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。我的短期目标是证明KAM（近可积）现象的普遍性。辅助目标包括显示温控器的观测特性，如结轨，源于KAM现象。安徒生提出了一个多粒子系统的哈密顿模型，该模型将哈密顿系统的每个轨道上的压力平均为预定值这一约束纳入其中。nos<s:1>（当时在渥太华的核管理委员会）修改了这一理论，创建了一个哈密顿系统，其轨道的平均预定温度为T=1/。胡佛通过在系统中加入一个类似摩擦的项来降低诺瑟斯的恒温器。胡佛的恒温器保留了恒温器的吉布斯测量，M = exp(- E) d dp dq （q,p）是位置和动量变量，是恒温器的控制变量，E = H + 1 / 2²是扩展的能量。这意味着吉布斯测量M（通过积分）投射到具有能量H.KAM理论的哈密顿系统。对还原论目标的一个巨大障碍是KAM理论，它预测了在各种条件下不变环面的正测度集的存在。Legoll, Luskin∞，具有周期势能的机械系统是kam -充分的。这是非常不直观的：提高温度会使材料冻结。与KAM现象相对应的是Mahdi和Valls的结果，即恒温谐振子是不可积的。本工作的一个目的是证明这一代数结果是由于某些结型的周期轨道的存在。进一步的目标是证明恒温系统具有马蹄形，因此具有正熵，并且在更强意义上是不可积的。恒温器是可访问的（恒温谐振子是一个只有2个二次项的3个多项式ode的耦合系统），并为探索物理学（统计力学基础）和数学（动力系统）的深层理论问题提供了丰富的游乐场。我的研究计划旨在对现有结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/. 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 dqwhere (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.