Quantum Diffusion in Fluctuating Media

波动介质中的量子扩散

基本信息

批准号：
1500386
负责人：
Jeffrey Schenker
金额：
$ 12万
依托单位：
Michigan State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-08-01 至 2019-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1500386&HistoricalAwards=false
关键词：
Quantum Diffusion Fluctuating Media

项目摘要

The main focus of this project is the analysis of wave motion in a disordered environment. In a broader context, the research is aimed at answering a basic scientific question: "What are the effects of disorder?" This is a fundamental question relevant to any mathematical model, even one in which disorder is not explicitly included. After all, any real world system is subject to a small amount of noise, and experience shows that even weak disorder may have a profound effect on the behavior of the system. The equations studied in this project arise in the theory of electrical conduction in disordered materials, but are of general interest because of the fundamental nature of both wave motion and disorder. Progress in understanding the solutions to these equations will improve basic understanding of models of theoretical physics and applied mathematics. In addition, a central goal of the research is pedagogical: to introduce undergraduate and graduate students to a fundamental subject and convey to them that mathematics is a vibrant, growing field. The project will proceed through a program of research on the effects of disorder in physical models. A key goal is to analyze the diffusion of waves in a weakly disordered medium over arbitrarily long times. There is a rich non-rigorous theory of the weakly disordered regime in the physics literature based on heuristic analyses and uncontrolled, renormalized perturbation theory which suggests that waves propagate diffusively, characterized by spreading of wave packets over a distance proportional to the square root of t in time t. However, we are far from having a rigorous analysis of the mathematics involved. A major challenge is that diffusive propagation does not occur for waves in a non-random medium. Thus, a naive approach in which the disorder is incorporated perturbatively has not worked, indeed in the physics literature the problem is attacked with renormalized perturbation theory. In recent years the PI and various post-doc and student collaborators have considered the problem of wave diffusion in time-dependent random media, with the time dependence generated by a Markov process. For such models the diffusive propagation, e.g., for the tight binding Schrödinger equation, can be established by spectral analysis. One aim of the present project is to approach the time independent equation as a perturbation of these time dependent equations, which have the virtue of sharing the expected qualitative behavior.

该项目的主要重点是对无序环境中波动运动的分析。在更广泛的背景下，该研究旨在回答一个基本的科学问题：“疾病的影响是什么？”这是与任何数学模型相关的基本问题，即使是疾病未明确包括的模型。毕竟，任何现实世界系统都会遇到少量噪音，并且经验表明，即使是弱的疾病也可能对系统的行为产生深远的影响。该项目中研究的方程式出现在无序材料的电导理论中，但由于波动和混乱的基本性质而引起了普遍的兴趣。了解这些方程的解决方案的进展将改善对理论物理和应用数学模型的基本理解。此外，这项研究的核心目标是教学：将本科生和研究生介绍给一个基本学科，并向他们传达数学是一个充满活力的，成长的领域。该项目将通过有关物理模型中疾病影响的研究计划进行。一个关键目标是分析在任意长时间内弱小的培养基中波的扩散。在物理文献中，基于启发式分析和不受控制的，重命名的扰动理论的物理文献中存在丰富的非矛盾理论，该理论表明，波浪在时间t的距离上散布在与T平方根成比例的距离上。但是，我们远没有对所涉及的数学进行严格的分析。一个主要的挑战是，在非随机介质中的波不会发生分化的传播。这是一种幼稚的方法，其中该疾病被扰动地纳入了疾病，实际上在物理文献中，该问题受到重新归一化的扰动理论的攻击。近年来，PI和各种博士后和学生合作者考虑了时间依赖性随机媒体的波浪差异问题，而Markov过程产生了时间依赖性。对于此类模型，可以通过光谱分析来建立差分传播，例如，对于紧密的结合schrödinger方程。本项目的目的之一是将时间独立方程式作为这些时间依赖方程的扰动，这具有共享预期的定性行为的优点。