The Fokker-Planck Equation

Abstract

In this chapter we derive the one-dimensional Fokker-Planck equation, which defines the change of the probability of a state variable x in dependence of time, i.e., $\frac{dP\left(x;t\right)}{dt}$. Starting from the Gaussian normal distribution, we formulate the deterministic drift term of the equation, which can be illustrated by the potential function of a fixed-point attractor (causation). Deterministic dynamics entailed by an attractor compresses state space. The stochastic diffusion term of the Fokker-Planck equation (chance) can be expressed by the variance of the Gaussian. Stochastic inputs expand state space and ultimately destroy attractors. The Fokker-Planck equation combines both dynamics and thus integrates causation with chance in a process model. Deterministic forces can be measured by the relaxation time of a system, i.e., the duration until the equilibrium state is realized again after some displacement of x. The location and drift of attractors can be estimated by the means of the values of the state variable.