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.

翻译：我们研究一个Boltzmann-Poisson系统的不确定性定量,这个系统在半导体中模拟电子传输和电荷的物理碰撞机制。我们使用Stochatic Galerkin 方法来处理与问题相关的随机性。Boltzmann 等式的主要不确定性涉及大量粒子的初始条件,这就是为什么问题是以相位空间的概率密度来表述的。第二个不确定性来源是碰撞操作员,直接与问题的数量性质有关,因为其结构来自与电子概率密度相关的波函数的量散射矩阵。更多的不确定性来源是迁移、边界数据等。在本研究中,我们首先选择磷能量作为随机变量,因为其价值会影响电荷散射相碰撞中出现的能量跳动。然后我们选择拉蒂温度作为随机变量,因为它界定了电荷散射的电荷操作员术语的价值,从而成为光分布的参数。这个案例的随机变量是我们的数字模型。