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Non-ergodicity of the Nosé-Hoover dynamics

Wednesday, May 13, 2009 - 11:15am - 12:15pm
Lind 409
Frédéric Legoll (École Nationale des Ponts-et-Chaussées)
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, that is coupled nonlinearly with the physical variables. The validity of the method depends on the dynamics being ergodic. It has been numerically observed for a long time that such a thermostat, applied to some model problems (including the one-dimensional harmonic oscillator), is actually not ergodic.

In this work, we first show that, for some multidimensional systems, the averaged dynamics, obtained in the limit of infinite thermostat mass, has many invariants, thus giving theoretical support for either non-ergodicity or slow ergodization. Next, in the case of one-dimensional Hamiltonian systems, we go further and prove non-ergodicity of the thermostat for large (but finite) thermostat masses.

Numerical experiments will illustrate the theoretical results.