Renormalization in Dynamical Systems and Statistical Mechanics

University Of Texas At Austin, Austin TX

Investigators

Abstract

Professor Koch is investigating critical behavior in Hamiltonian systems and models from statistical mechanics. Such behavior appears to be attributed not to individual systems, but to manifolds of systems. In the renormalization group approach to these phenomena, one attempts to identify these manifolds as the invariant manifolds of a transformation acting on the space of systems considered. Such a "renormalization group transformation", acting on Hamiltonians, was introduced in a previous project. It is expected to have a nontrivial fixed point, whose stable manifold describes the critical breakup of certain invariant tori. One of the main goals is to prove the existence of such a fixed point. The proposed method involves the use of a computer, to carry out the large number of estimates that will be needed. Another goal is to use the known or anticipated properties of the fixed point in order to describe critical invariant tori and nearby periodic orbits. Extensions of renormalization group methods to other (problems in) Hamiltonian systems will be investigated as well. Some interesting formal connections with quantum field theory and statistical mechanics also suggest new ways of approaching old problems in these areas. One of the long term goals in this area is to gain a better understanding of the critical behavior of Ising type models. The approach here is to find and study suitable improvements of the corresponding hierarchical model. Other interesting questions concern the hierarchical model itself, and two fixed point problems relevant to disordered media. Much of this project is related to the question of stability in classical Hamiltonian systems. Examples are coupled oscillators, certain plasma beams, and models from celestial mechanics. In the case of two degrees of freedom, quasiperiodic orbits that trace out smooth invariant tori divide phase space into regions from which other trajectories cannot escape, thus adding to the stability of the system. The breakup of such tori, as system parameter are varied, can cause the dynamics to become locally unstable or chaotic. Numerical studies reveal that this happens in a highly universal way, with certain measurable quantities taking exactly the same values, within a large class of systems. The standard explanation of these findings is based on the assumption that there exists a nontrivial Hamiltonian system that is invariant under a suitable "renormalization group transformation". The goal here is to prove that this assumption is correct. Part of the proof will be carried out by a computer. This work should yield valuable insight into the mechanism behind the observed phenomena, and at the same time, advance the state of the art in computer-assisted proofs -- a technique that will undoubtedly play an important role in the future of mathematical research. Similar universality phenomena are associated with phase transitions in condensed matter physics, and models from several other areas in mathematics and physics. A more general form of universality -- the fact that macroscopic descriptions are possible without the exact knowledge of microscopic details -- is in fact at the heart of physics and other sciences. The projects under investigation are part of a long term effort to understand such phenomena, and to describe them mathematically. A correct mathematical description can also be expected to lead to significant improvements in numerical algorithms.