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

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 13 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 〈x2n〉 (and the lateral variance 〈y2n〉) are given for small values of N. A closed-form expression is obtained for the (discrete Laplace) transform of 〈x2n〉 for general N. The asymptotic behaviors of 〈x2n〉 and 〈y2n〉 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.