Slowly Divergent Drift in the Field-Driven Lorentz Gas

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}.