RUI: Nonuniformly Hyperbolic Dynamical Systems out of Equilibrium

该奖项资助致力于研究失去平衡的混沌动力系统的研究。动力系统的许多研究都集中在动力学是自包含的封闭系统上。然而，在许多建模情况下，这样的全局视图是不可能的，因此有必要研究受其他未知系统影响的局部系统，可能在不同的尺度上。这些考虑促使在本奖项过程中要研究的许多系统：质量或能量可以通过确定性或随机机制进入或离开的系统，以及由较小的相互作用的组件交换质量或能量的大规模系统。这类系统已被用于理论物理和化学工程中，以模拟原子陷阱，探索固体中的热传导机制，并研究分子过程中的亚稳性。在获奖过程中获得的严谨的数学结果将有助于解释这些研究，并提出新的研究方向。该奖项还支持本科生参与数学研究。利用上述问题的高度直观性和物理动机，PI将在每个由该奖项资助的暑假期间招募本科生从事这些主题的研究。将特别重视从研究数学中代表性不足的群体中招收学生。学生将通过海报会议、会议报告和在同行评议期刊上发表文章来传播他们的研究成果。通过激发人们对数学研究事业的兴趣，并创建一个支持这种兴趣的同行社区，该奖项将有助于实现研究与教育相结合的重要目标。受上述问题的激励，该奖项围绕三个具体项目组织：第一个项目调查具有碰撞相互作用的经典和非平衡粒子系统的统计性质，这是统计力学中的一类重要模型；第二个项目涉及开放系统，一方面涉及允许质量或能量逃逸的物理系统的模型，另一方面涉及亚稳态的研究；第三个项目涉及由有限或无限数量的较小组件组成的扩展动力系统的行为，这些组件链接在一起，允许轨道或能量在它们之间传递。重要的例子包括非周期洛伦兹气体和热传导的力学模型。利用他最近关于分散粒子系统的转移算符的光谱分解的工作，PI将从统计力学中研究平衡和非平衡模型，并提供对重要物理量的严格分析，如熵产生。此外，PI将利用Birkhoff的投影锥以及马尔可夫扩张(马尔可夫划分的推广)的构造来适应和发展技术来研究这样的系统。这些技术都不需要动力学上的马尔可夫假设，这使得它们广泛适用于各种非一致双曲和物理上重要的系统。将这些工具应用于平衡和非平衡统计力学的中心模型，将代表着这类系统研究的重大进展。努力在一个背景下理解这些工具，可以加强它们，并有助于将它们应用于数学的其他领域。通过应用这些想法来解决物理文献中提出的和正式接近的问题，他们的智力兴趣得到了提高。这个奖项反映了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 award funds research dedicated to the study of chaotic dynamical systems out of equilibrium. Much research in dynamical systems focuses on closed systems in which the dynamics are self-contained. In many modeling situations, however, such a global view is not possible and it becomes necessary to study local systems that are influenced by other unknown systems, possibly on different scales. These considerations motivate many of the systems to be studied during the course of this award: systems in which mass or energy may enter or exit through deterministic or random mechanisms, and large-scale systems of smaller interacting components that exchange mass or energy. Such systems have been used in theoretical physics and chemical engineering to model atom traps, explore mechanisms for heat conduction in solids and investigate metastability in molecular processes. Rigorous mathematical results obtained during the course of this award will aid in the interpretation of these studies as well as suggest new directions of inquiry. This award also supports the involvement of undergraduates in mathematics research. Using the highly visual nature and physical motivation of the problems outlined above, the PI will recruit undergraduate students to work on these topics during each summer funded by the award. Special emphasis will be given to recruiting students from underrepresented groups in research mathematics. Students will disseminate results of their research via poster sessions, conference presentations and publications in peer-reviewed journals. By stimulating interest in research careers in mathematics and creating a peer community supportive of that interest, the award will contribute to the important goal of integrating research and education.Motivated by the problems outlined above, this award is organized around three specific projects: The first project investigates the statistical properties of both classical and non-equilibrium particle systems with collision interactions, an important class of models from statistical mechanics; the second concerns open systems, which relate on the one hand to models of physical systems in which mass or energy is allowed to escape, and on the other to the study of metastable states; the third project concerns the behavior of extended dynamical systems comprised of a finite or infinite number of smaller components linked together, with orbits or energy allowed to pass between them. Important examples include the aperiodic Lorentz gas and mechanical models of heat conduction. Using his recent work concerning spectral decompositions of transfer operators for dispersing particle systems, the PI will investigate equilibrium and non-equilibrium models from statistical mechanics and provide rigorous analysis of physically important quantities such as entropy production. In addition, the PI will adapt and develop techniques using projective cones due to Birkhoff as well as the construction of Markov extensions (a generalization of Markov partitions) to the study of such systems. None of these techniques require Markovian assumptions on the dynamics, making them widely applicable to a wide variety of nonuniformly hyperbolic and physically important systems. The application of these tools to central models from equilibrium and non-equilibrium statistical mechanics will represent significant advances in the study of such systems. Efforts to understand these tools in one context, strengthens them and aids in their application to other areas of mathematics. Their intellectual interest is enhanced by the application of these ideas to resolve problems posed and approached formally in the physics literature.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.