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 moments and the distribution of displacements in terms of the jump rates to first-, second-, and higher-nearest neighbors. Inasmuch as this approach to the distance distribution is not universally feasible, the latter is also evaluated directly as a combinatorial problem for diffusion by single and double jumps. Generating function techniques are then used to describe two-dimensional diffusion with contributions from long jumps on the (110) plane of the bcc lattice. Atomic jumps of different lengths are allowed along the close-packed directions, and jumps along the Cartesian coordinates are considered as well. The most challenging problem is to describe diffusion on the (111) plane of the fcc lattice, on which atoms can be bound at two different types of positions, one surrounded by the other, so that double jumps are no longer uncorrelated with single jumps. Lower moments of the displacements are developed from rather complicated generating functions, but combinatorial methods have to be used to derive the complete distribution function governing displacements. © 1990 American Institute of Physics.