  One of the central challenges in kinetic theory is the derivation of
macroscopic evolution equations--describing, for example, the dynamics of an
electron gas--from the underlying fundamental microscopic laws of classical or
quantum mechanics. An iconic mathematical model in this research area is the
Lorentz gas, which describes an ensemble of non-interacting point particles in
an infinite array of spherical scatterers. In the case of a disordered
scatterer configuration, the classical results by Gallavotti, Spohn and
Boldrighini-Bunimovich-Sinai show that the time evolution of a macroscopic
particle cloud is governed, in the limit of small scatterer density
(Boltzmann-Grad limit), by the linear Boltzmann equation. In this lecture I
will discuss the recent discovery that for a periodic configuration of
scatterers the linear Boltzmann equation fails, and the random flight process
that emerges in the Boltzmann-Grad limit is substantially more complicated. The
key ingredient in the description of the limiting stochastic process is the
renormalization dynamics on the space of lattices, a powerful technique that
has recently been successfully applied also to other open problems in
mathematical physics, including KAM theory and quantum chaos. This lecture is
based on joint work with Andreas Strombergsson, Uppsala.
