Ergodicity of Thermostat Family of Nos\'e--Hoover type

"Pedagogical derivations for Nos\'e's dynamics can be developed in two different ways, (i) by starting with a temperature-dependent Hamiltonian in which the variable $s$ scales the time or the mass, or (ii) by requiring that the equations of motion generate the canonical distribution including a Gaussian distribution in the friction coefficient $\zeta$. Nos\'e's papers follow the former approach. Because the latter approach is not only constructive and simple, but also can be...

In response to W. G. Hoover's comment [arXiv:1204.0312v2] on our work [arXiv:1203.5968], we show explicitly that the divergence of the velocity field associated with the Nos\'e-Hoover equations is nonzero, implying that those equations are not volume preserving, and hence, as often stated in the literature, are not Hamiltonian. We further elucidate that the trajectories {q(t)} generated by the Nos\'e-Hoover equations are generally not identical to those generated by Dettmann'...

Most deterministic schemes for controlling temperature by kinetic variables (NH thermsotat), configurational variables (BT thermostat) and all phase space variables (PB thermostat) are non-ergodic for systems with a few degrees of freedom. While for the NH thermostat ergodicity has been achieved by controlling the higher order moments of kinetic energy, the issues of nonergodicity of BT and PB thermostats still persist. In this paper, we propose a family of modifications for ...

We discuss the transient and steady state fluctuation relation for a mechanical system in contact with two deterministic thermostats at different temperatures. The system is a modified Lorentz gas in which the fixed scatterers exchange energy with the gas of particles, and the thermostats are modelled by two Nos\'e-Hoover thermostats applied at the boundaries of the system. The transient fluctuation relation, which holds only for a precise choice of the initial ensemble, is v...

Almost all Molecular Dynamics (MD) simulations are discrete dynamics with Newton's algorithm first published in 1687, and much later by L. Verlet in 1967. Discrete Newtonian dynamics has the same qualities as Newton's classical analytic dynamics. Verlet also published a first-order expression for the instant temperature which is inaccurate but presumably used in most MD simulations. One of the motivations for the present article is to correct this unnecessary inaccuracy in $N...

The popular method of Nose and Hoover to create canonically distributed positions and momenta in classical molecular dynamics simulations is generalized to a genuine quantum system of infinite dimensionality. We show that for the quantum harmonic oscillator, the equations of motion in terms of coherent states can easily be modified in an analogous manner to mimic the coupling of the system to a thermal bath and create a quantum canonical ensemble. Possible applications to mor...

This paper demonstrates sufficient conditions for the existence of KAM tori in a singly thermostated, integrable hamiltonian system with $n$ degrees of freedom with a focus on the generalized, variable-mass thermostats of order 2--which include the Nos\'e thermostat, the logistic thermostat of Tapias, Bravetti and Sanders, and the Winkler thermostat. It extends Theorem 3.2 of Legoll, Luskin & Moeckel, (Non-ergodicity of Nos\'e-Hoover dynamics, Nonlinearity, 22 (2009), pp. 167...

Some paradoxical aspects of the Nos\'e and Nos\'e-Hoover dynamics of 1984 and Dettmann's dynamics of 1996 are elucidated. Phase-space descriptions of thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, as is described here by a variety of three- and four-dimensional phase-space models. These findings illustrate some surprising consequences when Liouville's continuity equation is applied to Hamiltonian flows.

The relation between finite isokinetic thermostats and infinite Hamiltonian thermostats is studied and their equivalence is heuristically discussed.

Campisi, Zhan, Talkner, and Haenggi state, in promoting a new logarithmic computational thermostat [ arXiv 1203.5968 and 1204.4412 ], that (thermostated) Nose-Hoover mechanics is not Hamiltonian. First I point out that Dettmann clearly showed the Hamiltonian nature of Nose-Hoover mechanics. The trajectories {q(t)} generated by Dettmann's Hamiltonian are identical to those generated by Nose-Hoover mechanics. I also observe that when the (Hamiltonian) Campisi thermostat is appl...