Coarse-graining, Renormalization, and Fractal Homogenization

该项目的重点是发展新的数学，用于分析在大量长度尺度上表现出复杂行为的物理系统的统计行为。一个典型的例子包括湍流流体，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，通常被物理学家称为“临界现象”。物理学家已经开发出启发式的、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。该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。在过去的十年中，主要研究者（PI）和其他数学家的工作导致了某些偏微分方程的“定量均匀化”的严格理论。这些方程具有上面提到的复杂物理系统的一些性质，并且均匀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关键行为的物理系统上。这需要发展新的数学思想和概念，并需要分析、概率论、偏微分方程和数学物理学的投入。该项目为研究生提供研究培训机会。 该项目有两个主要目标。第一个是关于改进定量均匀化理论，使其更明确地依赖于方程中的重要参数（如椭圆率），并允许退化和可能无界的系数场。这是一个众所周知的开放问题，但PI和他的合作者Kuusi在这个问题上取得了最新进展，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开发的分析方法，作为物理学中形式化启发式重整化群参数的手段。这些方法出现在各种各样的环境中，但该项目有一些具体的问题。一个是在流体湍流中，涉及证明由粗糙矢量场平流的被动标量的异常扩散。PI和他的合作者Vicol最近在这个问题上取得了进展，他们使用均匀化来形式化重整化群的论点。这为进一步的可能性指明了方向，包括构建更真实的异常扩散实例。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NSF的法定使命，并通过使用基金会的智力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.