Preface

This section presents basic material about Fokker-Planck equation.

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Glossary

Fokker--Planck equation

The Fokker--Planck equation was first introduced by the Dutch physicist and musician Andriaan Fokker (1887--1972) and the German theoretical physicist Max Planck (1858--1947) to describe the Brownian motion of particles. It is also known as the Kolmogorov forward equation, after Andrey Kolmogorov, who independently discovered the concept in 1931. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski), and in this context it is equivalent to the convection–diffusion equation. The case with zero diffusion is known in statistical mechanics as the Liouville equation. The Fokker–Planck equation is obtained from the master equation through Kramers–Moyal expansion. This equation has been used in different fields in naturalsciences such as quantum optics, solid state physics, chemical physics, theoretical biologyand circuit theory. Fokker--Planck equations describe the erratic motions of small particlesthat are immersed in fluids, fluctuations of the intensity of laser light, velocity distributions of fluid particles in turbulent flows and the stochastic behavior of exchange rates.In general, Fokker--Planck equations can be applied to equilibrium and nonequilibriumsystems. The general form of Fokker--Plank equation is
\[ \frac{\partial u}{\partial t} + \frac{\partial }{\partial x} \,a(x)\,u(x,t) = \frac{\partial^2}{\partial x^2} \,b(x)\,u(x,t) , \]
with the initial condition
\[ u(x,0) = f(x) . \]
Here u(x,t) is unknown function, while a(x) and b(x > 0 are called diffusion and drift coeffficients. These coefficients may also depend on time variable and unknown function u(x,t).

 

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