Sedimentation and dispersion of Brownian particles in spatially periodic potential fields

Abstract

A generalized Taylor dispersion analysis is presented of the combined diffusion and sedimentation of a Brownian particle subjected to a spatially periodic potential plus a uniform force. At long times this transport process is macroscopically characterized by a uniform mean drift velocity vector of the Brownian particle, and by a dispersion dyadic that quantifies stochastic ‘‘spreading’’ about the (traveling) mean position. A general explicit integral formula is derived for the dispersion coefficient in the one-dimensional case, which supplements known results for the mean velocity. Illustrative calculations show that, for sufficiently large values of the drift force, the dispersivity exceeds the molecular diffusivity, which characterizes dispersion in the absence of any potential barriers. These calculations also lead to the surprising conclusion that an increase in the height of the potential barriers can sometimes lead to an increase in the dispersivity. A general asymptotic analysis is developed for the limiting case where the drift force tends to infinity. Although in one dimension the dominance of the drift force over the potential leads to a spatially uniform, steady-state, periodic distribution, the same behavior does not generally obtain in higher-dimensional cases. Asymptotic expressions are derived for the mean velocity and dispersivity characterizing one-dimensional systems and a class of two-dimensional systems. The analysis is subsequently applied to the related problem of sedimentation and dispersion of a particle within a medium for which its molecular diffusivity is a periodic function of position (in the absence of any potential). Although a periodic potential and a periodic diffusivity lead to equivalent mathematical problems in the absence of a sedimentation force, the presence of drift leads to important qualitative differences.