  The Lorentz gas is one of the simplest, most widely used models to study the
transport properties of rarified gases in matter. It describes the dynamics of
a cloud of non-interacting point particles in an infinite array of fixed
spherical scatterers. More than one hundred years after its conception, it is
still a major challenge to understand the nature of the kinetic transport
equation that governs the macroscopic particle dynamics in the limit of low
scatterer density (the Boltzmann-Grad limit). Lorentz suggested that this
equation should be the linear Boltzmann equation. This was confirmed in three
celebrated papers by Gallavotti, Spohn, and Boldrighini, Bunimovich and Sinai,
under the assumption that the distribution of scatterers is sufficiently
disordered. In the case of strongly correlated scatterer configurations (such
as crystals or quasicrystals), we now understand why the linear Boltzmann
equation fails and what to substitute it with. A particularly striking feature
of the periodic Lorentz gas is a heavy tail for the distribution of free path
lengths, with a diverging second moment, and superdiffusive transport in the
limit of large times.
