First return probabilities in a Lorentz gas

We investigate the distribution of first return times, J(t), for a two-dimensional, overlapping Lorentz gas model. This can be seen as a study of a model geminate recombination reaction. We compare simulation results with predictions from the Lorentz-Boltzmann equation and also from other kinetic models which include short-term memory effects. All these theories predict the correct value for J(t = 0) but the theories including memory effects are more accurate than Lorentz-Boltzmann theory at later times. For densities less than the percolation density, when the traveller moves in infinitely connected space, the long-time form of J(t) is given by the solution of the diffusion equation. Above this density, when the particle is trapped in a cage of finite area, simulation indicates that J(t) has a long-time algebraic decay proportional to t-2. We put forward a theory based on a rough circle model that predicts a t-3 decay at long times. As yet we have no explanation for the observed t-2 tail.