Mathematical exploration of the ergodicity of Nose-Hoover dynamics

8 June 2009
Mitchell Luskin
The accuracy of the Nos-Hoover thermostat to sample the Gibbs measure depends on the dynamics being ergodic. It has been observed for a long time that this dynamics is actually not ergodic for some simple systems, such as the harmonic oscillator. In this talk, we rigourously prove the non-ergodicity of the Nos-Hoover thermostat, for the one-dimensional harmonic oscillator. We will also show that, for some multidimensional systems, the averaged dynamics for the limit of infinite thermostat "mass" has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Our numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity. We also present numerical experiments for 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. Joint work with Frederic Legoll and Richard Moeckel
  • OxPDE Lunchtime Seminar