Diffusion in lattice Lorentz gases with a percolation threshold.

Abstract

A mean-field approximation for the diffusion coefficient in lattice Lorentz gases with an arbitrary mixture of pointlike stochastic scatterers in the low-density limit is proposed. In this approximation, the diffusion coefficient is directly related to the first return probability of the moving particle in the corresponding Cayley tree through an effective ring operator. A renormalization scheme for the approximate determination of the first return probability is constructed. The predictions of this mean-field theory and those of the repeated ring approximation (RRA) are compared with computer simulation results for models in which a fraction x(B) of the scatterers are deterministic backscatterers, so that the diffusion coefficient vanishes beyond a certain percolation threshold x(c)(B). The approximation proposed in this paper is seen to be in good agreement with the simulation results, in contrast to the RRA, which already fails to give the correct percolation threshold.