One-dimensional-particle kinetics in a multicomponent alloy. The asymptotic time dependence of the mean-square displacement

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Abstract
The mean-square displacement, 〈r2(t)〉, of a labeled particle hopping on a linear chain through a multicomponent dynamic background with vacancy concentration v is evaluated in the limit of extremely long times. We find 〈r2(t)〉=[2va2(1−v)12](J̃tπ)12, where J̃ is a collective hopping rate given by (ΣλxλJλ)−1. Here Jλ is the hopping rate and xλ the concentration of the λth atomic species in the background.