The Localization Transition of the Two-Dimensional Lorentz Model

Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
The European Physical Journal Special Topics (Impact Factor: 1.4). 03/2010; 189(1). DOI: 10.1140/epjst/e2010-01313-1
Source: arXiv


We investigate the dynamics of a single tracer particle performing Brownian motion in a two-dimensional course of randomly distributed hard obstacles. At a certain critical obstacle density, the motion of the tracer becomes anomalous over many decades in time, which is rationalized in terms of an underlying percolation transition of the void space. In the vicinity of this critical density the dynamics follows the anomalous one up to a crossover time scale where the motion becomes either diffusive or localized. We analyze the scaling behavior of the time-dependent diffusion coefficient D(t) including corrections to scaling. Away from the critical density, D(t) exhibits universal hydrodynamic long-time tails both in the diffusive as well as in the localized phase. Comment: 13 pages, 7 figures.

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    • "Percolation thresholds were obtained with high precision yielding a critical void porosity ϕ c = 0.0301(3) for spheres [106] [122] [123] and ϕ c = 0.323 652 5(6) for discs [124] [125]; numbers in parentheses indicate the uncertainty in the last digit. Simulations for the Lorentz model [22] [79] [115] [118] [126] [127] confirm the picture that the localisation transition is indeed driven by the percolation transition of the medium. For point tracers, the critical density of the localisation transition is defined by the percolation threshold. "
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