This book deals with the derivation of the Fokker-Planck equation, methods of solving it and some of its applications. Various methods such as the simulation method, the eigenfunction expansion, numerical integration, the variational method, and the matrix continued-fraction method are discussed. This is the first time that this last method, which is very effective in dealing with simple Fokker-Planck equations having two variables, appears in a textbook. The methods of solution are applied to the statistics of a simple laser model and to Brownian motion in potentials. Such Brownian motion is important in solid-state physics, chemical physics and electric circuit theory. This new study edition is meant as a text for graduate students in physics, chemical physics, and electrical engineering.

Klappentext

This is the first textbook to include the matrix continued-fraction method, which is very effective in dealing with simple Fokker-Planck equations having two variables. Other methods covered are the simulation method, the eigen-function expansion, numerical integration, and the variational method. Each solution is applied to the statistics of a simple laser model and to Brownian motion in potentials. The whole is rounded off with a supplement containing a short review of new material together with some recent references. This new study edition will prove to be very useful for graduate students in physics, chemical physics, and electrical engineering, as well as for research workers in these fields.

Inhalt

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 N 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 Schrüdinger 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 Complete Sets.- 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 Superionic Conductor.- 11.1.3 Josephson Tunneling Junctio…