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Non-ergodicity of the Nose-Hoover Thermostatted Harmonic Oscillator.pdf

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Non-ergodicity of the Nose-Hoover Thermostatted Harmonic Oscillator

a r X i v : m a t h / 0 5 1 1 1 7 8 v 2 [ m a t h .D S ] 1 7 M a y 2 0 0 6 NON-ERGODICITY OF THE NOSE?-HOOVER THERMOSTATTED HARMONIC OSCILLATOR FRE?DE?RIC LEGOLL, MITCHELL LUSKIN, AND RICHARD MOECKEL Abstract. The Nose?-Hoover thermostat is a deterministic dynamical system designed for comput- ing phase space integrals for the canonical Gibbs distribution. Newton’s equations are modified by coupling an additional reservoir variable to the physical variables. The correct sampling of the phase space according to the Gibbs measure is dependent on the Nose?-Hoover dynamics being ergodic. Hoover presented numerical experiments that show the Nose?-Hoover dynamics to be non-ergodic when applied to the harmonic oscillator. In this article, we prove that the Nose?-Hoover thermostat does not give an ergodic dynamics for the one-dimensional harmonic oscillator when the “mass” of the reservoir is large. Our proof of non-ergodicity uses KAM theory to demonstrate the existence of invariant tori for the Nose?-Hoover dynamical system that separate phase space into invariant regions. We present numerical experiments motivated by our analysis that seem to show that the dynam- ics is not ergodic even for a moderate thermostat mass. We also give numerical experiments of the Nose?-Hoover chain with two thermostats applied to the one-dimensional harmonic oscillator. These experiments seem to support the non-ergodicity of the dynamics if the masses of the reservoirs are large enough and are consistent with ergodicity for more moderate masses. 1. Introduction Equilibrium statistical properties of molecular systems [3,10] are given by phase space integrals of the form 〈A〉 = ∫ A(q, p) dμ(q, p), (1.1) where q = (q1, . . . , qM ) ∈ RnM and p = (p1, . . . , pM ) ∈ RnM denote a set of positions qi ∈ Rn and momenta pi ∈ Rn of M particles (n denotes the space dimension), and A(q, p) is an observable, a function defined over the phase space and related to the macroscopic quantity under stud