Integrable Structure of Interacting Particle Systems

相互作用粒子系统的可积结构

基本信息

批准号：
1809311
负责人：
Craig Tracy
金额：
$ 32万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1809311&HistoricalAwards=false
关键词：
Integrable Structure Interacting Particle Systems

项目摘要

Statistical mechanics is that area of theoretical physics that predicts macroscopic behavior of physical systems given the microscopic dynamics. The dynamics can be either classical or quantum. Equilibrium statistical mechanics (the foundational work going back to L. Boltzmann and J.W. Gibbs) has a well-formulated, in principle, procedure to go from the microscopic dynamics to the description of the macroscopic system. Carrying out this procedure can be extremely difficult; nevertheless, the broad outlines are well understood. Most physical systems are not in equilibrium and the main interest is their long-time behavior. The Gaussian distribution (the familiar bell-shaped curve) is important because of its universality; that is, it applies to a wide variety of seemingly unrelated problems. The underlying common theme for all these problems is the fact that the objects under study have some degree of "independence"; or stated in more physical terms, are non-interacting (or weakly interacting). These types of problems are well-understood both physically and mathematically. The foundations of nonequilibrium statistical physics are not as well understood as equilibrium statistical physics, in particular when the processes involve interacting particles. This project is directed at understanding the processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. Simple models occupy a bigger role in understanding systems far from equilibrium. The asymmetric simple exclusion process (ASEP), a model of interacting particles on a lattice, is one such model. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. This project, building on earlier work of the one-point correlation, will analyze a class of higher-order correlations. The first higher-order correlations are block probabilities for step initial conditions. By a block of length L we mean a contiguous block of L particles (starting, say, at position m on the lattice). Recently the probability of this event was computed. The project will generalize this block probability result to the case of step Bernoulli initial conditions. This same result is also the starting point of the project where similar higher-order correlations will be analyzed. For example, the correlation of the first particle with the particle at position m. The methods employed are both combinatorial and analytical. In particular new combinatorial identities generalizing those found for block probabilities are an essential element of the project. The possible long term impact of this project is to discover new universal limit laws that generalize the Tracy-Widom distributions that appear in the limit laws for the one-point fluctuations.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.

统计力学是考虑到微观动力学的物理系统的宏观行为的理论物理学领域。动力学可以是经典的或量子的。均衡统计力学（可以追溯到L. Boltzmann和J.W. Gibbs的基础工作）原则上具有良好的成型程序，可以从微观动力学转变为宏观系统的描述。执行此过程可能非常困难；然而，广泛的概述却很熟悉。大多数物理系统都不处于平衡状态，主要兴趣是它们的长期行为。高斯分布（熟悉的钟形曲线）很重要，因为它的普遍性。也就是说，它适用于多种看似无关的问题。所有这些问题的基本共同主题是，所研究的对象具有一定程度的“独立性”。或用更多的物理术语陈述，是非相互作用（或弱相互作用）。这些类型的问题在物理和数学上都被众所周知。非平衡统计物理学的基础不太理解为平衡统计物理，特别是当过程涉及相互作用的颗粒时。该项目旨在了解强烈互动的过程。因此，没有令人满意的一般理论，因为非交互情况存在。简单模型在理解远离均衡的系统中占据更大的作用。不对称的简单排除过程（ASEP）是一种在晶格上相互作用粒子的模型，就是这样的模型。 ASEP是最简单，非平凡的随机模型之一，在该模型中，它可以研究运输现象，因为它远离平衡。该项目建立在单点相关性的早期工作的基础上，将分析一类高阶相关性。第一个高阶相关性是步骤初始条件的块概率。按长度为l的块，我们是指一个连续的L颗粒块（例如，在晶格上的位置m开始）。最近，计算了此事件的概率。该项目将将此块概率结果推广到步骤Bernoulli初始条件的情况。同样的结果也是项目的起点，在该项目中将分析相似的高阶相关性。例如，第一粒子与位置m处的粒子的相关性。所采用的方法既是组合和分析方法。特别是，新的组合身份将发现的块概率的概括是该项目的重要组成部分。该项目的长期影响是发现新的通用限制定律，该法律概括了一点点波动的极限定律中出现的Tracy-Widom分布。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛的影响审查标准通过评估来进行评估的。