A Hamiltonian formulation for recursive multiple thermostats in a common timescale

Molecular dynamics trajectories that sample from a Gibbs distribution can be generated by introducing a modified Hamiltonian with additional degrees of freedom as described by Nose [S. Nose, Mol. Phys., 52 ( 1984), p. 255]. To achieve the ergodicity required for canonical sampling, a number of techniques have been proposed based on incorporating additional thermostatting variables, such as Nose - Hoover chains and more recent fully Hamiltonian generalizations. For Nose dynamics, it is often stated that the system is driven to equilibrium through a resonant interaction between the self-oscillation frequency of the thermostat variable and a natural frequency of the underlying system. In this article, we clarify this perspective, using harmonic models, and exhibit practical deficiencies of the standard Nose chain approach. As a consequence of our analysis, we propose a new powerful "recursive thermostatting" procedure which obtains canonical sampling without the stability problems encountered with Nose - Hoover and Nose - Poincare chains.