Crystalline Order in Classical and Quantum Mechanical Systems

建议：DMS-9970608首席研究员：托马斯·G·肯尼迪摘要：这项研究将集中在结晶和球体堆积等问题中如何产生周期性结构。理解晶体晶格结构的一种方法是假设量子力学在分子之间产生有效的经典相互作用。然后，人们研究了如何排列分子以最小化这种经典能量的经典问题。即使是这个经典问题也是相当困难的，将球包装问题作为特例包含在内。这项研究将使用受挫系统统计力学的思想来证明其中一些经典问题的最优配置。用这些方法可能在三维空间中给出球体堆积问题的一种新的、更简单的证明。在Falicov-Kimball模型中研究了量子力学在晶体形成中的作用。这项研究的另一个目的是了解这个模型什么时候有周期基态，什么时候没有。Falicov-Kimball模型定义在晶格上，但实际上晶体是在没有先验晶格结构的空间中形成的。我们将研究一个连续体的法里科夫-金博尔模型。在这个模型中，离子可能占据空间中的任何位置，而不仅仅是某个晶格中的位置，然后电子在离子之间跳跃。引入和研究这个模型的目的是得到一个(希望)呈现周期性基态的量子力学模型。这项研究将集中在像晶体这样的周期性结构是如何产生的。其中最著名的问题之一就是球体包装问题。给出大量相同大小的球体，怎样排列它们才能使它们占据最小的体积？例如，在内战战场上发现炮弹堆放的方式在这方面是最优的吗？虽然这个问题有几百年的历史，也很简单，但它只是在过去的一年里才得到解决。经典的晶体问题与此类似。一个人被赋予了一串原子和它们之间的力定律。问题是要找到使原子总能量最小的排列方式。球体堆积问题在编码理论中有重要的应用，而晶体问题是理解人们在自然界中观察到的晶体结构如何产生的核心。这些问题也可以看作是包含大量变量的优化问题。这项研究将开发建立此类优化问题的解决方案的技术。这些问题都是困难的，因为它们受挫了--对少数原子或球体的最佳排列与它们作为大群的一部分时的最佳排列不同。在统计力学中开发的研究这种受挫系统的技术将适用于这些问题。令人惊讶的是，在所有这些问题中，最好的安排似乎是相对简单的周期性安排，尽管人们允许所有可能的安排，无论多么复杂。该奖项由数学科学系的分析计划和物理系的数学物理计划共同资助。

Proposal: DMS-9970608 Principal Investigator: Thomas G. KennedyAbstract: This research will focus on how periodic structures arise in problems such as crystallization and sphere packing. One approach to understanding how the lattice structure of crystals comes about is to assume that the quantum mechanics produces an effective classical interaction between molecules. Then one studies the classical problem of how to arrange the molecules to minimize this classical energy. Even this classical problem is quite hard, containing the sphere packing problem as a special case. This research will use ideas from statistical mechanics for frustrated systems to prove the optimal configuration for some of these classical problems. A new, simpler proof of the sphere packing problem in three dimensions may be possible by these methods. The role of quantum mechanics in crystal formation has been investigated in the Falicov-Kimball model. Another goal of this research is to understand when this model has a periodic ground state and when it does not. The Falicov-Kimball model is defined on a lattice, but in nature crystals form in space which has no a priori lattice structure. A continuum Falicov-Kimball type model will be investigated. In this model the ions may occupy any site in space rather than just the sites in some lattice, and then the electrons hop between the ions. The goal of introducing and studying this model is to have a quantum mechanical model that (hopefully) exhibits periodic ground states. This research will focus on how periodic structures like crystals arise. One of the most famous of these problems is the sphere packing problem. Given a large number of spheres of the same size, what is the best way to arrange them so that they occupy the smallest volume. For example, is the way one finds cannonballs stacked at a Civil War battlefield optimal in this regard? Although this problem is hundreds of years old and quite simple to state, it was only solved in the past year. The classical crystal problem is similar. One is given a bunch of atoms and a force law between them. The problem is to find the arrangement which minimizes the total energy of the atoms. The sphere packing problem has important applications to coding theory, and the crystal problem is at the heart of understanding how the crystalline structures that one observes in nature arise. These problems may also be thought of as optimization problems in a large number of variables. This research will develop techniques for establishing the solutions of such optimization problems. The problems are all difficult because they are frustrated -- the best arrangment for a small number of atoms or spheres is not the same as their best arrangment when they are part of a larger group. Techniques developed in statistical mechanics to study such frustrated systems will be adapted to these problems. Surprisingly, it appears that in all these problems the best arrangement is a relatively simple periodic one, even though one allows all possible arrangements, however complicated. This award is jointly funded by the Analysis Program in the Division of Mathematical Sciences and the Mathematical Physics Program in the Division of Physics.