# Stochastic Processes in Physics and Chemistry

## 3rd Edition

**Authors:**N.G. Van Kampen

**eBook ISBN:**9780080475363

**Paperback ISBN:**9780444529657

**Imprint:**North Holland

**Published Date:**21st March 2007

**Page Count:**464

**View all volumes in this series:**North-Holland Personal Library

## Table of Contents

Dedication

PREFACE TO THE FIRST EDITION

PREFACE TO THE SECOND EDITION

ABBREVIATED REFERENCES

PREFACE TO THE THIRD EDITION

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 Itô–Stratonovich dilemma

6 Non-Gaussian white noise

7 Colored noise

Chapter 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

Chapter 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 fluctuations

Chapter XII: FIRST-PASSAGE PROBLEMS

1 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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

Chapter 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 Schrödinger–Langevin equation and the quantum master equation

6 A new approach to noise

7 Internal noise

SUBJECT INDEX

## Description

The third edition of Van Kampen's standard work has been revised and updated. The main difference with the second edition is that the contrived application of the quantum master equation in section 6 of chapter XVII has been replaced with a satisfactory treatment of quantum fluctuations. Apart from that throughout the text corrections have been made and a number of references to later developments have been included. From the recent textbooks the following are the most relevant.

C.W.Gardiner, Quantum Optics (Springer, Berlin 1991)
D.T. Gillespie, Markov Processes (Academic Press, San Diego 1992)
W.T. Coffey, Yu.P.Kalmykov, and J.T.Waldron, The Langevin Equation (2nd edition, World Scientific, 2004)

## Key Features

- Comprehensive coverage of fluctuations and stochastic methods for describing them
- A must for students and researchers in applied mathematics, physics and physical chemistry

## Readership

Students and researchers in applied mathematics, physics and physical chemistry

## Details

- No. of pages:
- 464

- Language:
- English

- Copyright:
- © North Holland 2007

- Published:
- 21st March 2007

- Imprint:
- North Holland

- eBook ISBN:
- 9780080475363

- Paperback ISBN:
- 9780444529657

## About the Authors

### N.G. Van Kampen Author

### Affiliations and Expertise

Institute of Theoretical Physics, University of Utrecht, The Netherlands