THE LONG-TIME BEHAVIOR OF THE VELOCITY AUTOCORRELATION FUNCTION IN A LORENTZ GAS

We report a numerical study of the integral of the velocity autocorrel ation function, R(t), in a two-dimensional overlapping Lorentz gas. It is relatively easy to study R(t) at times where the velocity autocorr elation function itself, C(t), is statistically indistinguishable from zero. At lower densities we study R(t) up to 300 mean collision times and nearer the percolation threshold up to 2000 mean collision times, so we can infer the behaviour of C(t) at times up to 10 times those p reviously reported. Our results can be successfully explained in terms of an asymptotic decay of C(t) proportional to t-2 at densities up to 70% of the percolation threshold. As the percolation threshold is app roached we see a rapid shift in the onset of this decay to longer time s. At densities near the percolation threshold we are able to describe what we believe to be pre-asymptotic decay with a single effective ex ponent of - 1.38 +/- 0.02. These observations are consistent with self -consistent kinetic theories and recent work on the lattice Lorentz ga s. We find that these theories give poor predictions for the constant of proportionality characterising the asymptotic decay. Our estimate f or the percolation threshold is a reduced density of 0.357 +/- 0.03, w hich is consistent with values calculated by other methods. Via the ca lculation of diffusion constants near the percolation threshold we are able to estimate the critical exponent with which the diffusion const ant vanishes to be 1.5 +/- 0.3.