Integrable Systems, Integral Operators, and Probabilistic Models

可积系统、积分算子和概率模型

• 批准号: 1400248
• 负责人: Harold Widom
• 金额: $ 12万
• 依托单位: University of California-Santa Cruz
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-15 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1400248&HistoricalAwards=false
• 关键词: Integrable; Systems; Integral; Operators; Probabilistic

Natural phenomena are complex. The purpose of mathematical modeling is to study admittedly simpler versions of what one sees in nature and extract exact results which could help explain, and sometimes predict, events in the real world. There are "nonequlibrium processes", in which the system is evolving, and "equlibrium processes" which model the behavior of the system after it has settled down. This project studies one of each, the "Ising model" (an equlibrium process) and the "asymmetric simple exclusion process" (a nonequlibrium process). The first is the simplest (although still quite complicated) description of ferromagnetism. It is the two-dimensional model that is studied, in particular its thermodynamic properties (in which the number of particles gets infinitely large). Magnetism and magnetic materials are pervasive throughout technology, so it is important to achieve exact mathematical results for them. The asymmetric simple exclusion process is the simplest model of phenomena in which particles interact with each other. Precisely, no two can occupy the same site at the same time. The model has far-ranging applications in nonequilibrium statistical physics, engineering, and biological systems. The one-dimensional case is what is studied (think of electrons flowing in a wire). Theoretical results of the principal investigator and collaborator have been confirmed by recent experiment. This project will result in a deeper understanding of this process and related random models in statistical physics.The asymmetric simple exclusion process (ASEP) is one of the simplest, nontrivial stochastic models in which to study transport phenomena as it models processes far from equilibrium. ASEP is closely related, in a certain scaling limit, to the Kardar-Parisi-Zhang (KPZ) equation, a nonlinear stochastic partial differential equation that is expected to describe a large class of stochastically growing interfaces. Recent work by Sasamoto, Spohn, Amir, Corwin and Quastel, using results of Tracy and the PI, have rigorously established this link between ASEP and the KPZ equation. This project builds on the work of Tracy and the PI and involves: (i) the rigorous operator-theoretic underpinnings of asymptotic results for ASEP with an open boundary; (ii) the study of a discrete time version of ASEP; and (iii) the role of Weyl groups in classifying interacting particle systems solvable by a Bethe Ansatz.

自然现象是复杂的。数学建模的目的是研究人们在自然界中看到的公认的更简单的版本，并提取准确的结果，这可以帮助解释，有时甚至预测现实世界中的事件。有“非平衡过程”，系统在其中不断发展，还有“平衡过程”，在系统稳定下来后对系统的行为进行建模。该项目研究“伊辛模型”（平衡过程）和“不对称简单排除过程”（非平衡过程）中的一个。第一个是对铁磁性最简单（尽管仍然相当复杂）的描述。研究的是二维模型，特别是其热力学性质（其中粒子数量无限大）。磁性和磁性材料在整个技术中无处不在，因此获得精确的数学结果非常重要。 非对称简单排除过程是粒子相互作用现象的最简单模型。准确地说，没有两个人可以同时占据同一地点。该模型在非平衡统计物理、工程和生物系统中具有广泛的应用。研究的是一维情况（想象一下电子在电线中流动）。主要研究者和合作者的理论结果已被最近的实验所证实。该项目将使人们更深入地了解这一过程以及统计物理学中的相关随机模型。非对称简单排除过程 (ASEP) 是研究传输现象的最简单、非平凡的随机模型之一，因为它模拟了远离平衡的过程。 ASEP 在一定比例限制下与 Kardar-Parisi-Zhang (KPZ) 方程密切相关，这是一种非线性随机偏微分方程，有望描述一大类随机增长的界面。 Sasamoto、Spohn、Amir、Corwin 和 Quastel 最近的工作利用 Tracy 和 PI 的结果，严格建立了 ASEP 和 KPZ 方程之间的这种联系。该项目建立在 Tracy 和 PI 的工作基础上，涉及： (i) 具有开放边界的 ASEP 渐近结果的严格算子理论基础； (ii) ASEP 离散时间版本的研究； (iii) Weyl 群在对可通过 Bethe Ansatz 求解的相互作用粒子系统进行分类中的作用。