流体动力学极限及相关主题讲义
Notes on Hydrodynamic Limits and Related Topics
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中文总结 AI 辅助
该讲义介绍随机相互作用粒子系统的流体动力学极限等内容，以排除过程和零程过程为研究对象，补充相关经典著作，为领域入门者介绍相关基础内容。
中文摘要 AI 辅助
本讲义中，我们讨论随机相互作用粒子系统中的各类“流体动力学大数定律（hydrodynamic LLN）”与“中心极限定理（CLT）”尺度极限等内容，将“微观”行为与连续介质定律相联系。通过“短篇故事”的形式，旨在为学生及该领域入门者介绍一些“基础内容”，作为Kipnis-Landim 1999、Komorowski-Landim-Olla 2012、Liggett 1985、Liggett 1999等著作的补充。具体而言，我们将研究范围限定于离散空间上的几个“质量守恒”系统，即排除过程与零程过程，这两类系统在不同现象研究中已被证明具有稳健性。在预备知识之后，我们将讨论“熵”与“相对熵”方法，用于证明有限体积下体质量的流体动力学极限，还会涉及无限体积系统的构造、其不变测度的结构，以及局部泛函的尺度极限，如位点占据时间与标记粒子的运动。最后一部分，我们还将讨论当过程从不变测度出发时体质量的平衡涨落。
英文摘要
In these lecture notes, we discuss various `hydrodynamic LLN' and `CLT' scaling limits, among others, in types of stochastic interacting particle systems, connecting `microscopic' behaviors to continuum laws. Via `short stories', the aim is to present some of the `basics' for students and those entering the field, as a complement to books such as Kipnis-Landim 1999, Komorowski-Landim-Olla 2012, Liggett 1985, Liggett 1999. To be concrete, attention is restricted to a few `mass conservative' systems on discrete spaces, namely exclusion and zero-range processes, that have proved robust in the study of different phenomena. After preliminaries, we discuss the `entropy' and `relative entropy' methods to prove hydrodynamic limits of the bulk mass in finite volume, as well as other items such as construction of systems in infinite volume and the structure of their invariant measures, and scaling limits of local functionals, such as occupation times of sites and the motion of a tagged particle. In the last part, we also discuss equilibrium fluctuations of the bulk mass when the process starts from an invariant measure.
