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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.

这一提议涵盖了大型相互作用系统数学分析的广泛主题，该系统由几个在空间和时间上随机相互作用的小组件组成。这一概念出现在许多领域，概率、统计物理和计算机科学文献中研究的范例包括渗滤、自旋系统以及随机和动态网络。严格的统计力学的发展及其对现代概率论的影响在上世纪下半叶成为了一个非凡的成功故事，不仅丰富了这两个领域，而且同时刺激和建立了概率论、复分析、动力系统等之间的新联系。双方都产生了许多强大的理论和技术，导致对平衡问题的深入理解，特别是对于在微观层面上局部相互作用产生弱相互作用的系统。 “宏观独立性”。与此同时，理论计算机科学、组合学和非平衡统计物理学的需求提供了一大类模型，其中局部微观相互作用要么在宏观层面产生强相关性，要么产生非平衡动力学，其行为随时间发生巨大变化，打破平稳性和遍历性。这阻碍了基于遍历理论和严格统计力学技术（例如能量与熵、有限能量和组合参数）的当前方法的应用，并使我们面临巨大的挑战。我们的总体目标是开发数学技术来分析这些重要而困难的模型，在该领域产生突破性的结果，与其他主题建立新的联系，并开辟未来的研究方向。为了朝着这个广泛的目标取得进展，我们将集中于四个特定的模型，它们本身很有趣，并表现出一大类系统所共有的重要且具有挑战性的特征和现象。