Structure and Evolution of Low Temperature Spin Systems: Entropic Repulsion and Metastability

低温自旋系统的结构和演化：熵斥力和亚稳态

• 批准号: 2054833
• 负责人: Eyal Lubetzky
• 金额: $ 30万
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-15 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054833&HistoricalAwards=false
• 关键词: Structure; Evolution; Low; Temperature; Spin

Spin systems are mathematical models for ferromagnetism and other properties of materials that are governed by interactions of nuclear magnetic moments (spins). This project aims to study a variety of problems addressing fundamental features of some of the most canonical interacting spin systems at low temperature. Specific models to be studied include the three dimensional Ising model (one of the most fundamental models in statistical mechanics) and the (2+1) dimensional Solid-On-Solid model. One feature that these models exhibit is the entropic repulsion, where an initially flat surface goes through a series of meta-stable states as it builds its height towards equilibrium. The main focus of this project is to study the entropic repulsion phenomenon for these models and its interplay with Glauber dynamics (the natural stochastic process that models the evolution of the system). To advance understanding of these problems the project will develop new methods in probability theory which would find applications in other studies of interacting spins systems. The project provides research training opportunities for graduate students.The first research project focuses on the 3D Ising model at low temperature on an infinite cylinder, with mixed boundary conditions—minus above height 0 and plus elsewhere. These give rise rise to an interface, a random surface separating the plus/minus phases, and the PI aims to study the entropic repulsion effect conditioned on this interface being positive. The main goal is to show that, in order to gain entropy, the surface gradually rises, and its level lines eventually form a single plateau with an a.s. scaling limit given by a Wulff shape; the level line exhibit cube-root fluctuations away from the boundary; and started at a flat initial state, the evolution of the surface towards equilibrium goes through a sequence of meta-stable plateaus, each with a waiting time doubly-exponential in its height. A related research direction aims to study a class of crystal models that approximate 3D Ising, and show that the scaling limit in terms of a Wulff shape and cube-root fluctuations are universal for that entire class. For the (2+1)D Solid-On-Solid, the goal is to refine our understanding and obtain the order of the fluctuations in the top level line. The final topic to be studied concerns crystal models such as Solid-On-Solid at low temperature when the boundary condition is tilted, which is known to cause the (otherwise flat) surface to become de-localized. Here the goal is to give quantitative bounds on the height fluctuations, and show that Glauber dynamics is no longer exponentially slow due to meta-stable states.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.

自旋系统是铁磁性的数学模型和受核磁矩（旋转）相互作用的材料的其他特性。该项目旨在研究各种问题，以解决低温下一些最规范相互作用的自旋系统的基本特征。要研究的特定模型包括三维ISING模型（统计力学中最基本的模型之一）和（2+1）维固体模型。这些模型表现出的一项特征是熵排斥，在该排斥力上，最初的平坦表面经过一系列元稳定状态，因为它朝着平衡的高度建立了高度。该项目的主要重点是研究这些模型的熵排斥现象及其与Glauber动力学的相互作用（建模系统演变的自然随机过程）。为了促进对这些问题的理解，该项目将开发概率理论中的新方法，该方法将在其他相互作用的旋转系统研究中找到应用。该项目为研究生提供了研究培训机会。第一个研究项目的重点是在无限缸上低温下的3D ISING模型，边界条件混合 - 高度高于高度0及其他地方。这些引起了界面的产生，一个随机的表面将加分/减，PI旨在研究以该界面为正的熵排斥效应。主要目标是表明，为了获得熵，表面逐渐上升，其水平线最终形成一个与A.S.沃尔夫形状给出的缩放极限；水平线展示了立方根远离边界的波动。并以平坦的初始状态开始，表面向相等brium的演变经过一系列元稳定的高原，每个平台的高度都在等待时间双指数。相关的研究方向旨在研究一类近似3D ISING的晶体模型，并表明整个类别的wulff形状和立方根波动的缩放极限是普遍的。对于（2+1）d固体固体，目标是完善我们的理解并在顶级线上获得波动的顺序。最终主题是研究涉及晶体模型，例如在边界条件倾斜时在低温下的固体固体，这已知会导致（否则平坦）表面被脱位。在这里，目标是在高度波动上进行定量界限，并表明由于元稳定状态而不再呈指数速度，这反映了NSF的法定任务，并被认为是通过基金会的知识分子优点和更广泛的影响审查标准来评估的。