One of the central problems synergetics is concerned with consists in the study of macroscopic qualitative changes of systems belonging to various disciplines such as physics, chemistry, or electrical engineering. When such transitions from one state to another take place, fluctuations, i.e., random processes, may play an im­ portant role. Over the past decades it has turned out that the Fokker-Planck equation pro­ vides a powerful tool with which the effects of fluctuations close to transition points can be adequately treated and that the approaches based on the Fokker­ Planck equation are superior to other approaches, e.g., based on Langevin equa­ tions. Quite generally, the Fokker-Planck equation plays an important role in problems which involve noise, e.g., in electrical circuits. For these reasons I am sure that this book will find a broad audience. It pro­ vides the reader with a sound basis for the study of the Fokker-Planck equation and gives an excellent survey of the methods of its solution. The author of this book, Hannes Risken, has made substantial contributions to the development and application of such methods, e.g., to laser physics, diffusion in periodic potentials, and other problems. Therefore this book is written by an experienced practitioner, who has had in mind explicit applications to important problems in the natural sciences and electrical engineering.

1. Introduction.- 1.1 Brownian Motion.- 1.1.1 Deterministic Differential Equation.- 1.1.2 Stochastic Differential Equation.- 1.1.3 Equation of Motion for the Distribution Function.- 1.2 Fokker-Planck Equation.- 1.2.1 Fokker-Planck Equation for One Variable.- 1.2.2 Fokker-Planck Equation for TV Variables.- 1.2.3 How Does a Fokker-Planck Equation Arise?.- 1.2.4 Purpose of the Fokker-Planck Equation.- 1.2.5 Solutions of the Fokker-Planck Equation.- 1.2.6 Kramers and Smoluchowski Equations.- 1.2.7 Generalizations of the Fokker-Planck Equation.- 1.3 Boltzmann Equation.- 1.4 Master Equation.- 2. Probability Theory.- 2.1 Random Variable and Probability Density.- 2.2 Characteristic Function and Cumulants.- 2.3 Generalization to Several Random Variables.- 2.3.1 Conditional Probability Density.- 2.3.2 Cross Correlation.- 2.3.3 Gaussian Distribution.- 2.4 Time-Dependent Random Variables.- 2.4.1 Classification of Stochastic Processes.- 2.4.2 Chapman-Kolmogorov Equation.- 2.4.3 Wiener-Khintchine Theorem.- 2.5 Several Time-Dependent Random Variables.- 3. Langevin Equations.- 3.1 Langevin Equation for Brownian Motion.- 3.1.1 Mean-Squared Displacement.- 3.1.2 Three-Dimensional Case.- 3.1.3 Calculation of the Stationary Velocity Distribution Function.- 3.2 Ornstein-Uhlenbeck Process.- 3.2.1 Calculation of Moments.- 3.2.2 Correlation Function.- 3.2.3 Solution by Fourier Transformation.- 3.3 Nonlinear Langevin Equation, One Variable.- 3.3.1 Example.- 3.3.2 Kramers-Moyal Expansion Coefficients.- 3.3.3 Itô's and Stratonovich's Definitions.- 3.4 Nonlinear Langevin Equations, Several Variables.- 3.4.1 Determination of the Langevin Equation from Drift and Diffusion Coefficients.- 3.4.2 Transformation of Variables.- 3.4.3 How to Obtain Drift and Diffusion Coefficients for Systems.- 3.5 Markov Property.- 3.6 Solutions of the Langevin Equation by Computer Simulation.- 4. Fokker-Planck Equation.- 4.1 Kramers-Moyal Forward Expansion.- 4.1.1 Formal Solution.- 4.2 Kramers-Moyal Backward Expansion.- 4.2.1 Formal Solution.- 4.2.2 Equivalence of the Solutions of the Forward and Backward Equations.- 4.3 Pawula Theorem.- 4.4 Fokker-Planck Equation for One Variable.- 4.4.1 Transition Probability Density for Small Times.- 4.4.2 Path Integral Solutions.- 4.5 Generation and Recombination Processes.- 4.6 Application of Truncated Kramers-Moyal Expansions.- 4.7 Fokker-Planck Equation for N Variables.- 4.7.1 Probability Current.- 4.7.2 Joint Probability Distribution.- 4.7.3 Transition Probability Density for Small Times.- 4.8 Examples for Fokker-Planck Equations with Several Variables.- 4.8.1 Three-Dimensional Brownian Motion without Position Variable.- 4.8.2 One-Dimensional Brownian Motion in a Potential.- 4.8.3 Three-Dimensional Brownian Motion in an External Force.- 4.8.4 Brownian Motion of Two Interacting Particles in an External Potential.- 4.9 Transformation of Variables.- 4.10 Covariant Form of the Fokker-Planck Equation.- 5. Fokker-Planck Equation for One Variable; Methods of Solution.- 5.1 Normalization.- 5.2 Stationary Solution.- 5.3 Ornstein-Uhlenbeck Process.- 5.4 Eigenfunction Expansion.- 5.5 Examples.- 5.5.1 Parabolic Potential.- 5.5.2 Inverted Parabolic Potential.- 5.5.3 Infinite Square Well for the Schrödinger Potential.- 5.5.4 V-Shaped Potential for the Fokker-Planck Equation.- 5.6 Jump Conditions.- 5.7 A Bistable Model Potential.- 5.8 Eigenfunctions and Eigenvalues of Inverted Potentials.- 5.9 Approximate and Numerical Methods for Determining Eigenvalues and Eigenfunctions.- 5.9.1 Variational Method.- 5.9.2 Numerical Integration.- 5.9.3 Expansion into a Complete Set.- 5.10 Diffusion Over a Barrier.- 5.10.1 Kramers' Escape Rate.- 5.10.2 Bistable and Metastable Potential.- 6. Fokker-Planck Equation for Several Variables; Methods of Solution.- 6.1 Approach of the Solutions to a Limit Solution.- 6.2 Expansion into a Biorthogonal Set.- 6.3 Transformation of the Fokker-Planck Operator, Eigenfunction Expansions.- 6.4 Detailed Balance.- 6.5 Ornstein-Uhlenbeck Process.- 6.6 Further Methods for Solving the Fokker-Planck Equation.- 6.6.1 Transformation of Variables.- 6.6.2 Variational Method.- 6.6.3 Reduction to an Hermitian Problem.- 6.6.4 Numerical Integration.- 6.6.5 Expansion into a Complete Set.- 6.6.6 Matrix Continued-Fraction Method.- 6.6.7 WKB Method.- 7. Linear Response and Correlation Functions.- 7.1 Linear Response function.- 7.2 Correlation Functions.- 7.3 Susceptibility.- 8. Reduction of the Number of Variables.- 8.1 First-Passage Time Problems.- 8.2 Drift and Diffusion Coefficients Independent of Some Variables.- 8.2.1 Time Integrals of Markovian Variables.- 8.3 Adiabatic Elimination of Fast Variables.- 8.3.1 Linear Process with Respect to the Fast Variable.- 8.3.2 Connection to the Nakajima-Zwanzig Projector Formalism.- 9. Solutions of Tridiagonal Recurrence Relations, Application to Ordinary and Partial Differential Equations.- 9.1 Applications and Forms of Tridiagonal Recurrence Relations.- 9.1.1 Scalar Recurrence Relation.- 9.1.2 Vector Recurrence Relation.- 9.2 Solutions of Scalar Recurrence Relations.- 9.2.1 Stationary Solution.- 9.2.2 Initial Value Problem.- 9.2.3 Eigenvalue Problem.- 9.3 Solutions of Vector Recurrence Relations.- 9.3.1 Initial Value Problem.- 9.3.2 Eigenvalue Problem.- 9.4 Ordinary and Partial Differential Equations with Multiplicative Harmonic Time-Dependent Parameters.- 9.4.1 Ordinary Differential Equations.- 9.4.2 Partial Differential Equations.- 9.5 Methods for Calculating Continued Fractions.- 9.5.1 Ordinary Continued Fractions.- 9.5.2 Matrix Continued Fractions.- 10. Solutions of the Kramers Equation.- 10.1 Forms of the Kramers Equation.- 10.1.1 Normalization of Variables.- 10.1.2 Reversible and Irreversible Operators.- 10.1.3 Transformation of the Operators.- 10.1.4 Expansion into Hermite Functions.- 10.2 Solutions for a Linear Force.- 10.2.1 Transition Probability.- 10.2.2 Eigenvalues and Eigenfunctions.- 10.3 Matrix Continued-Fraction Solutions of the Kramers Equation.- 10.3.1 Initial Value Problem.- 10.3.2 Eigenvalue Problem.- 10.4 Inverse Friction Expansion.- 10.4.1 Inverse Friction Expansion for K0(t), G0, 0(t) and L0(t).- 10.4.2 Determination of Eigenvalues and Eigenvectors.- 10.4.3 Expansion for the Green's Function Gn, m(t).- 10.4.4 Position-Dependent Friction.- 11. Brownian Motion in Periodic Potentials.- 11.1 Applications.- 11.1.1 Pendulum.- 11.1.2 Supenonic Conductor.- 11.1.3 Josephson Tunneling Junction.- 11.1.4 Rotation of Dipoles in a Constant Field.- 11.1.5 Phase-Locked Loop.- 11.1.6 Connection to the Sine-Gordon…