Slowly Divergent Drift in the Field-Driven Lorentz Gas

TL;DR

This paper investigates the dynamics of a driven Lorentz gas, revealing a universal t^{1/3} growth in particle speed across dimensions and deriving a scaling form for the velocity distribution, with special corrections for infinite horizon cases.

Contribution

It develops an effective medium theory for the Lorentz gas, predicting universal speed growth and velocity distribution forms, including logarithmic corrections for infinite horizon geometries.

Findings

Speed grows as t^{1/3} in all dimensions.

Velocity distribution follows a specific scaling form.

Logarithmic correction for infinite horizon Lorentz gases.

Abstract

The dynamics of a point charged particle which is driven by a uniform external electric field and moves in a medium of elastic scatterers is investigated. Using rudimentary approaches, we reproduce, in one dimension, the known results that the typical speed grows with time as t^{1/3} and that the leading behavior of the velocity distribution is exp(-|v|^3/t). In spatial dimension d>1, we develop an effective medium theory which provides a simple and comprehensive description for the motion of a test particle. This approach predicts that the typical speed grows as t^{1/3} for all d, while the speed distribution is given by the scaling form P(u,t)=<u>^{-1}f(u/<u>), where u=|v|^{3/2}, <u>~t^{1/2}, and f(z) is proportional to z^{(d-1)/3}exp(-z^2/2). For a periodic Lorentz gas with an infinite horizon, e. g., for a hypercubic lattice of scatters, a logarithmic correction to the effective medium result is predicted; in particular, the typical speed grows as (t ln t)^{1/3}.