Fokker–Planck equation

[
thumb|A_solution_to_the_one-dimensional_Fokker–Planck_equation,_with_both_the_drift_and_the_diffusion_term._The_initial_condition_is_a_Dirac delta function in "x" = 1, and the distribution drifts towards "x" = 0.] The Fokker–Planck equation describes the time evolution of the probability density function of the position of a particle, and can be generalized to other observables as well. [cite book | title = Statistical Physics: statics, dynamics and renormalization
author = Leo P. Kadanoff | publisher = World Scientific | isbn = 9810237642 | year = 2000 | url = http://books.google.com/books?id=22dadF5p6gYC&pg=PA135&ots=_yDpXsrPqY&dq=Fokker%E2%80%93Planck&sig=OgjxJK7nfTYTVDAmAhkP3bpqviU#PPA134,M1 ] It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation.The first use of the "Fokker–Planck" equation was the statistical description of Brownian motion of a particle in a fluid.

In one spatial dimension "x", the Fokker–Planck equation for a process with drift "D"₁("x","t") and diffusion "D"₂("x","t") is:$frac\{partial\}\{partial\; t\}f(x,t)=-frac\{partial\}\{partial\; x\}left\; [\; D\_\{1\}(x,t)f(x,t)\; ight]\; +frac\{partial^2\}\{partial\; x^2\}left\; [\; D\_\{2\}(x,t)f(x,t)\; ight]\; .$

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

:$frac\{partial\; f\}\{partial\; t\}\; =\; -sum\_\{i=1\}^\{N\}\; frac\{partial\}\{partial\; x\_i\}\; left\; [\; D\_i^1(x\_1,\; ldots,\; x\_N)\; f\; ight]\; +\; sum\_\{i=1\}^\{N\}\; sum\_\{j=1\}^\{N\}\; frac\{partial^2\}\{partial\; x\_i\; ,\; partial\; x\_j\}\; left\; [\; D\_\{ij\}^2(x\_1,\; ldots,\; x\_N)\; f\; ight]\; ,$

where $D^1$ is the drift vector and $D^2$ the diffusion tensor; the latter results from the presence of the stochastic force.

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

:$D^1\_i(mathbf\{x\},t)\; =\; mu\_i(mathbf\{x\},t)$

:$D^2\_\{ij\}(mathbf\{x\},t)\; =\; frac\{1\}\{2\}\; sum\_k\; sigma\_\{ik\}(mathbf\{x\},t)\; sigma\_\{kj\}^mathsf\{T\}(mathbf\{x\},t).$

Similarly, a Fokker–Planck equation can be derived for Stratonovich stochastic differential equations. In this case, noise-induced drift terms appear if the noise strength is state-dependent.

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

:$frac\{partial\; f(x,t)\}\{partial\; t\}\; =\; frac\{1\}\{2\}\; frac\{partial^2\; f(x,t)\}\{partial\; x^2\},$

that is the simplest form of diffusion equation.

A simple algebraic substitution shows that: $f(x,t)\; =\; frac\{e^\{-frac\{x^2\}\{2t\}\{sqrt\{2\; pi\; t,$is a solution to this equation.

Computational considerations

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.

olution

Being a partial differential equation, the Fokker–Planck equation can be solved analytically only in special cases. A formal analogy of the Fokker-Planck equation with the Schroedinger equation allows the use of advanced operator techniques known from quantum mechanics for its solution in a number of cases. In many applications, one is only interested in the steady-state probability distribution$f\_0(x)$, which can be found from $dot\{f\}\_0(x)=0$.The computation of mean first passage times and splitting probabilities can be reduced to the solution of an ordinary differential equation which is intimately related to the Fokker–Planck equation.

ee also

*Kolmogorov backward equation
*Boltzmann equation
*Navier–Stokes equations
*Vlasov equation
*Master equation

References

External links

* [http://members.aol.com/jeff570/f.html Fokker–Planck equation] on the [http://members.aol.com/jeff570/mathword.html 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.