Quantum Diffusion in Fluctuating Media

这个项目的主要焦点是无序环境中的波动分析。在更广泛的背景下，这项研究旨在回答一个基本的科学问题：“无序的影响是什么？”这是一个与任何数学模型相关的基本问题，即使是没有明确包含无序的数学模型。毕竟，任何现实世界的系统都会受到少量噪音的影响，经验表明，即使是微弱的无序也可能对系统的行为产生深远的影响。本项目中研究的方程源于无序材料中的电导理论，但由于波动和无序的基本性质而引起普遍关注。在理解这些方程的解方面的进展将提高对理论物理和应用数学模型的基本理解。此外，这项研究的一个中心目标是教育性的：向本科生和研究生介绍一门基础学科，并向他们传达数学是一个充满活力的、不断增长的领域。该项目将通过一项关于物理模型中无序影响的研究计划来进行。一个关键的目标是分析波在弱无序介质中任意长时间内的扩散。在基于启发式分析和不受控制的重整化微扰理论的物理文献中，有丰富的关于弱无序区域的非严格理论，该理论认为波是以扩散的方式传播的，其特征是波包在与时间t的平方根成正比的距离上传播。然而，我们远未对其中涉及的数学进行严格的分析。一个主要的挑战是波在非随机介质中不会发生扩散传播。因此，将无序纳入微扰的幼稚方法并不起作用，事实上，在物理学文献中，这个问题是用重整化微扰理论来攻击的。近年来，PI和许多博士后和学生合作者考虑了时间依赖随机介质中的波扩散问题，其中时间依赖由马尔可夫过程产生。对于这样的模型，例如紧束缚薛定谔方程，可以通过频谱分析来建立扩散传播。本项目的一个目标是将与时间无关的方程作为这些与时间相关的方程的扰动来处理，这些方程具有共享预期的定性行为的优点。

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.