Lattice walks by long jumps

Diffusion on surfaces has in the past been modeled as a random walk in continuous time between nearest‐neighbor sites on a lattice. In order to allow tests for the possible participation of long jumps in actual diffusion processes, we examine the properties of random walks in which transitions are not limited to jumps between nearest‐neighbor sites. Two features of such walks are of special interest: (a) the moments of the displacements, which are directly related to the diffusivity and the statistical uncertainties in its determination; (b) the distribution function governing the probability of displacements, which is an important indicator of the contributions from long jumps. The techniques used to develop expressions for these quantities are illustrated for random walks in one dimension, with transitions allowed between neighbors up to three spacings removed. The appropriate probability generating function is derived starting from the Kolmogoroff equation. This is then manipulated to yield both the mo...