Limit Theorems for Multiparticle Systems

多粒子系统的极限定理

• 批准号: 0652896
• 负责人: Nikolai Chernov
• 金额: $ 12.49万
• 依托单位: University of Alabama at Birmingham
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2010-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0652896&HistoricalAwards=false
• 关键词: Limit; Theorems; Multiparticle; Systems

The project is devoted to the study of chaotic billiards, which includes Sinai's dispersing billiards, Bunimovich's stadium and its modifications, gases of hard balls, and Lorentz gases. The goal is to investigate the ergodic and statistical properties of such systems, focusing on bounds for correlations and probabilistic limit theorems. A special emphasis is given to gases of particles (disks) of different masses and/or sizes, where the number and/or the masses of some particles grow to infinity. The approach involves rescaling time and space in order to apply the scheme known in statistical physics as the "hydrodynamical limit." This allows one to overcome unpleasant features of the underlying dynamics such as nonuniform hyperbolicity and very slow mixing rates. By using methods borrowed from averaging theory the principal investigator hopes to establish the convergence of the orbits to certain stochastic processes.The broader impact of this project derives from its interdisciplinary character: it is motivated by problems in physics and other sciences and its main goal is the development of adequate mathematical tools for addressing such problems. In particular, the project deals with Brownian motion, which arises in a variety of natural processes whose evolution is subject to many random factors. The principal investigator also studies the behavior of a piston in a cylinder, in an attempt to model what occurs in an automobile engine. The mathematical methods developed in this project should apply to wide classes of problems coming from the natural sciences.

该项目致力于混沌台球的研究，包括西奈的分散台球、布尼莫维奇的体育场及其改造、硬球气体和洛伦兹气体。目标是研究这种系统的遍历和统计性质，重点是相关性和概率极限定理的界限。特别强调的是不同质量和/或大小的粒子（盘）气体，其中一些粒子的数量和/或质量增长到无穷大。该方法涉及重新调整时间和空间，以便应用统计物理学中称为“流体动力极限”的方案。这允许人们克服潜在动力学的不愉快的特征，如不均匀双曲和非常缓慢的混合速率。通过借用平均理论的方法，主要研究者希望建立轨道对某些随机过程的收敛性。这个项目的广泛影响源于它的跨学科性质：它是由物理学和其他科学中的问题所驱动的，它的主要目标是开发适当的数学工具来解决这些问题。特别是，该项目涉及布朗运动，它出现在各种自然过程中，其演变受到许多随机因素的影响。首席研究员还研究气缸中活塞的行为，试图模拟汽车发动机中发生的情况。在这个项目中发展的数学方法应该适用于来自自然科学的各种各样的问题。