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

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.

该奖项由材料研究、物理学和数学科学部门支持。智力价值：该项目侧重于模型系统统计力学精确研究的持续研究。 研究集中在两个领域：1）二维临界渗流，一个重要的和非常广泛研究的模型系统，我们正在带来新的和意想不到的方法，和2）Farey分数自旋链的热力学，一组有趣的相变行为和连接到多重分形和动力系统的一维模型。该项目旨在这两个领域的新结果和见解。Farey模型的研究阐明了相变理论中一个有趣的边界情况，并激发了数论中的几个出版物。渗流研究已经导致了一些令人惊讶的功能，特别是模形式的适用性。正在进行的工作旨在扩展和深化这些联系。研究的一个重要特点是与纯数学的关系，包括与数学家的密切合作。这已经导致了一些新的发展，数论在上述两个领域，并协同发展，数论提供了新的工具，以获得洞察intophysical systems.We采用各种理论方法，包括共形场论，蒙特卡罗模拟，数学精确解的模型统计力学系统，和技术从数论。本研究扩展和发展了PI的统计力学基础研究计划，在两个维度和相关的主题。更广泛的影响，这个项目包括协同与数论已经提到，培养研究生在这些研究领域，并影响PI的教学方法。在非常实用的层面上，也有正在进行的重大副产品，涉及质谱仪（和类似仪器）的仪器。这尤其导致了斯蒂尔沃特科学仪器公司的成立。我们在这家公司的主要作用是利用我们对保形方法的熟悉。非技术摘要：在这项由材料研究、物理和数学科学部门资助的研究中，对粒子和小磁体在表面上运动的行为的研究将与对纯数的研究相联系。 表面物理和纯数学之间的这种联系为学生的教育和意外的数学和物理的发现提供了肥沃的土壤。