Stochastic and Quantum Dynamics of Large Systems

大型系统的随机和量子动力学

• 批准号: 0072098
• 负责人: Horng-Tzer Yau
• 金额: $ 35.06万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072098&HistoricalAwards=false
• 关键词: Stochastic; Quantum; Dynamics; Large; Systems

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.

0072098丘欧拉方程是由Morrey在60年代从微观牛顿动力学中推导出来的，最近由Olla-Varadhan-Yau严格推导出来的（假设玻尔兹曼假设）。 该项目的第一部分建议将基础动力学改为量子力学。 目的是表明，在量子情况下得到的欧拉方程类似于经典的，除了所有的物理量应该量子力学计算。 该方法将基于相对熵方法和大偏差理论对量子情况的某些扩展。 该项目的第二部分涉及电子与晶格结构相互作用的动力学，由声子场建模。 因此，它的动力学是由耦合到标量量子场的电子的薛定谔方程控制的。 目的是证明电子的相空间密度在弱耦合极限下收敛于玻尔兹曼方程。材料中电子的基本动力学由量子力学控制，或者更准确地说是薛定谔方程。 由于薛定谔方程无论是解析还是数值求解都很困难，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下的关键量子效应。 本项目的目的是从第一性原理--薛定谔方程出发研究输运行为。 特别是，欧拉和玻尔兹曼方程的有效性将建立从薛定谔方程。 这是理解量子效应在输运行为中的第一步。 长期目标是理解这些方程的量子修正。