Limit Theorems for Multiparticle Systems

该项目致力于混沌台球的研究，其中包括西奈半岛的分散台球，布尼莫维奇的体育场及其修改，硬球气体和洛伦兹气体。我们的目标是调查遍历和统计性质的系统，专注于相关性和概率极限定理的界限。特别强调的是不同质量和/或尺寸的颗粒（圆盘）的气体，其中一些颗粒的数量和/或质量增长到无穷大。这种方法涉及到重新调整时间和空间，以便应用统计物理学中称为“流体动力学极限”的方案。“这使得人们能够克服潜在动力学的不愉快特征，例如不均匀的双曲性和非常缓慢的混合速率。通过使用平均理论的方法，主要研究者希望建立轨道的收敛到某些随机过程。该项目的更广泛的影响来自其跨学科的特点：它是由物理学和其他科学的问题所激发的，其主要目标是发展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。首席研究员还研究了气缸中活塞的行为，试图模拟汽车发动机中发生的情况。在这个项目中开发的数学方法应该适用于来自自然科学的各种问题。

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.