Non-compact, random billiard systems; Quantum large deviations

非紧凑、随机台球系统；

• 批准号: 0405439
• 负责人: Marco Lenci
• 金额: $ 8.31万
• 依托单位: Stevens Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-07-01 至 2007-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405439&HistoricalAwards=false
• 关键词: Non; compact; random; billiard; systems

AbstractLenciThis research project consists of three parts: Part I deals withbilliards on non-compact tables, as models for hyperbolic dynamicalsystems with an infinite invariant measure. The plan is to extendto these dynamical systems certain important results available for compact billiards, including a suitable version of Pesin's Theory. This will require the rewriting of some notions of ergodic theory for the case of a non-probability measure. Part II concerns random billiards; more precisely, billiards with a random table (changing after each collision), a quenched random table (selected once and for all from a random ensemble), or a random law of reflection (in a fixed table). In the first two cases there is a physical invariant measure, and the PI will study its ergodic properties (including Lyapunov exponents and the like) for a typical realization of the random process. In the third case an equilibrium measure can be shown to exist, and its asymptotic properties will be investigated, together with the natural question of the stochastic stability at the zero-noise limit. Similar perturbative questions are considered for the other systems as well. Part III, in the realm of equilibrium statistical mechanics, considers the question of quantum large deviations. In a collaborative project, the PI sets out to initiate a theory of large deviations for a noncommutative quasi-local algebra on a d-dimensional lattice. Of primary interest is the convergence of the moment-generating function for an extensive observable, and the smoothness and physical significance of its limit.Billiards are a class of dynamical systems that has been mostextensively studied. This is so for a two-fold reason: On one hand,mathematically, they are relatively treatable, with their geometricfeatures often giving hints on how to prove a certain soughtresult. On the other hand, from the point of view of physics, they arefairly realistic models that have been applied to a variety of fields,from statistical mechanics to optics to scattering theory. Therefore the drive is natural to expand this class to cover new areas where it could have a remarkable impact: the family of open systems (every system with unbounded dynamics belongs to this category) and of random systems (which tries to justify when, and why, deterministic predictions still make sense in a noisy world). As for Part III, the theory of large deviations in statistical mechanics gives a theoretical understanding of the fact that large systems, which are deeply random from the observer's standpoint, yield nonetheless very accurate deterministic measurements. Although large deviations for classical systems have been the subject of massive and very successful research, a counterpart of quantum system (which should rightfully be even more fundamental) is conspicuously missing at the moment.

Lenci本研究项目包括三个部分：第一部分涉及非紧表上的台球，作为具有无穷不变测度的双曲动力系统的模型。该计划是extendto这些动力系统的某些重要成果可用于紧凑的台球，包括一个合适的版本Pesin的理论。这将需要重写遍历理论的一些概念的情况下，一个非概率措施。第二部分涉及随机台球;更准确地说，台球与随机表（每次碰撞后改变），淬火随机表（选择一次，从一个随机合奏），或随机反射定律（在一个固定的表）。在前两种情况下，有一个物理不变的措施，和PI将研究其遍历特性（包括李雅普诺夫指数等）的一个典型的实现随机过程。在第三种情况下，可以证明存在一个平衡措施，其渐近性质将被调查，连同自然问题的随机稳定性在零噪声限制。类似的微扰问题被认为是其他系统以及。第三部分，在平衡统计力学领域，考虑量子大偏差的问题。在一个合作项目中，PI开始着手研究d维格上的非交换准局部代数的大偏差理论。最主要的兴趣是矩母函数的收敛性及其极限的光滑性和物理意义。台球是一类被广泛研究的动力系统。这是因为两个方面的原因：一方面，在数学上，它们是相对可处理的，它们的几何特征经常给如何证明某个结果的提示。另一方面，从物理学的角度来看，它们是相当现实的模型，已被应用于各种领域，从统计力学到光学到散射理论。因此，很自然地，我们要扩展这一类别，以涵盖它可能产生显著影响的新领域：开放系统家族（每个具有无界动力学的系统都属于这一类别）和随机系统家族（试图证明确定性预测在嘈杂的世界中何时以及为什么仍然有意义）。至于第三部分，统计力学中的大偏差理论从理论上理解了这样一个事实，即从观察者的观点来看，大系统是非常随机的，但仍然产生非常精确的确定性测量。虽然经典系统的大偏差已经成为大规模和非常成功的研究的主题，但量子系统的对应物（它应该是更基本的）目前明显缺失。