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, and can be generalized to other observables as well. 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 D₁(x,t) and diffusion D₂(x,t) is

${\frac {\partial }{\partial t}}f(x,t)=-{\frac {\partial }{\partial x}}\left[D_{1}(x,t)f(x,t)\right]+{\frac {\partial ^{2}}{\partial x^{2}}}\left[D_{2}(x,t)f(x,t)\right]$

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

${\frac {\partial f}{\partial t}}=-\sum _{i=1}^{N}{\frac {\partial }{\partial x_{i}}}\left[D_{i}^{1}(x_{1},\ldots ,x_{N})f\right]+\sum _{i=1}^{N}\sum _{j=1}^{N}{\frac {\partial ^{2}}{\partial x_{i}\,\partial x_{j}}}\left[D_{ij}^{2}(x_{1},\ldots ,x_{N})f\right],$

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.

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

$\mathrm {d} \mathbf {X} _{t}={\boldsymbol {\mu }}(\mathbf {X} _{t},t)\,\mathrm {d} t+{\boldsymbol {\sigma }}(\mathbf {X} _{t},t)\,\mathrm {d} \mathbf {W} _{t},$

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

$D_{i}^{1}(\mathbf {x} ,t)=\mu _{i}(\mathbf {x} ,t)$

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

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

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

that is the simplest form of diffusion equation.

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 $(\mathbf {v} ,\mathbf {v} +d\mathbf {v} )$, when it starts its motion with $\mathbf {v} _{0}$ at time 0.

See also

• Kolmogorov backward equation

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.