Statistics of one-dimensional cluster motion

Abstract

The statistics of clusters, made up of metal atoms in adjacent one-dimensional diffusion channels, are developed quantitatively. Kolmogorov’s equation is used to find the mean square displacement for clusters capable of existing in energetically different configurations at the same position of the center of mass; this is done under steady state conditions, for which the probability of finding a specified configuration does not vary in time. Two systems are examined: (1) dimers capable of existing in an infinite number of states, a situation realized if dissociation is allowed, and (2) trimers diffusing on planes, such as W(211), on which nine distinct jump processes may contribute to the cluster motion. In dimer diffusion it is demonstrated that dissociation may be important even if the fraction dissociated is minor. For trimers, previous attempts to approximate the motion through the use of average transition rates are compared with the exact solutions and found wanting. Important statistical quantities beyond the mean square displacement are presented for simple dimers capable of existing in only two states. The generating function is derived, together with the higher moments of the displacement. Probability density functions for the number of jumps in an interval t, for the waiting time up to a specified jump, as well as for the displacements are all presented. These differ significantly from the density functions for an ordinary random walk. However, an averaging technique allows simple approximations for the behavior of dimers.