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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

基本信息

批准号：
1839358
负责人：
Matthew Kahle
金额：
$ 30万
依托单位：
Ohio State University
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=1839358&HistoricalAwards=false
关键词：
TRIPODS+X RES Collaborative Research Thermodynamic

项目摘要

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.

这个项目的目的是使用纯和应用拓扑学中开发的技术来研究与相变有关的未回答的问题。相变是物理科学中不寻常的现象;它们是我们日常经验的一部分（例如，水的冻结），但仍然拒绝完全令人满意的解释。相传统上被定义为所有物理性质在整个系统中有效地均匀的系统。当然，水转化为蒸汽的过程总是要有相变的。奇怪的是，事实并非如此;通过在水的压力-温度图中的临界点周围构造一条路径，液体可以转化为蒸汽，而不会发生密度的不连续变化。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呢？该项目通过考虑一个简化的系统来解决上述问题，其中分子是通过硬球势相互作用的球体（即，不重叠）。由于这通常被认为是简单流体的原型，因此这项研究可以为控制其他材料固液相变和玻璃化转变的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，或更好地了解地震时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学进行数据探索：配置空间上的函数控制热力学行为，但空间本身非常复杂，可能只有在几年前还不存在的计算和理论技术的帮助下才能充分表征。首先，重言式函数在组态空间上的Reeb图将使用映射器算法来近似。Reeb图提供了比文献中常见的不连通图更多的关于位形空间拓扑的信息。其次，虽然点的配置空间已经研究了几十年，给点正半径提出了新的和有趣的挑战代数拓扑。这些挑战有望刺激莫尔斯理论的新发展。第三，如果猜想认为相变发生在由扩散距离测量的位形空间的直径中存在非解析性的地方，那么这将提供一个显着不同的和更基本的关于相变本质的观点。任何与硬盘系统相关的结果都将在其他领域进行硬球测试，以评估结论的普遍性。该奖项反映了NSF的法定使命，并被认为值得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来支持。