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Criticality and Nonlinearity in Interacting Particle Systems and Stochastic Partial Differential Equations

相互作用粒子系统和随机偏微分方程中的临界性和非线性

基本信息

批准号：
1712575
负责人：
Li Cheng Tsai
金额：
$ 14.91万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2019-11-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712575&HistoricalAwards=false
关键词：
Criticality Nonlinearity Interacting Particle Systems

项目摘要

The project concerns the time evolution of large, stochastic systems, with a focus on their nonlinear and critical behaviors. The systems studied in this project are representative of natural phenomena, such as crystal growths, evolution of magnetic domains, paths evolving in a noisy environment, and randomly stirred fluids. Often these systems exhibit nonlinearity and criticality. Nonlinearity refers to the behavior that the randomness of the macroscopic observables are related to the underlying microscopic randomness in a nonlinear fashion. Criticality refers to the phenomenon that a given system exhibits drastically distinct macroscopic behaviors when certain parameters in the underlying microscopic model reach some critical values. Time-evolutionary stochastic systems form a large body of probability theory, yet the type of phenomena considered here sits on the frontiers of current standard theories. The goal of this project is to unveil the mathematical structure pertaining to the aforementioned scopes, and to refine the existing theories to study these systems.In concrete terms, the project studies three types of models: interacting particles with moving boundaries, stochastic partial differential equaitons (SPDEs) at their criticality, and stochastic six vertex-types models. Particle systems with moving boundaries give rise to Stefan?s problem in PDE, and in one particular case those systems relate to the critical point of a reaction-diffusion particle system. The principal investigator seeks to develop more robust tools that do not require explicit stationary distributions and apply to the aforementioned critical point. SPDEs exhibit criticality for certain parameters, where the solutions become non-measurable with respect to the driving noise. This research aims at studying the correlation functions, regularity, and local properties of a few specific examples of such SPDEs. The stochastic six-vertex model is a specialization of the ice-type models that can be formulated as a Markov process. It hosts a number of degenerations, including the totally asymmetric simple exclusion process. As a first step toward understanding the limiting shape of these models, the principal investigator plans to study the large deviations utilizing the Markov structures of these models.

该项目关注大型随机系统的时间演化，重点关注其非线性和临界行为。在这个项目中研究的系统是自然现象的代表，如晶体生长，磁畴的演变，在嘈杂的环境中演变的路径，和随机搅拌的流体。这些系统通常表现出非线性和临界性。非线性是指宏观观测值的随机性与微观随机性以非线性方式相关的行为。临界性是指当微观模型中的某些参数达到某个临界值时，给定系统表现出截然不同的宏观行为的现象。时间演化随机系统形成了大量的概率论，但这里考虑的现象类型位于当前标准理论的前沿。本项目的目标是揭示与上述范围相关的数学结构，并完善研究这些系统的现有理论。具体而言，本项目研究三种类型的模型：移动边界相互作用粒子模型、临界状态下的随机偏微分方程（SPDE）模型和随机六顶点模型。移动边界的粒子系统会产生斯特凡？的问题，并在一个特定的情况下，这些系统涉及的反应扩散粒子系统的临界点。首席研究员寻求开发更强大的工具，不需要明确的平稳分布，并适用于上述临界点。SPDE对某些参数表现出临界性，其中解决方案相对于驱动噪声变得不可测量。本研究的目的是研究的相关函数，规律性和局部性质的一些具体的例子，这样的SPDE。随机六顶点模型是冰型模型的一种特殊化，可以用马尔可夫过程表示。它有许多退化，包括完全不对称的简单排斥过程。作为理解这些模型的极限形状的第一步，主要研究者计划利用这些模型的马尔可夫结构来研究大偏差。