Non-ergodicity of Nosé-Hoover dynamics

Frédéric Legoll (1, 2) , Mitchell Luskin (3) , Richard Moeckel (3)

1 navier umr 8205 - Laboratoire Navier
2 MICMAC - Methods and engineering of multiscale computing from atom to continuum
3 UMN-MATH - School of Mathematics

The Nosé–Hoover dynamics is a deterministic method that is commonly used to sample the canonical Gibbs measure. This dynamics extends the physical Hamiltonian dynamics by the addition of a ‘thermostat' variable, which is coupled nonlinearly with the physical variables. The accuracy of the method depends on the dynamics being ergodic. Numerical experiments have been published earlier that are consistent with non-ergodicity of the dynamics for some model problems. The authors recently proved the non-ergodicity of the Nosé–Hoover dynamics for the one-dimensional harmonic oscillator. In this paper, this result is extended to non-harmonic one-dimensional systems. We 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. Numerical experiments for a two-dimensional central force problem and the one-dimensional pendulum problem give evidence for non-ergodicity.

Domaines: Analyse numérique [math.NA]

Frédéric Legoll, Mitchell Luskin, Richard Moeckel. Non-ergodicity of Nosé-Hoover dynamics. Nonlinearity, 2009, 22 (7), pp.1673-1694. ⟨hal-00534914⟩