Homogenization Methods in Statistical Physics

在复合材料的物理和工程中，使用基于具有高振荡系数的偏微分方程的模型是很自然的。数学的均匀化理论的形成是为了回答从研究这类方程中产生的一大类分析问题。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，但一些基本和基本的现象仍然无法进行严格的数学分析。首席研究员（PI）将开发新的技术和新的观点，这些剩余的问题，重点是统计物理学的问题，可以通过均匀化的透镜。这包括半导体、流体混合和粗糙表面上液滴形成的模型。该项目为研究生提供研究培训机会。PI将应用分析和概率技术来研究统计物理学中的问题。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。这将包括热和波动方程的随机系数，以及他们的晶格离散和相关的自由边界问题。许多被提议的项目的灵感来自于安德森本地化现象。另一个统一的主题是均匀化，其中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。主要目标是获得定性和定量估计的大规模行为的解决方案。PI将利用最近的进展，在均匀化和大规模的规律性理论，以攻击一些开放的问题。主要研究领域是安德森局部化、阿贝尔沙堆、流体的粒子模型、流体的混合和液滴的形成。该奖项反映了NSF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.