Renormalization of the diffusion coefficient in a fluctuating fluid I

Abstract

The diffusion of tagged particles through a fluctuating fluid is described in terms of a nonlinear Langevin-like equation, taking into account the convection by the fluid and the dependence of the “bare” diffusion coefficient on, e.g., the local density of the fluid. Upon averaging over the fluid fluctuations a macroscopic diffusion equation is obtained with a renormalized wave vector and frequency-dependent diffusion coefficient. A closed expression for the this quantity is derived in terms of the correlation functions of the fluid and the bare diffusion propagator. To lowest order in the fluid correlations this expression is evaluated explicitly as a function of the wave vector and the time and as a function of the wave vector and the frequency. For zero wave vector the usual long-time and small-frequency dependence is found. An alternative exact expression for the renormalized diffusion coefficient containing the dressed instead of the bare diffusion propagator is derived. It is argued that the last expression evaluated to lowest order in the fluid correlations completely describes the asymptotic time behavior of the diffusion memory kernel for zero wave vector. Finally the fluctuation properties of the random current in the original nonlinear Langevin equation are discussed.