MULTIPLE TIME-SCALE DERIVATION OF THE FOKKER-PLANCK EQUATION FOR 2 BROWNIAN SPHERES SUSPENDED IN A HARD-SPHERE FLUID

The Fokker-Planck equation for the distribution function of two massiv e Brownian spheres, suspended in a fluid of much lighter spheres, is d erived from the full hierarchy of exact kinetic equations for the time evolution of the full system consisting of two Brownian and N fluid s pheres. The separation of time scales is automatically achieved by a s ystematic multiple timescale analysis of the expansion in powers of th e square root of the fluid-to-Brownian particle mass ratio. This proce dure guarantees uniform convergence of the expansion and requires no e xtra physical assumptions to justify the separation of time scales. An exact expression is obtained for the mutual friction tensors, which n aturally split into a static (Enskog) part and a contribution due to d ynamical correlations. The present derivation of the two-particle Fokk er-Planck equation also leads to an expression for the fluid-induced, effective depletion force between two Brownian particles.