Stochastic and Quantum Dynamics of Large Systems

欧拉方程是由60年代Morrey和Olla-Varadhan-Yau分别在60年代和最近(假设玻尔兹曼假设)严格地从微观牛顿动力学导出的。该项目的第一部分建议将潜在的动力学改变为量子力学。其目的是证明在量子情况下得到的欧拉方程类似于经典的欧拉方程，只是所有的物理量都应该用量子力学计算。该方法将基于相对熵方法和大偏差理论的某种扩展到量子情况。该项目的第二部分涉及由声子场模拟的电子与晶格结构相互作用的动力学。因此，它的动力学由一个耦合到标量量子场的电子的薛定谔方程所支配。目的是证明在弱耦合极限下，电子的相空间密度收敛于玻尔兹曼方程。材料中电子的基本动力学受量子力学支配，更准确地说，是薛定谔方程。由于薛定谔方程无论是解析解还是数值解都很难求解，目前用于电子输运行为的模型大多是基于经典粒子和扩散的半经典模型。虽然这些半经典模型是大距离行为的有效近似，但它们忽略了小尺度上的关键量子效应。这个项目的目的是从第一原理-薛定谔方程出发来研究输运行为。特别是，欧拉和玻尔兹曼方程的有效性将从薛定谔方程建立起来。这是理解传输行为中的量子效应的第一步。长期目标是理解对这些方程的量子修正。

0072098Yau The Euler equations were derived from microscopic Newtonian dynamics heuristically by Morrey in the sixties and rigorously by Olla-Varadhan-Yau recently (assuming the Boltzmann hypothesis). The first part of the project proposes to change the underlying dynamics to quantum mechanics. The aim is to show that the Euler equations obtained in the quantum case are similar to the classical ones except all physical quantities should be computed quantum mechanically. The approach will be based on the relative entropy method and certain extension of large deviation theory to the quantum case. The second part of the project concerns the dynamics of an electron interacting with a lattice structure, modeled by a phonon field. Therefore, its dynamics is governed by a Schrodinger equation of an electron coupled to a scalar quantum field. The goal is to prove that the phase space density of the electron converges to a Boltzmann equation in the weak coupling limit. The underlying dynamics for electrons in a material are governed by quantum mechanics, or more precisely the Schrodinger equations. Since the Schrodinger equations are difficult to solve no matter analytically or numerically, most models in use nowadays for transport behavior of electrons are semi-classical ones based on classical particles and diffusions. Although these semi-classical models are valid approximations for large distance behavior, they neglect crucial quantum effects in small scales. The aim of this project is to study the transport behavior starting from the first principle---the Schrodinger equations. In particular, the validity of the Euler and Boltzmann equations will be established from the Schrodinger equations. This is the first step toward understanding the quantum effects in transport behavior. The long term goal is to understand quantum corrections to these equations.