Analytic calculation of the diffusion coefficient for random walks on strips of finite width: Dependence on size and nature of boundaries

Abstract

We study unbiased random walks in discrete time n on a square lattice, in the form of a strip of finite width N in the y direction, with a family of boundary conditions parametrized by a stay probability \ensuremath{\Gamma} per time step at the edge sites. The diffusion coefficient K=${\mathrm{lim}}_{\mathit{n}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈${\mathit{X}}_{\mathit{n}}^{2}$〉/n is computed analytically to exhibit its dependence on N and \ensuremath{\Gamma}. The result is generalized to the case of a strip with side branches attached to the boundary sites to simulate the effect of rough edges. A further generalization is made to obtain K for a random walk in d dimensions on a lattice bounded in one of the directions. Thus, K serves as a probe of both the transverse size of the region in which diffusion takes place and the nature of the bounding surfaces.