From Nosé–Hoover chain to Nosé–Hoover network: design of non-Hamiltonian equations of motion for molecular-dynamics with multiple thermostats

Abstract

A systematic algorithm to design multiple thermostat systems in the framework of the Nosé–Hoover type non-Hamiltonian formulation is presented. Using ‘non uniform’ time transformations in a generalised Hamiltonian equation, we develop the non-Hamiltonian equations of motion for multiple thermostat systems having an arbitrary number of thermostats and arbitrary connections between a physical system and thermostats (‘Nosé–Hoover network’). We then present the algorithm to construct the Nosé–Hoover network equations based on a simple diagram only. On the basis of this algorithm, recursively attached Nosé–Hoover thermostats are introduced as an example of the Nosé–Hoover network and its high efficiency in sampling the canonical distribution for an one-dimensional double-well system is illustrated by numerical calculations.

Keywords:

• Nosé–Hoover thermostat
• molecular dynamics
• canonical ensemble
• non-Hamiltonian dynamics

Acknowledgements

The author thanks Peter Daivis, Ian Snook, and Masuhiro Mikami for helpful discussion and careful reading of the manuscript. This work is partly supported by the Next Generation Super Computing Project, the Nanoscience Program, Japan.

Notes

Notes

1. In d (>1) dimensional systems of N particles, i denotes both the particle number and a spatial component (i.e. i = 1, …, dN).

2. We do not incorporate the auxiliary function f _aux in 6 for clarity. In fact, the generalised Nosé–Hoover dynamics can be constructed on the basis of H _gMT with f _aux, which results in extra harmonic oscillator-like terms in the dynamical equations of the thermostat variables [p _{η l} : Equation (22)]. These extra terms might obviate the resonance problem in multiple time step integrators (see 14 for details).

3. For simplicity, we assume that all particles are attached to the same set of thermostats (e.g. Figure 1), but the discussion here can be generalised to ‘individual’ or ‘massive’ thermostatting 13,14. In this case, the product of j and j ′ in Equations (12) and (13) depends on i as in Equation (14).

4. h _i (s) and f _i (s) are, in this case, given as h _i (s) = s ₁ s ₂ and f _i (s) = s ₃. u _l (s), on the other hand, depends on l; u ₁ = s ₂ s ₄, u ₂ = s ₅, u ₃ = s ₅, and u ₄ = u ₅ = 1 (i.e. u _l consists of the thermostat variables that control the ‘lth’ thermostat). Note that s ₂ is included both in s _j [Equation (12): h _i ] and s _{j ′′(1)} [Equation (14): u _l=1].

5. Instead, ‘Nosé–Poincaré chain thermostat’ is introduced via the Poincaré transformation of H _NC in 6 and 8.

6. F _i , G _i , h ₁, and h ₂ in Equations (3) and (5) in the original paper 9 are in this case taken to be , , h ₁ = ζ, and h ₂ = ξ. Note that we use p _η/Q _η and p _ξ/Q _ξ instead of ζ and ξ, respectively, in this paper.