  We study uncertainty quantification for a Boltzmann-Poisson system that
models electron transport in semiconductors and the physical collision
mechanisms over the charges. We use the stochastic Galerkin method in order to
handle the randomness associated with the problem. The main uncertainty in the
Boltzmann equation concerns the initial conditions for a large number of
particles, which is why the problem is formulated in terms of a probability
density in phase space. The second source of uncertainty, directly related to
the quantum nature of the problem, is the collision operator, as its structure
in this semiclassical model comes from the quantum scattering matrices
operating on the wave function associated to the electron probability density.
Additional sources of uncertainty are transport, boundary data, etc. In this
study we choose first the phonon energy as a random variable, since its value
influences the energy jump appearing in the collision integral for
electron-phonon scattering. Then we choose the lattice temperature as a random
variable, since it defines the value of the collision operator terms in the
case of electron-phonon scattering by being a parameter of the phonon
distribution. The random variable for this case is a scalar then. Finally, we
present our numerical simulations.
