The Connection between the Smoluchowski Equation and the Kramers-Chandrasekhar Equation

Abstract

It is shown that the Kramers and Chandrasekhar extension of the Fokker-Planck equation to phase space leads to a modified Telegraph or hyperbolic-type diffusion equation in configuration space. The solution of this equation at large times asymptotically approaches the solution of the Smoluchowski equation. The hyperbolic diffusion equation and the Smoluchowski equation differ by one term which is related to the fact that particle velocities exist in the former case whereas in the latter case they do not.