On the mean free path for a periodic array of spherical obstacles

We prove theorems pertaining to periodic arrays of spherical obstacles which show how the macroscopic limit of the mean free path depends on the scaling of the size of the obstacles. We treat separately the cases where the obstacles are totally and partially absorbing, and we also distinguish between two-dimensional arrays, where our results are optimal, and higher dimensional arrays, where they are not. The cubically symmetric arrays to which these results apply do not have finite horizon.