Overdamped Brownian motion in periodic symmetric potentials

The dynamics of an overdamped Brownian particle in the field of a one-dimen sional symmetric periodic potential U(x;alpha) have been studied by numeric al solution of the Smoluchowski diffusion equation and the Langevin equatio n using the Brownian Dynamics method. The parameter alpha controls the shap e and height of the potential barrier, which ranges from a sinusoidal spati al dependence for low barrier heights (alpha small) to a near delta-functio n appearance for barrier heights tending to infinity (alpha very large). Bo th the mean square displacement (MSD) d(alpha)(t), and the probability dens ity n(x,t\x(0)), where x(0) denotes the initial position, have been calcula ted. The MSD over a wide time domain has been obtained for a number of valu es of alpha. The exact asymptotic (t --> infinity) form of the diffusion co efficient has been exploited to obtain an accurate representation for d(alp ha)(t) at long times. The function, d(alpha)(t) changes its form in the ran ge alpha =8-10, with the appearance of a "plateau" which signals a transiti on in the particle's Brownian dynamics from a weakly hindered (but continuo us) mechanism to essentially jump diffusion. In the limit alpha --> infinit y, 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(al pha)(t) at alpha much greater than1. An effective "attraction" of the proba bility density towards the SW walls is observed for off-center initial star ting positions, and it is suggested that this could explain an observed cha nge in the analytic form of the SW MSD, d(sw)(t), at long times. Two approx imate analytic forms for d(sw)(t) at short times have been derived. The rel axation of the Brownian particle distribution n(x,t\x(0)) in the initial-we ll of U(x;alpha) has been studied. (C) 2000 American Institute of Physics. [S0021-9606(00)50246-3].