The generalized Langevin equation as a contraction of the description. An approach to tracer diffusion

Abstract

An effective Langevin equation for a tracer Brownian particle immersed in a macrofluid of other diffusing particles is derived as a contraction of the description involving the stochastic equations for the local concentration and the local current of the macrofluid particles. The resulting Langevin equation contains the effects of the interactions with the other diffusing particles in a temporally non-local friction term plus a fluctuating force representing the random, diffusion-driven departures from spherical symmetry of the distribution of macrofluid particles around the tracer. This fluctuating force satisfies a fluctuation–dissipation relation with the effective time-dependent friction. This program is fully developed here only in the absence of hydrodynamic interactions, although the formal aspects of its extension are also suggested. The results derived here, however, are found to provide a unifying framework to describe, for example, self-friction and electrolyte friction in suspensions of charged colloidal particles within the same theoretical scheme.