Klein–Kramers equation
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In physics and mathematics, the Klein–Kramers equation or sometimes referred as Kramers–Chandrasekhar equation[1] is a partial differential equation that describes the probability density function f (r, p, t) of a Brownian particle in phase space (r, p).[2][3] It is a special case of the Fokker–Planck equation.
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
Here
and
are the gradient operator with respect to r and p, and
is the Laplacian with respect to p.
The fractional Klein-Kramers equation is a generalization that incorporates anomalous diffusion by way of fractional calculus.[4]