 in statistical mechanics, the fokkerâplanck 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 fokkera. d. fokker, die mittlere energie rotierender elektrischer dipole im strahlungsfeld, ann. phys. 348 (4. folge 43), 810â820 (1914). and max planck m. planck, sitz.ber. preuÃ. akad. (1917). and is also known as the kolmogorov forward equation (diffusion), named after andrey kolmogorov, who first introduced it in a 1931 paper.andrei kolmogorov, "on analytical methods in the theory of probability", 448-451, (1931), (in german). when applied to particle position distributions, it is better known as the smoluchowski equation. the case with zero diffusion is known in statistical mechanics as liouville equation. the first consistent microscopic derivation of the fokkerâplanck equation in the single scheme of classical and quantum mechanics was performedn. n. bogolyubov (jr) and d. p. sankovich (1994). "n. n. bogolyubov and statistical mechanics". russian math. surveys 49(5): 19â49. by nikolay bogoliubov and nikolay krylov.n. n. bogoliubov and n. m. krylov (1939). fokkerâplanck equations generated in perturbation theory by a method based on the spectral properties of a perturbed hamiltonian. zapiski kafedry fiziki akademii nauk ukrainian ssr 4: 81â157 (in ukrainian). the smoluchowski equation (after marian smoluchowski) is the fokkerâplanck equation for the probability density function of the particle positions of brownian particles.dhont, an introduction to dynamics of colloids, p. 183 == one dimension == in one spatial dimension x, for an itÅ process given by the stochastic differential equation (sde) :dx_t = \mu(x_t,t)dt + \sqrt{2 d(x_t,t)}dw_t with drift \mu(x_t,t) and diffusion coefficient d(x_t,t), and wiener process w_t, the fokkerâ
Equations Parabolic_partial_differential_equations Stochastic_processes