Large-Scale Phenomena in Models of Statistical Mechanics

统计力学模型中的大规模现象

• 批准号: 0806198
• 负责人: Marek Biskup
• 金额: $ 27万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-15 至 2009-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0806198&HistoricalAwards=false
• 关键词: Large; Scale; Phenomena; Models; Statistical

The PI will investigate a variety of projects on the borderline of probability theory and statistical mechanics. A common feature of these is their focus on overcoming the technical obstacles that have insofar held progress to the mathematical understanding of the underlying physical phenomena. The first set of specific problems studies the role of spectral characteristics of random lattice Laplacians in the derivation of scaling limits of random walks in disordered media. The second project investigates the effects of introducing non-attractive potentials on the statics and dynamics of random gradient fields. The third project focuses on developing a detailed approach to localization of random walks in a random potential landscape based on eigenvalue extreme order statistics for lattice Anderson Hamiltonians. The fourth project utilizes new ideas to control the rigidity of interfaces in statistical mechanical models. The final project outlines a new approach based on exchangeability to quantify the effects of a singular interaction on the time evolution of a large system of interacting quantum indistinguishable particles.The project will impact our understanding of various systems of practical interest where the analytic techniques of homogenization theory, spectral analysis, differential equations as well as probabilistic methods, e.g., stochastic analysis, extreme order statistics and theory of disordered systems, etc., play an important role. A number of projects are devised to facilitate training, and inclusion in research, of postdocs and graduate students who have interest in Probability Theory and Mathematical Physics.

PI 将研究概率论和统计力学边缘的各种项目。它们的一个共同特点是专注于克服技术障碍，这些障碍迄今为止在对潜在物理现象的数学理解方面取得了进展。第一组具体问题研究随机晶格拉普拉斯算子的光谱特征在推导无序介质中随机游走的标度极限中的作用。第二个项目研究引入非吸引势对随机梯度场的静态和动态的影响。第三个项目侧重于开发一种基于格子安德森哈密顿量的特征值极端阶统计的随机潜在景观中随机游走定位的详细方法。第四个项目利用新的想法来控制统计力学模型中界面的刚性。最终项目概述了一种基于可交换性的新方法，用于量化奇异相互作用对相互作用的量子不可区分粒子的大型系统的时间演化的影响。该项目将影响我们对实际感兴趣的各种系统的理解，其中均质化理论、谱分析、微分方程以及概率方法（例如随机分析、极序统计和无序系统理论等）的分析技术发挥着重要作用。 角色。设计了许多项目来促进对概率论和数学物理感兴趣的博士后和研究生的培训和参与研究。