Quantitative Stochastic Homogenization and Renormalization Methods

该研究项目的重点是发展新的数学思想和工具，以理解具有许多小尺度不规则性的物理系统的宏观特性。例如，复合材料是通过混合两种或两种以上的组成材料制成的，并且通常与单个成分具有非常不同的物理特性（例如热传导或导电性）。令人惊讶的是，复合材料的性能通常不是其各组分性能的简单平均值——（微观）混合如何安排非常重要。我们想要精确地理解大量的微观相互作用是如何引起这样的系统的宏观行为的，能够通过仔细观察样品的微观排列来预测系统的宏观性质，并且确切地知道这样的样品在我们的预测准确之前必须有多大。我们继续开发和分析理想化的数学模型，这些模型显示了真实物理系统的关键特征和复杂性，但仍然足够简单，可以用数学方法进行研究。通常，我们的模型是带有随机系数的偏微分方程——也就是说，假设系统的潜在微观行为是随机的——目标是理解系统的统计数据。我们的目标是开发数学方法，这些数学方法(i)由系统的物理特性提供信息，（ii）鲁棒性-它们可以帮助我们解决诸如统计物理和概率论中出现的各种其他类似问题，以及（iii）定量-它们导致不确定性的定量估计和对模拟和预测的数值算法发展的洞察力。本项目为研究生提供研究训练机会。在数学文献中，获得一个“平均”偏微分方程来描述具有许多自由度的潜在的高度非均质方程（例如，具有随机系数的方程）的大规模行为被称为“均质化”。均匀化方程通常比“真正的”异质方程简单得多，因此更容易处理。因此，精确地理解均匀化方程与真实方程的近似程度是很重要的，这也是均匀化理论的目标。数学家们最近发展了一个完整的椭圆方程随机均匀化的定量理论，这个理论已经被证明在概率论和统计力学中有一些令人惊讶和重要的应用。目前的项目旨在通过应用和发展梯度晶格模型和统计物理中相关模型的均匀化技术，来研究运动理论（模拟气体和等离子体）中产生的次椭圆扩散，并朝着理解随机矢量场强制扩散的行为的方向发展。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 research project is focused on the development of new mathematical ideas and tools for understanding the macroscopic properties of physical systems with many small-scale irregularities. Composite materials, for example, are manufactured by intermingling two or more constituent materials and often have very different physical properties (heat or electrical conduction, for example) from the individual components. Surprisingly, the properties of the composite material are typically not a simple average of those of its components--it matters greatly how the (microscopic) intermingling is arranged. We want to understand precisely how the large number of microscopic interactions give rise to the macroscopic behavior of such systems, to be able to predict the macroscopic properties of the system by carefully looking at the microscopic arrangement of a sample, and to know exactly how large such a sample must be before our predictions are accurate. We proceed by developing and analyzing idealized mathematical models that display the key features and complexity of real physical systems, but still simple enough that they can be studied mathematically. Very often, our models are partial differential equations with stochastic coefficients--that is, the underlying microscopic behavior of the system is assumed to be random--and the goal is to understand the statistics of the system. Our objective is to develop mathematical approaches that are (i) informed by the physics of the systems, (ii) robust--they can help us solve a variety of other, similar problems arising in, for example, statistical physics and probability theory, and (iii) quantitative--they lead to quantitative estimates of the uncertainty and insight into the development of numerical algorithms for simulation and prediction. This project provides research training opportunities for graduate students.Obtaining an "averaged" partial differential equation that describes the large-scale behavior of an underlying highly heterogeneous equation with many degrees of freedom (for instance, one with random coefficients) is referred to as "homogenization" in the mathematical literature. The homogenized equation is typically much simpler than the "true" heterogeneous one, and hence easier to work with. It is therefore important to understand precisely how well the homogenized equation approximates the true equation, and this is the goal of homogenization theory. Mathematicians have recently developed a complete quantitative theory of stochastic homogenization for elliptic equations, and this theory has already been shown to have some surprising and important applications to probability theory and statistical mechanics. The present project aims to continue in this direction, by applying and developing homogenization techniques for gradient lattice models and related models in statistical physics, to the study of hypoelliptic diffusions arising in kinetic theory (modeling gases and plasmas) and toward understanding the behavior of diffusions forced by random vector fields.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.