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), 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.In the following, use σ = 2 D {displaystyle sigma ={sqrt {2D}}} . 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), 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. 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 W t {displaystyle W_{t}} and described by the stochastic differential equation (SDE) with drift μ ( X t , t ) {displaystyle mu (X_{t},t)} and diffusion coefficient D ( X t , t ) = σ 2 ( X t , t ) / 2 {displaystyle D(X_{t},t)=sigma ^{2}(X_{t},t)/2} , the Fokker–Planck equation for the probability density p ( x , t ) {displaystyle p(x,t)} of the random variable X t {displaystyle X_{t}} is While the Fokker–Planck equation is used with problems where the initial distribution is known, if the problem is to know the distribution at previous times, the Feynman–Kac formula can be used, which is a consequence of the Kolmogorov backward equation. The stochastic process defined above in the Itō sense can be rewritten within the Stratonovich convention as a Stratonovich SDE: