We study the deterministic diffusion coefficient of the two-dimensional
periodic Lorentz gas as a function of the density of scatterers. Results
obtained from computer simulations are compared to the analytical approximation
of Machta and Zwanzig [Phys.Rev.Lett. 50, 1959 (1983)] showing that their
argument is only correct in the limit of high densities. We discuss how the
Machta-Zwanzig argument, which is based on treating diffusion as a Markovian
hopping process on a lattice, can be corrected systematically by including
microscopic correlations. We furthermore show that, on a fine scale, the
diffusion coefficient is a non-trivial function of the density. We finally
argue that, on a coarse scale and for lower densities, the diffusion
coefficient exhibits a Boltzmann-like behavior, whereas for very high densities
it crosses over to a regime which can be understood qualitatively by the
Machta-Zwanzig approximation.Comment: 9 pages (revtex) with 9 figures (postscript
