FokkerPlanckGleichung

The FokkerPlanckGleichung, known in English as the Fokker–Planck equation, describes the time evolution of the probability density of a continuous stochastic process. It is commonly derived as the forward equation for systems governed by Langevin dynamics, where a set of variables x evolves via a drift term a(x,t) and a diffusion term characterized by a matrix D(x,t). The equation for the probability density p(x,t) is ∂t p(x,t) = -∑i ∂i [a_i(x,t) p(x,t)] + 1/2 ∑i∑j ∂i ∂j [D_{ij}(x,t) p(x,t)]. Here ∂i denotes partial differentiation with respect to x_i, and D = B B^T is the diffusion matrix obtained from the noise terms.

In one dimension with constant diffusion D, the equation reduces to ∂t p = -∂x (a(x,t) p) + (D/2) ∂x^2 p. The Fokker–Planck equation has a corresponding backward form used for computing expectations of functionals of the process. It is applicable to Markov processes in continuous state spaces and is closely related to the master equation in discrete settings.

Key properties include conservation of total probability and, under detailed balance, convergence to equilibrium distributions such as Boltzmann forms in physical systems. The equation is central in physics, chemistry, biology, and quantitative finance, where it models Brownian motion, diffusion, reaction-diffusion systems, and various stochastic processes. Analytic solutions are available for simple cases, while numerical methods such as finite differences and spectral approaches are used for more complex, high-dimensional problems.