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性质。例如，复合材料是通过混合两种或更多种组成材料而制造的，并且通常具有与单个组分非常不同的物理特性（例如热或电传导）。令人惊讶的是，复合材料的性能通常不是其组分的简单平均值-它非常重要（微观）混合如何安排。我们希望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，并且确切地知道在我们的预测准确之前这样的样品必须有多大。我们通过发展和分析理想化的数学模型来进行，这些模型显示了真实的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，我们的模型是具有随机系数的偏微分方程-也就是说，假设系统的基本微观行为是随机的-目标是了解系统的统计数据。我们的目标是开发数学方法，这些方法（i）由系统的物理学所告知，（ii）鲁棒性-它们可以帮助我们解决各种其他类似的问题，例如，统计物理和概率论，以及（iii）定量-它们导致对不确定性的定量估计，并深入了解模拟和预测的数值算法的发展。本项目为研究生提供研究训练的机会。数学文献中，将描述具有多个自由度（例如随机系数）的高度非均匀性方程的大尺度行为的“平均化”偏微分方程称为“均匀化”。均匀化方程通常比“真正的”非均匀方程简单得多，因此更容易处理。因此，重要的是要精确地理解均匀化方程近似真实方程的程度，这就是均匀化理论的目标。数学家最近发展了一个完整的定量理论的随机均匀化椭圆方程，这个理论已经被证明有一些令人惊讶的和重要的应用概率论和统计力学。本项目的目的是继续朝这个方向努力，在统计物理学中应用和发展梯度格点模型和相关模型的均匀化技术，动力学理论中亚椭圆扩散的研究（模拟气体和等离子体）该奖项反映了NSF的法定使命，并已被认为是值得通过评估使用的支持，基金会的学术价值和更广泛的影响审查标准。

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.