论文标题
大规模相互作用系统的拓扑结构通过统一功能和形式
Topological Structures of Large Scale Interacting Systems via Uniform Functions and Forms
论文作者
论文摘要
在本文中，我们通过构建合适的同胞学来研究无限图上的大规模相互作用系统的拓扑结构，我们称为统一的共同体。构建的主要思想是引入一类称为统一功能的功能。统一的共同体学为从微观系统中识别宏观观察物提供了一种新的观点。作为我们理论的直接应用，当基础图具有一个群体的自由作用时，我们证明了对换档不变的闭合均匀形式的一定分解定理。该结果是一个非常通用的分解结果的统一版本，用于瓦拉达汉（Varadhan）最初提出的Shift-Invariant封闭$ l^2 $ forms，该版本在证明不扩大大规模交互系统的流体动力限制的证明中反复发挥了关键作用。在随后的一篇文章中，我们将此结果用作证明瓦拉达汉（Varadhan）的分解定理的一般类别类别相互作用系统的关键。
In this article, we investigate the topological structure of large scale interacting systems on infinite graphs, by constructing a suitable cohomology which we call the uniform cohomology. The central idea for the construction is the introduction of a class of functions called uniform functions. Uniform cohomology provides a new perspective for the identification of macroscopic observables from the microscopic system. As a straightforward application of our theory when the underlying graph has a free action of a group, we prove a certain decomposition theorem for shift-invariant closed uniform forms. This result is a uniform version in a very general setting of the decomposition result for shift-invariant closed $L^2$-forms originally proposed by Varadhan, which has repeatedly played a key role in the proof of the hydrodynamic limits of nongradient large scale interacting systems. In a subsequent article, we use this result as a key to prove Varadhan's decomposition theorem for a general class of large scale interacting systems.
