Renormalization in Dynamical Systems and Statistical Mechanics

科赫教授 从统计力学的角度研究哈密顿系统和模型的临界行为。 这样的行为似乎不是归因于单个系统，而是归因于系统的流形。 在重整化群中 为了研究这些现象，人们试图把这些流形确定为作用于所考虑的系统空间的变换的不变流形。 这样一个“重整化群 在以前的一个项目中，引入了作用于哈密顿量的“变换”。 期望它有一个非平凡的不动点，其稳定流形描述了某个不变量的临界破裂 桃丽 主要目标之一就是证明这样一个不动点的存在性。 所提出的方法包括使用计算机进行大量的估计，这将是有益的。 另一个目标是使用固定点的已知或预期属性， 描述临界不变环面和附近的周期轨道。 重正化群方法在其他Hamilton系统中的推广 也将被调查。 与量子场论和统计力学的一些有趣的形式联系也表明了新的 在这些领域解决老问题的方法。 这方面的长期目标之一是更好地理解Ising类型模型的关键行为。 这里的方法是寻找和研究相应的层次模型的适当改进。 其他有趣的问题涉及层次模型本身，以及与无序介质相关的两个不动点问题。这在很大程度 该项目涉及到古典主义的稳定性问题， Hamilton系统 例如耦合振荡器、某些等离子体束和模型 从天体力学。 在两个自由度的情况下，跟踪光滑不变环面的准周期轨道将相空间划分为其他轨道无法从其处划分的区域 逃脱，从而增加了 的稳定性。 当系统参数发生变化时，这种环面的破裂会导致系统的局部不稳定或混沌。 数值研究表明， 以一种高度普遍的方式发生，在一个大类的系统中，某些可测量的量取完全相同的值。 这些发现的标准解释是基于这样的假设，即存在一个非平凡的哈密顿系统，该系统在适当的“重整化群变换”下是不变的。 这里的目标是证明这个假设是正确的。 部分校对工作将由计算机完成。 这项工作应该能提供有价值的见解， 观察到的现象背后的机制，并在同一时间，推进最先进的计算机辅助证明-一种技术，无疑将在未来的数学研究中发挥重要作用。 相似的普遍性现象 与凝聚态物理学中的相变，以及数学和物理学中其他几个领域的模型。 普遍性的一种更普遍的形式--宏观的 描述是可能的 没有微观细节的精确知识没有微观细节的精确知识 在 物理学和其他科学的核心。 正在调查的项目是长期努力的一部分，以了解这种现象，并以数学方式描述它们。 一个正确的数学描述也可以预期导致显着改善数值算法。

Professor Koch is investigating critical behavior in Hamiltoniansystems and models from statistical mechanics. Such behavior appearsto be attributed not to individual systems, but to manifolds ofsystems. In the renormalization group approach to these phenomena,one attempts to identify these manifolds as the invariant manifolds ofa transformation acting on the space of systems considered. Such a"renormalization group transformation", acting on Hamiltonians, wasintroduced in a previous project. It is expected to have a nontrivialfixed point, whose stable manifold describes the critical breakup ofcertain invariant tori. One of the main goals is to prove theexistence of such a fixed point. The proposed method involves the useof a computer, to carry out the large number of estimates that will beneeded. Another goal is to use the known or anticipated properties ofthe fixed point in order to describe critical invariant tori andnearby periodic orbits. Extensions of renormalization group methodsto other (problems in) Hamiltonian systems will be investigated aswell. Some interesting formal connections with quantum field theoryand statistical mechanics also suggest new ways of approaching oldproblems in these areas. One of the long term goals in this area isto gain a better understanding of the critical behavior of Ising typemodels. The approach here is to find and study suitable improvementsof the corresponding hierarchical model. Other interesting questionsconcern the hierarchical model itself, and two fixed point problemsrelevant to disordered media. Much of this project is related to the question of stability inclassical Hamiltonian systems. Examples are coupled oscillators,certain plasma beams, and models from celestial mechanics. In thecase of two degrees of freedom, quasiperiodic orbits that trace outsmooth invariant tori divide phase space into regions from which othertrajectories cannot escape, thus adding to the stability of thesystem. The breakup of such tori, as system parameter are varied, cancause the dynamics to become locally unstable or chaotic. Numericalstudies reveal that this happens in a highly universal way, withcertain measurable quantities taking exactly the same values, within alarge class of systems. The standard explanation of these findings isbased on the assumption that there exists a nontrivial Hamiltoniansystem that is invariant under a suitable "renormalization grouptransformation". The goal here is to prove that this assumption iscorrect. Part of the proof will be carried out by a computer. Thiswork should yield valuable insight into the mechanism behind theobserved phenomena, and at the same time, advance the state of the artin computer-assisted proofs -- a technique that will undoubtedly playan important role in the future of mathematical research. Similaruniversality phenomena are associated with phase transitions incondensed matter physics, and models from several other areas inmathematics and physics. A more general form of universality -- thefact that macroscopic descriptions are possible without the exactknowledge of microscopic details -- is in fact at the heart ofphysics and other sciences. The projects under investigation are partof a long term effort to understand such phenomena, and to describethem mathematically. A correct mathematical description can also beexpected to lead to significant improvements in numerical algorithms.