Fokker-Planck equation

The Fokker-Planck equation describes the time evolution of the probability density function of position and velocity of a particle.

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 differrent stochastic forcings with results being averaged (the Monte Carlo Method).

However, instead of this computationally intensive approach, one can use the Fokker-Planck equation and consider <math>\mathbf{W}(\mathbf{v}, t)</math>, that is, the probability density function of the particle having a velocity in the interval <math>(\mathbf{v}, \mathbf{v} + \mathbf{dv})</math>, when it starts its motion with <math>\mathbf{v_0}</math> at time <math>t_0</math>.

The general form of the Fokker-Planck equation for N variables is

<math>\frac{\partial \mathbf{W}}{\partial t} = \left[-\sum_{i=1}^{N} \frac{\partial}{\partial x_i} D_i^1(x_1, \ldots, x_N) + \sum_{i=1}^{N} \sum_{j=1}^{N} \frac{\partial^2}{\partial x_i \partial x_j} D_{ij}^2(x_1, \ldots, x_N) \right] \mathbf{W}</math>,

where <math>D^1</math> is the drift vector and <math>D^2</math> the diffusion tensor, the latter of which results from the presence of the stochastic force.