  This work is related to developing entropy-stable positivity-preserving
Discontinuous Galerkin (DG) methods as a computational scheme for
Boltzmann-Poisson systems modeling the probability density of collisional
electronic transport along semiconductor energy bands. In momentum coordinates
representing spherical / energy-angular variables, we pose the respective
Vlasov-Boltzmann equation with a linear collision operator and a singular
measure, modeling scatterings as functions of the band structure appropriately
for hot electron nanoscale transport.
  We show stability results of semi-discrete DG schemes under an entropy norm
for 1D-position (2D-momentum) and 2D-position (3D-momentum), using dissipative
properties of the collisional operator given its entropy inequality. The latter
depends on an exponential of the Hamiltonian rather than the Maxwellian
associated with only kinetic energy. For the 1D problem, knowing the analytic
solution to the Poisson equation and convergence to a constant current is
crucial to obtaining full stability (weighted entropy norm decreasing over
time). For the 2D problem, specular reflection boundary conditions and
periodicity are considered in estimating stability under an entropy norm.
Regarding the positivity-preservation proofs in the DG scheme for the 1D
problem, inspired by \cite{ZhangShu1}, \cite{ZhangShu2}, and \cite{CGP},
\cite{EECHXM-JCP}, we treat collisions as a source and find convex combinations
of the transport and collision terms which guarantee positivity of the cell
average of our numerical probability density at the next time. The positivity
of the numerical solution to the probability density in the domain is
guaranteed by applying the limiters in \cite{ZhangShu1} and \cite{ZhangShu2}
that preserve the cell average modifying the slope of the piecewise linear
solutions to make the function non-negative.
