The first-order concentration dependence of the mutual diffusion coefficient of Brownian particles

Abstract

The first-order concentration dependence of the mutual diffusion coefficient of spherical Brownian particles in a suspension has previously been calculated essentially within the framework of the Smoluchowski equation in the configuration space of a pair of particles. After these calculations are reviewed, the same quantity is here treated on the basis of the Fokker–Planck equation in the phase space of a pair of particles which is considered to have a wider range of validity than the Smoluchowski equation. Two methods are used in the treatment. In one method the mobility of the particles is determined by solving the Fokker–Planck equation in a stationary state which is reached by application of the same external force on each particle. In the other method a known expression for the mobility in terms of time-correlation functions is evaluated by solving the time-dependent Fokker–Planck equation in the absence of external forces. The mobility thus obtained is transformed into the diffusion coefficient with the aid of the generalized Einstein relation. It is found that both methods lead to the same result for the diffusion coefficient as derived from the Smoluchowski equation. Implications of the present calculations are also discussed.