Fokker-Planck
Equation| Fokker-Planck
Equation |
In General > s.a. 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 
= – a
J a
for the stochastic evolution of a system on a manifold M of states,
where
is
a scalar density interpreted as probability density, and Ja = va – b ( Kab)
the probability current, with
va := limt to
0 t–1
p(
; x, t)
a d ,
Kab :=
limt to 0 t–1
p(; x, t) a b d
(we assume that similar expressions with more 's
vanish), p(; x, 
)
being the probability that the system will evolve from x to x +
in a
time ;
Notice that v does not transform like a vector.
* Applications: 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.
References > s.a. brownian
motion; phase
transitions.
@ Texts: Papoulis 65; Soize 94 [non-linear];
Risken 96.
@ General articles:
Desloge AJP(63); 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].
@ Non-linear: Donoso et al JPA(99)
[short-time propagator]; Kiessling
& Lancellotti mp/04-in;
Plyukhin PhyA(05)
[higher-order corrections]; Donoso et al JPA(05),
Donoso & Salgado JPA(06)
[propagator].
@ Quantum version: Neumann & Sparber a0707-CMS
[for bosons and fermions].
@ Other generalized: Schertzer
et al JMP(01)m.AP/04 [fractional];
Chavanis
PhyA(04)
[generalized thermodynamics]; Khan & Reynolds PhyA(05)
[generalized Langevin dynamics]; Rybicki ApJ(06)ap/05
[for resonance line scattering].
Online Resources > see Physics Daily page.
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
21 jun 2008