Superdiffusion in a honeycomb billiard

We investigate particle transport in the honeycomb billiard which consists of connected channels placed on the edges of a honeycomb structure. The spreading of particles is superdiffusive due to the existence of ballistic trajectories which we term perfect paths. Simulations give a time exponent of 1.72 for the mean-square displacement and a starlike, i.e., anisotropic, particle distribution. We present an analytical treatment based on the formalism of continuous-time random walks and explain the anisotropic distribution under the assumption that the perfect paths follow the directions of the six lattice axes. Furthermore, we derive a relation between the time exponent and the exponent of the distribution function for trajectories close to a perfect path. In billiards with randomly distributed channels, conventional diffusion is always observed in the long-time limit, although for small disorder transient superdiffusional behavior exists. Our simulation results are again supported by an analytical analysis.