Crystalline Order in Classical and Quantum Mechanical Systems

摘要：本研究将重点研究结晶和球体填充等问题中周期性结构的产生机制。理解晶体晶格结构是如何产生的一种方法是假设量子力学产生了分子之间有效的经典相互作用。然后研究经典问题如何排列分子使经典能量最小化。即使是这个经典问题也是相当困难的，因为球面填充问题是一个特例。本研究将利用统计力学的思想来证明这些经典问题的最优配置。利用这些方法，可以得到一个新的、更简单的三维球面填充问题的证明。在Falicov-Kimball模型中研究了量子力学在晶体形成中的作用。本研究的另一个目标是了解该模型何时具有周期性基态，何时没有。Falicov-Kimball模型是在晶格上定义的，但实际上晶体是在没有先验晶格结构的空间中形成的。一个连续的Falicov-Kimball型模型将被研究。在这个模型中，离子可以占据空间中的任何位置，而不仅仅是晶格中的某个位置，然后电子在离子之间跳跃。引入和研究这个模型的目的是有一个量子力学模型（希望）显示周期性基态。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。这些问题中最著名的一个是球体填充问题。给定许多相同大小的球体，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的方式是否最佳？虽然这个问题已经有几百年的历史了，而且表述起来很简单，但直到去年才得到解决。经典的晶体问题也是类似的。一个是给定一堆原子和它们之间的作用力定律。问题是找到使原子总能量最小的排列方式。球体填充问题在编码理论中有着重要的应用，而晶体问题是理解人们在自然界中观察到的晶体结构是如何产生的核心。这些问题也可以被认为是包含大量变量的优化问题。本研究将发展建立此类优化问题的解决方案的技术。这些问题之所以困难，是因为他们感到沮丧——少数原子或球体的最佳排列方式与它们在较大群体中的最佳排列方式是不一样的。统计力学中用来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，最好的排列似乎是一个相对简单的周期性排列，尽管一个问题允许所有可能的排列，无论多么复杂。该奖项由数学科学部的分析项目和物理部的数学物理项目共同资助。

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.