The Localization Transition of the Two-Dimensional Lorentz Model

TL;DR

This paper studies how a tracer particle's motion in a 2D environment with randomly placed obstacles transitions from normal to anomalous behavior at a critical obstacle density, revealing a percolation transition affecting diffusion.

Contribution

It provides a detailed analysis of the critical dynamics and scaling behavior near the percolation transition in the 2D Lorentz model.

Findings

Anomalous diffusion occurs near the critical obstacle density.

Universal hydrodynamic tails are observed in the diffusion coefficient.

The dynamics are governed by an underlying percolation transition.

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.