Langevin and Fokker-Planck Equations and their Generalizations: Descriptions and Solutions

Description

This invaluable book provides a broad introduction to a rapidly growing area of nonequilibrium statistical physics. The first part of the book complements the classical book on the Langevin and Fokker–Planck equations (H. Risken, The Fokker–Planck Equation: Methods of Solution and Applications (Springer, 1996)). Some topics and methods of solutions are presented and discussed in details which are not described in Risken's book, such as the method of similarity solution, the method of characteristics, transformation of diffusion processes into the Wiener process in different prescriptions, harmonic noise and relativistic Brownian motion. Connection between the Langevin equation and Tsallis distribution is also discussed. Due to the growing interest in the research on the generalized Langevin equations, several of them are presented. They are described with some details. Recent research on the integro-differential Fokker–Planck equation derived from the continuous time random walk model shows that the topic has several aspects to be explored. This equation is worked analytically for the linear force and the generic waiting time probability distribution function. Moreover, generalized Klein-Kramers equations are also presented and discussed. They have the potential to be applied to natural systems, such as biological systems. Preface Contents 1. Introduction 2. Langevin and Fokker-Planck equations 2.1 Introduction 2.2 General Langevin equation for one variable 2.3 Kramers-Moyal expansion coefficients 2.4 Ito, Stratonovich and other prescriptions 2.5 Kramers-Moyal expansion and Fokker-Planck equation 2.6 General Langevin equation for several variables 2.7 Appendices 2.7.1 Colored noise 3. Fokker-Planck equation for one variable and its solution 3.1 Introduction 3.2 Time-independent drift and diffusion coefficients 3.2.1 Stationary solution 3.2.2 Method of separation of variables and eigenfunction expansion 3.2.3 Solution for the harmonic potential 3.3 Solution by the method of transformation of variables 3.3.1 Solution for constant diffusion coefficient and the general linear force 3.3.2 Solution for time dependent drift coefficient and variable diffusion coefficient 3.4 Langevin equation with multiplicative white noise: Transformation of diffusion processes into the Wiener process in different prescriptions 3.4.1 Deterministic drift h₁(x,t) and multiplicative noise term h₂(x,t) are separable in time and space 3.4.2 General drift h₁(x,t) and multiplicative noise term h₂(x,t) 3.4.3 Applications 3.5 Similarity solution 3.5.1 Similarity solution 3.6 Solution in a finite interval and first passage time 3.6.1 First passage time distribution and mean first passage time 3.6.2 Solution for vanishing drift coefficient and constant diffusion coefficient 3.6.3 Solution for power-law diffusion coefficient 3.7 Langevin equation with multiplicative noise in different orders of prescription and its connection with the Tsallis distribution 3.7.1 Tsallis distribution and the atom-laser interaction in the optical lattice 3.7.2 Tsallis distribution and a class of populationgrowth models with linearly coupled noise 4. Fokker-Planck equation for several variables 4.1 Introduction 4.2 Ornstein-Uhlenbeck process 4.3 The Klein-Kramers equation for a linear force in a one dimensional space 4.4 Harmonic oscillator driven by colored noise in a one dimensional space 4.4.1 Stationary solution 4.4.2 Time-dependent solution 4.5 Fokker-Planck equations of the relativistic Brownian motion 4.6 Appendices 4.6.1 Numerical inversion of Laplace transforms 5. Generalized Langevin equations 5.1 Introduction 5.2 Derivation of the generalized Langevin equation from the Hamiltonian formalism 5.2.1 General analysis: First two moments, variances, covariance, PDF and Fokker-Planck equation 5.3 Fractional Langevin equation 5.4 Appendices 5.4.1 Generalized Mittag-Leffler function 6. Continuous Time Random Walk model 6.1 Introduction 6.2 Uncoupled Continuous Time Random Walk model 6.3 Integro-differential equations 6.3.1 Integro-differential diffusion equation and Integro-differential Fokker-Planck equation 6.3.2 Generalized Chapman-Kolmogorov equation and integro-differential Klein-Kramers equation 6.4 Appendices 6.4.1 Integro-differential diffusion equation with external force 6.4.2 Riesz space fractional derivative 7. Uncoupled Continuous Time Random Walk model and its solution 7.1 Introduction 7.2 Integro-differential diffusion equation for force-free 7.2.1 The first two moments and PDF 7.2.2 First passage time density and mean first passage time 7.2.3 Full decoupled case 7.3 Integro-differential Fokker-Planck equation or integro-differential diffusion equation with external force 7.3.1 Correlation function, mean square displacement, intermediate scattering function and dynamic structure factor 7.4 Generalized Rayleigh equation 7.5 Integro-differential Klein-Kramers equation 7.6 Generalized Klein-Kramers equation 7.6.1 First two moments for velocity and displacement in the force-free case 7.6.2 Solution of the generalized Klein-Kramers equation 7.7 Applications 7.8 Appendices 7.8.1 Wright function Index A B C D E F G-Gen Gen H I J K L M N O P R S-Sim Sim T V W