Approximation by diffusion of semiconductor Boltzmann equation

In this paper, we deal with the asymptotic behaviour of a semiconductor Boltzmann equation by using the sigma convergence method. We first prove that the scaled model is well-posed in the usual Lebesgue space of square integrable functions, and we perform the a priori estimates. Then, assuming that the coefficients of the model are highly oscillating in space variable, we show that in the non-vanishing flux case, the homogenized problem is equivalent at first order, to a hyperbolic process modified by a perturbation of viscosity, and the diffusion term appears at second order. In the vanishing flux case, we obtain a diffusion model.