The long-time behaviour of the velocity autocorrelation function in a Lorentz gas

We report a numerical study of the integral of the velocity autocorrelation function. R(t), in a two-dimensional overlapping Lorentz gas. It is relatively easy to study R(t) at times where the velocity autocorrelation 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 previously 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 approached we see a rapid shift in the onset of this decay to longer times. At densities near the percolation threshold we are able to describe what we believe to be pre-asymptotic decay with a single effective exponent of -1.38 ± 0.02. These observations are consistent with self-consistent kinetic theories and recent work on the lattice Lorentz gas. We find that these theories give poor predictions for the constant of proportionality characterising the asymptotic decay. Our estimate for the percolation threshold is a reduced density of 0.357 ± 0.03, which is consistent with values calculated by other methods. Via the calculation of diffusion constants near the percolation threshold we are able to estimate the critical exponent with which the diffusion constant vanishes to be 1.5 ± 0.3.