 Fokker-Planck Equation

In General > s.a. Kramers Equation; stochastic processes.
* Idea: An equation describing stochastic diffusion processes, or the time evolution of a non-equilibrium probability distribution.
* More general formulation: The continuity equation

t ρ = − ∂a J a

applied to the stochastic evolution of a system on a manifold M of states, where ρ is a scalar density interpreted as probability density, and J a = ρ va − ∂b (ρKab) the probability current, with

va := limt → 0 t−1 ∫ p(ξ; x, t) ξa dξ ,

Kab := limt → 0 t−1  p(ξ; x, t) ξa ξb dξ ,

where p(ξ; x, τ) is the probability that the system will evolve from x to x + ξ in a time τ; we assume that similar expressions with more ξs vanish; notice that v does not transform like a vector.
* Applications: It is a central equation in the theory of brownian motion; Also used, e.g., by astronomers to find the general evolution of the orbits of stars in clusters or galaxies, and by population geneticists to describe random genetic drift.
* Note: In the theory of continuous-time, continuous-state Markov processes it is also known as the Kolmogorov forward equation.

References > s.a. brownian motion; phase transitions.
@ Texts: Papoulis 65; Soize 94 [non-linear]; Risken 96.
@ General articles: Desloge AJP(63)apr; Miyazawa JMP(99), JMP(00) [Green function]; Wei JPA(00) [approach to solution]; Kleinert AP(01) [from the forward-backward path integral]; Sparber et al mp/02 [quantum, long-time behavior];
Lo PLA(03) [propagator]; Oron & Horwitz mp/03 [covariant Brownian motion]; Lubashevsky et al mp/06 [boundary conditions]; Lucia PhyA(13) [and entropy generation]; Ryter a1403 [uniqueness].
@ Approaches: Jafari & Aminataei PS(09) [homotopy perturbation method]; Verevkin TMP(11) [Euler-Darboux transformation and solutions].
@ Non-linear: Donoso et al JPA(99) [short-time propagator]; Kiessling & Lancellotti TTSP(04)mp-conf; Plyukhin PhyA(05) [higher-order corrections]; Donoso et al JPA(05), Donoso & Salgado JPA(06) [propagator]; Tsekov IJTP(09)-a0808.
@ Quantum version: Neumann & Sparber CMS(07)-a0707 [for bosons and fermions].
@ Fractional: Schertzer et al JMP(01)m.AP/04; Tarasov & Zaslavsky PhyA(08); Zhang & Chen PhyA(14) [stochastic stability].
@ Other generalized: Chavanis PhyA(04) [generalized thermodynamics]; Khan & Reynolds PhyA(05) [generalized Langevin dynamics]; Rybicki ApJ(06)ap/05 [for resonance-line scattering]; Alcántara & Calogero KRM(11)-a1011, JMP(13) [relativistic]; > s.a. diffusion [in curved spacetime]; heat equation [relativistic].
> Related topics: see hamilton-jacobi theory; Maxwell-Boltzmann Distribution.

Online Resources > see Physics Daily page; Wikipedia Fokker-Planck equation page, and Kolmogorov backward equation page.