Fokker–Planck equation

The Fokker -Planck equation (FPG, by Adriaan Daniël Fokker ( 1887-1972 ) and Max Planck ( 1858-1947 ) )

Describes the time evolution of a probability distribution of the effects of drift and diffusion. The equation is in probability theory also known as the Kolmogorov forward equation. It is a linear partial differential equation. For vanishing drift, it proceeds in the diffusion (or heat conduction ) equation.

In D dimensions is the Fokker -Planck equation

The FPG is evident for Markov processes from the Kramers- Moyal development, truncated after the second order.

The FPG is a partial differential equation that only for some special cases (in terms of simple body geometry and linearity of the boundary conditions and the drift and diffusion coefficients) can be analytically solved exactly. Of great importance is the equivalent description of problems by Langevin equations describing FPG compared to the microscopic dynamics of stochastic systems and - in contrast - are generally not linear.

Derivation

The FPG can be from the continuous Chapman - Kolmogorov equation, a more general equation for the time evolution of probabilities in Markov processes derived. If is a continuous variable and the jumps are small. In this case, a Taylor series (in this case it is referred to as Kramer - Moyal development ) of the Chapman - Kolmogorov equation

Possible and gives the FPG. Here, the probability that a state of the state transitions. You can also directly start the development of the master equation, then the Taylor expansion is no longer necessary after the time.

Under the assumption that the transition probability at large distances is small ( just only small jumps take place ) you can use the following Taylor expansion (using the summation convention ):

By performing integration ( as not dependent, it may be pulled out from the integrals ) is then obtained

With

Stationary solution

The stationary solution of the one-dimensional FPG, ie all, is given by

Wherein said normalization constant may be determined by the condition. It should be noted that the integral for the bottom disappears. In the case of higher dimensions can generally no longer find steady-state solution; here one has to rely on various approximation methods.

Context of stochastic differential equations

Is for the functions and the stochastic differential equation ( the interpretation Ito ) in the form

Given, with a -dimensional Wiener process ( Brownian motion ) refers. Then satisfies the density function of a FPG, are given in the drift and diffusion coefficients and.

Fokker-Planck equation and path integral

Each Fokker -Planck equation is equivalent to a path integral. This example follows the fact that the general Fokker -Planck equation for D variables q = { qi }

Same structure as the Schrödinger equation has. The Fokker - Planck operator F corresponds to the Hamiltonian, the probability distribution P corresponds to the wave function ψ. The equivalent to the Fokker -Planck equation path integral is accordingly (see path integral )

Where N is a constant normalization factor. Path integrals of this type are in the critical dynamics starting point for perturbation theory and renormalization group. The variables q are doing, for example, for Fourierkomponenen of the order parameter. The variables glad Response variables. The Lagrangian L contains the response variable only in a square shape. In contrast to quantum mechanics, it is here but not zweckmäß? Strength to perform the integrations.

Fokker -Planck equation in the plasma physics

The Fokker - Planck equation in plasma physics, especially of importance, since the collision term of the kinetic equation ( Boltzmann equation ) can be written for plasmas as a Fokker-Planck term. The equation is also called the Landau equation, since it was first established by Lev Davidovich Landau, but not. Within your Fokker-Planck form

The reason that the shock term can be written as a Fokker-Planck term is that the movement of the particles in the plasma is dominated by the numerous collisions with distant partners which carry only a small change in the speed. Strong collisions with nearby particles are comparatively rare and therefore often negligible.

The single-particle density distribution in velocity space for particles of type, specifies how many particles it at a certain speed there. In a plasma, to which no external forces, the change in the density distribution by collisions with particles of the type can be approximated by the equation

With

And

Are described. It is the Coulomb logarithm, the larger the value, the more easily the dominance of many collisions and the better the Landau Fokker -Planck equation is valid. and the electric charges of the particle types, their mass. Since the particles in the plasma collide with the particles of the same species, which is normally non-linear equation.

This equation is replaced by the number of particles, the momentum and energy. They also met the H-theorem, that is, Shocks lead to a Maxwell -Boltzmann velocity distribution.