Exact Results in Model Statistical Systems

该奖项由材料研究、物理和数学科学部支持。智力价值：这个项目专注于对模型系统统计力学的精确研究的持续研究。这项研究集中在两个领域：1)二维的临界渗流，这是一个重要的和非常广泛研究的模型系统，我们正在为其带来新的和意想不到的方法；2)Farey分数自旋链的热力学，一组具有有趣的相变行为和与多重分形之间的联系的一维模型，以及动力学系统。本项目旨在这两个领域的新结果和新见解。对Farey模型的研究揭示了相变理论中一个有趣的边界情况，并启发了数论中的几篇论文。渗流研究已经导致了一些令人惊讶的特征，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。这已经导致了数论在上述两个领域的一些新的发展，以及协同的发展，其中数论提供了新的工具来洞察物理系统。我们使用了各种理论方法，包括保形场理论，蒙特卡罗模拟，模型统计机械系统的数学精确解，以及来自数论的技术。本研究在两个维度和相关主题上扩展和发展了PI的统计力学基础研究项目。该项目的广泛影响包括与前面提到的数论的协同作用，这些研究领域的研究生培养，以及对PI的教学方法的影响。在非常实际的层面上，也有正在进行的重大衍生产品，涉及质谱仪(和类似仪器)的仪器。这尤其导致了Stillwater科学仪器公司的成立。我们在该公司的主要角色是利用我们对保角方法的熟悉。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，对粒子和小磁铁在表面运动的行为的研究将与纯数的研究相联系。表面物理和纯数学之间的这种联系为学生的教育和发现意想不到的数学和物理提供了一个肥沃的领域。

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.