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 different 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 
, that is, the probability density function of the particle having a velocity in the interval 
, when it starts its motion with 
at time t0.
The general form of the Fokker-Planck equation for N variables is
![\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},](a/../upload/math/c343daed270336b99991fd764e3cf216.png)
where D1 is the drift vector and D2 the diffusion tensor, the latter of which results from the presence of the stochastic force.
Last updated: 10-18-2005 03:20:03
