Renormalization in Statistical Mechanics and Dynamical Systems

9705095科赫教授正在研究临界点附近的连续自旋模型，经典哈密顿流的准周期轨道，以及与非周期瓦片和准晶有关的晶格气体。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，一些基本问题在一个简化的层次结构中得到了回答。正在进行的工作现在涉及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先前项目中引入的一类新的重整化群变换。其中一个目标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闭合轨道在不变环面的积累，以及对应于不同频率向量的重整化群变换之间的相互作用。第三个项目涉及一类很有希望但却鲜为人知的统计力学模型：具有非周期吉布斯态的晶格气体。目前的研究集中在几个例子上，基于非周期瓦片，其最小能量组态是众所周知的。目标是确定和了解这些模型的低温特性，并开发适当的分析方法。科赫教授对铁磁自旋模型的研究是为凝聚态物理学中临界现象的现代理论奠定数学基础的长期努力的一部分。一个引人注目的现象是，在大类不同的系统中，有一些可观测的量(临界指数)似乎与所考虑的系统无关。从一些基本假设出发，目前的理论允许对这些宇宙量进行近似计算。但要证明这些假设在一类相当现实的模型中是成立的，这仍然是一个悬而未决的问题。这一领域正在进行的研究使用计算机辅助校样，并涉及这些技术的进一步发展。这包括经过验证的数值计算--一种在工程学和现代工业设计中也越来越受关注的技术。本项目研究的另一个关键现象是经典哈密顿系统中准周期轨道的稳定性丧失。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。在上述意义上，这一过程似乎是普遍的，但人们还不了解这一点。对这个问题的兴趣源于天体力学和等离子体物理，稳定性问题在其中扮演着重要的角色。当然，所涉及的数学本身也很有趣。第三个项目涉及最小能量构型为非周期的统计力学模型。其中一些模型被认为描述了准晶，另一些模型可能具有类似自旋玻璃的特征。以目前的标准来看，它们是不寻常的，但并不例外。事实上，在一类标准的统计力学模型中，非周期性是基态的一种普遍性质。到目前为止，对这类模型的研究几乎都局限于对零温态的研究。现在的目标是利用(大规模)数值模拟和分析技术相结合的方法，获得有关正温度下行为的一些有用信息。

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.