Mathematical Problems from Statistical Mechanics

摘要本文研究的问题包括自避免随机漫步、量子海森堡铁磁体中的液滴状态以及二维和三维的覆盖和堆积问题。自我回避行走是一个模型，它展示了统计力学中最重要的两个思想——临界现象和普遍性。理解二维和三维模型是数学物理中的一个主要问题。液滴状态将在量子海森堡铁磁体中进行研究，这是一种晶体中电子自旋如何排列以产生铁磁性的模型。液滴指的是自旋与主流方向排列方向不同的区域。这种液滴已经用经典统计力学进行了广泛的研究。然而，电子自旋只能用量子力学来恰当地描述，并且在超过一个维度的量子力学模型上没有数学结果。我们将在二维空间中研究这些液滴的量子力学行为。第三个研究领域是经典几何，特别是覆盖和包装问题。包装问题的一个例子是，你应该如何安排两个不同大小的圆盘在平面上，使它们不重叠，但覆盖尽可能多的平面。像这样的问题表述起来很简单，但解决起来却是出了名的困难。在提出的研究中，数学技术将用于研究源自统计力学和凝聚态物理的问题，而在物理问题研究中发展起来的思想将用于解决经典几何中的问题。自我回避行走的研究将影响物理学和物理化学，因为这是一个很好的稀溶液聚合物模型。对自避行走模型的仿真可以改进该模型和其他领域的模型的算法。在这些模拟中开发的代码将公开提供。量子海森堡铁磁体中液滴的研究将影响我们对物理学的理解，但它也将有助于我们理解更多的数学问题，包括简并微扰理论和连续谱的存在性。在包装和覆盖问题的研究中，在受挫自旋系统的研究中发展起来的思想将影响一个完全不同的领域。挫折是指旋转系统需要做一件事来最小化总能量的一部分，但需要做一些不一致的事情来最小化另一部分。数学物理学家开发了一种技术来研究这样的系统，我们将适应于包装和覆盖问题。

AbstractKennedyThe problems to be studied include self-avoiding random walks, droplet states in the quantum Heisenberg ferromagnet and covering and packing problems in two and three dimensions. A self-avoiding walk is a model that exhibits critical phenomena and universality, two of the most important ideas in statistical mechanics. Understanding such models in two and three dimensions is a major problem in mathematical physics. Droplet states will be studied in the quantum Heisenberg ferromagnet, a model of how the spins of the electrons in a crystal can line up to produce ferromagnetism. Droplets refer to domains where the spins are aligned in a different direction from the prevailing direction. Such droplets have been studied extensively using classical statistical mechanics. However, electron spins are only properly described with quantum mechanics, and there are no mathematical results on quantum mechanical models in more than one dimension. We will study the quantum mechanical behavior of these droplets in two dimensions. The third area of research is in classical geometry, specifically covering and packing problems. An example of a packing problem is to ask how you should arrange discs of two different sizes in the plane so that they do not overlap, but cover as much of the plane as possible. Problems like this are simple to state, but notoriously difficult to solve.In the research proposed, mathematical techniques will be used to study problems whose origins are in statistical mechanics and condensed matter physics, and ideas developed in the study of physical problems will be used to attack questions in classical geometry. The work on self-avoiding walks will impact physics and physical chemistry since this is a good model for polymers in dilute solution. Simulations of self-avoiding walk models can lead to improvements in the algorithms, both for this model and for models in other fields. The code developed in these simulations will be made publicly available. The study of droplets in the quantum Heisenberg ferromagnet will impact our understanding of the physics, but it should also contribute to our understanding of more mathematical questions including degenerate perturbation theory and the existence of continuous spectrum. In the research on packing and covering problems, ideas developed in the study of frustrated spin systems will impact a completely different field. Frustration refers to spin systems which need to do one thing to minimize one part of their total energy but need to do something inconsistent to minimize another part. Mathematical physicists developed a technique for studying such systems which we will adapt to the packing and covering problems.