Hydrodynamic Equations from Fokker-Planck Equations –Multiple Time Scale Method–

Abstract

It is well-known that a Fokker-Planck equation, which governs time evolution of a distribution function of a Brownian particle in µ-space, can be reduced to a diffusion equation, the so-called Smoluchowski equation, in coordinate space after relaxation in momentum space is established. In this paper new derivations of the Smoluchowski equation are given. One is based on a multiple time scale (MTS) method and the other on a projection operator (PO) method. Emphasis is put on the MTS method since it has the advantage of displaying the physics of the relaxation processes contained in the kinetic equation. This aspect of the MTS method is further elucidated by our derivation of a (linearized) Navier-Stokes equation from a conserving Fokker-Planck equation.