Fokker–Planck equation

FokkerPlanck

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The Fokker–Planck equation (also known as the Kolmogorov forward 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 forces. It is widely used in statistical mechanics and stochastic processes.

Formulation

The Fokker–Planck equation can be written in the general form: <math> \frac{\partial f(x,t)}{\partial t} = -\frac{\partial}{\partial x} \left[ A(x) f(x,t) \right] + \frac{\partial^2}{\partial x^2} \left[ B(x) f(x,t) \right] </math> where:

• \( f(x,t) \) is the probability density function of the variable \( x \) at time \( t \),
• \( A(x) \) represents the drift coefficient,
• \( B(x) \) represents the diffusion coefficient.

Applications

The Fokker–Planck equation is used in various fields such as:

• Physics: To describe the behavior of particles in a fluid or plasma.
• Finance: To model the evolution of stock prices and other financial variables.
• Biology: To study the dynamics of populations and the spread of diseases.

Special Cases

There are several special cases of the Fokker–Planck equation, including:

• The Ornstein–Uhlenbeck process, which describes the velocity of a particle undergoing Brownian motion with friction.
• The Smoluchowski equation, which is a simplified version of the Fokker–Planck equation for overdamped systems.

Numerical Solutions

Solving the Fokker–Planck equation analytically is often challenging, so numerical methods are frequently employed. Common techniques include:

• Finite difference method
• Monte Carlo simulation
• Spectral method

See Also

• Langevin equation
• Brownian motion
• Stochastic differential equation
• Diffusion equation