Brownian particles in shear flow and harmonic potentials: A study of long-time tails.

Abstract

In this paper we present the results of a study of the mean-square displacement of a Brownian particle in a harmonic potential and of a Brownian particle in shear flow. We have focused on the long-time behavior of the mean-square displacement. In contrast with earlier results, presented by others who studied the Stokes limit of these problems, we have studied this problem using the time-dependent linearized incompressible Navier-Stokes equations to describe the fluid motion. Then we see that the mean-square displacement is strongly influenced by backflow effects in the fluid, resulting, among other things, in long-time tails of correlation functions. We have compared our results with those calculated in the Stokes limit; important differences exist between them. The main differences are the long-time tails in correlation functions and, related with them, the larger time scales that should be considered to obtain diffusive behavior in the case of a Brownian particle in a harmonic potential or to obtain the cubic regime in the mean-square displacement of a Brownian particle in shear flow. Furthermore, we have studied the velocity autocorrelation function of a Brownian particle in a harmonic potential. In the overdamped case we have shown a ${\mathrm{\ensuremath{\tau}}}^{\mathrm{\ensuremath{-}}7/2}$ long-time tail instead of the exponential tail that can be obtained in the Stokes limit. Also the sign of both tails differ.