RUI: Equilibrium and Nonequilibrium Dynamics for Systems of Physical Origin

在动力系统中，如行星运动等随时间变化的系统，大部分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，并且有必要研究受其他未知系统影响的系统，可能在不同的尺度上。这个项目将研究混沌动力系统的性质，这些系统要么是由于外力的作用，要么是由于质量或能量的逃逸。在本项目过程中要研究的系统包括质量或能量可以通过确定性或随机机制进入或退出的系统，以及由较小的相互作用组件交换质量或能量的大型系统。这些问题与统计力学有着强烈的联系，并试图促进我们对与能量传输和扩散相关的基本问题的理解。该项目还将支持本科生参与数学研究。上述问题的高度视觉性和物理动机将使首席研究员能够在项目过程中招募本科生参与相关研究项目。特别强调的是，将招收来自代表性不足的研究数学群体的学生。学生将通过海报会议、会议演讲和在同行评审的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，这项资助将有助于实现整合研究和教育的重要目标。受上述问题的启发，该项目围绕三个具体项目进行组织：第一个项目研究具有碰撞相互作用的经典和非平衡粒子系统的统计和热力学性质，这是统计力学中的一类重要模型；第二种是开放系统，它一方面涉及允许质量或能量逸出的物理系统，另一方面涉及亚稳态的研究；第三个项目将开放系统概括为包括由两个或多个组件组成的连接和扩展的动力系统，这些组件通过确定性或随机机制交换质量或能量。重要的例子包括非周期洛伦兹气体和热传导的力学模型。研究者将带来他为这类系统开发的几种分析技术，包括他最近关于分散粒子系统转移算子的谱分解，由于Birkhoff引起的射影锥的收缩，以及适应性马尔可夫扩展（马尔可夫分区的推广）的构造。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均匀双曲和物理上重要的系统。将这些技术应用于平衡和非平衡统计力学的中心模型将代表这类系统研究的重大进展。在一个环境中理解这些工具的努力加强了它们，并有助于将它们应用于数学的其他领域。通过应用这些思想来解决物理文献中提出和正式处理的问题，提高了他们的智力兴趣。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。

In dynamical systems, systems that change with time such as planetary motion, much of the research is focused 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 systems that are influenced by other unknown systems, possibly on different scales. This project will investigate properties of chaotic dynamical systems that are out of equilibrium, either due to the application of external forces or because mass or energy is allowed to escape. Systems to be studied during the course of this project include 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. These problems are strongly motivated by connections with statistical mechanics and seek to advance our understanding of fundamental questions related to energy transport and diffusion. This project will also support the involvement of undergraduate students in mathematics research. The highly visual nature and physical motivation of the problems outlined above will enable the principal investigator to recruit undergraduate students to participate in related research projects during the course of the project. 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 grant will contribute to the important goal of integrating research and education.Motivated by the problems outlined above, this project is organized around three specific projects: The first project investigates the statistical and thermodynamic 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 physical systems in which mass or energy is allowed to escape, and on the other to the study of metastable states; the third project generalizes open systems to include linked and extended dynamical systems comprised of two or more components that exchange mass or energy through deterministic or random mechanisms. Important examples include the aperiodic Lorentz gas and mechanical models of heat conduction. The investigator will bring to bear several analytical techniques that he has been instrumental in developing for these classes of systems, including his recent work concerning the spectral decomposition of transfer operators for dispersing particle systems, contractions in projective cones due to Birkhoff, and the construction of adapted Markov extensions (a generalization of Markov partitions). 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 techniques 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.