Coarse-graining, Renormalization, and Fractal Homogenization

这个项目专注于发展新的数学来分析物理系统的统计行为，这些系统在大量的长度尺度上表现出复杂的行为。一个典型的例子包括湍流流体，如地球大气，它在人类尺度(一阵风)和大陆尺度(天气模式)上都有波动，以及介于两者之间的每一种尺度。其他例子包括统计力学和量子场论中的重要模型。这种混沌的物理系统具有有趣的行为，这些行为是通过这些非常不同的长度尺度的相互作用而出现的，物理学家通常将其称为“临界现象”。物理学家已经发展出启发式的、非严格的方法来理解和分析许多这样的物理系统，其中一些被称为“重整化群”论证。这个项目的主要目标之一是开发这些非正式论点的精确版本，这些版本在数学上是严格的。在过去的十年里，首席调查员(PI)和其他数学家的工作导致了某些偏微分方程量化齐次化的严格理论。这些方程具有上述复杂物理系统的一些性质，而齐次化理论在重要方面类似于重整化群类型的论点。然而，它目前只适用于具有少量长度比例的问题。该项目建议提高同质化方法的复杂程度，直到该理论可以更灵活地部署在表现出关键行为的物理系统上。这需要发展新的数学思想和概念，需要分析、概率论、偏微分方程式和数学物理的投入。该项目为研究生提供了研究培训机会。该项目有两个主要目标。第一个是关于改进定量齐化理论，使其对方程中的重要参数(如椭圆率)的依赖性更加明确，并允许退化的和可能的无界系数场。这是一个众所周知的子领域的悬而未决的问题，但Pi和他的合作者Kuusi最近在这个问题上取得了进展，这个项目将继续发展这些新的想法。该项目的第二个重点是使用为均匀化开发的这些分析方法，作为形式化物理学中启发式重整化群论点的手段。这些方法出现在各种各样的环境中，但该项目考虑到了几个具体的问题。一种出现在流体湍流中，它涉及到证明被粗糙矢量场平流的被动标量的反常扩散。PI和他的合作者Vicol最近在这个问题上取得了进展，他们使用齐次化来形式化一个重整化群论点。这为进一步的可能性指明了方向，包括构建更多物理上真实的反常扩散例子。另一个潜在的应用在于欧几里德场论，遵循随机量化的方法来研究吉布斯测量。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is focused on the development of new mathematics for analyzing the statistical behavior of physical systems which exhibit complex behavior across a large number of length scales. A typical examples include turbulent fluids, such as the earth's atmosphere, which have fluctuations on the human scale (a gust of wind) and on the continental scale (weather patterns), and every scale in between. Other examples include important models in statistical mechanics and quantum field theory. Such chaotic physical systems have interesting behaviors which emerge through the interaction of these very different length scales, often called "critical phenomena" by physicists. Physicists have developed heuristic, non-rigorous ways of understanding and analyzing many such physical systems, some of which are called "renormalization group" arguments. One of the main goals of this project is to develop precise versions of these informal arguments which are mathematically rigorous. In the past decade, the work of the Principal Investigator (PI) and other mathematicians have led to a rigorous theory of "quantitative homogenization" of certain partial differential equations. These equations have some of the properties of the complex physical systems mentioned above, and the homogenization theory resembles renormalization group-type arguments in important ways. However, it currently works well only for problems with a small number of length scales. The project proposes to increase the level of sophistication of the homogenization methods until the theory can be deployed more flexibly on physical systems exhibiting critical behavior. This requires the development of new mathematical ideas and concepts and will require input from analysis, probability theory, partial differential equations and mathematical physics. The project provides research training opportunities for graduate students. The project has two main goals. The first one concerns improving the quantitative homogenization theory, so that it is more explicit in its dependence on important parameters in the equation (like the ellipticity ratio) and allows for degenerate and possibly unbounded coefficient fields. This is a well-known open problem in the subfield, but the PI and his collaborator Kuusi have made recent progress on this question, and this project will continue to develop these new ideas. A second focus of the project is to use these analytic methods developed for homogenization as means of formalizing heuristic renormalization group arguments in physics. Such methods arise in a wide variety of contexts, but the project has a few specific problems in mind. One arises in fluid turbulence, and concerns proving the anomalous diffusion of a passive scalar advected by a rough vector field. The PI and his collaborator Vicol have made recent progress on this question by using homogenization to formalize a renormalization group argument. This points the way to further possibilities, including the construction of more physically realistic examples of anomalous diffusion. Another potential application lies in Euclidean field theory, following a stochastic quantization approach to study Gibbs measures.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.