TRIPODS+X:RES: Collaborative Research: Thermodynamic Phases and Configuration Space Topology

TRIPODS X:RES：协作研究：热力学相和构型空间拓扑

• 批准号: 1839370
• 负责人: Jeremy Mason
• 金额: $ 30万
• 依托单位: University of California-Davis
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-10-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1839370&HistoricalAwards=false
• 关键词: TRIPODS+X; RES; Collaborative; Research; Thermodynamic

The purpose of this project is to use techniques developed within pure and applied topology to study unanswered questions relating to phase transitions. Phase transitions are unusual phenomena in the physical sciences; they are at once part of our everyday experience (e.g., the freezing of water) and yet continue to resist entirely satisfactory explanation. A phase is traditionally defined as a system throughout which all physical properties are effectively uniform. Certainly then, the transformation of water into steam should always involve a phase transition. Curiously, this is not the case; by constructing a path around the critical point in the pressure-temperature diagram of water, the liquid can be converted into the vapor without any discontinuous change in the density. How then can the definition of a phase be made more precise, and the presence or absence of a phase transition be understood? This project approaches the above question by considering a simplified system where molecules are spheres that interact by a hard sphere potential (i.e., do not overlap). Since this is often regarded as a prototype for simple fluids, this research could offer valuable insight into the factors governing solid-liquid phase transitions and glass transitions in other materials. Possible applications include prediction of the metallic alloy compositions most likely to form bulk metallic glasses, or a better understanding of the conditions leading to soil liquefaction during earthquakes.More specifically, the project will study the statistical topology of the configuration space of the hard disk system. What is unique about this project is the emphasis on the use of topology for data exploration: functions on the configuration space govern the thermodynamic behavior, but the space itself is fantastically complicated and likely can only be adequately characterized with the help of computational and theoretical techniques that did not exist even a few years ago. First, the Reeb graph of the tautological function on the configuration space will be approximated using the mapper algorithm. The Reeb graph provides more information about the configuration space topology than the disconnectivity graphs common in the literature. Second, while configuration spaces of points have been studied for decades, giving the points positive radius presents new and interesting challenges for algebraic topologists. These challenges are expected to stimulate new developments in Morse theory.. Third, if the conjecture holds that a phase transition occurs where there is a non-analyticity in the diameter of the configuration space as measured by the diffusion distance, that would provide a dramatically different and more fundamental view of the nature of phase transitions in general. Any results pertaining specifically to the hard disk system will be tested for hard spheres in other domains to evaluate the generality of the conclusions.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.

该项目的目的是利用纯拓扑学和应用拓扑学中开发的技术来研究与相变有关的未解决的问题。相变是物理科学中的罕见现象。它们立即成为我们日常经验的一部分（例如，水的结冰），但仍然无法得到完全令人满意的解释。一个相传统上被定义为一个系统，在这个系统中，所有的物理性质都是一致的。当然，水到蒸汽的转变应该总是包含一个相变。奇怪的是，事实并非如此；通过在水的压力-温度图的临界点附近构造一条路径，可以使液体在密度没有间断变化的情况下转化为蒸汽。那么，如何才能使相的定义更精确，以及如何理解相变的存在或不存在？这个项目通过考虑一个简化的系统来解决上述问题，其中分子是由硬球势相互作用的球体（即，不重叠）。由于这通常被视为简单流体的原型，因此这项研究可以为其他材料的固-液相变和玻璃相变的控制因素提供有价值的见解。可能的应用包括预测最有可能形成大块金属玻璃的金属合金成分，或者更好地了解地震期间导致土壤液化的条件。更具体地说，该项目将研究硬盘系统配置空间的统计拓扑。这个项目的独特之处在于强调使用拓扑进行数据探索：组态空间上的函数控制热力学行为，但空间本身非常复杂，可能只有在计算和理论技术的帮助下才能充分表征，而这些技术在几年前还不存在。首先，用映射算法逼近同义函数在构形空间上的Reeb图。与文献中常见的断连图相比，Reeb图提供了更多关于配置空间拓扑的信息。其次，虽然点的组态空间已经研究了几十年，但给予点正半径对代数拓扑学家来说是一个新的有趣的挑战。这些挑战有望刺激莫尔斯理论的新发展。第三，如果该猜想认为相变发生在由扩散距离测量的构型空间直径中存在非解析性的地方，这将为一般相变的本质提供一个截然不同的、更基本的观点。任何专门与硬盘系统有关的结果都将在其他领域的硬球上进行测试，以评估结论的一般性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Configuration spaces of hard spheres

硬球的构型空间

• DOI: 10.1103/physreve.104.055304
• 发表时间: 2021
• 期刊: Physical Review E
• 影响因子: 2.4
• 作者: Eriçok, O. B.;Ganesan, K.;Mason, J. K.
• 通讯作者: Mason, J. K.

Foundations of a finite non-equilibrium statistical thermodynamics: extrinsic quantities

有限非平衡统计热力学的基础：外在量

• DOI: 10.1088/1751-8121/ac798a
• 发表时间: 2022
• 期刊: Journal of Physics A: Mathematical and Theoretical
• 作者: Ericok, O B;Mason, J K
• 通讯作者: Mason, J K

Topological Constraint Theory for Network Glasses and Glass-Forming Liquids: A Rigid Polytope Approach

网络玻璃和玻璃形成液体的拓扑约束理论：刚性多面体方法

• DOI: 10.3389/fmats.2019.00213
• 发表时间: 2019
• 期刊: Frontiers in Materials
• 影响因子: 3.2
• 作者: Sen, Sabyasachi;Mason, Jeremy K.
• 通讯作者: Mason, Jeremy K.

Continuous and optimally complete description of chemical environments using Spherical Bessel descriptors

• DOI: 10.1063/1.5111045
• 发表时间: 2020-01-01
• 期刊: AIP ADVANCES
• 影响因子: 1.6
• 作者: Kocer, Emir;Mason, Jeremy K.;Erturk, Hakan
• 通讯作者: Erturk, Hakan

Quotient maps and configuration spaces of hard disks

硬盘的商图和配置空间

• DOI: 10.1007/s10035-022-01235-5
• 发表时间: 2022
• 期刊: Granular Matter
• 影响因子: 2.4
• 作者: Ericok, Ozan B.;Mason, Jeremy K.
• 通讯作者: Mason, Jeremy K.