A description of random walks with collision anisotropy and with a nonconstant mean free path

Abstract

We study various characteristics of a particle's random walk both analytically and with the help of Monte-Carlo simulation techniques. With the analytical approach, we derive the expression[Formula: see text]which relates the mean-square displacement [Formula: see text] to (i) the number of steps N in the walk, (ii) the mean-square displacement [Formula: see text] on each of the steps, and (iii) a coefficient of collision anisotropy A defined as the average value of the cosine of the scattering angle θ. This expression is general in the sense that it holds for any value of N and A. It is, however, restricted to cases where the mean free path is constant throughout the random walk. The results of the simulations allow a further generalization to random walks with a nonconstant mean free path. They also allow the study of the radial distribution f(r) of particles after the walk. We find that a set of six functions f_i(r) is necessary to give a satisfactory description of the particles' radial distribution for arbitrary values of N and A.