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

这个项目的目的是使用在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回答的问题。相变在物理科学中是不寻常的现象；它们立即成为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。传统上，相被定义为其中所有物理性质实际上都是一致的系统。当然，水到水蒸气的转变应该总是涉及到相变。奇怪的是，情况并非如此；通过在水的压力-温度图中的临界点周围建立一条路径，液体可以转化为蒸汽，而不会造成密度的不连续变化。那么，如何才能使一个阶段的定义更精确，以及是否存在一个阶段转变才能被理解呢？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，这项研究可以为控制其他材料的固-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块状金属玻璃的金属合金成分，或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结构。这个项目的独特之处在于强调使用拓扑学进行数据探索：组态空间上的函数控制热力学行为，但空间本身极其复杂，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描述。首先，使用映射器算法逼近位形空间上重言式函数的Reeb图。与文献中常见的不连通性图相比，Reeb图提供了更多关于配置空间拓扑的信息。其次，虽然点的构形空间已经被研究了几十年，但给点以正的半径给代数拓扑学家带来了新的有趣的挑战。这些挑战预计将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组态空间直径中存在非解析性的情况下，这将为相变的本质提供一个截然不同的更基本的观点。任何与硬盘系统有关的具体结果都将在其他领域的硬体领域进行测试，以评估结论的一般性。该奖项反映了NSF的法定使命，并已通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.