Diffusion limit of a non-linear kinetic model without the detailed balance principle

The aim of this paper is the rigorous derivation of a non-linear drift-diffusion model from the Boltzmann equation in a semiconductor device. Collisions are taken into account through the nonlinear Pauli operator, without assuming any relation concerning the cross-section (such as the so-called detailed balance principle). The proof of the convergence relies on a formal Hilbert expansion and the estimation of the remainder. The same program is applied to the derivation of a non-linear SHE (Spherical Harmonics Expansion) model, when elastic collisions are considered, and is rigorously carried through in the case of zero electric field.