Stable Formality Quasi-isomorphisms for Hochschild Cochains
Stable Formality Quasi-isomorphisms for Hochschild Cochains
复制标题
Hochschild Cochains 的稳定形式准同构
DOI：
10.24033/msmf.477
复制
发表时间：
2016
期刊：
影响因子：
--
通讯作者：
C. Erignoux
中科院分区：
文献类型：
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作者：
C. Erignoux
在许多动物物种中可以观察到集体动力学，并在过去几十年中形成了一个活跃的跨学科研究领域。这样的行为通常由活动物质来模拟，在活动物质中，每个个体都是自我驱动的，并倾向于根据其邻居之一更新其速度。在Vicsek等人提出的经典模型中，以及在许多相关的活性物质模型中，可以观察到高温下的混沌行为和低温下的全局有序化之间的相变。尽管对于集体动力学已经获得了这些相变的充分证据，但从数学的角度来看，这样的活跃系统还没有完全被理解。近些年来，在平均场相互作用的假设下取得了显著的进展，然而到目前为止，对于纯局域相互作用的模型，几乎没有得到严格的结果。在这篇文章中，作为对活动微观动力学数学理解的第一步，我们描述了一个晶格活动粒子系统，在该系统中，粒子通过局部相互作用来调整它们的速度。利用为晶格气体流体动力学极限发展的公式，我们严格地得到了这个非平衡系统的标度极限。本文建立在Quastel介绍的多类型排斥模型的基础上，详细介绍了他的证明，并结合了几个概括，增加了重大的技术和现象学困难。
Collective dynamics can be observed among many animal species, and have given rise in the last decades to an active and interdisciplinary field of study. Such behaviors are often modeled by active matter, in which each individual is self-driven and tends to update its velocity depending on the one of its neighbors. In a classical model introduced by Vicsek and al., as well as in numerous related active matter models, a phase transition between chaotic behavior at high temperature and global order at low temperature can be observed. Even though ample evidence of these phase transitions has been obtained for collective dynamics, from a mathematical standpoint, such active systems are not fully understood yet. Significant progress has been achieved in the recent years under an assumption of mean-field interactions, however to this day, few rigorous results have been obtained for models involving purely local interactions. In this paper, as a first step towards the mathematical understanding of active microscopic dynamics, we describe a lattice active particle system, in which particles interact locally to align their velocities. We obtain rigorously, using the formalism developed for hydrodynamic limits of lattice gases, the scaling limit of this out-of-equilibrium system. This article builds on the multi-type exclusion model introduced by Quastel by detailing his proof and incorporating several generalizations, adding significant technical and phenomenological difficulties.