Fokker-Planck equation

Fokker-Planck Equation

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, originating from statistical physics and mathematical physics. It is a fundamental equation in the study of stochastic processes.

Overview

The Fokker-Planck equation is used to describe the dynamics of systems under the influence of noise and is a generalization of the Smoluchowski equation, which is itself a limit of the Fokker-Planck equation for large friction limits. It is named after Adriaan Fokker and Max Planck, who independently developed the equation in the early 20th century.

Mathematical Formulation

The standard form of the Fokker-Planck equation for the probability density function \\(P(x,t)\\) of a variable \\(x\\) at time \\(t\\) is given by:

\[ \frac{\partial P(x,t)}{\partial t} = -\frac{\partial}{\partial x}[A(x)P(x,t)] + \frac{\partial^2}{\partial x^2}[B(x)P(x,t)] \]

where \\(A(x)\\) represents the drift term, which is the deterministic part of the motion, and \\(B(x)\\) represents the diffusion term, which accounts for the random part of the motion.

Applications

The Fokker-Planck equation has a wide range of applications across various fields of science and engineering. It is extensively used in statistical mechanics, quantum mechanics, biophysics, economics, and financial mathematics to model the evolution of systems over time under the influence of random forces.

Numerical Solutions

Due to the complexity of the Fokker-Planck equation, analytical solutions are only available for a limited number of cases. Therefore, numerical methods, such as finite difference methods, Monte Carlo simulations, and finite element methods, are often employed to solve the equation for specific initial and boundary conditions.

See Also

• Stochastic differential equations
• Brownian motion
• Langevin equation