2001/01/09 by Olivier Bénichou, O. Benichou, Gleb Oshanin +1
Materials Science · Mathematics · Physics and Astronomy · #Material Dynamics and Properties #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.66.031101
25 pages, one figure, TeX, submitted to J. Stat. Phys
arxiv created 2001/01/09 · openalex publication_date 2002/09/04 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Xn; the latter is shown to obey an anomalous, logarithmic law |Xn|=\ensuremathα0(|E|)ln(n). Comparing our results with earlier predictions by Brummelhuis and Hilhorst [J. Stat. Phys. 53, 249 (1988)] for the TP diffusivity Dn in the unbiased case, we infer that the Einstein relation \ensuremathμn=\ensuremathβDn between the TP diffusivity and the mobility \ensuremathμn=limop_|E\stackrel\ensuremath→|0(|Xn|/|E|n) holds in the leading n order, despite the fact that both Dn and \ensuremathμn are not constant but vanish as \stackrel\ensuremath→n\ensuremath∞. We also generalize our approach to the situation with very small but finite vacancy concentration \ensuremathρv, in which case we find a ballistic-type law |Xn|=\ensuremathπ\ensuremathα0(|E|)\ensuremathρvn. We demonstrate that here, again, both Dn and \ensuremathμn, calculated in the linear in \ensuremathρv approximation, do obey the Einstein relation.