Novel phenomena in Lorentz lattice gases

In Lorentz Lattice Gases a point particle moves along the bonds of a lattice, whose lattice sites are occupied by randomly placed scatterers, which scatter the particle according to certain a priori given scattering rules. We consider here mainly the simplest case of a square lattice fully or partly covered with two kinds of scatterers, which scatter the particle to its right or its left, respectively, either according to probabilistic or strictly deterministic scattering rules. Although the global probability of the particle to be scattered to its right or left is equal for the two scattering rules, the local probability is very different, leading to a completely different diffusive behavior for the two cases. While the probabilistic scattering rules lead to Gaussian diffusion, the deterministic rules lead either to anomalous or no-diffusion, depending on the concentrations of the scatterers on the lattice. In the first case the mean square displacement is still proportional to the time, so that a diffusion coefficient can be defined, but the diffusion is non-Gaussian, while in the second case the mean square displacement is bounded and all particles are trapped in a finite time. In both cases this behavior is due to the existence of closed (periodic) orbits, which in the first case can be of infinite extent (extended closed orbits), while in the second case they can only be of finite extent. The lengths of the extended closed orbits show, for a lattice fully occupied equally by right and left scatterers, a scaling behavior identical to that of percolation clusters at the bond percolation threshold of a related bond percolation problem. For special concentrations on a not fully occupied lattice the same scaling behavior is found, leading to two critical lines, but no known related percolation-like problem exists in this case. This critical behavior of the closed orbits in anomalous diffusion appears to obtain for all two dimensional lattices studied so far. A number of extensions of the above results are indicated.