1‐D random walk in presence of a totally reflecting barrier

Abstract

AbstractA rigorous analysis of counting paths for 1‐D random walk in the presence of a reflecting barrier is presented. This paper defines and distinguishes between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. An exact combinatorial formula is proven which describes the probability distribution of a diffusing particle at a totally reflecting barrier, allowing computation of any random walk redistribution of a diffusing species near or at the totally reflecting barrier. The analysis shows that for a particle starting its random walk at the barrier, the probability of finding it at the interface is diminishing with the number of diffusion steps N as 1/(N/2 + 1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as . This analysis has implications on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.