Stochastic Processes in Physics and Chemistry

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