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