Dynamics and Kinetics

拟议的研究解决了动力系统和统计力学理论中的一些长期存在的问题，以及一些新的自然问题，这些问题从一般的角度来看和应用都很重要。该项目开发了一种新的方法来设计双曲台球，这将允许人们证明双曲性更一般的台球类。这种方法是基于一个新的一般特征的绝对聚焦曲线，这是唯一允许的聚焦组件的双曲台球，在连续的分数。一个长期存在的问题，即人们是否可以平滑球场台球的边界将得到解决。该项目还将揭示新的问题之间的边界位于完全混沌台球凸域和台球划分成混沌和规则组件的相空间。两个自然的问题提出的主要研究人员对开放系统的动力学将得到解决。第一个问题是如何通过一个洞逃逸依赖于相空间中的位置的孔。第二个问题是关于从一个洞逃跑和从多个洞逃跑的关系。这些问题揭示了组合数学和数论之间的一些微妙联系。非凸多边形中有限尺寸台球粒子的动力学将被证明是双曲线的。这将被应用到统计力学中的经典Escherifest周期风树模型，并证明，从自然物理的角度来看，这个模型超越了周期洛伦兹模型在其动态的丰富性。该项目将提供新的可视化和相对简单的模型，台球动力系统的混沌以及混合（混沌和规则动态共存的区域）的行为。（注：“台球”是一个技术数学概念，并不是指那个名字的室内游戏。此外，主要研究者引入的一些模型将（有些已经）被理论家和实验家用于物理学，他们实际上已经建造了这样的设备，因此将促进跨学科的合作。一个问题，找到一个最佳的（以确保最快/最慢的逃逸）位置的一个洞将有一个潜在的各种各样的应用开放系统。这个问题，以及关于通过多个孔逃逸的问题，是受到原子台球实验的启发。此外，这种方法开辟了对动态进行有限时间（而不是时间上的渐近）预测的可能性（例如，预测一个时刻，在该时刻之后，通过特定的孔逃逸比通过相同尺寸的任何其他孔逃逸更有可能）。有限尺寸粒子风树模型的分析在统计力学中有应用。 该项目将通过与美国、墨西哥、加拿大和欧洲的研究人员合作，加强研究和教育的基础设施。研究生已经参与了这项研究，本科生的参与是预料之中的。该项目的结果将被广泛传播，以提高科学和技术的理解，通过参与的主要研究者（通常与全体会议），他的合作者，和他的学生在跨学科的会议与物理学家，生物学家和工程师的广泛参与。

The proposed research addresses some long-standing problems in the theory of dynamical systems and statistical mechanics, as well as some new natural questions that are important from a general point of view and for applications. The project develops a new approach to the design of hyperbolic billiards that will allow one to prove hyperbolicity for more a general class of billiards. This approach is based on a new general characterization of absolutely focusing curves, which are the only admissible focusing components of hyperbolic billiards, in terms of continued fractions. A long-standing problem on whether one can smooth the boundary of a stadium billiard will be resolved. The project will also shed new light on the question of where the border lies between completely chaotic billiards in convex domains and billiards with divided phase spaces into chaotic and regular components. Two natural questions raised by the principal investigator on the dynamics of open systems will be addressed. The first one asks how the escape through a hole depends on the position of a hole in phase space. The second question is about the relationship between escape through one hole and escape through multiple holes. These questions reveal some subtle connections between combinatorics and number theory. The dynamics of a finite-size billiard particle in nonconvex polygon will be shown to be hyperbolic. This will be applied to the classical Ehrenfest periodic wind-tree model in statistical mechanics and demonstrate that, from a natural physical point of view, this model surpasses the periodic Lorentz model in the richness of its dynamics.The project will provide new visual and relatively simple models of billiard dynamical systems with chaotic as well as with mixed (coexisting regions with chaotic and with regular dynamics) behavior. (N.B."Billiards" is a technical mathematical concept that does not refer to the parlor game of that name.) Moreover, some of the models introduced by the principal investigator will be (and some already have been) used in physics by both theoreticians and experimentalists, who have actually built such devices, and therefore will foster interdisciplinary collaborations. A problem of finding an optimal (to ensure the fastest/slowest escape) placement of a hole will have a potentially large variety of applications for open systems. This question, as well as the one on escape through multiple holes, was inspired by experiments on atomic billiards. Moreover, this approach opens up the possibility of making finite-time (rather than asymptotic in time) predictions of dynamics (e.g., predicting a moment after which escape through a specific hole is more likely than escape through any other hole of the same size). The analysis of the wind-tree model with a finite-size particle will have applications in statistical mechanics. The project will enhance the infrastructure for research and education through collaborations with researchers in the US, Mexico, Canada, and Europe. Graduate students are already involved in this research, and the involvement of undergraduates is anticipated. The results of the project will be broadly disseminated to enhance scientific and technological understanding via participation of the principal investigator (often with plenary talks), his collaborators, and his students in interdisciplinary conferences with a broad participation of physicists, biologists, and engineers.