Integrable Structure of Interacting Particles Systems and Quantum Spin Chains

相互作用粒子系统和量子自旋链的可积结构

• 批准号: 1207995
• 负责人: Craig Tracy
• 金额: $ 96.17万
• 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: 2012
• 资助国家: 美国
• 起止时间: 2012-08-01 至 2019-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1207995&HistoricalAwards=false
• 关键词: Integrable; Structure; Interacting; Particles; Systems

he asymmetric simple exclusion process (ASEP) is a model of interacting particles on a lattice. ASEP is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. As such the model has attracted much attention from both mathematicians and physicists. This award will support (1) the exploration of the underlying integrable structure of ASEP using spectral methods, (2) the generalization of ASEP to multi-species where there are now second-class, third-class, etc. particles, and (3) ASEP on the half-line. The integrable structure referred to commonly goes under the name ``Bethe Ansatz''; and in (3) this project will support the analysis of the new structures required in restricting to a half-line. (Here the familiar sum over permutations in Bethe Ansatz is replaced by a sum over signed permutations.) A closely related model is the quantum spin model called the Heisenberg-Ising model. This project will support the analysis of the domain wall problem in the Heisenberg-Ising chain. A long-term goal of this project is the study of limit laws and their universality in these various models.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. Current research involves processes which are strongly interacting; and as such, there is no satisfactory general theory as there exists for the non-interacting cases. In the search for new universal laws, the ideas, methods and results of random matrix theory, interacting particle systems and their associated limit theorems; in particular, the Tracy-Widom (TW) distributions, have found an ever widening impact in science and engineering. Since their initial discovery in random matrix theory, the TW distributions have been shown to describe the properties of a number of random systems including the fluctuations of a large class of growing interfaces and directed polymer systems. These same TW distributions are now regularly used in multivariate statistical analysis where they have become a standard statistical tool for inferring population structure from genetic data. Their use in engineering involving signal analysis and wireless communications is just beginning. This project extends the scope of these theoretical investigations; and as such, will make available to a wider audience the resulting mathematical results.

非对称简单排除过程 (ASEP) 是晶格上相互作用的粒子的模型。 ASEP 是研究传输现象的最简单、最重要的随机模型之一，因为它模拟远离平衡的过程。 因此，该模型引起了数学家和物理学家的广泛关注。 该奖项将支持（1）使用光谱方法探索 ASEP 的底层可积结构，（2）将 ASEP 推广到现在存在二类、三类等粒子的多物种，以及（3 ) ASEP 就在半线上。 这种可积结构通常被称为“Bethe Ansatz”； (3) 该项目将支持限制半线所需的新结构的分析。 （这里，Bethe Ansatz 中熟悉的排列之和被符号排列之和所取代。）一个密切相关的模型是称为海森堡-伊辛模型的量子自旋模型。 该项目将支持海森堡-伊辛链中域壁问题的分析。 该项目的长期目标是研究极限定律及其在这些不同模型中的普遍性。高斯分布（熟悉的钟形曲线）因其普遍性而很重要；也就是说，它适用于各种看似无关的问题。 所有这些问题的基本共同主题是，所研究的对象具有某种程度的“独立性”； 或者用更物理的术语来说，是非相互作用的（或弱相互作用的）。 这些类型的问题在物理和数学上都是很好理解的。 当前的研究涉及强相互作用的过程；因此，对于非相互作用的情况，没有令人满意的一般理论。 在寻找新的普遍规律时，随机矩阵理论、相互作用粒子系统及其相关极限定理的思想、方法和结果；特别是 Tracy-Widom (TW) 发行版，在科学和工程领域产生了越来越广泛的影响。 自从在随机矩阵理论中最初发现以来，TW 分布已被证明可以描述许多随机系统的属性，包括一大类生长界面和定向聚合物系统的波动。 这些相同的 TW 分布现在经常用于多元统计分析，它们已成为从遗传数据推断种群结构的标准统计工具。 它们在涉及信号分析和无线通信的工程中的应用才刚刚开始。 该项目扩展了这些理论研究的范围；因此，将向更广泛的受众提供由此产生的数学结果。