Second-Order Investigations in Particle and Trapping Systems

本研究的目的是研究两类系统的涨落行为：质量保守粒子系统和布朗俘获模型。第一种类型包括排除和零范围过程，而第二种类型涵盖了“维也纳香肠”模型的变体。在这些系统中，重点研究了三类问题：保守粒子系统中的标记粒子问题、不相容模型中固定位置的电流波动问题以及陷阱设置中长时间存活布朗路径的局部化结构问题。跟随粒子集合运动的随机动力系统已经成功地用于模拟各种物理现象，如流体和交通流、排队、化学捕获行为等。一些关于“一阶”计算的理论研究，描述了感兴趣对象的主要模式，例如，平均速度或陷阱生存概率。这个项目的目的是提供这些流体和交通模型的“二阶”特征。二阶计算的意义在于，它们在应用中提供了一阶近似鲁棒性的度量。除了这个一般的概念之外，这个项目中涉及的计算将阐明所研究的随机介质模型的内在物理特性。此外，从数学的角度来看，这些具有挑战性的问题的解决技术本身对未来的努力也有很大的价值。

The goal of this research is to investigate the fluctuation behaviors in two types of systems: Mass conservative particle systems, and Brownian trapping models. The first type includes the exclusion and zero-range processes, while the second type covers variants of the "Wiener Sausage" model. Three categories of problems are focused upon in these systems: Tagged particle problems in conservative particle systems, current fluctuations at a fixed location in the exclusion model, and localization structure of long time surviving Brownian paths in trap settings.Stochastic dynamical systems following the motion of a collection of particles have been successful in the modeling of diverse physical phenomena such as fluid and traffic flow, queuing, chemical trapping behavior, etc. Several theoretical studies have been made on "first-order" calculations, which describe the dominant modes of objects of interest, for instance, average velocities or trap survival probabilities. The intent of this project is to supply the "second-order" characteristics in these fluid and traffic models. The significance of second-order calculations is that they provide a measure of the robustness of the first-order approximations in applications. Aside from this general notion, the computations involved in this project would shed light on the intrinsic physics of the random media models studied. Also, from the mathematical view, the solution techniques of these challenging problems would be of great value themselves for future endeavors.