Statistical Properties of Hyperbolic Systems with Singularities

该奖项是根据2009年美国复苏和再投资法案(公法111-5)资助的。该项目致力于对动力系统的统计性质及其在物理科学中的应用进行严格的调查。主要目的是研究具有奇点的双曲型系统。该项目中的大多数研究对象都属于混沌台球，其中包括洛伦茨和硬球气体等流行的物理模型，这些模型提供了统计力学的数学基础。这些系统具有类似于负曲流形上的测地线流和公理A微分同胚和流的性质，但它们的奇性导致了相空间的令人不快的碎片化，这使得研究它们变得更加困难。特别是，具有弱统计性质(如关联的缓慢衰减)的动力系统在物理科学的应用中扮演着极其重要的角色。这类系统的动力学在规则和混沌之间是间歇性的。然而，用于分析慢混合速率体系的数学方法是最近才发展起来的，仍然很难应用于实际的模型。基于Young，Dolgopyat和Chernov的最新结果，首席研究员建议开发新的方法来估计关联的衰变，这种方法可以应用于更一般的具有奇点的系统，即现有技术无法处理的系统。这个项目的第一个主要组成部分涉及代表物理现象的最一般数学模型的奇点系统。对这个项目特别感兴趣的是“混沌台球”的主题，这个术语被数学家和物理学家用来描述在研究玻尔兹曼遍历假说时观察到的某些现象。该项目的第二个关键部分集中在遍历理论上，这是概率和动力系统学科结合在一起的一个重要的数学研究领域。通过与费尔斯的合作，首席研究员将探索随机台球(同样是数学概念，而不是室内游戏)中气体-表面碰撞的散射特性。这表明了对抽象奇异系统的严格研究对化学工程具体领域的贡献。事实上，这样的研究最终可能导致工业过程中气体分离的新技术。预计这个项目在理解动力系统统计性质方面取得的进展将对我们对自然界中许多重要事件的知识产生影响，并在其他科学和工程学科中找到应用。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).This project is devoted to the rigorous investigation of statistical properties of dynamical systems and their applications to the physical sciences. The primary goal is to study hyperbolic systems with singularities. Most of those under study in the project fall under the chaotic billiards heading, which includes popular physical models such as the Lorentz and hard-ball gases that provide the mathematical foundations of statistical mechanics. These systems have properties similar to those of geodesic flows on negatively curved manifolds and Axiom-A diffeomorphisms and flows, but their singularities give rise to an unpleasant fragmentation of phase space, which makes them much harder to study. In particular, dynamical systems with weak statistical properties (such as slow decay of correlations) play extremely important roles in applications to physical sciences. The dynamics in such systems are intermittent between regular and chaotic. However, mathematical methods for the analysis of systems with slow mixing rates were developed only recently and are still difficult to apply to realistic models. Based upon recent results of Young, Dolgopyat, and Chernov, the principal investigator proposes to develop new approaches to estimating the decay of correlations that can be applied to more general systems with singularities, systems that could not be handled by existing techniques.The first major component of this project deals with systems with singularities that represent the most general mathematical models of physical phenomena. Of special interest in the project is the subject of "chaotic billiards," a term used by mathematicians and physicists to describe certain phenomena that are observed in the study of Boltzmann's ergodic hypothesis. The second key component of the project focuses on ergodic theory, which is an important mathematical research field in which the subjects of probability and dynamical systems come together. Through a collaboration with Feres, the principal investigator will explore the scattering properties of gas-surface collisions through random billiards (again the mathematical concept, not the parlor game). This demonstrates the contribution of the rigorous study of abstract singular systems to the concrete field of chemical engineering. Indeed, such a study may ultimately lead to new techniques for gas separation in industrial processes. It is expected that the progress in understanding statistical properties of dynamical systems that will be made in this project will have an impact on our knowledge of many important events in nature and find application in other scientific and engineering disciplines.