In one spatial dimension, f is a function of three independent variables: the scalars x, p, and t. In this case, the Klein–Kramers equation is
where V(x) is the external potential, m is the particle mass, ξ is the friction (drag) coefficient, T is the temperature, and kB is the Boltzmann constant. In d spatial dimensions, the equation is
The physical model underlying the Klein–Kramers equation is that of an underdamped Brownian particle.[3] 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 r and momentum p, which evolve in time according to the Langevin equations
The dynamics can also be described in terms of a probability density function f (r, p, t), which gives the probability, at time t, of finding a particle at position r and with momentum p. By averaging over the stochastic trajectories from the Langevin equations, f (r, p, t) can be shown to obey the Klein–Kramers equation.
Solution in free space
The d-dimensional free-space problem sets the force equal to zero, and considers solutions on that decay to 0 at infinity, i.e., f (r, p, t) → 0 as |r| → ∞.
For the 1D free-space problem with point-source initial condition, f (x, p, 0) = δ(x - x')δ(p - p'), the solution which is a bivariate Gaussian in x and p was solved by Subrahmanyan Chandrasekhar (who also devised a general methodology to solve problems in the presence of a potential) in 1943:[3][5]
where
This special solution is also known as the Green's functionG(x, x', p, p', t), and can be used to construct the general solution, i.e., the solution for generic initial conditions f (x, p, 0):
Similarly, the 3D free-space problem with point-source initial condition f (r, p, 0) = δ(r - r') δ(p - p') has solution
with , , and and defined as in the 1D solution.[5]
Asymptotic behavior
Under certain conditions, the solution of the free-space Klein–Kramers equation behaves asymptotically like a diffusion process. For example, if
The 1D, time-independent, force-free (F = 0) 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 p domain: the positive half (p > 0) at the left boundary and the negative half (p < 0) at the right.[7] For a semi-infinite problem defined on 0 < x < ∞, boundary conditions may be given as:
for some function g(p).
For a point-source boundary condition, the solution has an exact expression in terms of infinite sum and products:[8][9] 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 z = 0 is g(w) = δ(w - w0), where w0 > 0, 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 w close to 0.[10]