The drift velocity of a hard-sphere Lorentz gas

Studies the Boltzmann equation for a Lorentz gas with scattering on stationary hard spheres in the presence of a constant field E. The exact initial and asymptotic time evolutions are given and compared with numerical calculations. Starting with an initial equilibrium velocity distribution the authors study the influence of the initial temperature T on the drift velocity of the Lorentz gas. The drift velocity quickly reaches a maximum and then decreases slowly towards zero. In particular an upper bound, close to 0.8 E^1/2 lambda ^1/2, exists for the drift velocity. Here lambda =( pi a²n)^-1 is the mean free path, related to the density n and radius a of the scatterers. In an initially cool gas the drift velocity slows down as t^-1/2 soon after the maximum is passed. In an initially hot gas, however, there are two asymptotic regimes. After a time of order lambda ^1/2E^-1/2 the drift velocity stays constant for a time interval whose length is proportional to T^3/2, and eventually decays as t^-1/2.