Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified

Abstract

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 $\mathbf{E}$, (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 ${\bar{\mathbf{X}}}_n$; the latter is shown to obey an anomalous, logarithmic law $|{\bar{\mathbf{X}}}_n|={\alpha }_0(|\mathbf{E}|)\ln (n)$. 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 ${\mu }_n=\beta D_n$ between the TP diffusivity and the mobility ${\mu }_n={\lim }_{|\mathbf{E}|\to 0}(|{\bar{\mathbf{X}}}_n|/|\mathbf{E}|n)$ holds in the leading in $n$ order, despite the fact that both $D_n$ and ${\mu }_n$ are not constant but vanish as $n\to \infty$. We also generalize our approach to the situation with very small but finite vacancy concentration $\rho$, in which case we find a ballistic-type law $|{\bar{\mathbf{X}}}_n|=\pi {\alpha }_0(|\mathbf{E}|)\rho n$. We demonstrate that here, again, both $D_n$ and ${\mu }_n$, calculated in the linear in $\rho$ approximation, do obey the Einstein relation.

arXiv comment25 pages, one figure, TeX, submitted to J. Stat. Phys

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