Tracer diffusion in small periodic systems

We study tracer diffusion in lattices consisting of identical replicas of smaller systems. We develop an exact enumeration method for computing the diffusion coefficient, which involves mapping the system into a single-particle random walk in a different geometry in configuration space. In one dimension the simulations show a crossover to normal diffusion, in agreement with mode-coupling results. In two dimensions the diffusion coefficient is enhanced by finite-size effects: we show this for a 3×{}3 periodic system. We find exact solutions for the single-vacancy case; our simulations agree with theory essentially to machine precision. We also develop an approximate theory for the periodic system for arbitrary concentrations of background particles, in close agreement with the numerical results.