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 $\mathbf{E}$ (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 ${\overline{\mathbf{X}}}_n;$ the latter is shown to obey an anomalous, logarithmic law $|{\overline{\mathbf{X}}}_n|={\alpha }_0\left(|\mathbf{E}|\right)\ln \mathrm{}\left(n\right).$ 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 ${\mu }_n=\beta D_n$ between the TP diffusivity and the mobility ${\mu }_n={limop}_{|\mathbf{E}\overrightarrow{|}0}(|{\overline{\mathbf{X}}}_n|/|\mathbf{E}|n)$ holds in the leading n order, despite the fact that both $D_n$ and ${\mu }_n$ are not constant but vanish as $\overrightarrow{n}\infty .$ 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 $|{\overline{\mathbf{X}}}_n|=\pi {\alpha }_0\left(|\mathbf{E}|\right){\rho }_{v}n.\mathrm{}$ We demonstrate that here, again, both $D_n$ and ${\mu }_n,$ calculated in the linear in ${\rho }_v$ approximation, do obey the Einstein relation.