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

点击翻译按钮获取中文摘要

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.