Microscopic Theory of Brownian Motion: The Multiple-Time-Scale Point of View

We present a microscopic derivation of a Fokker-Planck equation for the distribution function of a heavy Brownian particle in a dense fluid from the Liouville equation. The usual perturbation theory suffers from the presence of secular terms familiar from non-linear mechanics. We employ the "multiple time scale" technique developed by Frieman and Sandri to eliminate the secular terms and render the expansions uniformly valid in time. The method introduces explicit time variables to exploit the existence of a multiplicity of time scales inherent in the problem. Using this formalism we derive the Fokker-Planck equation for spatially inhomogeneous Brownian motion.