A Drift-Collision Balance for a Boltzmann--Poisson System in Bounded Domains

We consider a low density approximation to a Boltzmann--Poisson system for electrons in a semiconductor in regimes where strong forcing balances the collision terms. We compute the hydrodynamics corresponding to this regime by the Chapman--Enskog method and by a closure of moment equations as well. In both cases we obtain reaction diffusion systems, with anisotropic diffusions and convection coefficients that depend on the spatial derivatives of the anisotropic diffusion. In the case of a relaxation collision operator there is a recursion formula such that this self-consistent closure computes all the moments at all orders in the expanding parameter. The recursion procedure can be applied to field dependent relaxation time approximations, yielding a velocity saturated mobility. In addition we discuss boundary conditions for both the distribution function and the moments. These nonequilibrium hydrodynamical regimes have been observed in the modeling of submicron doped semiconductor silicon devices and nume...

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