Statistical Properties of Hyperbolic Dynamical Systems and Applications to Statistical Physics

由于任何微小的测量误差都会在相对较短的时间内导致巨大的不确定性，因此预测混沌动力系统中某一轨迹的未来行为实际上是不可能的。在数学和物理文献中都提出了许多这样的混沌系统来模拟一些现实生活中的现象。例如，相互作用的粒子的确定性系统可以模拟电子的微观运动。本研究项目研究的是这样的系统。目标是用严格的数学方法证明，从长远来看，这些系统的行为就像它们是随机的一样。因此，可以应用随机过程理论的思想从微观无序中推导出统计宏观有序的出现，即当系统规模或观测时间较大且初始状态为典型时。对于上述例子，这种方法提供了从一些微观确定性模型推导热方程的方法，这是统计物理数学理论中的一个中心问题。具体来说，将研究三个项目。在第一章中，我们考虑了一些大的确定性相互作用球系统。受物理中常见的时间尺度分离现象的驱使，将研究一种所谓的稀有相互作用极限，即当系统可以由相互独立的西奈台球粒子之间的任意一对相互作用来近似时。第二个项目是关于双曲系统在有限和无限测度情况下的高级统计性质。例如，局部中心极限定理，作为一个非常有用的工具，在包括方案一在内的许多应用中，以及它与无限测度混合的联系，将被研究用于双曲映射和流。第三个项目研究相互作用粒子的随机系统。将要研究的问题包括被迫打破平衡的系统的局部平衡的出现以及质量和能量的联合传输。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

It is practically impossible the predict the future behavior of one trajectory in a chaotic dynamical system as any small measurement error leads to huge uncertainty in relatively short amount of time. Many such chaotic systems have been proposed in both the mathematics and physics literature to model some real-life phenomena. For example, a deterministic system of interacting particles could model the microscopic motion of electrons. This research project studies such systems. The goal is to prove, in a mathematically rigorous way, that these systems on the long run behave as if they were random. Consequently, ideas from the theory of random processes can be applied to derive the emergence of statistical macroscopic order from microscopic disorder, that is when the system size or the time of observation is large and the initial state is typical. For the above-mentioned example, this approach provides a derivation of the heat equation from some microscopic deterministic models, which is a central question in the mathematical theory of statistical physics.Specifically, three research projects will be studied. In the first one, some large system of deterministic interacting balls is considered. Motivated by the common separation of time scale phenomenon is physics, a so called rare interaction limit will be studied, that is when the system can be approximated by independent Sinai billiard particles between any pair interactions. The second project is about advanced statistical properties of hyperbolic systems in both finite and infinite measure case. For example, the local central limit theorem, as a very useful tool in many applications including project one, as well as its connections to infinite measure mixing will be studied for hyperbolic maps and flows. The third project studies stochastic systems of interacting particles. The problems to be studied include the emergence of local equilibrium for systems forced out of equilibrium and joint transport of mass and energy.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.