Exact Results in Aperiodic Systems

非周期系统中的精确结果

• 批准号: EP/X012239/2
• 负责人: Felix Flicker
• 金额: $ 41.15万
• 依托单位: University of Bristol
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2023
• 资助国家: 英国
• 起止时间: 2023 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FX012239%2F2
• 关键词: Exact; Results; Aperiodic; Systems

It is increasingly true that our modern world runs on solving problems of optimization subject to constraints. Examples include the scheduling of public transport, electrical power distribution, and the flow of internet traffic. Condensed matter physics - the study of that which emerges when huge numbers of particles interact - is an essential example: individual particles minimise their energies subject to the constraints of their mutual interactions and local environments. Understanding the resulting matter has enabled the technologies which run our lives.An elegant description of this interplay of constraints and emergence is provided by dimer models. Dimers were originally conceived of as two-atom molecules landing on the surfaces of materials. How densely the dimers can pack onto the surface depends on the arrangement of surface atoms and the bonds connecting them. Understanding this process -- adsorption -- has never been more urgent. That is because it is an essential component in many carbon capture technologies (`dimers' in this case being carbon dioxide molecules, which contain three atoms lying in a straight line). Adsorption also forms the basis of catalysis, the process of reducing the energy required to carry out chemical reactions: if a molecule temporarily sticks to a surface the chance of it reacting with a second stuck molecule can be increased. Catalysis is used in the generation of over a trillion dollars' worth of products annually. The study of dimer models is therefore of pressing industrial and environmental concern. The mathematical study of dimers has an exalted history within statistical physics, where it has led to some profound results. For example, 'conformal invariance' is a property appearing in diverse physical models from quantum gravity to water boiling under pressure --- but the first formal proof of conformal invariance in statistical physics was found only recently, in a dimer model. Dimers were used to prove that there are exactly 12,988,816 ways to tile a chessboard with dominoes, and can be used to prove the following remarkable statement: if you deal a pack of cards into thirteen piles of four, it is always possible to choose one card from each pile so as to select one of each number.Dr Flicker recently initiated the study of dimer models in a fundamentally new context: quasicrystals. The discovery of these remarkable materials in 1982 led to a Nobel prize, and forced a redefinition of what it means for something to be a crystal. Until then, all solids were believed to be either disordered on the atomic scale, or to have their atoms lined up in regular periodic structures, like the squares of a chessboard. Quasicrystals lack periodic structure, but nor are they disordered; instead they feature remarkable and beautiful symmetries: their structures can be described mathematically as slices through higher-dimensional crystals. While many quasicrystals have now been grown artificially, only three naturally occurring quasicrystals have ever been found, all in the same Siberian meteorite. Quasicrystals have potential advantages for adsorption: they are brittle, meaning they can be broken into particles only a few hundred atoms across, maximising their surface area; they remain solid at high temperatures, facilitating increased reaction rates; and their surface atoms feature a wide range of angles between neighbouring atomic bonds, which could be exploited by bendy molecules landing on these surfaces.The aim of this project is to establish exact mathematical results in dimer models, and other statistical models of constrained optimization, on quasicrystals and related structures. Working with project partners - experimental physicists, machine-learning startups, and visual artists and composers - these results will be put to practical and industrial use, and will bring the inherent aesthetic appeal of these intriguing problems to a wider audience.

越来越真实的是，我们的现代世界运行在解决受约束的优化问题上。例子包括公共交通的调度、电力分配和互联网流量。凝聚态物理——研究大量粒子相互作用时产生的现象——是一个重要的例子：单个粒子在相互作用和局部环境的约束下，使它们的能量最小化。对由此产生的物质的理解使我们生活中的技术得以发展。二聚体模型对约束和涌现之间的相互作用提供了一个优雅的描述。二聚体最初被认为是落在材料表面的双原子分子。二聚体在表面上的密度取决于表面原子的排列和连接它们的键。理解这个过程——吸附——从未像现在这样紧迫。这是因为它是许多碳捕获技术的重要组成部分（在这种情况下，“二聚体”是二氧化碳分子，它包含三个原子在一条直线上）。吸附也构成了催化作用的基础，这一过程减少了进行化学反应所需的能量：如果一个分子暂时粘在表面上，它与第二个粘在表面的分子发生反应的机会就会增加。催化作用每年用于生产价值超过一万亿美元的产品。因此，二聚体模型的研究具有迫切的工业和环境问题。二聚体的数学研究在统计物理学中有着崇高的历史，它导致了一些深刻的结果。例如，“共形不变性”是一种出现在从量子引力到水在压力下沸腾的各种物理模型中的特性，但统计物理学中首次正式证明共形不变性是最近才发现的，是在一个二聚体模型中。二聚体被用来证明有12,988,816种方法可以用多米诺骨牌铺棋盘，并且可以用来证明以下重要的陈述：如果你把一副牌分成13摞，每摞4张，总是可以从每摞中选择一张牌，以便从每个数字中选择一张牌。弗利克博士最近开始在一个全新的背景下研究二聚体模型：准晶体。1982年，这些非凡材料的发现获得了诺贝尔奖，并迫使人们重新定义了晶体的含义。在此之前，人们认为所有的固体要么在原子尺度上是无序的，要么它们的原子排列成规则的周期性结构，就像棋盘上的方块一样。准晶体缺乏周期结构，但也不是无序的；相反，它们具有非凡而美丽的对称性：它们的结构可以用数学方法描述为高维晶体的切片。虽然现在人工培育出了许多准晶体，但只发现了三种自然产生的准晶体，而且都是在同一块西伯利亚陨石中发现的。准晶体具有潜在的吸附优势：它们很脆，这意味着它们可以被分解成只有几百个原子大小的颗粒，从而最大化它们的表面积；它们在高温下保持固体状态，有利于提高反应速率；它们的表面原子在相邻的原子键之间具有很大的角度，这可以被落在这些表面上的弯曲分子所利用。本项目旨在建立准晶体及其相关结构的二聚体模型和其他约束优化统计模型的精确数学结果。与实验物理学家、机器学习初创公司、视觉艺术家和作曲家等项目合作伙伴合作，这些结果将被用于实际和工业用途，并将这些有趣问题的内在美学吸引力带给更广泛的受众。