Renormalization in Statistical Mechanics and Dynamical Systems

统计力学和动力系统中的重整化

• 批准号: 9705095
• 负责人: Hans Koch
• 金额: $ 7.04万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-15 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705095&HistoricalAwards=false
• 关键词: Renormalization; Statistical; Mechanics; Dynamical; Systems

9705095 Koch Professor Koch is investigating continuous-spin models near criticality, quasiperiodic orbits for classical Hamiltonian flows, and lattice gases related to aperiodic tilings and quasicrystals. The first two of these projects involves the use, and further development, of renormalization group techniques. In a previous renormalization group analysis of ferromagnetic spin models, some fundamental questions were answered in a simplified hierarchical setting. Ongoing work now deals with the problem of restoring some of the ingredients that have been missing. The investigation of quasiperiodic motion for Hamiltonian flows is based on a new class of renormalization group transformations that was introduced in a previous project. One of the goals is to develop these transformations into tools for studying non-perturbative phenomena, such as the breakup of smooth invariant tori. Other interesting questions concern the accumulation of closed orbits at invariant tori and the interplay between renormalization group transformations that correspond to different frequency vectors. The third project deals with a promising but largely unknown class of statistical mechanics models: Lattice gases with non-periodic Gibbs states. The current investigation focuses on a few examples, based on aperiodic tilings, for which the minimum energy configurations are well known. The goal is to identify and understand the low temperature properties of these models and to develop the appropriate methods for analyzing them. Professor Koch's study of ferromagnetic spin models is part of a long-term effort toward a mathematical foundation of the modern theory of critical phenomena in condensed matter physics. One of the striking phenomena is that there are observable quantities (critical indices) which seem to be independent of the system considered, within large classes of different systems. Starting from some basic assumptions, the current theory allows an approximate computation of these universal quantities. But it is still an open problem to show that these assumptions hold, within a class of reasonably realistic models. Ongoing investigations in this area use computer-assisted proofs and involve further development of these techniques. This includes validated numerics -- a technique which is of increasing interest also in engineering and modern industrial design. Another critical phenomenon investigated in this project is the loss of stability of quasi- periodic orbits in classical Hamiltonian systems. In certain cases, this loss of stability is believed to be associated with a significant increase in "chaotic" motion. The process appears to be universal, in the sense described above, but it is not yet understood. Interest in this problem stems from celestial mechanics and plasma physics, where questions of stability play an important role. And of course, the mathematics involved is interesting in itself. The third project deals with statistical mechanics models whose minimum energy configurations are non-periodic. Some of these models are believed to describe quasicrystals, and others may have features similar to spin glasses. They are unusual by current standards, but not exceptional. In fact, non-periodicity is a generic property for ground states in a standard class of statistical mechanics models. So far, almost all investigations of such models have been limited to studying the zero temperature state. The goal now is to obtain some useful information about the behavior at positive temperatures, using a combination of (large scale) numerical simulations and analytical techniques.

9705095 科赫 科赫教授正在研究临界附近的连续自旋模型，经典哈密顿流的准周期轨道，以及与非周期镶嵌和准晶体相关的晶格气体。前两个项目涉及重整化群技术的使用和进一步发展。在以前的铁磁自旋模型的重整化群分析中，一些基本问题在简化的层次结构中得到了回答。正在进行的工作现在处理恢复一些已经丢失的成分的问题。哈密顿流的准周期运动的研究是基于在以前的项目中引入的一类新的重整化群变换。目标之一是将这些变换发展成研究非微扰现象的工具，例如光滑不变环面的分裂。其他有趣的问题涉及在不变环面上闭合轨道的积累和对应于不同频率向量的重整化群变换之间的相互作用。 第三个项目涉及一个有前途的，但在很大程度上未知的统计力学模型：格子气体与非周期吉布斯状态。 目前的调查集中在几个例子，基于非周期性平铺，其中的最低能量配置是众所周知的。 我们的目标是识别和理解这些模型的低温特性，并开发适当的方法来分析它们。 科赫教授对铁磁自旋模型的研究是凝聚态物理学中临界现象现代理论的数学基础的长期努力的一部分。其中一个引人注目的现象是，在大类不同的系统中，存在着似乎与所考虑的系统无关的可观测量（临界指数）。从一些基本假设出发，目前的理论允许对这些普适量进行近似计算。但是，要证明这些假设在一类相当现实的模型中成立，仍然是一个悬而未决的问题。在这一领域正在进行的调查使用计算机辅助证据，并涉及进一步发展这些技术。 这包括验证数字-一种在工程和现代工业设计中也越来越感兴趣的技术。另一个重要的现象是经典哈密顿系统中拟周期轨道的失稳。在某些情况下，这种稳定性的丧失被认为与“混沌”运动的显著增加有关。这个过程似乎是普遍的，在上述意义上，但它还没有被理解。对这个问题的兴趣源于天体力学和等离子体物理学，其中稳定性问题起着重要作用。当然，其中涉及的数学本身就很有趣。第三个项目涉及统计力学模型，其最小能量配置是非周期性的。其中一些模型被认为描述了准晶体，而另一些模型可能具有类似于自旋玻璃的特征。以目前的标准来看，它们是不寻常的，但并不例外。事实上，非周期性是标准统计力学模型中基态的一般性质。到目前为止，几乎所有对这类模型的研究都局限于研究零温状态。现在的目标是利用（大规模）数值模拟和分析技术的结合，获得一些关于正温度下行为的有用信息。