Slowly divergent drift in the field-driven Lorentz gas

Abstract

The dynamics of a point charged particle that moves in a medium of elastic scatterers and is driven by a uniform external electric field 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 ${e}^{\ensuremath{-}|v{|}^{3}/t}.$ In spatial dimension $d>1,$ we develop an effective-medium theory that 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 $P(u,t)=〈u{〉}^{\ensuremath{-}1}f(u/〈u〉),$ where $u=|v{|}^{3/2},$ $〈u〉\ensuremath{\sim}\sqrt{t},$ and $f(z)\ensuremath{\propto}{z}^{(d\ensuremath{-}1)/3}{e}^{\ensuremath{-}{z}^{2}/2}.$ 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 which the typical speed grows as $(t\mathrm{ln}{t)}^{1/3}.$

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