Ultraslow vacancy-mediated tracer diffusion in two dimensions: the Einstein relation verified.
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(n); the latter is shown to obey an anomalous, logarithmic law /X(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(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(n)/=pi(alpha)(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.