Ultraslow vacancy-mediated tracer diffusion in two dimensions:: The Einstein relation verified -: art. no. 031101

We study the dynamics of a charged tracer particle (TP) on a two-dimensional lattice, all sites of which except one (a vacancy) are filled with identical neutral, hard-core particles. The particles move randomly by exchanging their positions with the vacancy, subject to the hard-core exclusion. In the case when the charged TP experiences a bias due to external electric field E (which favors its jumps in the preferential direction), we determine exactly the limiting probability distribution of the TP position in terms of appropriate scaling variables and the leading large-n (n being the discrete time) behavior of the TP mean displacement (X) over bar (n); the latter is shown to obey an anomalous, logarithmic law \(X) over bar (n)\=alpha(0)(\E\)ln(n). Comparing our results with earlier predictions by Brummelhuis and Hilhorst [J. Stat. Phys. 53, 249 (1988)] for the TP diffusivity D-n in the unbiased case, we infer that the Einstein relation mu(n)=betaD(n) between the TP diffusivity and the mobility mu(n)=lim(\E\-->0)(\(X) over bar (n)\/\E\n) holds in the leading n order, despite the fact that both D-n and mu(n) are not constant but vanish as n-->infinity. We also generalize our approach to the situation with very small but finite vacancy concentration rho(v), in which case we find a ballistic-type law \(X) over bar (n)\=pialpha(0)(\E\)rho(v)n. We demonstrate that here, again, both D-n and mu(n), calculated in the linear in rho(v) approximation, do obey the Einstein relation.