Ultra-Slow 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 case when the charged TP experiences a bias due to external electric field ${\bf E}$, (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 $\bar{\bf X}_n$; the latter is shown to obey an anomalous, logarithmic law $|\bar{\bf X}_n| = α_0(|{\bf E}|) \ln(n)$. On comparing our results with earlier predictions by Brummelhuis and Hilhorst (J. Stat. Phys. {\bf 53}, 249 (1988)) for the TP diffusivity $D_n$ in the unbiased case, we infer that the Einstein relation $μ_n = βD_n$ between the TP diffusivity and the mobility $μ_n = \lim_{|{\bf E}| \to 0}(|\bar{\bf X}_n|/| {\bf E} |n)$ holds in the leading in $n$ order, despite the fact that both $D_n$ and $μ_n$ are not constant but vanish as $n \to \infty$. We also generalize our approach to the situation with very small but finite vacancy concentration $ρ$, in which case we find a ballistic-type law $|\bar{\bf X}_n| = πα_0(|{\bf E}|) ρn$. We demonstrate that here, again, both $D_n$ and $μ_n$, calculated in the linear in $ρ$ approximation, do obey the Einstein relation.