Coarse-graining, Renormalization, and Fractal Homogenization

这个项目的重点是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，例如地球的大气层，它在人类尺度（一阵风）和大陆尺度（天气模式）以及两者之间的每一个尺度上都有波动。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，这些行为通过这些非常不同的长度尺度的相互作用而出现，物理学家通常称之为“临界现象”。物理学家已经开发了启发式的、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证的精确版本，它们在数学上是严格的。在过去的十年中，首席研究员（PI）和其他数学家的工作已经导致了某些偏微分方程的“定量均匀化”的严格理论。这些方程具有上述复杂物理系统的一些性质，并且均质化理论在许多重要方面类似于重整化群型论证。然而，它目前只适用于少数长度尺度的问题。该项目建议提高均匀化方法的复杂程度，直到该理论可以更灵活地部署在表现出临界行为的物理系统上。这需要发展新的数学思想和概念，并需要从分析、概率论、偏微分方程和数学物理中输入。本项目为研究生提供研究训练机会。该项目有两个主要目标。第一个是改进定量均匀化理论，使其对方程中重要参数（如椭圆率）的依赖更加明确，并允许退化和可能的无界系数场。这是子领域中一个众所周知的开放问题，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想法。该项目的第二个重点是使用这些为均匀化而开发的分析方法作为形式化物理中启发式重整化群论证的手段。这些方法出现在各种各样的环境中，但是这个项目有一些特定的问题。一种出现在流体湍流中，涉及到证明被动标量被粗糙矢量场平流的异常扩散。PI和他的合作者Vicol最近在这个问题上取得了进展，他们使用均匀化形式化了一个重整化群论证。这为进一步的可能性指明了道路，包括构造更实际的反常扩散例子。另一个潜在的应用是欧几里得场论，遵循随机量化方法来研究吉布斯测度。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。

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.