Eur. Phys. J. Special Topics 189, 103-118 (2010)
https://doi.org/10.1140/epjst/e2010-01313-1

Regular Article

The localization transition of the two-dimensional Lorentz model

¹ Arnold Sommerfeld Center for Theoretical Physics (ASC) and Center for NanoScience (CeNS), Fakultät für Physik, Ludwig-Maximilians-Universität München, Theresienstraße 37, 80333 München, Germany
² Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford OX1 3NP, UK
³ Max-Planck-Institut für Metallforschung, Heisenbergstraße 3, 70569 Stuttgart, Germany
⁴ Institut für Theoretische Physik, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
⁵ Institut für Theoretische und Angewandte Physik, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany

Received: 1 September 2010
Revised: 14 September 2010
Published online: 12 November 2010

Abstract

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.