Slowly divergent drift in the field-driven Lorentz gas

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 and that the leading behavior of the velocity distribution is e2uvu 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 for all d , while the speed distribution is given by the scaling form P(u ,t)5^u& f (u/^u&), where u5uvu, ^u&;At , and f (z)}z e 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 lnt). @S1063-651X~97!09609-8#