STOCHASTIC PROCESSES IN PHYSICS AND CHEMISTRY

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1 STOCHASTIC PROCESSES IN PHYSICS AND CHEMISTRY Third edition N.G. VAN KAMPEN Institute for Theoretical Physics of the University at Utrecht ELSEVIER Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

2 TABLE OF CONTENTS PREFACE TO THE FIRST EDITION PREFACE TO THE SECOND EDITION ABBREVIATED REFERENCES PREFACE TO THE THIRD EDITION VÜ ix X xi I. STOCHASTIC VARIABLES 1. Definition 1 2. Averages 5 3. Multivariate distributions Addition of stochastic variables Transformation of variables The Gaussian distribution The central Limit theorem 26 O. RANDOM EVENTS 1. Definition The Poisson distribution Alternative description of random events The inverse formula The correlation functions Waiting times Factorial correlation functions 47 III. STOCHASTIC PROCESSES 1. Definition Stochastic processes in physics Fourier transformation of stationary processes The hierarchy of distribution functions The vibrating string and random fields Brandung processes 69 IV. MARKOV PROCESSES 1. The Markov property The Chapman-Kolmogorov equation Stationary Markov processes The extraction of a subensemble Markov chains The decay process 93 xiii

3 xiv TABLE OF CONTENTS V. THE MASTER EQUATION 1. Derivation The class of W-matrices The long-time limit Closed, isolated, physical Systems The increase of entropy Proof of detailed balance Expansion in eigenfunctions The macroscopic equation The adjoint equation Other equations related to the master equation 129 VI. ONE-STEP PROCESSES 1. Definition; the Poisson process Random walk with continuous time General properties of one-step processes Examples of linear one-step processes Natural boundaries Solution of linear one-step processes with natural boundaries Artificial boundaries Artificial boundaries and normal modes Nonlinear one-step processes 161 VH. CHEMICAL REACTIONS 1. Kinematics of chemical reactions Dynamics of chemical reactions The stationary Solution Open Systems Unimolecular reactions Collective Systems Composite Markov processes 186 Vffl. THE FOKKER-PLANCK EQUATION 1. Introduction Derivation of the Fokker-Planck equation Brownian motion The Rayleigh particle Application to one-step processes The multivariate Fokker-Planck equation Kramers' equation 215 IX. THE LANGEVIN APPROACH 1. Langevin treatment of Brownian motion Applications 221

4 TABLE OF CONTENTS xv 3. Relation to Fokker-Planck equation The Langevin approach Discussion of the Itö-Stratonovich dilemma Non-Gaussian white noise Colored noise 240 X. THE EXPANSION OF THE MASTER EQUATION 1. Introduction to the expansion General formulation of the expansion method The emergence of the macroscopic law The linear noise approximation Expansion of a multivariate master equation Higher Orders 267 XI. THE DIFFUSION TYPE 1. Master equations of diffusion type Diffusion in an external field Diffusion in an inhomogeneous medium Multivariate diffusion equation The limit of zero fluctuations 287 Xn. FIRST-PASSAGE PROBLEMS 1. The absorbing boundary approach The approach through the adjoint equation - Discrete case The approach through the adjoint equation - Continuous case The renewal approach Boundaries of the Smoluchowski equation First passage of non-markov processes Markov processes with large jumps 322 XIII. UNSTABLE SYSTEMS 1. The bistable system The escape time Splitting probability Diffusion in more dimensions Critical fluctuations Kramers' escape problem Limit cycles and fluctuations 355 XIV. FLUCTUATIONS IN CONTINUOUS SYSTEMS 1. Introduction Diffusion noise The method of compounding moments 367

5 xvi TABLE OF CONTENTS 4. Fluctuations in phase space density Fluctuations and the Boltzmann equation 374 XV. THE STATISTICS OF JUMP EVENTS 1. Basic formulae and a simple example Jump events in nonlinear Systems Effect of incident photon statistics Effect of incident photon statistics - continued 392 XVI. STOCHASTIC DIFFERENTIAL EQUATIONS 1. Definitions Heuristic treatment of multiplicative equations The cumulant expansion introduced The general cumulant expansion Nonlinear stochastic differential equations Long correlation times 416 XVII. STOCHASTIC BEHAVIOR OF QUANTUM SYSTEMS 1. Quantum probability The damped harmonic oscillator The elimination of the bath The elimination of the bath - continued The Schrödinger-Langevin equation and the quantum master equation A new approach to noise Internal noise 451 SUBJECT INDEX 457

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