  We show that the time evolution of a quantum wavepacket in a periodic
potential converges in a combined high-frequency/Boltzmann-Grad limit, up to
second order in the coupling constant, to terms that are compatible with the
linear Boltzmann equation. This complements results of Eng and Erd\"os for
low-density random potentials, where convergence to the linear Boltzmann
equation is proved in all orders. We conjecture, however, that the linear
Boltzmann equation fails in the periodic setting for terms of order four and
higher. Our proof uses Floquet-Bloch theory, multi-variable theta series and
equidistribution theorems for homogeneous flows. Compared with other scaling
limits traditionally considered in homogenisation theory, the Boltzmann-Grad
limit requires control of the quantum dynamics for longer times, which are
inversely proportional to the total scattering cross section of the single-site
potential.
