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Analysis and Computational Studies of the Ergodicity of the Nose-Hoover Thermostat

Monday, July 23, 2007 - 4:00pm - 4:30pm
Mitchell Luskin (University of Minnesota, Twin Cities)
The Nose-Hoover thermostat is a deterministic dynamical system
designed for computing 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. We have proven 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 dynamics is not ergodic even for a moderate thermostat mass.
We also give numerical experiments of the Nose-Hoover chain (proposed by Martyna, Klein, and Tuckerman) 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.

Joint work with Frederic Legoll and Richard Moeckel.