CAREER: The Nature of SRB Measures for Nonequilibrium Hyperbolic Systems

平衡双曲动力系统理论已经发展到现在它提供了强大的数学工具来解决物理学中的公开问题。首席研究员将重点了解西奈-鲁埃尔-鲍恩(SRB)措施的性质和非平衡现象的相关观测统计，使用与混沌系统研究相关的创新方法。虽然关于平衡系统的复杂技术已经发展并取得了很好的结果，但非平衡系统中的随机和混沌行为的研究课题仍然是开放的。这样做的原因是，这些环境中的混沌现象与均衡环境中的混沌现象有着截然不同的味道。最近开发的双曲型系统的数学工具为取得重大进展带来了希望。这个项目试图通过使用光谱分析、耦合方法和其他创新方法来解决两个具有挑战性的问题，从而获得对数学思想与各种复杂现象联系起来的理论理解和新方法。第一个涉及非平衡双曲系统的统计性质(例如，证明描述定态的SRB测度的存在性；研究与输运过程中扩散矩阵有关的时间关联函数；验证其他摄动双曲系统的极限定理)。具体模型包括受一般力作用的洛伦兹气体、具有慢速运动散射体的理想气体和具有微观结构的随机系统。第二个问题涉及SRB度量的性质和相关的物理定律(例如，了解SRB度量的性质，包括它们的熵和Hausdorff维度；获得欧姆定律、爱因斯坦关系和物理中出现的某些非平衡双曲型系统的其他物理定律)。这个项目将使用数学工具来解决物理、化学工程和其他科学中的应用问题。双曲系统理论为理解随机或随时间变化的系统中的混沌和扩散过程提供了极好的模型或范例。目标是捕捉这些系统的主要复杂性，从可积性到混沌行为，而不是整合运动方程的困难。由于对物理不变度量及其渐近统计性质的研究提供了对稳态和输运现象本质的新见解，该项目将对现代统计物理和化学工程做出贡献。因此，它将在物理科学中的数学之外以及更广泛的科学界内产生广泛的影响。首席研究员计划通过指导研究生的研究，将与研究相关的概念和非平衡系统的模拟纳入新的本科课程，从而增强本科生(REU)的研究经验，并通过开展K-12外联活动来激励和激发学生对混沌系统的研究。这项研究将被纳入研究生和本科生的研究项目，以及课程开发。首席研究员将寻求机构批准为研究生开设一门关于随机微分方程的新专题课程，以及一门新的本科课程，该课程将侧重于与混沌和分形学相关的主题。她将在阿默斯特地区高中开设一系列客座讲座，主题与混沌和分形几何有关，以吸引当地高中生参与这一跨学科的前沿研究。她还将为高中女生组织年度研讨会，其中将包括动手科学活动和与寻求数学科学职业的个人讨论职业，无论是学术上的还是其他方面的。该项目将扩大妇女和代表性不足的少数群体的参与，帮助他们将数学之美形象化。

The theory of equilibrium hyperbolic dynamical systems has been developed to the point where it now provides powerful mathematical tools to address open problems in physics. The principal investigator will focus on understanding the nature of Sinai-Ruelle-Bowen (SRB) measures and the statistics of relevant observations for nonequilibrium phenomena, using innovative approaches related to the study of chaotic systems. While sophisticated techniques have been developed and beautiful results obtained for equilibrium systems, the research topics of random and chaotic behavior in nonequilibrium systems have remained wide open. The reason for this is that chaotic phenomena in these settings have quite a different flavor from those in the equilibrium context. Recently developed mathematical tools in hyperbolic systems offer hope for significant progress. This project seeks to obtain both a theoretical understanding of and new ways to connect mathematical ideas to a variety of complex phenomena by addressing two challenging issues through the use of spectral analysis, the coupling method, and other innovative approaches. The first concerns statistical properties of nonequilibrium hyperbolic systems (e.g.,to prove the existence of SRB measures that characterize the steady states; to study the time correlation functions that relate to the diffusion matrix in the transport processes; to verify other limiting theorems for perturbed hyperbolic systems). Specific models include Lorentz gases under general forces, an ideal gas with slow-moving scatterers, and random systems with microstructure. The second issue has to do with properties of SRB measures and related physical laws (e.g., to understand the nature of SRB measures, including their entropy and Hausdorff dimensions; to obtain Ohm's law, the Einstein relation, and other physical laws for certain nonequilibrium hyperbolic systems that arise in physics).This project will use mathematical tools to address applied problems in physics, chemical engineering, and other sciences. The theory of hyperbolic systems has provided excellent models or paradigms for understanding chaos and diffusion processes in systems that are random or changing over time. The goal is to capture the major complexity of these systems, from integrability to chaotic behavior, without the difficulty of integrating the equations of motion. As the study of physical invariant measures and their asymptotic statistical properties provides new insight into the nature of steady states and transport phenomena, the project will contribute to modern statistical physics and chemical engineering. Thus, it will have a broad impact outside of mathematics in the physical sciences and within the broader scientific community. The principal investigator plans to motivate and excite students about research on chaotic systems by directing graduate student research, by incorporating research-related concepts and simulations of nonequilibrium systems into new undergraduate courses, thereby enhancing research experiences for undergraduates (REUs), and by carrying out K-12 outreach activities. The research will be integrated into both graduate and undergraduate research projects, as well as into curriculum development. The principal investigator will seek institutional approval for a new topics course for graduate students on stochastic differential equations and a new undergraduate course that will focus on topics related to chaos and fractals. She will initiate a Guest Lecture Series at the Amherst Regional High School on topics related to chaos and fractal geometry, in order to engage local high school students in this interdisciplinary, cutting-edge research. She will also organize annual workshops for high school girls that will include hands-on scientific activities and career discussions with individual pursuing careers in the mathematical sciences, academic or otherwise. The project will broaden participation of women and underrepresented minorities and help them to visualize the beauty of mathematics.