Quantum Diffusion in Fluctuating Media

这个项目的主要重点是分析无序环境中的波动。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：“紊乱的影响是什么？”这是一个与任何数学模型相关的基本问题，即使是没有明确包括无序的数学模型。毕竟，任何真实世界的系统都受到少量噪声的影响，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。在这个项目中研究的方程出现在无序材料的导电理论中，但由于波动和无序的基本性质而引起普遍的兴趣。理解这些方程的解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学：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、不断发展的领域。该项目将通过一个关于物理模型中无序影响的研究项目进行。一个关键的目标是分析波在弱无序介质中任意长时间的扩散。在物理文献中，基于启发式分析和不受控制的、重整化的微扰理论，有一个丰富的非严格的弱无序状态理论，它表明波是扩散的，其特征是波包在时间t中以与根号t成比例的距离传播。然而，我们远远没有对所涉及的数学进行严格的分析。一个主要的挑战是波在非随机介质中不会发生扩散传播。因此，将无序以微扰的方式纳入其中的朴素方法是行不通的，事实上，在物理文献中，这个问题是用重整化微扰理论来解决的。近年来，PI和各种博士后和学生合作者研究了时变随机介质中的波扩散问题，其时间依赖性由马尔可夫过程产生。对于这类模型，可以通过谱分析建立扩散传播方程，例如紧结合Schrödinger方程。本项目的目的之一是将时间无关方程作为这些时间相关方程的扰动来处理，这些时间相关方程具有共享预期定性行为的优点。

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.