Statistical mechanics of two-dimensional interfaces

二维界面的统计力学

• 批准号: 0805493
• 负责人: Richard Kenyon
• 金额: $ 24万
• 依托单位: Brown University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805493&HistoricalAwards=false
• 关键词: Statistical; mechanics; two; dimensional; interfaces

The study of random surfaces and random interfaces has long held the interest of physicists and mathematicians. Only recently, however, have there arisen mathematical techniques for understanding simple interfaces of more than one dimension. The next simplest case,that of two dimensional interfaces in three space,is already quite difficult. The two-dimensional interfaces we study are called 'stepped surfaces'. Under the simplest choice of measure on these surfaces, the uniform measure for a fixed boundary, the large-scale shapes taken by these surfaces has begun to be worked out by the PI and Okounkov, using techniques from PDEs, analysis and tropical geometry. This model is essentially the only mathematically 'solved' model of random interfaces. Moreover it contains a great deal of mathematical connections with other areas: to random matrix theory, integrable systems, string theory and Gromov-Witten theory. For these reasons it is worth understanding this model better, and also worth looking for generalizations.We are studying mathematical models of crystal surfaces. On an atomic scale, the surface of a crystal, such as a salt crystal or diamond, is rough and 'random', but at large scales it is typically smooth and facetted. How these large scale features arise from the microscopic interactions of the atoms comprising the crystal is, to a large extent, still mysterious. However we can make models of crystal surfaces which are computationally tractable in a mathematical sense, and display the same behavior as real crystals: in particular they display facetting and large-scale shape formation. By studying these models we hope to gain understanding not just of crystal surfaces but of the general phenomenon of how local interactions among a large number of constituents can develop into large-scale behavior.

随机表面和随机界面的研究长期以来一直具有物理学家和数学家的兴趣。然而，直到最近才出现数学技术来理解一个多个维度的简单接口。下一个最简单的情况是三个空间中的两个维度接口的情况，已经很困难。我们研究的二维界面称为“阶梯表面”。在这些表面上最简单的度量选择下，固定边界的均匀度量，这些表面所采用的大规模形状已经由PI和Okounkov使用PDES，分析和热带几何形状来制定。该模型本质上是随机接口的唯一数学上“解决”模型。此外，它包含与其他领域的数学联系：与随机矩阵理论，可集成系统，弦理论和Gromov-witten理论。由于这些原因，值得更好地理解该模型，而且值得寻找概括。我们正在研究晶体表面的数学模型。在原子尺度上，晶体的表面（例如盐晶体或钻石）是粗糙且“随机”的，但在大尺度上，它通常是光滑且方形的。这些大规模特征是由包含晶体的原子的微观相互作用而产生的，这在很大程度上仍然是神秘的。但是，我们可以制作晶体表面的模型，这些模型从数学意义上讲是可以在计算上进行的，并显示与真实晶体相同的行为：特别是它们显示刻面和大规模的形状形成。通过研究这些模型，我们希望不仅了解晶体表面的理解，而且希望对大量成分之间的局部相互作用如何发展为大规模行为的一般现象。