  In this paper we derive a semiclassical hydrodynamic system for electron
densities and currents in the two energy bands of a semiconductor. We use the
semiclassical Wigner equation with a k.p Hamiltonian and a BGK dissipative term
to construct the first two moment equations. The closure of the moment system
is obtained using the Maximum Entropy Principle, by minimizing a Gibbs
free-energy functional under suitable constraints. We prove that the constraint
equations can be uniquely solved, i.e. that the local equilibrium state can be
parametrized by the density and velocity field. Some BGK-like models are
proposed to mimic the quantum interband migration.
