Mathematical analysis of strongly correlated processes on discrete dynamic structures

离散动态结构强相关过程的数学分析

基本信息

批准号：
EP/N004566/1
负责人：
Alexandre Stauffer
金额：
$ 113.01万
依托单位：
University of Bath
依托单位国家：
英国
项目类别：
Fellowship
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN004566%2F1
关键词：
Mathematical analysis strongly correlated processes

项目摘要

This proposal embraces the broad theme of mathematical analysis of large interacting systems, which consist of several small components that randomly interact with one another over space and time.This concept arises in many fields, and paradigm examples studied in the probability, statistical physics and computer science literature include percolation, spin systems, and random and dynamic networks. The development of rigorous statistical mechanics and its influence on modern probability theory turned into a remarkable success story in the second half of the last century, not only enriching both fields, but at the same time stimulating and establishing new connections between probability theory, complex analysis, dynamical systems etc.Many powerful theories and techniques were produced on both sides, which led to deep understanding of equilibrium problems, in particular for systems whose local interactions at microscopic level give rise to weak ``macroscopic independence''.In parallel, demands from theoretical computer science, combinatorics and non-equilibrium statistical physics offer a large class of models where local microscopic interactions either produce strong correlations at macroscopic levels, or generate non-equilibrium dynamics, whose behavior changes drastically in time, breaking stationarity and ergodicity. This prevents current methods based on ergodic theory and rigorous statistical mechanics techniques (e.g., energy vs. entropy, finite energy and combinatorial arguments) to be applied, and puts us in front of great challenges. Our overall objective is to develop mathematical techniques to analyze such important and difficult models, producing ground-breaking results in this area, establishing new connections with other topics, and opening up future directions of research.In order to make progress towards this broad goal,we will concentrate on four specific models, which are interesting in their own right, and exhibit important and challenging characteristics and phenomena that are common to a large class of systems.

该提案包含了大型交互系统数学分析的广泛主题，这些组件由几个小组件组成，这些组件在空间和时间上相互随机相互作用。这在许多领域都产生，并且在概率，统计学物理学和计算机科学文献中研究的范式示例包括渗透率，旋转系统以及随机和动态网络。严格的统计力学及其对现代概率理论的影响的发展变成了上个世纪下半叶的一个了不起的成功故事，不仅丰富了这两个领域，而且同时刺激并建立了概率理论，复杂分析，动态系统等的概率理论之间的新联系和建立新的联系。 ``宏观独立性''。同时，理论计算机科学，组合和非平衡统计物理学的要求提供了大量模型，在这些模型中，局部微观相互作用要么在宏观级别上产生强烈的相关性，要么在非平衡动力学上产生非平衡动力学，其行为在时间上发生了巨大的变化，在时间上发生了巨大的变化。这防止了基于厄尔贡理论和严格的统计力学技术（例如能量与熵，有限的能量和组合参数）的当前方法，并将我们置于巨大的挑战面前。我们的总体目标是开发数学技术来分析这种重要和困难的模型，在该领域产生突破性的结果，与其他主题建立新的联系，并打开未来的研究方向。为了在朝着这一广泛的目标方面取得进展，我们将专注于四个特定模型，这些模型在其自身的权利中很有趣，并展现了重要的，具有挑战性的特征和现象，这些模型是一个大型系统，这些模型是一个大型的系统。