Thermostat Dynamics

我的长期研究计划的目标是以定性和定量的方式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。与每个加拿大人都熟悉的恒温器类似，这些恒温器控制着一个经典系统，以重现恒温下物质的统计性质。简而言之，数学恒温器旨在将统计力学简化为经典力学。我的短期目标是证明KAM(近可积)现象的普遍性。辅助目标包括显示观察到的恒温器的属性，如打结轨道，源自KAM现象。安德森提出了一个多粒子系统的哈密顿模型，该模型包含了这样的约束，即沿着哈密顿系统的每个轨道，平均而言，压力是一个预定值。Nosé(当时在渥太华的核研究中心)对此进行了修改，创建了一个哈密顿系统，其轨道平均具有预定的温度T=1/β。胡佛通过将一个类似摩擦的项耦合到系统中，简化了诺塞的S恒温器。胡佛的恒温器保留了恒温器的吉布斯度量，*M=EXP(-βE)dξDP dq*其中(q，p)是位置和动量变量，ξ是恒温器控制变量，E=H+1/2ξ？是扩展能量。这意味着吉布斯度量M个投影(通过积分ξ)到具有能量H*KAM理论的哈密顿系统的投影。简化论目标的一大障碍是KAM理论，它预言在各种条件下不变环面的正测集的存在。Legoll、Luskin和Moeckel证明，当系统与加热器绝缘时，恒温一维机械系统是KAM充分的。我推广了这些结果：当加热器的温度T→∞时，具有周期势能的机械系统是Kam充分的。这是非常不直观的：升高温度会冻结材料。*不可积。与KAM现象相反的是Mahdi&Valls的结果，即恒温谐振子是不可积的。这项工作的一个目的是证明这个代数结果是由于某些纽结类型的周期轨道的存在。另一个目的是证明恒温系统具有马蹄形，因此具有正熵，并且在更强的意义上是不可积的。恒温器是可以使用的(恒温谐振子是一个由3个多项式组成的耦合系统，只有2个二次项)，并为探索物理学(统计力学基础)和数学(动力学系统)中的深层理论问题提供了丰富的场地。我的研究计划旨在为现有成果和未来发展提供深入的见解。

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/β. 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(-β E) dξ dp dq******where (q,p) are position and momentum variables, ξ is the thermostat control variable and E = H + ½ ξ² is an extended energy. This implies that the Gibbs measure M projects (by integrating out ξ) 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 & Moeckel prove that when the system is insulated from the heater, a thermostated 1-d mechanical system is KAM-sufficient. I extend these results: when the heater's temperature T → ∞, 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.