Renormalization in Dynamical Systems and Statistical Mechanics

科赫教授正在研究统计力学中哈密顿系统和模型中的临界行为。这种行为似乎不能归因于单个系统，而是归因于系统的流形。在这些现象的重整化群方法中，人们试图将这些流形识别为作用于所考虑的系统空间的变换的不变流形。这种作用于哈密顿量的“重整化群变换”在之前的一个项目中被引入。期望有一个非平凡的不动点，其稳定流形描述了某不变环面的临界破裂。主要目标之一就是证明这样一个不动点的存在性。拟议的方法涉及使用计算机来进行所需的大量估计。另一个目标是利用不动点的已知或预期性质来描述临界不变环面和附近的周期轨道。将重整化群方法推广到哈密顿系统的其他问题也将被研究。与量子场论和统计力学的一些有趣的形式联系也提出了解决这些领域老问题的新方法。该领域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。这个项目的大部分内容与经典哈密顿系统的稳定性问题有关。例如耦合振荡器、某些等离子体束和天体力学模型。在二自由度的情况下，沿光滑不变环面的准周期轨道将相空间划分为其他轨迹无法逃脱的区域，从而增加了系统的稳定性。当系统参数变化时，这种环面的破裂会导致动力学变得局部不稳定或混沌。数值研究表明，这是以一种高度普遍的方式发生的，在一大类系统中，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一个假设，即存在一个在合适的“重整化群变换”下不变的非平凡哈密顿系统。这里的目标是证明这个假设是正确的。部分证明工作将由计算机完成。这项工作应该对观察到的现象背后的机制产生有价值的见解，同时，推进人工计算机辅助证明的状态-一种在未来数学研究中无疑将发挥重要作用的技术。类似的普遍性现象与凝聚态物理中的相变以及数学和物理中其他几个领域的模型有关。事实上，物理学和其他科学的核心是一种更为普遍的普遍性，即不需要对微观细节的确切了解就可以对宏观进行描述。正在调查的项目是理解这些现象并以数学方式描述它们的长期努力的一部分。正确的数学描述也有望导致数值算法的重大改进。

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.