On the effects of a totally reflecting barrier on an unbiased 1D random walk

Abstract

AbstractThis paper is devoted to discussing the behavior of an unbiased 1D random walk in a semi‐infinite lattice confined by a certain type of boundary, termedtotally reflecting, and in particular to proving the validity of the so‐called “reflection principle” for computing the probability mass function in such systems. This is motivated by the publication of an earlier paper by [Orlowski, Phys. Status Solidi B239, 158–161 (2003)], which denied the validity of the reflection principle, and proposed a different, inconsistent method of computing the walker's probability mass, and also by the fact that earlier authors published arguments for the reflection principle that themselves contained errors or were not entirely rigorous. This paper provides a new, rigorous argument for the validity of the reflection principle, and also shows where the error in Orlowski's analysis lies. It further contextualizes these arguments with respect to some existing literature on similar systems where the probability of reflection may be less than unity, and discusses the proper relationship between the discrete random walk, the diffusion equation, and approches which employ a different, master equation‐based discretization technique. The spatial relationship between the discrete and continuous formulations is discussed, and an existing derivation [van Kampen and Oppenheim, J. Math. Phys.13, 842–849 (1972)] is extended. Finally, it is shown that the reflection principle, as outlined in this paper, preserves initially uniform concentrations for all time (in contrast to Orlowski's proposed method).