Mathematical Sciences: Mathematical Analysis of Kinetic Quantum Transport Models for Semiconductors

提出的研究对象是固体物理中量子动力学方程的数学和数值分析。我们将首先研究松弛时间维格纳-泊松模型（量子玻尔兹曼方程）。该非线性演化方程的存在唯一性分析将基于密度矩阵算子的重新表述。大时间行为和热力学平衡稳态的分析将基于紧致性方法和量子熵泛函。一维Wigner方程的吸收边界条件将扩展到二维情况，即耦合Wigner- poisson和松弛时间模型。适定性分析将基于把维格纳方程作为有限维双曲系统的极限来考虑。量子力学输运方程的模型已经成为精确模拟新型、超集成半导体器件（量子器件、电子波导）的最重要基础。对演化方程的合理的数学理解将构成数值实现的基础。高超声速气体在刚性物体（如航天飞机在其重返大气层的高空阶段）周围流动的数值模拟通常是通过数值耦合两种不同的传输模型来完成的。这里提出了一种精细的耦合策略，它将同样适用于气体动力学应用，以及超集成半导体器件的有效模拟。我们将对其与现有的耦合策略进行数值比较，并研究其稳定性。

DMS-9500852 Arnold The object of the proposed research is the mathematical and numerical analysis of quantum kinetic equations in solid state physics. We will first investigate relaxation-time Wigner-Poisson models (quantum Boltzmann equation). The existence and uniqueness analysis will be based on the reformulation of this nonlinear evolution equation in terms of the density matrix operator. The analysis of the large time behavior and the steady states in thermodynamic equilibrium will be based on compactness methods and a quantum entropy functional. Absorbing boundary conditions for the 1D Wigner equation will be extended to 2D situations, the coupled Wigner-Poisson and relaxation-time models. The well- posedness analysis will be based on considering the Wigner equation as a limit of finite dimensional hyperbolic systems. The models of quantum mechanical transport equations have become the most important basis for accurate simulations of novel, ultra-integrated semiconductor devices (quantum devices, electron wave guides). A sound mathematical understanding of the evolution equation will form the basis for numerical implementations. The numerical simulation of hypersonic gas flows around rigid objects (e.g., space shuttles in the high altitude phase of their re-entry into the atmosphere) is usually accomplished by numerically coupling two different transport models . A refined coupling strategy is proposed here, which will be equally relevant for gas dynamics applications, and for efficient simulations of ultra-integrated semiconductor devices. We will numerically compare it to existing coupling strategies and investigate its stability.