4. Fokker-Planck Equation

4. Fokker-Planck Equation

1996 | H. Risken
The chapter discusses the derivation of the Fokker-Planck equation, which is a differential equation describing the evolution of the distribution function in systems with Brownian motion. The derivation begins with the Kramers-Moyal expansion, which expresses the transition probability in terms of its moments. For nonlinear Langevin equations, the Kramers-Moyal coefficients for $n \geq 3$ vanish, simplifying the expansion to only include the drift and diffusion coefficients. This results in the Fokker-Planck equation, which reduces the problem of obtaining averages to solving this equation. The chapter then outlines two methods for deriving the transition probability: one using the characteristic function and another using a Taylor series expansion of the delta function.The chapter discusses the derivation of the Fokker-Planck equation, which is a differential equation describing the evolution of the distribution function in systems with Brownian motion. The derivation begins with the Kramers-Moyal expansion, which expresses the transition probability in terms of its moments. For nonlinear Langevin equations, the Kramers-Moyal coefficients for $n \geq 3$ vanish, simplifying the expansion to only include the drift and diffusion coefficients. This results in the Fokker-Planck equation, which reduces the problem of obtaining averages to solving this equation. The chapter then outlines two methods for deriving the transition probability: one using the characteristic function and another using a Taylor series expansion of the delta function.
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