Renormalization in Statistical Mechanics and Partial Differential Equations

统计力学和偏微分方程的重整化

• 批准号: 1954357
• 负责人: Scott Armstrong
• 金额: $ 36万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-05-01 至 2024-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1954357&HistoricalAwards=false
• 关键词: Renormalization; Statistical; Mechanics; Partial; Differential

A great variety of mathematical models involve multiple scales: an explicit description of a system's microscopic properties is given, and a challenge is to describe the effective behavior induced by it on much larger scales. In order to predict the correct large-scale effective behavior theoretically, physicists introduced the idea of "renormalization," a systematic technique for implementing a progressive coarsening of the system of interest. The broad goal of research supported by this award is to widen and strengthen the mathematical understanding of this idea. The project provides research training opportunities for graduate students. The past few years have seen great progress on fundamental models that can be represented as partial differential equations with random coefficients. In the present project, the investigator plans to test the versatility and power of the methods developed there on new classes of models, including Langevin dynamics and systems of particles in interaction. The project will take inspiration from renormalization ideas to study mean-field models with disordered interactions, such as spin glasses.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

各种各样的数学模型涉及多个尺度：给出了系统微观属性的明确描述，而挑战是在更大的尺度上描述由它引起的有效行为。为了从理论上预测正确的大规模有效行为，物理学家引入了“重整化”的概念，这是一种用于实现感兴趣系统的渐进粗化的系统技术。该奖项支持的研究的总体目标是扩大和加强对这一想法的数学理解。该项目为研究生提供研究培训机会。过去几年，基本模型取得了巨大进展，这些模型可以表示为具有随机系数的偏微分方程。在目前的项目中，研究人员计划测试在新型模型上开发的方法的多功能性和威力，包括朗之万动力学和相互作用的粒子系统。该项目将从重整化思想中汲取灵感，研究具有无序相互作用的平均场模型，例如自旋玻璃。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Free energy upper bound for mean-field vector spin glasses

• DOI: 10.1214/22-aihp1292
• 发表时间: 2020-10
• 期刊: Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
• 作者: J. Mourrat

Dynamical Fractional and Multifractal Fields

动态分形场和多重分形场

• DOI: 10.1007/s10955-021-02867-2
• 发表时间: 2022
• 期刊: Journal of Statistical Physics
• 影响因子: 1.6
• 作者: Apolinário, Gabriel B.;Chevillard, Laurent;Mourrat, Jean-Christophe
• 通讯作者: Mourrat, Jean-Christophe

Local versions of sum-of-norms clustering

• DOI: 10.1137/21m1448732
• 发表时间: 2021-09
• 期刊: SIAM J. Math. Data Sci.
• 作者: Alexander Dunlap;J. Mourrat

Quantitative homogenization of interacting particle systems

• DOI: 10.1214/22-aop1573
• 发表时间: 2020-11
• 期刊: The Annals of Probability
• 作者: A. Giunti;Chenlin Gu;J. Mourrat

Statistical inference of finite-rank tensors

有限秩张量的统计推断

• DOI: 10.5802/ahl.146
• 发表时间: 2022
• 期刊: Annales Henri Lebesgue
• 作者: Chen, Hongbin;Mourrat, Jean-Christophe;Xia, Jiaming
• 通讯作者: Xia, Jiaming