Radiative transport in a periodic structure with band crossings

We use the Wigner transformation and asymptotic analysis to systematically derive the semi-classical model for the Schrödinger equation in arbitrary spatial dimensions, with any periodic structure. Our particular emphasis lies in addressing the diabatic effect, i.e., the impact of Bloch band crossings. We consider both deterministic and random scenarios. In the former case, we derive a coupled Liouville system, revealing lower-order interactions among different Bloch bands. In the latter case, a coupled system of radiative transport equations emerges, with the scattering cross-section induced by the random inhomogeneities. As a specific application, we deduce the effective dynamics of a wave packet in graphene with randomness.