Kinetic transport in quasicrystals

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g. at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In this talk I will report on recent joint work with Andreas Strombergsson (Uppsala) on quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of the Penrose tiling. Our main result proves the existence of a limit distribution for the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed in the random setting. I will also discuss related results for other aperiodic scatterer configurations, such as the union of pairwise incommensurate lattices. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner's measure classification theorem.