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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 f of a Brownian particle in phase space.
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 is the external potential, m is the particle mass, ξ is the friction coefficient, T is the temperature, and kB is the Boltzmann constant. In d spatial dimensions, the equation is
Here ∇ r {\displaystyle \nabla _{\mathbf {r} }} and ∇ p {\displaystyle \nabla _{\mathbf {p} }} are the gradient operator with respect to r and p, and ∇ p 2 {\displaystyle \nabla _{\mathbf {p} }^{2}} is the Laplacian with respect to p.