Brownian motion of two interacting particles through logarithmic and lennard-jones potentials

Abstract

The Brownian motion of two interacting particles through the logrithmic and Lennard–Jones potentials has been treated based on the Smoluchowski equation to check the validity of Einstein's relation between the mean-square displacement and time. The former case is considered analytically to obtain moments as well as the probability density exactly with the orientational motion, and the latter numerically. It is shown that for both cases the mean-square displacement is proportional to time for long times; however, the slope depends on the strength of the potential, showing a deviation from Einstein's relation, and the slope depends mainly on the repulsive interaction. Asymptotic expressions at long time for the logarithmic potential have been obtained in the form of t^–P, where t is time. P for the moments for the interparticle distance r;〈r^u〉(µ > 0) were found to be –µ/2 independent of the potential parameter, therefore P is the same as the free Brownian motion, whereas those for the probability density and the orientational correlation functions depend strongly upon the potential parameter.