FOKKER–PLANCK EQUATION

The 'Fokker–Planck equation' (named after Adriaan Fokker and Max Planck; also known as the 'Kolmogorov forward equation') describes the time evolution of the probability density function of position and velocity of a particle, but it can be generalized to any other observable, too.^[1] It applies to systems that can be described by a small number of "macrovariables", where other parameters vary so rapidly with time that they can be treated as noise.
The first use of the ''Fokker–Planck'' equation was the statistical description of Brownian motion of a particle in a fluid.
Brownian motion follows the Langevin equation, which can be solved for many different stochastic forcings with results being averaged (the Monte Carlo method, canonical ensemble in molecular dynamics).
However, instead of this computationally intensive approach, one can use the Fokker–Planck equation and consider $f(mathbf\{v\},\; t)$, that is, the probability density function of the particle having a velocity in the interval $(mathbf\{v\},\; mathbf\{v\}\; +\; dmathbf\{v\})$, when it starts its motion with $mathbf\{v\}\_0$ at time 0.
More generally, the time-dependent probability distribution may depend on a set of $N$ macrovariables $x\_i$. The general form of the Fokker–Planck equation is then
:
ight] f,
where $D^1$ is the drift vector and $D^2$ the diffusion tensor, the latter of which results from the presence of the stochastic force.

Contents

Relationship with stochastic differential equations

Examples

See also

References

External links

Books

Relationship with stochastic differential equations

The Fokker–Planck equation can be used for computing the probability densities of stochastic differential equations. Consider the Itō stochastic differential equation
:
where $mathbf\{X\}\_t\; in\; mathbb\{R\}^N$ is the state and $mathbf\{W\}\_t\; in\; mathbb\{R\}^M$ is a standard M-dimensional Wiener process. If the initial distribution is $mathbf\{X\}\_0\; sim\; f(mathbf\{x\},0)$, then the probability density $f(mathbf\{x\},t)$ of the state $mathbf\{X\}\_t$ is given by the Fokker–Planck equation with the drift and diffusion terms
:$D^1\_i(mathbf\{x\},t)\; =\; mu\_i(mathbf\{x\},t)$
:

Examples

A standard scalar Wiener process is generated by the stochastic differential equation
:$mathrm\{d\}X\_t\; =\; mathrm\{d\}W\_t.$
Now the drift term is zero and diffusion coefficient is 1/2 and thus the corresponding Fokker–Planck equation is
:$$
rac{partial f(x,t)}{partial t} = rac{1}{2} rac{partial^2 f(x,t)}{partial x^2},

that is the simplest form of diffusion equation.

See also

★ Kolmogorov backward equation

References

1. Statistical Physics: statics, dynamics and renormalization, Leo P. Kadanoff, , , World Scientific, 2000,

External links

★ Fokker–Planck equation on the Earliest known uses of some of the words of mathematics

Books

★ Hannes Risken, "The Fokker–Planck Equation: Methods of Solutions and Applications", 2nd edition, Springer Series in Synergetics, Springer, ISBN 3-540-61530-X.

★ Crispin W. Gardiner, "Handbook of Stochastic Methods", 3rd edition (paperback), Springer, ISBN 3-540-20882-8.