Fokker-Planck Equation| 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.
main page
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