The description of catastropic changes in tagged particle dynamics by the self-consistent repeated ring equation

The nature of tagged particle motion in random media can change catastrophically as some parameter, most notably the scatterer density, is varied. In some systems, the self-diffusion constant vanishes above a critical density, providing a dynamic analog of the static percolation problem. Good theoretical treatments of these phenomena are given by solutions of the nonlinear equations generated by the “self-consistent repeated ring” approximation. In this paper, we work out the repeated ring approximation for three systems: a random walk on a lattice where randomly chosen sites are excluded to the walker, the Lorentz gas, and the motion of a light particle in a real fluid.