Overdamped Brownian motion in periodic symmetric potentials

摘要

The dynamics of an overdamped Brownian particle in the field of a one-dimensional symmetric periodic potential U(x;alpha) have been studied by numerical solution of the Smoluchowski diffusion equation and the Langevin equation using the Brownian Dynamics method. The parameter alpha controls the shape and height of the potential barrier, which ranges from a sinusoidal spatial dependence for low barrier heights (alpha small) to a near delta-function appearance for barrier heights tending to infinity (alpha very large). Both the mean square displacement (MSD) d(alpha)(t), and the probability density n(x,tx(0)), where x(0) denotes the initial position, have been calculated. The MSD over a wide time domain has been obtained for a number of values of alpha. The exact asymptotic (t --> infinity) form of the diffusion coefficient has been exploited to obtain an accurate representation for d(alpha)(t) at long times. The function, d(alpha)(t) changes its form in the range alpha =8-10, with the appearance of a "plateau" which signals a transition in the particle's Brownian dynamics from a weakly hindered (but continuous) mechanism to essentially jump diffusion. In the limit alpha --> infinity, each well of U(x;alpha) becomes similar to the classical square well (SW), which we have revisited as it provides a valuable limiting case for d(alpha)(t) at alpha much greater than1. An effective "attraction" of the probability density towards the SW walls is observed for off-center initial starting positions, and it is suggested that this could explain an observed change in the analytic form of the SW MSD, d(sw)(t), at long times. Two approximate analytic forms for d(sw)(t) at short times have been derived. The relaxation of the Brownian particle distribution n(x,tx(0)) in the initial-well of U(x;alpha) has been studied. (C) 2000 American Institute of Physics. [S0021-9606(00)50246-3]. [References: 35]