Langevin equation

A Langevin equation ( by Paul Langevin ) is a stochastic differential equation describing the dynamics of a subset of the degrees of freedom of a physical system. These are typically ' slow ' ( macroscopic ) degrees of freedom, the ' fast ' ( microscopic ) degrees of freedom are responsible for the stochastic nature of the differential equation.

Conventional Langevin equation

The following equation for the velocity v (t) of particles suspended in a liquid particle is based on a heuristic approach of Paul Langevin:

Here, a coefficient of friction, and F (x, t) an external force caused, for example, by an external potential. The magnitude f ( t) is the so-called fluctuating speed, typically a Gaussian white noise.

General Langevin equation

The following equation is a formal generalization of the conventional Langevin equation:

The rate of the slow variables here consists of the M sizes.

Further, it is assumed to be a deterministic function, it contains the friction.

The size is a coupling matrix and describes the correlations of the various components of the stochastic noise.

One distinguishes two cases:

General scheme

Basic approach here is the addition of a fluctuation term to mean in a valid relationship:

For the noise is assumed:

Along with a usually assumed normal distribution is then a white Gaussian noise.

Relationship with the Fokker -Planck equation

In general, a solution of a Langevingleichung to a particular realization of the fluctuating force is uninteresting. Instead, we are interested in correlation functions of the slow variables after averaging over the fluctuating force. Such averages can also be obtained by other means, eg by means of the corresponding one of the Langevingleichung Fokker -Planck equation. A Fokker -Planck equation is a deterministic equation for the time-dependent probability distribution of a stochastic variable y. For example, is the Fokker -Planck equation

Equivalent to the above Langevin equation.

Examples

Brownian motion

Has the classic case for the Brownian motion of a particle in a fluid and Stokes - Einstein - Smoluchowski law and the relationship equation with damping coefficient.

Thermal noise

Thermal noise across a capacitor follows

With.