Deriving the Fokker-Planck equation

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In the theory of dynamic systems, Fokker-Planck equation is used to describe the time evolution of the probability density function. It is a partial differential equation that describes how the density of a stochastic process changes as a function of time under the influence of a potential field. Some common application of it are in the study of Brownian motion, Ornstein–Uhlenbeck process, and in statistical physics. Here I attempt to note the formal derivation of the partial differential equation by deriving a master equation and using Taylor series to obtain the Kramers-Moyal expansion. A special case of the expansion with finite sum is called the Fokker-Planck equation.

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