Size-dependent diffusion coefficient in a myopic random walk on a strip

Abstract

We consider a random walk in discrete time (n = 0, 1, 2, …) on a square lattice of finite width in the y-direction, i.e., {j.m|j ε Z. M=1.2.3.... N} . A myopic walker at (j,1) or (j, N) jumps with probability fraction 1/3 to any of the available nearest-neighbor sites at the end of a time step. This couples the motions in the x- and y-directions, and leads to several interesting features, including a coefficient of diffusion in the x-direction that depends on the transverse size N of the strip. Explicit solutions for (x²_n) (and the lateral variance (y²_n)) are given for small values of N. A closed-form expression is obtained for the (discrete Laplace) transform of (x²_n) for general N. The asymptotic behaviors of (x²_n) and (y²_n)are found, the corrections falling off exponentially with increasing n. The results obtained are generalized to a myopic random walk in d dimensions, and it is shown that the diffusion coefficient has an explicit geometry dependence involving the surface-to-volume ratio. This coefficient can therefore serve as a probe of the geometry of the structure on which diffusion takes place.