Exact Results in Model Statistical Systems

模型统计系统中的精确结果

• 批准号: 0536927
• 负责人: Peter Kleban
• 金额: $ 19.8万
• 依托单位: University of Maine
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2011-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0536927&HistoricalAwards=false
• 关键词: Exact; Results; Model; Statistical; Systems; Exact; Results; Model; Statistical; Systems

This award is supported by the Divisions of Materials Research, Physics, and Mathematical Sciences.Intellectual merit: This project focuses on continued research on the exact study of the statistical mechanics of model systems. The research concentrates on two areas:1) critical percolation in two dimensions, an important and very extensivelystudied model system, to which we are bringing new and unexpected approaches, and2) the thermodynamics of the Farey fraction spin chain, a set of onedimensional models with interesting phase transition behavior and connections tomultifractals, and dynamical systems.This project aims at new results and insights in both these areas.Research on the Farey models illuminates an interesting borderline case in thetheory of phase transitions and has inspired several publications in number theory.The percolation research has already led to some surprising features, especially theapplicability of modular forms. Ongoing work aims at extending and deepeningthese connections.An important feature of the research is the relation to pure mathematics,including close collaboration with mathematicians. This has already led to somenew developments in number theory in both areas mentioned, and to synergisticdevelopments, where the number theory provides new tools to gain insight intophysical systems.We employ a variety of theoretical methods including conformal fieldtheory, Monte Carlo simulations, mathematically exact solutions of modelstatistical mechanical systems, and techniques from number theory. This researchextends and develops the PI's program of basic research in statistical mechanics intwo dimensions and related topics.Broader impacts of this project include the synergy with number theoryalready mentioned, training of graduate students in these research areas, andimpacts on the PI's teaching methods. There is also ongoing significant spinoff ona very practical level, involving instrumentation for mass spectrometers (andsimilar instruments). This has led, in particular, to the formation of StillwaterScientific Instruments, Inc. Our main role in this company exploits our familiaritywith conformal methods.Non-technical Abstract:In this research funded by the Divisions of Materials Research, Physics, and Mathematical Sciences, the study of the behavior of particles and small magnets moving on surfaces will be connected to the study of pure numbers. This connection between surface physics and pure mathematics provides a fertile area for the education of students and the discovery of unexpected mathematics and physics.

该奖项得到了材料研究，物理学和数学科学的划分的支持。智能优点：该项目着重于对模型系统统计力学的确切研究的持续研究。 研究集中在两个领域：1）在两个维度上进行临界渗透，这是一个重要且非常广泛的模型系统，我们正在为此带来新的和意外的方法，并且2）Furey分数旋转链的热力学能力，一套偏离的相匹配模型，具有有趣的阶段过渡行为和连接的模型，以进行新的型号。在相位过渡的三个方面阐明了一个有趣的边界案例，并激发了数量理论的几个出版物。渗透研究已经导致了一些令人惊讶的特征，尤其是模块化形式的征服性。正在进行的工作旨在扩展和深度的联系。该研究的重要特征是与纯数学的关系，包括与数学家的密切合作。这已经导致在提到的两个领域和协同开发方面的数字理论方面的发展，其中数字理论提供了新的工具来获取洞察力的无物质系统。我们采用了各种理论方法，包括共形的田间理论，蒙特卡洛模拟，数学上精确的方法，这些方法是模型机械系统和技术理论的数学精确解决方案。这项研究延期并开发了PI在统计力学方面的基础研究计划和相关主题。该项目的影响力的影响包括协同作用，其中提到了数字理论，对这些研究领域的研究生进行了培训，以及对PI教学方法的影响。还在不断进行的大量衍生水平非常实用，涉及质谱仪（和仿效仪器）的仪器。这尤其是导致了Stillwaterscientific Instruments，Inc。的形成。我们在这家公司中的主要作用利用了我们的熟悉度与整合方法。没有技术摘要：在这项研究中，由材料研究，物理学，物理学和数学科学的划分资助，对表面的行为和小磁铁的研究将与该研究相关。 表面物理学与纯数学之间的这种联系为学生的教育以及意外数学和物理学的发现提供了一个肥沃的领域。