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 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 〈 x 2 n 〉 (and the lateral variance 〈 y 2 n 〉) are given for small values of N. A closed-form expression is obtained for the (discrete Laplace) transform of 〈 x 2 n 〉 for general N. The asymptotic behaviors of 〈 x 2 n 〉 and 〈 y 2 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.