物理和化學中的隨機過程

前言

The interest in fluctuations and in the stochastic methods for describing themhas grown enormously in the last few decades.The number of articles scatteredin the literature of various disciplines must run to thousands，and special journalsare devoted to the subject.Yet the physicist or chemist who wants to becomeacquainted with the field cannot easily find a suitable introduction.He reads theseminal articles of Wang and Uhlenbeck and Of Chandrasekhar.which are almostforty years old，and he culls some useful information from the books ofFeller，Bharucha.Reid.Stratonovich，and a few others.Apart from that he is confrontedwith a forbidding mass of mathematical literature.much of which is of littlerelevance to his needs.Tllis book is an attempt to fill this gap in the literature.The first part covets the main points of the classical material.Its aim is to provide physicists and chemists with a coherent and sufficiently complete frame-work，in a language that is familiar to them.A thorough intuitive understandingofthe material is held to be a more important tool for research than mathemat-ical rigor and generality.A physical system at best only approximately flulfillsthe mathematical conditions on which rigorous proofs are built，and a physicistshould be constantly aware of the approximate nature of his calculations.（Forinstance.Kolmogorov's derivation of the Fokke卜.Planck equation does not tellhim for which actual systems this equation may be used.）Nor is he interestedin the most generai formulations，but a thorough insight in special cases willenablehimto extendthetheorytoother caseswhentheneed arises.Accordinglythe theory is here developed in close connection with numerous applications andexamples.The second part，starting with chapter IX『now chapter Xl，is concerned wimfluctuations in nonlinear systems.This subject involves a number of conceptualdimculties.first pointed out bv D.K.C.MacDonald.They are of a Phy'sical ratherthan a mathematical nature.Much confusion is caused by the still prevailing viewthat nonlinear fluctuations can be approached from the same physical starting point as linear ones and merely require more elaborate mathematics.In actuaifact.what is needed is a firmer physical basis and a more detailed knowledge ofthe physical system than required for the study oflinear noise.This is the subject of the second part，which has more the character of a monograph and inevitablycontains much of my own work.

內(nèi)容概要

本書1981年初版，1992年第2版，1997年第2版修訂出版，2007年第3版，2008年重印出版。這是第3版，較第2版的最大不同是，取消了原來第17章中第6節(jié)量子主方程的應用，取而代之的是量子波動的介紹；除此之外，本書做了不少修訂。并且也增加了許多最近發(fā)展成果。目次：隨機變量；隨機事件；隨機過程；馬爾科夫過程；主方程；一步過程；化學反應；Fokker-Planck過程；Langevin方法；主方程的展開；擴散型；不穩(wěn)定系統(tǒng)；連續(xù)系統(tǒng)中的波動；隨機微分方程；量子系統(tǒng)的隨機行為?！　∽x者對象：數(shù)學專業(yè)、統(tǒng)計物理專業(yè)以及理論物理化學交叉學科的研究生和科研人員。

書籍目錄

PREFACE TO THE FIRST EDITION PREFACE TO THE SECOND EDITION ABBREVIATED REFERENCES PREFACE TO THE THIRD EDITION I. STOCHASTIC VARIABLES 1. Definition 2. Averages 3. Multivariate distributions 4. Addition of stochastic variables 5. Transformation of variables 6. The Gaussian distribution 7. The central limit theorem II. RANDOM EVENTS 1. Definition 2. The Poisson distribution 3. Alternative description of random events 4. The inverse formula 5. The correlation functions 6. Waiting times 7. Factorial correlation functions .III. STOCHASTIC PROCESSES 1. Definition 2. Stochastic processes in physics 3. Fourier transformation of stationary processes 4. The hierarchy of distribution functions 5. The vibrating string and random fields 6. Branching processes IV. MARKOV PROCESSES 1. The Markov property 2. The Chapman-Kolmogorov equation 3. Stationary Markov processes 4. The extraction of a subensemble 5. Markov chains 6. The decay process V. THE MASTER EQUATION 1. Derivation 2. The class of W-matrices 3. The long-time limit 4. Closed, isolated, physical systems 5. The increase of entropy 6. Proof of detailed balance 7. Expansion in eigenfunctions 8. The macroscopic equation 9. The adjoint equation 10. Other equations related to the master equation VI. ONE-STEP PROCESSES 1. Definition; the Poisson process 2. Random walk with continuous time 3. General properties of one-step processes 4. Examples of linear one-step processes 5. Natural boundaries 6. Solution of linear one-step processes with natural boundaries 7. Artificial boundaries 8. Artificial boundaries and normal modes 9. Nonlinear one-step processes VII. CHEMICAL REACTIONS 1. Kinematics of chemical reactions 2. Dynamics of chemical reactions 3. The stationary solution 4. Open systems 5. Unimolecular reactions 6. Collective systems 7. Composite Markov processes VIII. THE FOKKER-PLANCK EQUATION 1. Introduction 2. Derivation of the Fokker-Planck equation 3. Brownian motion 4. The Rayleigh particle 5. Application to one-step processes 6. The multivariate Fokker-Planck equation 7. Kramers' equation IX. THE LANGEVIN APPROACH 1. Langevin treatment of Brownian motion 2. Applications 3. Relation to Fokker-Planck equation 4. The Langevin approach 5. Discussion of the Itt--Stratonovich dilemma 6. Non-Gaussian white noise 7. Colored noise X. THE EXPANSION OF THE MASTER EQUATION 1. Introduction to the expansion 2. General formulation of the expansion method 3. The emergence of the macroscopic law 4. The linear noise approximation 5. Expansion of a multivariate master equation 6. Higher orders XI. THE DIFFUSION TYPE 1. Master equations of diffusion type 2. Diffusion in an external field 3. Diffusion in an inhomogeneous medium 4. Multivariate diffusion equation 5. The limit of zero fluctua6ons XII. FIRST-PASSAGE PROBLEMS I. The absorbing boundary approach 2. The approach through the adjoint equation - Discrete case 3. The approach through the adjoint equation - Continuous case 4. The renewal approach 5. Boundaries of the Smoluchowski equation 6. First passage of non-Markov processes 7. Markov processes with large jumps XIII. UNSTABLE SYSTEMS 1. The bistable system 2. The escape time 3. Splitting probability 4. Diffusion in more dimensions 5. Critical fluctuations 6. Kramers' escape problem 7. Limit cycles and fluctuations. XIV. FLUCTUATIONS IN CONTINUOUS SYSTEMS 1. Introduction 2. Diffusion noise 3. The method of compounding moments 4. Fluctuations in phase space density 5. Fluctuations and the Boltzmann equation XV. THE STATISTICS OF JUMP EVENTS 1. Basic formulae and a simple example 2. Jump events in nonlinear systems 3. Effect of incident photon statistics 4. Effect of incident photon statistics - continued XVI. STOCHASTIC DIFFERENTIAL EQUATIONS 1. Definitions 2. Heuristic treatment of multiplicative equations 3. The cumulant expansion introduced 4. The general cumulant expansion 5. Nonlinear stochastic differential equations 6. Long correlation times, XVII. STOCHASTIC BEHAVIOR OF QUANTUM SYSTEMS 1. Quantum probability 2. The damped harmonic oscillator 3. The elimination of the bath, 4. The elimination of the bath - continued 5. The Schr'odinger-Langevin equation and the quantum master equation 6. A new approach to noise 7. Internal noise SUBJECT INDEX

章節(jié)摘錄

插圖：A "random number" or "stochastic variable" is an object X defined bya. a set of possible values （called "range", "set of states", "sample space"or "phase space"）;b. a probability distribution over this set.Ad a. The set may be discrete, e.g.: heads or tails; the number of electronsin the conduction band of a semiconductor; the number of molecules of acertain component in a reacting mixture. Or the set may be continuous in agiven interval: one velocity component of a Brownian particle （interval-∞+∞）; the kinetic energy of that particle （0,∞）; the potential differencebetween the end points of an electrical resistance （-∞, +∞）. Finally the setmay be partly discrete, partly continuous, e.g., the energy of an electron inthe presence of binding centers. Moreover the set of states may be multidimen-sional; in this case X is often conveniently written as a vector X. Examples:X may stand for the three velocity components of a Brownian particle; orfor the collection of all numbers of molecules of the various components ina reacting mixture; or the numbers of electrons trapped in the various speciesof impurities in a semiconductor.For simplicity we shall often use the notation for discrete states or for acontinuous one-dimensional range and leave it to the reader to adapt thenotation to other cases.

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