Interest in time-modulated, stochastic, nonlinear systems has been increasing in recent years. Such systems are important in studies on dynamical bifurcation phenomena, for example, phase transitions occurring under the simultaneous influence of noise- and time-dependent parameters as well as various transport and phase-locking properties of multistable systems [l]-[llj. Stochastic resonance (SR) is a somewhat specialized example of the result of combined periodic and stochastic forcing in a multistable nonlinear system. It has an interesting history as well as a number of modern applications spanning a range of science and technology, from laser applications to superconducting quantum interference devices (SQUIDs) for the detection and characterization of noisy 205

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magnetic fields, noisy digital information processing, and the noise-induced information flow in sensory neurons in living systems. The physical phenomenon of SR in one dimension is simply explained. Consider, for example, an overdamped particle moving in a double well potential subject to a random force. The system may be in contact with a heat bath, so that the random forces result from thermal fluctuations. This simple construction is the starting point for many molecular dynamical models which form the microscopic basis of modern chemical rate theory, as well as a variety of other physical applications [12]-[16]. Moreover, we suppose that the system is also exposed to a periodic external modulation which is additive and weak. "Additive" means that the physical action of the modulation is to "rock" the potential to and fro. This has the effect of alternately raising and lowering each potential well with respect to the barrier separating the wells. In contrast, "multiplicative," or "parametric" modulation, as it is sometimes called, has the effect of modulating only the height of the barrier, leaving the relative levels of the two wells unchanged. "Weak" means that the amplitude of the modulation is always smaller than the barrier height. The result of these actions is that in the absence of noise, or stochastic forces, the particle will be forever confined to whichever well it happened to start out in. The motion of the particle will be a small amplitude periodic trajectory confined to one well. By contrast, when noise is present, and we will always assume the noise to be Gaussian-distributed, there will always be some nonzero probability that a switching event, or transition from one well to the other, will take place during some interval. A schematic depiction of this dynamic is shown in Fig. 5.1, where it is clear that the particle remains in the right-hand well in the absence of a stochastic force, but can switch states when such a force is present, as indicated by the solid and dotted ball and arrows. We can now look at the transition or switching rate. SR arises because of an interplay between the modulation frequency and the famous Kramers rate [17], TO, which specifies the mean switching rate in the unmodulated system, as depicted by the middle glyph in Fig. 5.1. The Kramers rate is given by

where [/"(O) is the curvature of the potential barrier located (at x = 0) between the two wells, and U"(c) is the curvature of the bottom of the wells (at x = ±c). The double prime indicates second differentiation with respect to .r. The unmodulated barrier height is At/o = \U(x = 0) — U(x = c)}. The noise (or thermal fluctuation) intensity, D, is defined by its correlation function. In the case of "white" noise, or noise of infinite bandwidth, the correlation is a delta function,

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FlG. 5.1. Depiction of a periodically modulated, bistable potential where the forcing is additive. The amplitude of the modulation is assumed to be small compared to the unperturbed barrier height AC/o- In the absence of noise, the state point remains in a single well determined by an initial condition. With Gaussian noise, however weak, noise-induced switching occurs.

whereas for "colored" noise, there exists a nonzero correlation time, r,

The power spectrum of the colored noise is the Fourier transform of this correlation function, as specified by the Wiener Khintchine theorem, and is given by the usual line shape,

where the "cutoff" frequency is given by uc = I/T. Such noise would be obtained in the real world by passing noise of wide bandwidth through a linear filter

where W(t) is the so-called Wiener process, that is, a Gaussian, white noise of unit variance. It is to be emphasized that in all real physical processes, the noise is colored. Nevertheless, in this review, we shall frequently refer to white noise, or make use of theories strictly valid only for white noise. This will have the significance in real physical systems, that the noise correlation time, rn, is much smaller than the characteristic time scale, TJ, of the dynamical system to which the noise is applied.

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We can connect these expressions immediately to physical measurements made in the real world by noting that in the limit t —> s, equation (5.2b) results in

which is just the mean square noise amplitude. In the analog simulators to be discussed later, \/{£2} is just the root mean square (r.m.s.) noise voltage, which can be measured by any "true r.m.s." alternating current (a.c.) voltmeter. Moreover, PlancherePs theorem, sometimes called the "energy" theorem, connects the mean square noise amplitude to the power spectrum,

The mean of this noise will always be zero: {£(£)} = 0. In the absence of modulation, the particle makes transitions over the barrier at random times and resides in one or the other well for a random length of time. The motion within each well is stochastic; however, if we are interested only in which well the particle is to be found, the intrawell stochastic motions can be filtered out, replacing the coordinate x(€) with a signed constant, say ±1, with the sign indicating which well. In this review, we shall often make use of this filtering process, which we shall call the "two-state filter." Assuming that the switching time is short compared to the mean intrawell residence time, after two-state filtering the trajectories are given approximately by the random telegraph signal. An example of such a trajectory is shown in Fig. 5.2. The probability density of the intrawell residence times, TJ, for the random telegraph signal is a decreasing exponential function,

FIG. 5.2. A typical switching response, where the intrawell motion has been eliminated by the two-state filter. The times at which switching events occur are random, but to some extent coherent with the periodic modulation.

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and its power spectrum is a Lorentzian,

Good and readable discussions of these noise theorems and the properties of the random telegraph signal can be found in [18] and [19]. Now we can further imagine that the particle is subject to an additional force: a weak periodic modulation of frequency uj. As stated above, "weak" means that the periodic force alone is not sufficiently strong to induce the particle to undergo a transition from one potential well to the other. However, in the additional presence of the noise, the particle makes transitions, which are now to some extent coherent with the modulation. The potential is now time-dependent,

and if e
It is this periodic modulation of the rate which accounts for the coherence between the response and the modulation functions. For vanishingly small noise intensity, D —>• 0, the switching rate approaches zero, and consequently the coherence vanishes. For indefinitely large noise, the coherence again becomes vanishingly small as the system response becomes completely randomized. Between these two limits, there is an optimum noise intensity which maximizes the coherence. Early theorists called this a "resonance." Though the phenomenon is clearly distinct from deterministic resonances, the term has a well-defined statistical meaning. Deterministic resonances, for example in electrical circuits, are defined to occur at parameter values (typically the frequency) such that the amplitude of the periodic energy exchange between two reservoirs (for example, a capacitor and an inductor) is a maximum. Resonance activation, which occurs when an underdamped system with inertia is driven by an external frequency which is comparable to its natural frequency [1], [2], represents such a process. However, even in overdamped systems, we can understand that, in the case of periodic modulation plus noise, the probability densities in each well, which represent the respective particle populations, are also periodically modulated. The population maximum thus alternates between the two wells, representing an energy exchange in a very real sense. Since the amplitude of this exchange passes through a maximum for a specific noise intensity, the process meets the requirements for being termed a resonance.

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A number of characteristics of SR have been observed experimentally, and satisfactorily explained by modern theory, as we will outline in the subsequent sections. The dynamics of SR can be approached, both in theory and experiment, on two levels. One can look at a reduced, two-state dynamics, wherein the only information required is which well the particle resides in at a given moment, or the full dynamics which has been passed through a two-state filter. Alternatively, one can observe the complete dynamics which, in addition to the switching events, includes the stochastic and regular motions within the individual wells, that is, the complete intrawell motion. The most frequently observed physical quantity in SR experiments is the power spectrum P(u). In measurements on real physical systems with symmetric potentials U(x), the power spectrum shows a sequence of sharp peaks (in theory they are delta functions) of decreasing amplitude located at odd integer multiples of the modulation frequency riding on a broad Lorentzian noise background. If the symmetry of U(x) is destroyed, weaker peaks at the even integer multiples of UQ appear. The signal-to-noise ratio (SNR) is the ratio of the amplitude of the signal peak to the amplitude of the noise background, both determined at the fundamental frequency. The signal peaks represent the coherent part of the response. SR is demonstrated by observing that the SNR increases from zero and passes through a maximum, located at an optimum noise intensity, in the two-state dynamics. For the complete dynamics, in addition, the SNR —> oc in the limit D —> 0, due to the coherent intrawell motion. This becomes more transparent when we consider the noise-free modulated system. While no switching events can take place, the particle executes a periodic motion of small amplitude within the well due to the "rocking" motion of the potential caused by the modulation. This motion has the same frequency as the modulation. The total power contained in the noise and the signal can be determined by integration of the power spectra. In the two-state dynamics, it is observed that the total power is a constant, that is, as the power of the modulation is increased the power in the signal peak increases while the noise power decreases to maintain the total constant. For the complete dynamics, the relation between these two powers is more complicated, but generally, the signal power grows at the expense of the noise power in the response. An alternative quantity, which also clearly demonstrates the coherence of the stochastic response with the modulation, is the probability density of residence times, P(T). This quantity shows a sequence of strong. Gaussianlike peaks centered at odd integer multiples of the modulation half-period. T/2 = TT/O;O, or, depending on the sequence of time intervals counted, at all integer multiples of T. In both cases, the peaks are characterized by exponentially decaying maximum amplitudes at long times T. At shorter times, memory effects cause the peaks at small T to increase in amplitude to a maximum. These effects are fully discussed in a later section of this

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paper. There are no corresponding features at subharmoriics of CJQ in the power spectra. In §5.2 of this paper, we will sketch the historical development of SR theory and an early realization in an electronic circuit. In §§5.3-5.5 we will outline the modern theories of SR. We will discuss the three experiments which have been done and summarize their results in §§5.6, 5.7, and 5.8. In §5.9 the technique of physical measurements on analog simulators of SR systems will be introduced and outlined. Some representative data is presented in this section and compared with the theory. Several applications will also be discussed here. Section 5.10 will show some current, practical applications. In particular, we will discuss the use of SR in conjunction with a SQUID to detect, with much improved SNR, a low-frequency magnetic signal contaminated with noise. In this section we will also speculate on future practical developments, including very large scale integration (VLSI) applications. Section 5.11 will present the probability density of residence times as an alternative to the power spectrum. The only two symmetries available to any bistable system will be discussed here and examples of the probability density of residence times for the two sequences will be shown. The closely related topic of periodically modulated random walks will be discussed in §5.12. and the similarities to the residence time probabilities will be drawn. In §5.13, I will introduce the only existing biological application closely related to SR. Finally, in §5.14, I will summarize and collect the more significant of these results and make some speculations on future opportunities for research and the possibility for the discovery of SR in real natural systems. 5.2. Historical development of stochastic resonance The mechanism of SR was first propounded and investigated by Vulpiani and his coworkers [20] as an interesting stochastic effect in nonlinear dynamics which might have useful applications in a variety of fields. The chief theoretical difficulty in SR problems is that in the presence of an external temporal modulation, stochastic processes are not stationary, and consequently the Fokker Planck (F P) equation corresponding to any Langevin equation (which represents the dynamics of the physical process) is not solvable for the probability density exactly [7]. An historically important early consideration of nonstationary processes was, however, already introduced in 1975 by Hanggi and Thomas [21]. Nevertheless, it remains true that nonstationary F--P equations must be attacked with approximation, a fact which has given rise to a flourishing industry continuing even today, as we shall see below. These early authors, therefore, concentrated on estimates of the mean residence or "sojourn" time, demonstrating that this time became closely comparable to the half-period of the external periodic modulation for the optimum value of the noise intensity. A residence time theory for periodically modulated stochastic systems, which avoided the problems posed by the nonstationary F-P equation, had

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been developed earlier by Eckmann, Thomas, and Wittwer and was successfully applied to SR early on [22], [23]. It can be mentioned at this point that residence time measurements have only recently been made on experimental SR systems, as we shall discuss below in §5.11. The first study of SR in a partial differential system was also carried out on the stochastically perturbed Ginzburg-Landau equation by Vulpiani and company [24]. However, earlier, in 1982, these same authors, together with Parisi, proposed SR as a possible explanation of the observed periodicities in the recurrences of the earth's ice ages [25]-[27j. This suggestion was made independently by C. and G. Nicolis, the former of whom also introduced the adiabatic approximation (which continues to be useful today) to the SR problem [28], [29]. In this view, the earth's climate is represented by a one-dimensional bistable potential, one (meta)stable state of which represents a largely icecovered earth. The instability which accounts for the bistability is well known. It results from increased climatic cooling when the polar ice sheets expand and reflect more of the solar radiation back into space. External noise due to annual fluctuations in solar radiation drives this potential system, causing switching between normal and ice age climates [30]. A time series representing approximately 200 such events and extending backwards in time for about 3.5 million years is available from measurements of the 18O/16O isotopic ratios found in the fossils of sea animals in ocean core samples. This ratio changes during an ice age, largely because of the preferential evaporation rate of water with 16O from the ocean surfaces. The 160 water then becomes locked in the massive ice sheets, leaving the remaining ocean water enriched in 18O. The external noise is assumed to come from short-term fluctuations in the solar insolation which modulates the balance between radiative and transport processes, and the periodic modulation is most often supposed to originate from variations in the insolation resulting from a small observed oscillation in the eccentricity of the earth's orbit having a period of 100,000 years. Moreover, the power spectrum of the dynamical coordinate of the system was introduced here for the first time in SR systems in order to make useful comparisons with the power spectra of the isotope records in the core samples. In 1983, Fauve and Heslot made detailed measurements on a noise-driven, periodically modulated, bistable electronic system [31]. The system they used was a two-state threshold device, a Schmitt trigger. The switching logic of the Schmitt trigger is that a state transition A —>• B is executed, if the state is not already 5, whenever a lower threshold x\ is crossed from below by the noise. The transition B —>• A is executed, if the state is not already A, whenever an upper threshold £2 is crossed from above. The output of the noise-driven Schmitt trigger is an accurate representation of the dynamics of an ideal two-state system. Since the voltages in the two states are fixed regardless of the input, there can be no intrawell motions. They measured the power spectrum of the output from which they extracted

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the SNR, and observed that this quantity passed through a maximum with increasing noise intensity, thus demonstrating SR for the first time in a laboratory experiment. The location of the maximum in the SNR was identified (roughly) with the specific value of the noise intensity for which the Kramers rate in the unperturbed potential becomes comparable to twice the modulation frequency. No theory was put forth by these authors. Instead, their experiment served to clearly demonstrate the observable, physical aspects of SR. The circuit diagram of a Schmitt trigger together with its input-output (transfer) characteristic arid representative input and output time series, are shown in Fig. 5.3. Interest in SR seems to have subsequently waned until 1988 when McNamara, Wiesenfeld, and Roy demonstrated it in an ingenious experiment with a ring laser [32]-[34], which I will briefly discuss in §5.6. This experiment immediately stimulated a rash of theoretical activity [35] [46] as well as several analog simulations [42] [44], [47] -[52], a new experiment in a bistable electron

FIG. 5.3. Noise-induced switching in the simplest electronic two-state device: the Schmitt trigger, (a) The transfer (input, Vs; output, VR) characteristic of the trigger; (b) An example input signal composed of periodic and noisy components; (c) The response of the trigger showing switching between states A and B and the only two possible consecutive time interval sequences available to any two-state system; and (d) A circuit diagram of a Schmitt trigger composed of an operational amplifier (AD712) and two resistors. The ratio of the resistances shown in (d) and the supply voltage of the operational amplifier determine the threshold (switching) levels as shown by the dotted lines in (b). The switching logic is such that a transition A —> B occurs, if the state is not already B, whenever Vs crosses the lower threshold from above, and B —> A if not A for Vs crossing the upper threshold from below.

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paramagnetic resonance system [53], [54], and an experiment on a free-standing magnetic beam [55], which I will discuss further in §§5.7-5.9. 5.3.

The adiabatic theory of McNamara and Wiesenfeld

We will only briefly discuss the theories outlined in this and the following two sections, endeavoring to highlight their particular strengths and practical utilities from the point of view of an experimentalist. The finer details are left to the original literature, which the interested theorist will no doubt consult. The general category of the problem is the classical statistical theory of nonequilibrium, nonlinear dynamical systems driven by white noise. A collection of reviews in this topical area, as well as applications in a variety of fields, can be found in [56]. Two dynamical models have been considered, as mentioned before: the two-state and the complete dynamics models. Considering these models, contemporary theories fit into two categories: the adiabatic approximations [35], [36], [39], [42]-[46], and the nonadiabatic calculations [37], [38], [40], [41]. Though originally the means by which F-P systems could be treated within adiabatic approximations were put forth by Carolli et. al. [57], introduced to the SR problem by C. Nicolis [29], and more recently by Bryant, Wiesenfeld, and McNamara [58], the first contemporary use of this approximation for SR theory was due to McNamara and Wiesenfeld [35] (MW). In the general spirit of this approximation, the word "adiabatic" is used to indicate infinitesimally slow time variations in the potential, so that stochastic stationarity can be continuously achieved. The probability density is thus imagined to "adiabatically" adjust to the changing potential. In practice, this means that the frequency of the external periodic modulation must be small compared to some other inverse time scale or scales. One obvious choice is the unperturbed Kramers rate TQ. Another inverse time scale is established by the curvature of the potential near the top of the barrier and near the bottoms of the wells. Under the adiabatic assumption, stationary F-P dynamics can be used to calculate slowly time-varying quantities such as the probability densities and correlation functions. The object is to calculate the power spectrum of the motion of a particle moving in a generic bistable potential within the framework of the twostate model. Following MW, discrete variables x± are chosen to denote the location of the particle in either the right (+) or left (— ) potential well with corresponding probabilities n±, for which n+ = I — n_. A rate equation can then be written in terms of W±, the transition rates out of the ± states:

We can note that already this is a two-state dynamics, since the only dynamical variables are the particle populations within the wells. The probability density is, consequently, effectively reduced to a pair of delta

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functions located at the minima of the two potential wells and weighted by n+ and n_ respectively, and from this density the moments can be computed. Specifically, the second moment, (x 2 ). is needed. In order to solve (5.11), some approximate form for W± is required, and MW use an expansion in terms of a small parameter r/ocosi^ot, where 770 = e/D:

where QQ and the product a 1770 are treated as parameters of the system. The form of this may be obtained directly from (5.10) if one identifies QQ with the unperturbed Kramers rate TQ. Then for small air/o = cc/D, (5.10) reduces to r(t) ~ ro[l + (ec/D) sinu;t], that is. it reduces to (5.12) to within a phase factor. From (5.11), a solution for n+ can be obtained, and from that the autocorrelation function (x(t)x(t 4- r)|xo,to) and its to —» ex: asymptotic limit (x(t)x(t4-r)}. This quantity leads directly to the power spectrum through the Wiener Khintchine theorem:

This result makes two notable predictions, both borne out by experiment, as we shall show in §5.9: first, the shape of the power spectrum is a delta function contributed by the modulation riding on a Lorentzian noise background: and second, the total power, signal plus noise, is a constant. This latter remarkable property means that the power in the signal part of the response grows at the precise expense of the noise power, a property which is true only of the twostate model. This result demonstrates that, in such a bistable system, the proper application of noise at the input can result in more order at the output. This, of course, could not be possible with a linear system. Moreover, the nonlinear system must also be a noriequilibriuni system. Such systems have received a great deal of attention over the last few decades [59]. [60]. MW have generalized their theory to include the complete dynamics of an overdamped particle in the standard quartic potential:

where, for the unperturbed (e = 0) potential. AC/o = « 2 /4£> and the minima are located at ±c — ±^a/b. The Langevin equation therefore reads

In this case, the parameters are known: QQ = (\/2o/7r) exp( —A[/o/D): fti = 2o;o and 770 = ec/2D, which can be substituted into (5.13) in order to obtain

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the power spectrum

where r = exp(—AUo/D). This leads to the signal-to-noise ratio

A useful approximate expression for the SNR, valid for small modulation strength, is

Some results of the McNamara-Wiesenfeld theory based on (5.14)-(5.17) are shown in Fig. 5.4.

D(Hz)

D(Hz)

D(H2)

D(Hz)

FIG. 5.4. Theoretical results from [35]: the output power versus input noise variance, D, (a) in the signal feature computed at four different values of the signal frequency U)Q; (b) of the noise with no signal input; (c) with signal input; and (d) the signal-to-noise ratio versus D showing the maximum at an optimum value of D. (Note also the small additional peak at very low signal frequency.}

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It only remains to summarize briefly the approximations used by MW. First, the adiabatic approximation in the MW theory is LJQ
where the one-dimensional state variable x represents the membrane potential. Moreover, for this model the MW theory has been generalized to include multiplicative noise:

where £a and £m are the additive arid multiplicative noises, respectively. For the details of this calculation the reader is referred to [49]. The results of an analog simulation of this neuron potential for both overdamped and inertia! dynamics are discussed in §5.9. 5.4.

The nonadiabatic theory of Hanggi and Jung

The only theoretical treatment to date which has avoided the adiabatic approximation is that of Hanggi and Jung [38]. They begin with a generic Langevin equation based on the overdamped motion of a particle in the standard quartic potential, which reads

where 0 is a random phase uniformly distributed over one cycle [0.2?r]. and where £(t) is the usual Gaussian, white noise. The time-dependent

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Fokker-Planck equation, analogous to (5.19), reads:

This nonstationary F-P equation has no eigenfunctions or eigenvalues. Instead, the periodicity of the drift is used, in analogy with procedures used in periodically forced quantum mechanical systems, to write down Floquet-type solutions:

The functions p^ are periodic in T = IK/UQ , so that the Floquet characteristic exponent // can be determined by a Fourier series:

The Floquet exponents, which are complex, are identified with the eigenvalues, Amn = Mm + inu>Q, with n = 0, ±1,±2,..., of an analogous stationary twodimensional F-P equation:

which corresponds to the process

with

and where periodic boundary conditions in 6 are required. W(x,0,t) over 9 yields a cycle-aver aged probability density

Integrating

which, with 0 uniformly distributed on [0,2?r], approaches a stationary distribution P(x) for large times. The behavior of the time-periodic, asymptotic probability density Wst(x, 0) is clearly shown in Fig. 5.5, where contours of constant probability are plotted in the (x, 6} plane, for various values of A, the magnitude of the periodic forcing [37]. The most probable trajectories across the saddle points, shown by the solid dots, are traced as the system switches between the wells located at positive and negative x values. The + and — signs locate local maxima and minima, respectively, in Wst- The periodic boundaries on 6 at 0 and 2?r radians are clear from the figure.

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FIG. 5.5. Contours of constant probability density Wst(x,6] for D = 0.1, UJQ = I at several signal intensities: A = (a) 0.1; (b) 0.5; (c) 0.75; and (d) 1.0. The solid dots indicate saddle points, and the +( —) signs denote regions of high (low] probability density. The contours are depicted for a set of equidistant values of the probability density. Reprinted, by permission, from Figs, la-d of [P. Jung, Thermal activation in bistable systems, Z. Phys. B, 76 (1989), p. 521], Springer- Verlag, Heidelberg, Germany. © 1989 Springer- Verlag.

It is important to note here that this (averaging over a uniformly distributed phase 0) is precisely the way in which virtually all experiments [32], [33], [53] and analog simulations [43], [47], [48], [51], [52] have been done. The distributions P(x,t] are sampled from many sets of experimentally generated time series, x(t), obtained by opening the gate of the analog-to-digital (ADC) converter at random times, that is, at random phases of the periodic forcing. After the many samples are averaged, it is observed that P(x,t) —> -P(x), that is, a stationary value which has been obtained by averaging over . These stationary experimental distributions are found to be in excellent agreement with the theory. Both theorists and experimentalists should note that to do otherwise (i.e., to avoid averaging over ) the experimentalist or analog simulationist would have to externally trigger his ADC on a particular phase of the periodic forcing function for each set x(t}. Alternatively, one could compare the time function x(t) to the forcing Asimuot in order to extract the phase difference A<5, and average this quantity for every noise intensity. One experiment has been performed (see [54]) in order to reveal this phase difference. The results revealed that A<3> —» vr/4 as the SNR —» maximum, in the

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same way that deterministic resonators do, thus lending further credence to the characterization of SR as a true resonance phenomenon. The periodicity of the Floquet solutions implies that the autocorrelation function of x(t) is a nondecaying periodic function. This function is calculated explicitly from the asymptotic probability density obtained as the solution of the nonstationary F-P equation. It is shown that the very same autocorrelation function can be obtained from the nonstationary F-P equation by averaging over the random phase (p. The predictions of this theory for the real onesided (that is, experimentally measurable) power spectrum are twofold: first, the predicted signal features in the power spectrum are a sequence of delta functions, and second, for symmetric potentials like the standard quartic, they occur only at odd multiples of the modulation frequency. These predictions have been borne out by both the laser experiment simulation [33] and more recently by measurements on analog simulators [48], where the measured line widths in the signal are completely accounted for by the instrumental resolution and bandwidth (that is, the signal features in an ideal system with infinite resolution and bandwidth would be delta functions). Moreover, the sequence of peaks at odd multiples of UQ for symmetric potentials is readily observed. The connection of the potential symmetry to the observed sequence of peaks in the power spectra has been demonstrated by destroying the symmetry and observing the consequent growth of weak peaks at the even harmonics [29]. This technique has been extended in [40] to include damped inertial systems described by a Langevin equation:

where k is the damping factor and £(£) is the usual white noise. The procedure leads again to an asymptotically stationary, periodic probability density Was(x, v, 0, t), which is the solution of a two-dimensional F-P equation in v, the velocity, and 9 = Mot. This density can be averaged over 0 to yield a stationary probability density (p(x, v}}0, a contour plot of which is shown in Fig. 5.6 for various forcing frequencies o>o- As the frequency increases, we note that the density, initially with a single saddle lying between the two peaks, develops a hole in the center. That is, the single saddle bifurcates into a pair of saddles, and the most probable trajectories connecting the peaks (which locate the two wells) now avoid the origin. This phenomenology has been observed previously in a different context [67], [68]. We can conclude that the Floquet technique provides a powerful tool for handling higher-dimensional F-P systems. 5.5.

The perturbation theory of Marchesoni and coworkers

Finally, we conclude this segment with a brief outline of the theory developed by Marchesoni and coworkers [39], [43]. They first treated the case of an overdamped particle in the standard quartic potential within the framework of perturbation theory, also in analogy to techniques developed for treating

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FlG. 5.6. Contours of constant probability density of the phase-averaged, twodimensional asymptotic probability (p(x,v))o plotted for D = 0.1. A = 0.1. A = 0.5 for various values of the signal frequency, LJQ = (a) 0.2; (b) 0.5; (c) 0.6; and (d) 0.7. The dots denote hyperbolic points and the + or — signs denote local maxima or minima, respectively. Reprinted, by permission, from Figs. 1, 3, and 4 of [P. Jung and P. Hdnggi, Resonantly driven Brownian motion: Basic concepts and exact results, Phys. Rev. A. 41 (1990), />. 2977]. American Physical Society, New York. © 1990 American Physical Society.

periodically forced quantum mechanical systems. Thus they also avoided the limitations of the adiabatic approximation, but were instead confined by the limitations of perturbation theory. While the results of these calculations are in qualitative agreement with analog simulations performed by the same group, the theory does not predict the delta function form for the signal contribution to the power spectrum, nor does it predict the sequence of decaying peaks at odd harmonics of LJQ. In a third paper, [44], the case of damped inertial motion is treated, however now within the confines of the adiabatic approximation. First, the case of the overdamped motion, which begins with the standard Langevin equation: for which the potential is decomposed into a time-independent part plus a (weak) perturbation: U(x, t] — U(x) — Ah(x] cos(u;ot + 0), which leads to the usual F---P equation:

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Solutions of the form

are assumed and substituted into (5.27). The result can be mapped to the Schrodinger equation with the corresponding Hamiltonian:

This Hamiltonian is then separated into an unperturbed part H°(x] which depends only on the spatial coordinate, and a perturbation Hl(x,t] which is time-dependent. The calculation then proceeds along the lines of those found in quantum perturbation theory. An autocorrelation function is found for two cases of coupling even and odd, which do not make or break, respectively. the symmetry of the potential. The authors point out that only symmetrybreaking perturbations lead to SR. This conclusion becomes more clear when it is realized that symmetry-preserving perturbations can lead to no net probability flow between wells and hence no SR. Perturbations which, for example, modulate only the height of the barrier without altering the depth of one well with respect to the other can cause no net change in the population of one well at the expense of the other; only the shape, but not the symmetry, of the asymptotic probability density is altered. However, the perturbation theory here does not explicitly result in the delta function feature of the power spectrum now well established by both analog and numerical simulations. Indeed, it cannot because the predicted correlation functions —> 0 in the longtime limit. Nor does it predict the odd harmonic sequence of these features, which have now come to be regarded as the signature of SR. A second, and important, contribution by this group has been to treat the case of inertial motion within the adiabatic approximation, and to show that the same approximation can be used to obtain some predictions about the behavior of SR systems in the presence of colored noise. The relevant Langevin equation is

where 7 is the damping constant, and £(£) is now a colored noise defined by

with T the noise correlation time. The results of this theory are in reasonably good qualitative agreement with SNR measurements made by this group on an analog simulation of (5.30) and (5.31) for various values of both 7 and T considered as parameters. They thus contributed the first data on the effects of inertia and damping as well as colored noise.

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5.6.

223

Stochastic resonance demonstrated in a ring laser

In 1988, McNamara, Wiesenfeld, and Roy [32], [33] designed and carried out a remarkable experiment which offered the first demonstration of SR in a device that was not an electronic circuit. Their apparatus consisted of a dye-ring laser wherein the bistability was the result of the degeneracy associated with the two counter-rotating lasing modes. In a real laser perfect degeneracy is impossible due to slight imperfections in the mirrors, etc., so that the lasing mode settles into one or the other sense of rotation after startup. Roy and his coworkers had developed a novel method for modulating the asymmetry between the two counter-rotating modes. When modulated with noise, the laser then switched randomly between the modes. When a weak periodic function was added to the noise, the noisy switching events became to some extent coherent with the periodic function. These are, of course, the necessary ingredients for SR. A diagram of the experimental apparatus is shown in Fig. 5.7. The ring-dye laser is pumped with an argon-ion laser. Operation at a single frequency was established with intracavity etalons and a Fabry-Perot interferometer which also monitored the longitudinal mode structure. One of the output beams, that is, a beam traveling in one direction, was extracted and applied to a photodiode which provided the output signal. A bistable potential, similar to that shown in Fig. 5.1 was introduced with an acousto-optic modulator (AOM). It had earlier been shown that such a device could be used to control the direction of propagation in a bidirectional laser [69]. The potential, and hence the direction of propagation, can be modulated with the AOM by modulating the frequency of a radio frequency

FIG. 5.7. The experimental apparatus of Roy et al. [32] and [33] showing the bistable, ring, dye laser and its acouso-optic modulator driven by a combination of noise plus a periodic, signal.

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(r. f. ) driver. The r. f. driver was in turn driven by an audio function which was made up of the sum of a periodic signal and noise. The directional switching signal from the photodiode was then passed to a digital oscilloscope, and the digitized signal to a computer for computation of the power spectrum. The power spectrum was itself a noisy quantity, and so it was averaged. Figure 5.8 shows four examples of the averaged power spectrum for small, intermediate, and large noise intensity, respectively. The features contributed by the signal in these spectra are first of all the large amplitude, narrow spike at the modulation frequency c± 2.0 kHz, and a weaker one at the second harmonic ~ 4.0 kHz. The SNR was obtained as the ratio of the amplitude of the spike at the fundamental frequency to the amplitude of the noise background at the same frequency. Figure 5.8 clearly shows a small SNR at small noise intensity. This is because for small enough noise intensity, there is very little switching, and so the random component in the output is much larger than the coherent part. The SNR becomes larger at intermediate noise intensities, as shown in Figs. 5.8(b) and (c) and again weaker for very large noise, for which the system again loses coherence due to the strong randomization. The now famous curve for SNR versus noise intensity is shown in Fig. 5.9. The experimentally measured points, together with their error bars, are shown by the open squares. The solid curve was calculated from the approximate

FIG. 5.8. Experimental results from the bistable laser of Roy et al. clearly showing the growth and subsequent decline of the relative signal and noise amplitudes in the measured power spectra of the laser output as the injected noise intensity increases (a) to (d). The maximum signal-to-noise ratio is achieved in (c).

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FIG. 5.9. Measured SNR versus noise intensity from the ring laser experiment [32] compared to the adiabatic theory (solid line) o/[35].

result of the McNamara-Wiesenfeld theory, (5.17b). with and At/o = 1-22 x 10~3 used as fitting constants. The fit is remarkably good, in fact, rather better than would be expected based on the approximations underlying the theory. Because of this, these authors point out. the generic result, that is, the nearly 12 dB generic improvement in SNR. should be robust enough to be easily observable in a wide variety of natural physical phenomena. The insight associated with this speculation has been well borne out by further experimental developments, as the next twro sections show. In [33] Roy and his coworkers offer additional insight into the SR experiment with a bistable laser by digitally integrating the semiclassical laser equations. Included in the equations was a model for the action of the AOM in introducing the modulated potential into the cavity. The resulting power spectra from their simulation are shown in Fig. 5.10 for small, intermediate, and large injected noise intensities. Clearly, the same features show up as in the laser experiment, and in fact, the digital simulation yields a SNR versus noise intensity curve very similar to the experimental one. These results lend much confidence not only to the validity of the semiclassical laser equations, but also to the AOM and noise models introduced by the authors. We also show the three power spectra here to call attention to two features. First, the weak peak at the second harmonic, which was also observed in the laser experiment, is reproduced by the simulation. However.

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FIG. 5.10. Numerically generated power spectra using the semiclassical laser equations with increasing noise intensity (a)-(c) due to Roy et al. Note the weaker peaks at the second and third harmonics of the signal frequency apparent in (b). It is now known that the third harmonic peak always occurs, but the second harmonic one shows up only for spatially asymmetric potentials.

there is also a third peak, somewhat stronger than the second harmonic one at the third harmonic. Both the Hanggi and Jung theory [37], [38] and analog simulations [47], [48] have established that: (1) the peak at the second harmonic is the result of an asymmetric potential, that is, it vanishes for a perfectly symmetric potential (see also [39]); and (2) the peak at the third harmonic is generic to the SR phenomenology. This third peak was not reported in the laser experiment, perhaps because the power spectra were not plotted or measured far enough out in the frequency to observe the third harmonic. These results established SR on a firm experimental foundation for the first time.

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227

. . . and in electron paramagnetic resonance

Recently, an SR experiment using a detuned electron paramagnetic resonance system has been reported [53], [54]. In this experiment, polypyrrole paramagnetic samples were inserted into a standard paramagnetic resonance spectrometer. It had been previously shown that with suitable concentrations of the polypyrrole paramagnetic centers, together with suitable choices for the cavity resonance compared to the sample resonance, bistable operation of the cavity could be achieved [70], [71]. That is, the reflectivity of the spectrometer cavity had two minima with driving frequency. Normally, a feedback loop from the r. f. detector to the r. f. oscillator frequency control is used to lock onto one of these minima. However, if noise is introduced at the oscillator frequency control circuit input, the system will randomly switch between the two minima. Coherence can be introduced into this random switching by adding a weak periodic signal to the noise. The SR experiment can then proceed by measuring the SNR while varying the noise intensity. The system was shown to be well described by a dynamical equation:

where v is the microwave source frequency, vc is the cavity resonant frequency. and U(v,t] = (K/r 2 ) J R(z/) is a frequency-dependent potential obtained from the bistable reflectivity R. The remaining terms in this Langevin equation are AsmujQt, which is the periodic forcing at the audio frequency u;o, and £(/), which is the usual Gaussian noise. In this case the noise was characterized by a correlation time r = 500 ^.s. Whether or not the noise can be considered to be white or colored depends on the comparison of the noise correlation and the system response times. In this case the system response time, which was not specified, would be determined as usual from the closed loop step function response of the entire system measured, for example, at the input to the r. f. oscillator frequency control. Three examples of the experimental time series, showing the bistable system response (upper graphs) compared to the noise input (lower graphs). are shown in Fig. 5.11, where the sequence (a) through (c) represents small. intermediate, and large noise strength, respectively. These responses were recorded also in the presence of a weak periodic signal of 3.9 Hz. The signal-induced coherence is clearly evident in (b). and the suitably averaged power spectrum was reported to show a strong peak at U;Q, as is the usual case for SR. Instead of the usual SNR. however, the authors of [53] have simply measured the amplitude of the signal peak feature relative to that of the noise background. The results are shown in Fig. 5.12. where a very clear and sharp peak is evident at an optimum noise variance a2 . This was the second demonstration of SR in an experimental system other than an electronic circuit. Moreover, in further measurements with

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FIG. 5.11. Experimental noisy bistable outputs from the electron paramagnetic resonance system for increasing noise intensity (a)-(c) showing a near optimum response in (b).

this apparatus reported in [54], Gammaitoni and his colleagues have provided an entirely new experimental result: the phase relation between the signal and the system response. It was observed that the phase shift for vanishingly small noise variance was ~ Tr/2, passed through ~ 7T/4 at the optimum noise variance (where the maximum amplitude of the signal feature in the power spectrum is located), and finally approached very small values for large variance. This result, identical to the behavior observed in inductive-capacitative (LC) circuits, cavities, and other resonant structures, lends graphic credence to the characterization of the SR phenomenon as a true resonance. 5.8.

. . . and in a free-standing magnetoelastic beam

We turn now to a third experimental demonstration of stochastic resonance: that in a magnetically forced magnetoelastic ribbon standing upright in the gravitational field. In this apparatus, designed by Ditto and his coworkers [55], [72], an elastic ribbon made of an exotic amorphous magnetic material, Fe8iBi3.5Si3.sC2, is fixed at its base and allowed to stand vertically in the

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r i G . 5.12. Amplitude of the signal feature at o/'o m me measured spectrum versus noise variance a . from me electron, paramagnetic resonance experiment reported in [53]

earth's gravitational field. The elastic properties of the ribbon, in particular Young's modulus, can be controlled reversibly over a wide range by an external steady or time-varying magnetic field. Euler buckling can thus be either induced or prevented by adjusting this external field through the field dependence of the modulus E(H}. When the field is large enough to prevent buckling, the ribbon stands straight up. When the field is smaller than some critical value, that is, when the Euler critical height becomes smaller than the total length of the ribbon, the beam buckles and tilts toward one side or the other, say the right- or left-hand side. The resulting Instability can be driven with the external magnetic field, which can be generated from any desired time function plus a direct-current (d.c.) componen' to establish the desired quiescent operating point on the E(H) curve. In practice, the steady part of the field is biased near the critical value and a small time-dependent component is added, resulting in an external field given by

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where H(t) could be any time function but, in the SR experiments, was given by a periodic function plus random noise:

The resulting configuration behaved like a generic, parametrically driven, damped Duffing oscillator:

where U(x,t] is a time-dependent double well potential which can be modulated by the time-varying part of the external magnetic field [73]. Ditto and his colleagues have used this remarkable apparatus in a breathtaking series of experiments to demonstrate a variety of theoretical and/or numerically predicted results in contemporary nonlinear dynamics: crisis-induced intermittency [72], strange nonchaos [74], experimental control of chaos [75], and the scaling of noise-induced crises [76]. A recent conference proceedings will be helpful to those interested in experimental chaos [77]. In the present experiment [55], which depends on the aforementioned bistable property [73], they have demonstrated SR in the driven, magnetoelastic ribbon. A diagram of the apparatus is shown in Fig. 5.13. The inset in this figure shows the behavior of the modulus E, as a function of the external, steady magnetic field. The ribbon was biased at H^c ~ 0.7 Oe near the middle of the positive sloped portion of the E(H] characteristic. The ribbon, which is 3 mm wide by 100 mm long and 25 //m thick, was clamped at its base and mounted between a pair of Helmholtz coils which were driven by the time-dependent function shown in (5.34). An optical sensor was used to detect the horizontal position of the ribbon near its base. The output of the optical sensor was digitized into a sequence of position measurements which formed a noisy, bistable time series, which was later analyzed by computer. The data were passed through a two-state filter, as discussed in §5.1, in order to eliminate intrawell motions. The resulting time series shows a sequence of switching events representing only the right- or left-hand position. An example of this time series is shown in Fig. 5.14. It is evident that the potential is asymmetric, as indicated by the tendency for the system to reside preferentially in the well located near V = —1.0 volts. Nevertheless, measurements of the power spectra of such time series, carried out over weeks and under varying environmental conditions, yield SNRs which clearly show the maximum at an optimum noise intensity characteristic of SR and further demonstrate the robust nature of the phenomenon. The SNR data are shown in Fig. 5.15. An increase of around 10 to 11 dB in the SNR, which is consistent with virtually all other experimental realizations of SR, is evident. Moreover, Ditto et al., using this experiment, have been the first to make residence time measurements in an experimental SR system [55]. These

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FIG. 5.13. The elasto-magnetic free-standing beam apparatus designed by Ditto and his colleagues, as reported in [55] and [73].

FIG. 5.14. filtering.

A time series from the magnetic beam experiment after two-state

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232

FIG. 5.15.

The SNR versus input noise intensity from the magnetic beam

experiment.

time sequences, assembled into a histogram, will be discussed further in §5.12. The two main features of these data, the locations of the peaks at integer multiples of the modulation period, and the exponential decay of the peak amplitudes, which had previously been predicted by analog simulations, were well established by this experiment. An example of the residence time histogram is shown in Fig. 5.16. The exponential decay of peak amplitudes is demonstrated, to within experimental error, by the linear behavior of the semilogarithmic plot shown in the inset. Analog results on the residence time probability density are discussed and illustrated further in §§5.11-5.13. 5.9. Analog simulations of stochastic resonance The technique of simulating the behavior of differential equations by analog methods has a venerable history. The first analog simulator was invented by Lord Kelvin in 1876 based on the design of an integrator, consisting of a rolling ball on a rotating table, by his brother, the engineer James Thompson. These devices, which were entirely mechanical and were later called "differential analyzers," became highly developed during the first half of the next century, finally leading to important developments in military technology during the early part of World War II. Their advanced development in this country was led by Vannevar Bush at MIT in the early 1940s. Mechanical computing machines

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FIG. 5.16. The probability density of residence times for a magnetic beam experiment. The inset shows the data replotted on a semilogarithmic scale. The linear decay of maximum amplitudes in the inset indicates an exponential law.

were largely superseded by the electronic analog computer developed in the later part of the 1940s and carrying on until well into 1960. Electronic analog computation is based on integration by the collection of charge, q(t) = / i(t) dt, on a capacitor. The voltage across the capacitor is thus

Voltage multiplication can be accomplished electronically by the summation of logarithms. The logarithmic voltage-current characteristics of transistors and junction diodes are the source of log and antilog amplifiers which make up voltage multipliers. An excellent and enjoyably readable account of early calculating and computing machines can be found in [78]. Modern analog simulation is not done with a single machine (analog computer) which is "programmed" by the interconnection of various summing amplifiers, integrators, multipliers, and the like. Instead, one constructs a single electronic circuit for each application. This has been made feasible by the development of a variety of very accurate (typically 0.05%) single chips specialized for the performance of the following functions (Analog Devices and Burr-Brown chip numbers are stated, respectively, in parenthesis).1 1 These specialized electronic components can be obtained from Analog Devices, Inc., Box 9106, Norwood, MA 02062, USA or Burr-Brown, Inc., Box 11400, Tucson, AZ 85734, USA.

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multiplication and division (AD534; MPY534), raise a voltage to an arbitrary power (AD538; 4302), true rms-to-dc conversion (AD536; 4340), Log-antilog amplification (755/759; 4127), trigonometric functions (AD639).

Thus, using 1% accurate resistors and capacitors, an analog simulator, having a comparable accuracy, of any number of coupled ordinary differential equations of arbitrary order can be constructed. For example, a single overdamped Duffing oscillator, defined by the following dynamics,

can be constructed in less than one hour. In (37) the term, — (dU(x)/dx) is the bistable spatial force, and f ( t ) is an arbitrary external time-dependent forcing. Here the "standard quartic" bistable potential function is given by U(x] = —x2/2 -\- x4/4, which has been widely used to represent a variety of "generic" bistable physical systems. The dynamics of the equation are mimicked with the electronic circuit by integrating:

The schematic diagram of such a circuit is shown in Fig. 5.17. Such simulators are useful, for example, when f(t] is a stochastic function (or "noise," as it is widely called), in which case the analog simulator is often faster than digital simulation using an ordinary PC, even though the accuracy is limited. Accurately Gaussian, wideband noise voltages can be obtained for these simulations from commercially available noise generators.2'3 When f(i] = Asiuuot + £(£), SR can, of course, be investigated in this very simple system, as has been detailed in [48]. Figure 5.18 shows some example results measured on the overdamped Duffing circuit driven by noise and a periodic function of frequency /o = 500 Hz. Figures 5.18 (a) and (b) show the noisy and two-state filtered time series, respectively, and (c) shows the average power spectrum obtained from a sequence of two-state filtered time series. It is easy to note in the power spectrum the appearance of the first odd harmonic at /s = 1.5 kHz and the signal-induced features are very narrow peaks (delta functions, as first predicted by the theory of [37] and [40] for an ideal measuring system of infinite bandwidth and indefinitely 2

In my simulation laboratory, we use the Quan-Tech model 420 noise generator exclusively. Alas, the generators are no longer available new, but excellent reconditioned and calibrated models can be obtained from Tucker Electronics Co., P.O. Box 551419, Dallas, TX 75355. 3 In the simulations described here, filtered, real noise, having a nonzero correlation time rn, is applied to the electronic model. However, we wish to compare our results to a white noise (rn —> 0) theory. In the simulations we therefore keep the ratio rn/Ti = 0.1, which is a reasonable approximation to white noise within the limits of accuracy of the analog simulator. The integrator time constant TI is the system characteristic time by which all other times are scaled. See [47] and [48] for further details.

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FIG. 5.17. Schematic diagram of the analog simulator of an overdamped Duffing system. The crosses represent AD531 voltage multiplier chips. The integrator is an operational amplifier with a capacitor as the sole feedback element. Summation is accomplished by a single operational amplifier with multiple input resistors, the voltages on which are summed.

small frequency resolution). The widths of the measured peaks are. in fact, fully accounted for by the known bandwidth and resolution of the measuring system. Moreover, the noise background upon which ride the signal features is accurately Lorentzian as predicted in [35]. The strength of the primary peak at /o in Fig. 5.18 (c) can be measured, either by digitally integrating the data shown and then finding the amplitude of the step, or by measuring the amplitude of the peak itself and multiplying this by the bin width in Hz. It should be noted that the latter procedure requires that the entire power in the signal peak reside in one single frequency bin of the power spectrum. This, in turn, will depend on the stability of the signal generator used to supply the periodic modulation and the resolution of the digitized and Fourier transforming apparatus. In any case, if this strength is called S, and the amplitude of the noise background at the signal frequency is measured and called TV, then the SNR is defined as

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FIG. 5.18. Example outputs from the Duffing simulator of Fig. 5.17. (a) The response showing the intrawell motion (the complete motion): (b) The two-state filtered output from which intrawell motion has been eliminated: (c) The average power spectrum of the output shown in (b) showing the signal peak at 500 Hz and the third harmonic peak at 1.5 kHz.

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237

in decibels. The SNRs measured on the overdamped Duffing shown in Fig. 5.17 are shown in Fig. 5.19 by the asterisks. The plus signs in Fig. 5.19 show measurements of the noise alone, that is, with the input signal amplitude set to zero. The solid curve in Fig. 5.19 is calculated from the McNamara- Weisenfeld adiabatic theory [35]. It is not difficult to extend the design to include the effects of inertia and damping. A driven Duffing Oscillator is shown in Fig. 5.20. Note that two integrations are now necessary to mimic this second-order equation. Other simulators can be constructed. For example, we have recently been interested in the dynamics of the flux inside the loop of a r.f. SQUID (superconducting quantum interference device), which obeys an equation of the form [79]

where /cxt is a forcing in the form of an external magnetic flux incident on the SQUID loop [80]. Though this forcing is multistable, an external bias flux can be imposed such that a potential barrier is located at 0 = 0, and (3 can be adjusted such that two symmetric wells appear on either side of the origin. Obviously, when /ext — AsinuJot + £(£), an SR experiment can be done with this apparatus. Producing SR in such a device would be the first step toward practical utilization of the effect for signal detection by means of nonlinear filtering. At present, only an analog simulation of (5.40) exists. The simulator is shown in Fig. 5.21.

FIG. 5.19. The amplitude of the signal feature in the power spectrum at the fundamental frequency (asterisks] and the noise alone measured at the signal frequency (plus signs) versus noise intensity. The solid line is the adiabatic theory o/[35].

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FIG. 5.20. Schematic diagram of the inertial, damped Duffing oscillator.

FlG. 5.21. Schematic diagram of the analog simulator of a system for measuring the magnetic flux within a SQUID loop.

In order to study the advantage of such a technique for signal detection, the SNR is measured (using all the same parameters in the digitizing the Fourier transforming processes) before and after filtering. Before filtering, the SNR is measured on the signal plus the noise alone and this situation is called the linear case. Data on the linear case are shown in the following figures by the solid lines. Some representative data are shown in Fig. 5.22, where it is noted that SR used as a noise filter offers a significant advantage over the linear case at low frequency.

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239

FIG. 5.22. SNR results for the SQUID simulator shown in Fig. 5.21 measured at the output for two signal frequencies: 100 Hz (plus signs], 12 Hz (asterisks). The solid curve shows the measured SNR at the input of the simulator for the same conditions (frequency resolution and bandwidth) of the power spectrum measuring apparatus. The low frequency data show a significant advantage achieved with this essentially nonlinear filtering technique. Finally, we have designed and operated a simulator of a single neuron model [49]. This is essentially an inertial system with a "soft" potential:

that is, one for which the force (= —dU/dx) increases no more strongly than linearly as x —> oo. The Langevin equation is thus

where the nonlinearity parameter, 77 > 1.0, is the condition for bistability. This simulator was used to study the effects both of the stiffness of the potential and the damping factor. Some results are shown in Figs. 5.23 and 5.24. Since these experiments were done at low frequency (/o = 12.6 Hz), all the data show a significant advantage over linear filtering, as shown by the solid line, which represents our SNR measurements of the signal plus the noise alone before filtering. 5.10.

Some speculations on applications

In the last section it was demonstrated that SR offers an 8 to 12 dB advantage over linear measurements of the SNR before signal averaging. It should be noted that SR is an essentially nonlinear signal processing technique. It does

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FIG. 5.23. SNR results measured at a signal frequency of 12.6 Hz on an analog simulator of a single neuron model, that is, a damped, inertial system with a soft potential for two different damping factors: k = 1.0 (asterisks) and k = 0.5 (plus signs).

FlG. 5.24. SNR results measured at a signal frequency of 12.6 Hz on the neuron simulation for k = 5.0 (asterisks) and k = 1.0 (plus signs) plotted on an expanded scale.

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241

not faithfully reproduce the wave shape of the original noisy signal. It does, however, extract the frequency of the signal to high accuracy. Indeed, so long as the noise is additive, that is, so long as it enters a Langevin equation as an additive term, the width of the signal feature in the power spectrum is determined only by the bandwidth and frequency resolution of the (linear) measuring system. Moreover, SR works best when detecting low frequency signals. There is no theoretical lower limit on the frequency. An SR filter is most suitable for use before averaging. Possibly the first practical application will emerge as a result of the development of bistable SQUIDs specifically designed as stochastic resonators for use in determining the frequency of noisy, weak, low frequency magnetic signals. Recently, SR has been investigated in multielement systems [81], [82] wherein every element is coupled to every other element with equal strength. Such "globally coupled" systems, as they are called, have been the object of a number of recent studies both theoretical [81]--[85] and experimental [86]. The simplest possible system of coupled two-state devices might be realized with Schmitt triggers. There is a choice of coupling for the devices: ferromagnetic or antiferromagnetic. In the former case, a positive output of one device results in a positive response from a second one coupled to the first, so that in the absence of noise, a chain will result in outputs TTTT • • •; while in the latter case, a positive output results in a negative output of the coupled device, tltl For incoherent noise in each device, disorder results in large noise intensity. It has recently been shown by a simple, fully statistical theory based on the master equation, as well as by numerical simulations, that ferromagnetically coupled systems exhibit an ordering transition for noise intensity smaller than a well-defined critical value. The two-state master equation leads to a simple expression for the transition rates between + and — wells:

where g is the coupling strength, D is the noise intensity, and TQ — z/exp(—AU/D] is the usual Kramers rate of a single element with an unmodulated potential well. The populations in the jth pairs of wells, the Pj (t), are given by a nonlinear master equation,

A mean field approximation can then be used to obtain an equation of motion for the order parameter. The solutions of this equation show the ordering transition at a critical value of D as well as critical slowing down near the transition. Moreover, close to the transition but on the disordered side, the system exhibits SR with a significant enhancement of the SNR compared to that shown by a single element [81]. These two characteristics, an extremely simple device which, when coupled in an array, is capable of enhanced signal processing, are those most favorable

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to exploitation with VLSI technology. One may speculate that SR will find a number of applications in the form of single chip signal processors composed of a large number of globally coupled Schmitt triggers. 5.11. The probability density of residence times as an alternative to the power spectrum The residence time is defined as the time spent in one of the potential wells. For example, Fig. 5.18(b) shows the two-state filtered output of an overdamped Duffing system. The times spent in the positive (or negative) well can be tabulated and assembled into a histogram called the probability density of residence times. An example is shown in Fig. 5.25 measured on the analog simulator shown in Fig. 5.17 for u>o = 50 Hz (To = 20 ms). Note the sequence of peaks located at odd integer multiples of To/2. The amplitudes of the peaks also decay exponentially as shown in Fig. 5.26, where the same data are replotted on a semilogarithmic scale. The straight line indicates the exponential decay law Pmax(T) oc exp[—XT]. This object contains all the same information as the power spectrum. The signal frequency is determined by the locations of the peaks and the noiseto-signal intensity ratio by the decay constant. This dependence has been calculated within the framework of an adiabatic theory [42]. We shall not outline that theory here, but instead refer the interested reader to [42]. One can physically understand the To/2 sequence as follows: In the twostate system, the maximum probability that a transition, say from state A

FIG. 5.25. The probability density of residence times measured on the overdamped Duffing simulator for a signal frequency of 50 Hz ((To/2) = 10 ms) showing the characteristic sequence of peaks located at odd integer multiples ofTg/2.

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FIG. 5.26. The same data as in Fig. 5.25 replotted on a semilogarithmic scale showing the exponential decay of the maximum peak amplitudes.

to state 5, will take place occurs at, say, the first positive maximum of the external signal, that is, at 7T/2. Let us mark time at this point. The most probable time for the return transition, B —> A, will occur at 3?r/2, or exactly ^ period later. A sequence of such events is the most probable for large noise intensity, and such a sequence has TO/2 for the most probable residence time. But the transitions are statistical, so suppose the transition B —> A does not occur at its first most probable time (at 3?r/2, that is, at the first minimum in the signal). Since the state point remains in B, the next transition must be B —> A. The next most probable time for the next B —» A transition, however, does not occur until the next minimum, that is, at 7?r/2 (in other words at a time 3To/2 after the time mark at the first A —> B transition). The second most probable events (the second-order events) then contribute to the peak at 3To/2. Thus the odd integer sequence of maxima in P(T] are to be understood on the basis of the switching logic of any two-state system. Therefore, we would expect these patterns to be generic. Any noise-driven, periodically modulated, two-state system should show this same sequence. Indeed, in §5.13 we show the sequence as measured on the simplest electronic two-state system known: the Schmitt trigger. That the second most probable events are (at least for very large noise intensities) exponentially less probable than the first most probable is less obvious, but can be understood on the basis of the statistical theory of escapes from a potential well [42]. If the noise intensity is lowered sufficiently, however, switching events at every maximum followed by return events at every subsequent minimum is not the most probable sequence. After all, for zero noise intensity the probability is unity that an escape will require infinite time. So at some noise intensity,

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the A —> B events at every other maxima followed by the B —* A returns at every other minima, becomes the most probable sequence. The second peak in the probability density, located at 3To/2, then becomes the peak of maximum amplitude. Subsequent peaks thus pass through their own maxima as the noise intensity is lowered further. This effect is shown in Fig. 5.27, where the maximum amplitudes of the second, third, and forth peaks are plotted versus noise intensity. Clearly, there is an optimum noise intensity for each peak representing the maximum probability for that particular order of switching event. This effect could be termed "stochastic resonance" in the order of the most probable sequence of switching events. In §5.13, we will discuss two symmetries available to the generic two-state system, and show that they lead to two distinct sequences of peaks in the residence time probability density, one of which has significance in neurophysiology. 5.12.

Stochastic resonance in the periodically modulated random walks of Weiss and coworkers

Recently, Weiss and his coworkers have studied the problem of escape from a one-dimensional interval by a random walker under the influence of a weak external periodic field [87]. That this very simple model, based as it is on discrete random processes, very clearly demonstrates stochastic resonance, contributes a strong insight into the general problem of periodically modulated, nonlinear, stochastic processes. Moreover, the model results are of importance

FIG. 5.27. Stochastic resonance from the residence time probability densities. The symbols represent the maximum amplitudes of the second (asterisks), third (plus signs), and forth (crosses) peaks versus noise intensity. Each peak, representing a specific order of the noise-induced switching process, passes through its own maximum at a specific optimum noise intensity.

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in modern problems in electrophoresis and other separation processes. This work also places some much earlier numerical results relating to periodically stimulated sensory neurons [89] on a solid theoretical foundation. In this problem, a random walker is initially located on an interval [0, L] with absorbing boundaries at 0 and L, and with a site location distribution, p 0 (r) = 1/(L — 1). At time zero, the walkers are allowed to jump among discrete lattice sites on the interval according to the site r, and iterate n, probability:

where pn(r] is the probability that a walker will occupy the site r at the nth iteration, and e(n) = As'mujQn is the periodic field. Of interest are the mean residence time (n(u;o)}, that is, the time or the total number of iterations, before the walker is absorbed at one or the other of the boundaries, and the survival probability defined as

where TO is the initial position of the walker. Numerical results on the residence times are shown in Fig. 5.28 for three different values of A. the intensity of the periodic field. The sharp and distinctive minimum at a particular characteristic frequency cur is evidence of stochastic resonance. That is, the walkers can escape in minimum time due to coherence effects induced by the periodic field when its time scale is "resonant" with a time scale established by the random process (the "noise"). Moreover, the survival probability shows periodic effects similar to those shown by the residence time probability densities displayed in §5.11 (for a continuous bistable process). In particular. S(n TO) as defined by (5.46), is a cumulative probability obtained by summing the discrete probabilities p n (r). In some sense, this must be related to the integral of the residence time probability densities for the continuous processes of §5.11. Figure 5.29 shows this striking result. The periodic structure, with steps located at integer multiples of some characteristic period TQ. is clearly evident and is quite similar to what one would expect from integrating the probability density of residence times as, for example, shown in Fig. 5.25. These results, which were obtained from an extremely simple, fully discrete model, demonstrate that stochastic resonance is a generic and ubiquitous process, likely to be found in any natural setting where periodic modulation of a stochastic process in a nonlinear system which allows trapping obtains. An additional significance of these results is that they are based on exact theory, in contrast to the adiabatic, small noise, and small field intensity approximations which were necessary in studies on the continuous problem. 5.13.

Noise-induced switching in periodically stimulated neurons

Stochastic resonance per se, by which one means a self-generated optimization process with noise as a parameter, has not been discovered in a biological

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FIG. 5.28. Stochastic resonance in the escape from an interval in the random walk process studied by Weiss and his colleagues [87]. Plotted is the mean residence time on the interval versus frequency of the external field for three field amplitudes: A = 0.05 (diamonds), 0.1 (circles), andO.3 (triangles). The minimum in the residence time occurs when times scales set by the field frequency and the random walk process become comparable.

FIG. 5.29.

The survival probability versus time in the random walk problem.

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system, nor indeed in any natural system, as discussed in §5.14. Nevertheless, the presence of noise-induced switching in periodically stimulated sensory neurons in various animals has certainly been well established [62], [89]. It has been long known that the most coarse dynamical behavior of neurons is as discrete two-state systems [90], that is, firing or not firing. Moreover, it is also well known that after a neuron fires once (and after some absolute refractory period) it must be reset before it can fire again. The reset mechanism is well known to be the repolarization of the membrane potential after a strong depolarization which accompanies the firing potential spike. Thus the twostate system logic, as discussed in §5.11, obtains in a real neuron. Moreover, neuroscientists have known for decades that sensory information is somehow encoded in the time intervals between the spikes (or some variation representing this same quantity such as some suitably averaged, short-time firing rate], though the exact encoding or encodings is as yet unknown. Furthermore, it is also well known that in any sample of interspike intervals, a large fraction of the intervals is simply random, that is, induced by some noise process. Consequently, we might expect that when sensory neurons are periodically stimulated, the time intervals between firings, when suitably averaged and assembled into a histogram, would bear some relation to the probability density of residence times shown in §5.11. That is exactly what happens; however, as we show below, one must pay attention to the symmetry of the time interval sequence. We reproduce in Fig. 5.30 a histogram of time intervals obtained from the auditory nerve of a squirrel monkey whose ear was stimulated by a 600 Hz tone (To = 1.666 ms) [91]. Note that the peaks in this neurophysiological data are located at all integer multiples of TO, in contrast to the residence time probability densities shown in §5.11. This sequence is one of only two possible symmetries available to a generic two-state system. The peaks do, however, decay exponentially as shown by the inset, which depicts the same data plotted on a semilogarithmic scale. Similar results have been obtained from recordings in the visual cortex of the cat with periodic optical stimulation [92]. These data can be reproduced almost exactly by measuring the probability density of residence times in the simplest two-state system (the Schmitt trigger as shown on Fig. 5.3). However, instead of measuring the time sequence shown at the top of Fig. 5.3 (c), that is, instead of measuring the residence time in a single well, the bottom sequence, which represents the total time of residence in both wells, must be measured. These two possible sequences represent the two symmetries. In the language of the two-state system switching logic, the time between successive A —> B transitions must be measured, ignoring the B —> A reset events. When this is done, the probability density of residence times (where the residence time is now the total time spent in both states) displays peaks located at all integer multiples of TO as shown in Fig. 5.31. This is in exact correspondence with the physiological data as shown in Fig. 5.30. Indeed, in the real physiological experiments, the reset events the membrane

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FIG. 5.30. An interspike interval histogram measured on action potentials taken from the auditory nerve of a squirrel monkey with the ear stimulated by a 600 Hz tone. The inset shows the data replotted on a semilogarithmic scale. Note the peak sequence at all integer multiples of TQ = 1.666 ms. Data reproduced with permission from [91]. repolarizations—are normally not accessible to the experimenter. Indeed, the sequences shown in Figs. 5.30 and 5.31, with peaks located at all integer multiples of TO, are direct evidence of the existence of a hidden event (B —> A) which must occur between every firing event (A —>• B). By contrast, when we reprogram our digitizer to measure the time interval between the A —+ B and the following B —» A (reset) events on our Schmitt trigger, we obtain probability densities with the peak sequence shown by the inset in Fig. 5.31, that is, with peaks located at only the odd integer multiples of To/2. Data from the Schmitt trigger (or indeed any analog two-state device) can be produced which nearly perfectly match the physiological data if two requirements are met: (1) the measurements must, obviously, be made at the same stimulus frequency, and (2) either the noise intensity or the stimulus intensity must be adjusted to obtain the corresponding peak amplitude decay constant A. This has been done for the Schmitt trigger data shown in Fig. 5.31. These experiments demonstrate that noise-induced switching plays a significant (perhaps the most significant) role in information transmission in stimulated sensory neurons. 5.14.

Summary and speculations on future developments

It is clear that SR is generic enough that it should be observable in a wide variety of nonlinear noisy systems. Moreover, the indications are that it may be important in applied science, particularly for low frequency signal detection and processing when the frequency of the signal is of prime importance. There are two main topical areas which we have not touched on in this review: (1)

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FIG. 5.31. Probability density of residence times measured on the Schmitt trigger shown in Fig. 5.3 for the total interval between successive A —> B events. The stimulus frequency is 600 Hz corresponding to the physiological data shown in Fig. 5.30. Note the peak sequence at all integer multiples o/To = 1.666 ms. The inset shows the same measurement made for all the same conditions, except now the interval is between A —> B and its subsequent B —> A reset event. In the inset, the peak sequence is at odd integer multiples o/To/2.

the question of the existence of an analog of SR in quantum mechanics, and (2) the substitution of chaos for the noise in the SR mechanism. The former question is similar to that arising in studies on the existence of quantum chaos. Classical chaos is a characteristic of individual trajectories, yet trajectories are not observable in quantum systems [93]. In reference to the latter, it is easy to conceive a variety of man-made SR experiments wherein chaos would replace the high-dimensional noise that has been used to date. However, the notion of chaos driving an SR-type system may find widest interest in biology, specifically in neurophysiology. All the ingredients are present: noise-induced switching has already been demonstrated, and a large discussion of the possibilities of chaos in living systems is currently in progress [94] [98]. These topics are currently receiving attention, and the reader is referred to the proceedings of a recent workshop for references [99]. It may be interesting to conclude with a speculation. Stochastic resonance, as an information transmitting or processing phenomenon which exploits noise as a parameter in a self-optimizing process, has not yet been discovered in any natural system, though, as the above review shows, it has been demonstrated

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in a variety of man-made experiments. A necessary ingredient—noise-induced switching—has, however, been demonstrated, or at least strongly indicated, in at least two natural systems: the ice age recurrences and the phase locked spike intervals in stimulated sensory neurons. So where might one look for the selfoptimization? It seems to me that living systems offer the best hope [64], and within those, perhaps the optic, auditory, or tactile sensory systems are the most promising complexes for such research. After all, these systems, which are among the oldest in the evolutionary sense, were forced to evolve around inherently noisy rudimentary transducer cells. It does not seem unreasonable that, having had to deal with the noise originally, the evolving system may just have found a means to optimize its information transmitting and processing abilities by exploiting that noise. And if it has done this, may not such a system have found it useful also to control the noise and perhaps even to generate additional noise or chaos for the purpose? Note added in proof The premise that the recurrences of the Earth's Ice Ages are indeed periodic has recently been questioned. New data on the Earth's climatic variations over a continuous 500,000 year period, based on vein calcite cores obtained from Devil's Hole, Nevada, seem to indicate that the periods between deglaciations, marking the terminations of the ice ages, were steadily increasing during the specified period [100]. If firmly established, this result would decouple the Ice Age dynamics from periodicities in the solar insolation and thus from the Earth's orbital characteristics. Moreover, new data from Greenland ice cores indicate that the interstadials (rapid, more-or-less random, switchings between glacial and moderate climates which characterize the terminations) are probably driven by changes in direction and/or intensity of the North Atlantic Current [101]. Acknowledgments I am deeply grateful to the following colleagues, students and former students for discussions and collaborations and for the inspiration without which a large body of the work which has emerged from my laboratory in recent years would not have been possible: Peter Hanggi, Peter Jung, Ting Zhou, Adi Bulsara, Leone Fronzoni, Gabor Schmera, Eleni Pantazelou, and David Pierson. Grateful thanks are also due Michael Shlesinger at the Office of Naval Research for his constant and enthusiastic encouragement and funding of the majority of the work (not only my own) herein reviewed. Hanggi, Jung, Fronzoni, and Lhave also benefited substantially from NATO financial support. References [1] M. H. Devoret, J. M. Martinis, D. Esteve, and J. Clarke, Phys. Rev. Lett., 53 (1984), p. 1260. [2] H. M. Devoret, D. Esteve, J. M. Martinis, A. Cleland, and J. Clarke, Phys. Rev., B36 (1987), p. 58.

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