Klein–Kramers equation

In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation is a partial differential equation that describes the probability density function of a Brownian particle in phase space. It is a special case of the Fokker–Planck equation.
In one spatial dimension, is a function of three independent variables: the scalars,, and. In this case, the Klein–Kramers equation is
where is the external potential, is the particle mass, is the friction coefficient, is the temperature, and is the Boltzmann constant. In spatial dimensions, the equation is
Here and are the gradient operator with respect to and, and is the Laplacian with respect to.
The fractional Klein-Kramers equation is a generalization that incorporates anomalous diffusion by way of fractional calculus.

Physical basis

The physical model underlying the Klein–Kramers equation is that of an underdamped Brownian particle. Unlike standard Brownian motion, which is overdamped, underdamped Brownian motion takes the friction to be finite, in which case the momentum remains an independent degree of freedom.
Mathematically, a particle's state is described by its position and momentum, which evolve in time according to the Langevin equations
Here is -dimensional Gaussian white noise, which models the thermal fluctuations of in a background medium of temperature. is a row vector, making an outer product; is the by identity matrix.
These equations are analogous to Newton's second law of motion, but due to the noise term are stochastic rather than deterministic.
The dynamics can also be described in terms of a probability density function, which gives the probability, at time, of finding a particle at position and with momentum. By averaging over the stochastic trajectories from the Langevin equations, can be shown to obey the Klein–Kramers equation.

Solution in free space

The -dimensional free-space problem sets the force equal to zero, and considers solutions on that decay to 0 at infinity, i.e., as.
For the 1D free-space problem with point-source initial condition,, the solution which is a bivariate Gaussian in and was solved by Subrahmanyan Chandrasekhar in 1943:
where
This special solution is also known as the Green's function, and can be used to construct the general solution, i.e., the solution for generic initial conditions :
Similarly, the 3D free-space problem with point-source initial condition has solution
with,, and and defined as in the 1D solution.

Asymptotic behavior

Under certain conditions, the solution of the free-space Klein–Kramers equation behaves asymptotically like a diffusion process. For example, if
then the density satisfies
where is the free-space Green's function for the diffusion equation.

Solution near boundaries

The 1D, time-independent, force-free version of the Klein–Kramers equation can be solved on a semi-infinite or bounded domain by separation of variables. The solution typically develops a boundary layer that varies rapidly in space and is non-analytic at the boundary itself.
A well-posed problem prescribes boundary data on only half of the domain: the positive half at the left boundary and the negative half at the right. For a semi-infinite problem defined on, boundary conditions may be given as:
for some function.
For a point-source boundary condition, the solution has an exact expression in terms of infinite sum and products: Here, the result is stated for the non-dimensional version of the Klein–Kramers equation:
In this representation, length and time are measured in units of and, such that and are both dimensionless. If the boundary condition at is, where, then the solution is
where
This result can be obtained by the Wiener–Hopf method. However, practical use of the expression is limited by slow convergence of the series, particularly for values of close to 0.