Fokker-Planck Equation

As shown in Sects. 3.1, 2 we can immediately obtain expectation values for processes described by the linear Langevin equations (3.1, 31). For nonlinear Langevin equations (3.67, 110) expectation values are much more difficult to obtain, so here we first try to derive an equation for the distribution function. As mentioned already in the introduction, a differential equation for the distribution function describing Brownian motion was first derived by Fokker [1.1] and Planck [1.2]: many review articles and books on the Fokker-Planck equation now exist [1.5 – 15].

[1]  F. Dyson.  The Radiation Theories of Tomonaga, Schwinger, and Feynman , 1949 .

[2]  Adalbert Duschek,et al.  Grundzüge der Tensorrechnung in Analytischer Darstellung , 1950 .

[3]  Dr. phil. A. Duschek (o. Prof. a. d. Techn. Hochschule Wien) und Dr. techn. A. Hochrainer, Grundzüge der Tensorrechnung in analytischer Darstellung. II. Teil: Tensoranalysis. VII + 338 S. m. 64 Abb. Wien 1950. Springer‐Verlag. Preis 24,80 DM . , 1951 .

[4]  Lars Onsager,et al.  Fluctuations and Irreversible Processes , 1953 .

[5]  R. F. Pawula,et al.  Approximation of the Linear Boltzmann Equation by the Fokker-Planck Equation , 1967 .

[6]  好村 滋洋.  G. Hohler 編: Springer Tracts in Modern Physics, Vol. 64; Tasso Springer: Quasielastic Neutron Scattering for the Investigation of Diffusive Motions in Solids and Liquids, Springer-Verlag, Berlin and Heidelberg, New York, 1972, 100ページ, 24×16cm, 4,370円 , 1973 .

[7]  Hermann Haken,et al.  Generalized Onsager-Machlup function and classes of path integral solutions of the Fokker-Planck equation and the master equation , 1976 .

[8]  H. Leschke,et al.  Operator orderings and functional formulations of quantum and stochastic dynamics , 1977 .

[9]  Robert Graham,et al.  Path integral formulation of general diffusion processes , 1977 .

[10]  Robert Graham,et al.  Covariant formulation of non-equilibrium statistical thermodynamics , 1977 .

[11]  U. Weiss.  Operator ordering schemes and covariant path integrals of quantum and stochastic processes in Curved space , 1978 .

[12]  C. Wissel,et al.  Manifolds of equivalent path integral solutions of the Fokker-Planck equation , 1979 .

[13]  Robert Graham,et al.  Fluctuations and nonlinear irreversible processes. II , 1979 .

[14]  H. Risken,et al.  On the application of truncated generalized Fokker-Planck equations , 1979 .

[15]  Two particle model for the diffusion of interacting particles in periodic potentials , 1979 .