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 , that is, the probability density function of the particle having a velocity in the interval , when it starts its motion with at time .
The general form of the Fokker-Planck equation for N variables is
where is the drift vector
and the diffusion tensor
, the latter of which results from the presence of the stochastic force.