The Fokker–Planck Equation

Abstract

The Langevin equation is a stochastic differential equation and the position and velocity of the particle are random variables. It is thus natural to define the probability distribution function or p.d.f. of the position of the particle at time t for the case of the overdamped Langevin equation, and similarly, a probability distribution for the position and momentum of the particle for the underdamped Langevin equation. This p.d.f. satisfies a differential equation, called the Smoluchowski or the Fokker–Planck equation (depending on the presence of the mass term in the Langevin Equation). This chapter considers the overdamped case, the underdamped case being a simple generalization. The case of a Brownian particle with no external force can be easily solved for both the over- and underdamped cases. One of the main uses of the probability distribution is that it allows to compute thermal averages as a function of time.