Fokker-Planck equation

The Fokker-Planck equation is a partial differential equation that describes the time evolution of the probability density function of the velocity of a particle under the influence of drag forces and random forces, as in Brownian motion. The equation can be generalized to other observables as well. It is named after Adriaan Fokker and Max Planck and is also known as the Kolmogorov forward equation (diffusion), named after Andrey Kolmogorov, who first introduced it in a 1931 paper. When applied to particle position distributions, it is better known as the Smoluchowski equation (after Marian Smoluchowski). The case with zero diffusion is known in statistical mechanics as the Liouville equation.

The first consistent microscopic derivation of the Fokker-Planck equation in the single scheme of classical and quantum mechanics was performed by Nikolay Bogoliubov and Nikolay Krylov. The Smoluchowski equation is the Fokker-Planck equation for the probability density function of the particle positions of Brownian particles.

See also

Dynamical systems, Statistical Physics, Stochastic processes

Papers

• Risken, H. (1984). Fokker-planck equation. In The Fokker-Planck Equation. Springer Berlin Heidelberg, 63-95.
• Jordan, R., Kinderlehrer, D., & Otto, F. (1998). The variational formulation of the Fokker–Planck equation. SIAM journal on mathematical analysis, 29(1), 1-17.
• Barkai, E., Metzler, R., & Klafter, J. (2000). From continuous time random walks to the fractional Fokker-Planck equation. Physical Review E, 61(1), 132.

Books

• Gardiner, C. W. (1985). Handbook of stochastic methods (Vol. 3). Berlin: Springer.
• Zubarev, D. N., Morozov, V., & Röpke, G. (1996). Statistical mechanics of nonequilibrium processes (Vol. 1). Berlin: Akademie Verlag.
• Risken, H. (1989). The Fokker-Planck Equation. Methods of Solution and Applications, vol. 18 of. Springer Series in Synergetics.