Homogenization Methods in Statistical Physics

在复合材料的物理和工程中，使用基于高振荡系数的偏微分方程的模型是很自然的。齐次化的数学理论是为了回答由于研究这类方程而产生的大量分析问题而形成的。在过去的十年里，一般学科领域取得了巨大的进步。然而，尽管过去取得了许多进展，一些基本和基本现象仍然无法进行严格的数学分析。首席研究员（PI）将为这些遗留问题开发新技术和新视角，重点关注统计物理中的问题，这些问题可以通过均质化的视角来看待。这包括半导体、流体混合和粗糙表面上液滴形成的模型。本项目为研究生提供研究训练机会。PI将应用分析和概率的技术来研究统计物理中的问题。主要的焦点将放在具有高振荡系数的偏微分方程解的大尺度行为上。这将包括带有随机系数的热和波动方程，以及它们的晶格离散化和相关的自由边界问题。许多被提议的项目都受到了安德森本地化现象的启发。另一个统一的主题是均质化，即粗糙的微观结构在大尺度上“平均”，并产生非平凡的光滑宏观效应。主要目标是获得解决方案大规模行为的定性和定量估计。PI将利用均匀化和大规模正则性理论的最新进展来解决一些开放问题。主要研究领域包括安德森定位、阿贝尔沙堆、流体粒子模型、流体混合和液滴形成。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

In the physics and engineering of composite materials, it is natural to use models based on partial differential equations with highly oscillatory coefficients. The mathematical theory of homogenization was formed to answer the large class of analytic questions that arise from studying such equations. The general subject area has seen tremendous progress in the last ten years. However, despite the many past advances, some basic and fundamental phenomena still elude rigorous mathematical analysis. The principal investigator (PI) will develop new techniques and new perspectives for some of these remaining problems, with a focus on problems in statistical physics that can be viewed through the lens of homogenization. This includes models of semiconductors, fluid mixing, and droplet formation on rough surfaces. The project provides research training opportunities for graduate students. The PI will apply techniques from analysis and probability to study problems in statistical physics. The main focus will be on the large scale behavior of solutions of partial differential equations with highly oscillatory coefficients. This will include heat and wave equations with random coefficients as well as their lattice discretizations and related free boundary problems. Many of the proposed projects are inspired by Anderson localization phenomena. Another unifying theme is homogenization, where rough microscopic structure “averages” over large scales and gives rise to non-trivial smooth macroscopic effects. The primary goal is to obtain both qualitative and quantitative estimates for the large-scale behavior of solutions. The PI will utilize recent advances in homogenization and large scale regularity theory in order to attack a number of open problems. The primary areas of interest are Anderson localization, the Abelian sandpile, particle models for fluids, the mixing of fluids, and droplet formation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.