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28 March 2024
 
  » arxiv » cond-mat/0101117

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Ultra-Slow Vacancy-Mediated Tracer Diffusion in Two Dimensions: The Einstein Relation Verified
O.Benichou ; G.Oshanin ;
Date 9 Dec 2000
Subject Statistical Mechanics | cond-mat.stat-mech
Affiliation LPMC, College de France, Paris, France; LPTL, University of Paris VI, Paris, France
AbstractWe 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 ${f E}$, (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 $ar{{f X}}_n$; the latter is shown to obey an anomalous, logarithmic law $|ar{{f X}}_n| = alpha_0(|{f E}|) ln(n)$. On comparing our results with earlier predictions by Brummelhuis and Hilhorst (J. Stat. Phys. {f 53}, 249 (1988)) for the TP diffusivity $D_n$ in the unbiased case, we infer that the Einstein relation $mu_n = eta D_n$ between the TP diffusivity and the mobility $mu_n = lim_{|{f E}| o 0}(|ar{{f X}}_n|/| {f E} |n)$ holds in the leading in $n$ order, despite the fact that both $D_n$ and $mu_n$ are not constant but vanish as $n o infty$. We also generalize our approach to the situation with very small but finite vacancy concentration $ ho$, in which case we find a ballistic-type law $|ar{{f X}}_n| = pi alpha_0(|{f E}|) ho n$. We demonstrate that here, again, both $D_n$ and $mu_n$, calculated in the linear in $ ho$ approximation, do obey the Einstein relation.
Source arXiv, cond-mat/0101117
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