Statistical Mechanics and Field Theory

统计力学和场论

• 批准号: 0244884
• 负责人: John Imbrie
• 金额: $ 11.36万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-06-01 至 2007-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0244884&HistoricalAwards=false
• 关键词: Statistical; Mechanics; Field; Theory

1. The purpose of this project is to enhance our understanding of fundamental problems instatistical mechanics and probability theory through the use of mathematical methodsinvolving supersymmetry and the renormalization group. Of particular interest arestatistical models with a geometrical flavor, such as branched polymers and self-avoidingwalks. Basic questions for these models involve, for example, determining the rate atwhich the size of a sample grows with the number of monomers or steps. Such divergencesare governed by critical exponents. The values of the exponents may be determined in oneof two ways, either through dimensional reduction (which relates exponents to knownexponents in related systems in fewer dimensions), or through the renormalization group(which calculates the flow of couplings with the length scale as the microscopicstructures of the problem are averaged out). Supersymmetry plays an important role inboth methods.Specific goals are: (a) to continue to develop a mathematical theory of dimensionalreduction and apply it to new models, such as directed branched polymers; and (b) toconstruct an exact renormalization group flows for such systems.2. Research activities in this proposal will involve collaborations and exchanges ofideas with mathematicians and physicists in a wide range of fields, including probability,analysis, statistical mechanics, and quantum field theory. Graduate student support, andgraduate student participation in scientific meetings, will integrate research andeducation and help train a new generation of mathematicians in the methods of fieldtheory, supersymmetry, and the renormalization group. These activities will in the longrun lead to enhanced interaction between mathematics and physics and to new ideas in bothfields.

1. 本项目的目的是通过使用涉及超对称性和重正化群的数学方法来增强我们对统计力学和概率论基本问题的理解。特别令人感兴趣的是具有几何风格的统计模型，例如支化聚合物和自回避行走。例如，这些模型的基本问题涉及确定样本大小随单体或步骤数量增长的速率。这种分歧是由临界指数决定的。指数的值可以通过两种方式之一确定，或者通过降维（​​将指数与相关系统中的已知指数以更少的维度联系起来），或者通过重正化群（当问题的微观结构被平均时计算与长度尺度的耦合流）。超对称性在这两种方法中都发挥着重要作用。具体目标是：（a）继续发展降维数学理论并将其应用于新模型，例如定向支化聚合物； (b) 为此类系统构造精确的重整化群流。 2．该提案中的研究活动将涉及与数学家和物理学家在广泛领域的合作和思想交流，包括概率、分析、统计力学和量子场论。研究生支持以及研究生参加科学会议将把研究和教育结合起来，帮助培养场论、超对称和重正化群方法方面的新一代数学家。从长远来看，这些活动将增强数学和物理之间的相互作用，并在这两个领域产生新的想法。