Microscopic derivation of zero-flux boundary condition in one-dimensional random walk in presence of a totally reflecting barrier

Abstract

In this paper a rigorous probabilistic two-probability-parameter model of a diffusion barrier is investigated that describes comprehensively reflection, absorption, and segregation phenomena at a diffusion barrier. As a special case, a rigorous analysis of counting paths for 1D random walk in the presence of a reflecting barrier is presented. This paper defines and makes distinction between partially and totally reflecting barriers. So far, in the literature only a special case of partially reflecting barrier has been dealt with. A combinatorial formula is derived showing that in the presence of a totally reflecting barrier (at m b=0) the probability of a particle departing from position m=2 j and arriving at position m=2 k on the positive axis after N=2 M steps is given by W 2 j (2 k, N) = [ C(2 M, M− j+ k)− C(2 M, M− j− k− l)]/[ C(2 M, M)+2 ∑ i=0 j C(2 M, M+ i)], where C( n, m) denotes the binomial coefficient. This formula enables easy 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=2 M as 1/( M+1) and that the peak of the probability distribution is moving away from the barrier with the increasing number of steps as 4M . Thus, the subsurface region is progressively depleted. The present analysis has bearing on the treatment of diffusion of impurities and point defects in thin films and in subsurface layers.