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* *v*^{a}
− ∂_{b}
(*ρK*^{ab}) the probability current, with

*v*^{a} := lim_{t
→ 0} *t*^{−1}
∫ *p*(*ξ*;* x*, *t*)
*ξ*^{a} d*ξ* ,

*K*^{ab} :=
lim_{t → 0} *t*^{−1}
∫ *p*(*ξ*; *x*, *t*)
*ξ*^{a}
*ξ*^{b} d*ξ* ,

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 *ξ*s 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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 27 oct 2018