Diffusion, percolation, and trapping in the Lorentz gas via variational kinetic theory

A recently developed variational principle is applied to the solution of a self-consistent repeated-ring kinetic theory of the Lorentz gas in one, two, and three dimensions. Calculated values of the diffusion constant $D$, are in excellent agreement with molecular-dynamics simulation results for $d=2\mathrm{} \mathrm{and} 3$. The theory predicts the existence of "critical scatterer densities," ${\rho }_c^{*}$, above which $D=0\mathrm{}$; for $d=1-3\mathrm{}$, ${\rho }_c^{*}=0,{\pi }^{-1\mathrm{}},\mathrm{and}\frac{3}{2\pi }$, respectively. The theory behaves well above ${\rho }_c^{*}$. The behavior of $D$ near ${\rho }_c^{*}$ is examined, as is the "long-time tail" at densities close to ${\rho }_c^{*}$, and a comparison of the results to those of other authors is given.