Fluctuations, Resonances, and Critical Phenomena

波动、共振和关键现象

• 批准号: 1104596
• 负责人: Michael Aizenman
• 金额: $ 57.3万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2014-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104596&HistoricalAwards=false
• 关键词: Fluctuations; Resonances; Critical; Phenomena

This research will address fundamental issues concerning quantum dynamics in the presence of disorder, and topics related to critical phenomena in statistical mechanics. One of the outstanding challenges is to develop mathematical tools for the analysis of the extended states and conduction in the presence of disorder. Of particular interest is the possibility of the localization - delocalization transition in systems of interacting particles. The work will build on the progress which was recently made by the PI which showed that in certain situations extended states emerge through tunneling facilitated by resonances between well separated localization centers. In the area of Statistical Mechanics, research will be resumed into critical phenomena, in particular at critical dimensions. The goal is to resolve some unsettled questions, in particular in relation to the phi(4d) field theory. The well known non-perturbative results on this topic did not yet address conclusively the critical dimension d = 4. Research will also be directed at clarifying the possible existence of a novel type of transition, at which the nature of the coupling of the local order parameter to the quenched disorder in d = 2 dimensional systems may change as the disorder strength is varied. The PI's past work on this subject included proof, jointly with J. Wehr, of the rounding effect of the disorder, and the recent extension of this to quantum systems, done jointly with R. Greenberg and J. L. Lebowitz. The envisioned research is expected to combine the perspectives of Probability (fluctuation analysis and large deviation theory) and Analysis (Hp function spaces and operator theory) for results which are of interest for Physics. As was demonstrated in the past, the challenge of shedding light on issues of physics enriches also fields of mathematics. This grant includes funds for a graduate student and a postdoc, and for the continuation of fruitful collaborations. These are highly valuable both for the proposed research and for the training component of the work. The PI's past work has contributed to the professional development of a number of men and women who are now active researchers in related fields, and this project will continue to serve that role as well. The research results will be presented through publications, lectures and workshops. The PI has assisted in the organization of meetings.

这项研究将解决有关在存在障碍存在下量子动态的基本问题，以及与统计力学中的批判现象有关的主题。出色的挑战之一是开发数学工具，以分析扩展状态并在存在障碍的存在下进行传导。特别感兴趣的是在相互作用粒子系统中定位的可能性 - 定位转变。这项工作将建立在PI最近取得的进展的基础上，该进展表明，在某些情况下，通过良好的分离定位中心之间的共鸣来促进的隧道出现了扩展的状态。在统计力学领域，研究将恢复为关键现象，特别是在临界维度下。目的是解决一些未解决的问题，特别是与PHI（4D）田间理论有关。关于该主题的众所周知的非扰动结果尚未最终解决临界维度d = 4。研究还将针对阐明一种新型的过渡类型的存在，在这种情况下，局部顺序参数与d = 2维度系统中猝灭障碍的耦合的性质可能会随着障碍的强度而变化。 PI过去在该主题上的工作包括与J. Wehr共同证明该疾病的舍入效应，以及与R. Greenberg和J. L. Lebowitz共同完成的量子系统的最新扩展为量子系统。预期的研究有望结合概率（波动分析和大偏差理论）和分析（HP功能空间和操作员理论）的观点，以实现物理学感兴趣的结果。正如过去所证明的那样，阐明物理学问题的挑战也是数学领域。该赠款包括用于研究生和博士后的资金，以及继续进行富有成果的合作。这些对于拟议的研究和工作的培训部分都是非常有价值的。 PI的过去工作为许多现在是相关领域的积极研究人员的男女的专业发展做出了贡献，该项目也将继续担任该角色。研究结果将通过出版物，讲座和讲习班介绍。 PI已协助组织会议。