Molecular Theory of Brownian Motion for Several Particles

Abstract

A molecular derivation is presented for the coupled Langevin equations that describe the motion of heavy particles in a fluid. In contrast to the case of a single heavy particle, the frictiontensors which appear depend upon the instantaneous separations between the particles. A Fokker–Planck equation describing the reduced distribution function for the heavy particles is also obtained. Calculations with these equations require evaluation of the frictiontensors. The frictiontensors are evaluated in two ways, by an approximate macroscopic hydrodynamic calculation and an approximate hydrodynamic fluctuation calculation. Both calculations lead to identical expressions for the frictiontensors which are shown to have a long‐range character at large interparticle separations. Finally, it is shown that these long‐range effects cancel in the calculation of the diffusion constant for each particle but remain in the calculation of the relative diffusion constant.