Thermostat Dynamics

我的长期研究计划的目标是用定性和定量的方法描述可积和不可积动力系统之间的区别。被称为恒温器的保守动力系统为所涉及的问题提供了特别好的例子。类似于每个加拿大人都熟悉的恒温器，这些恒温器控制着一个经典系统，以便在恒定温度下再现物质的统计特性。简单来说，数学恒温器旨在将统计力学简化为经典力学。

The goal of my long-term research program is to delineate, in both qualitative and quantitative ways, the distinction between integrable and non-integrable dynamical systems. Conservative dynamical systems called thermostats provide particularly good examples of the issues involved. Similar to the thermostat that is familiar to every canadian, these thermostats control a classical system in order to reproduce the statistical properties of matter at a constant temperature. In simplistic terms, a mathematical thermostat aims to reduce statistical mechanics to classical mechanics. My short-term objective is to demonstrate the pervasiveness of KAM (near-integrable) phenomena. Subsidiary goals include showing that observed properties of thermostats, like knotted orbits, originate from KAM phenomena. THERMOSTATS. Andersen proposed a hamiltonian model of a many-particle system which incorporated the constraint that along each orbit of the hamiltonian system, on average, the pressure is a predetermined value. Nosé (while at the NRC in Ottawa) modified this to create a hamiltonian system whose orbits, on average, had a predetermined temperature T=1/. Hoover reduced Nosé's thermostat by coupling a friction-like term to the system. Hoover's thermostat preserves the Gibbs measure of the thermostat, M = exp(- E) d dp dq where (q,p) are position and momentum variables, is the thermostat control variable and E = H + ½ ² is an extended energy. This implies that the Gibbs measure M projects (by integrating out ) to that of the hamiltonian system with energy H. KAM THEORY. A great barrier to the reductionist goal is KAM theory, which predicts the existence of positive measure sets of invariant tori under a variety of conditions. Legoll, Luskin , a mechanical system with a periodic potential energy is KAM-sufficient. This is very un-intuitive: raising the temperature freezes the material. NON-INTEGRABILITY. A counterpoint to KAM phenomena is the result of Mahdi & Valls that the thermostated harmonic oscillator is non-integrable. A goal of this work is show that this algebraic result is due to the existence of periodic orbits of certain knot types. A further goal is to demonstrate that the thermostated system possesses a horseshoe, therefore has positive entropy and is non-integrable in a stronger sense. SUMMARY. Thermostats are accessible (the thermostated harmonic oscillator is a coupled system of 3 polynomial ODEs with only 2 quadratic terms), and provide a rich playground for exploring deep theoretical issues in physics (foundations of statistical mechanics) and mathematics (dynamical systems). My research program aims to provide deep insights into both the existing results and future developments.