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.