Fokker–Planck equation
In statistical mechanics, 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, 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). 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.
In one spatial dimension x, for an Itō process driven by the standard Wiener process and described by the (SDE)
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