Reply to W. G. Hoover [arXiv:1204.0312v2]

TL;DR

This paper clarifies that Nosé-Hoover equations are not Hamiltonian and not volume-preserving, contrasting them with a new Hamiltonian thermostat that maintains standard kinetic energy form and is experimentally compatible.

Contribution

The authors explicitly demonstrate the non-Hamiltonian nature of Nosé-Hoover equations and introduce a new Hamiltonian thermostat with standard kinetic energy terms.

Findings

Nosé-Hoover equations are not volume preserving.

Dettmann's Hamiltonian matches Nosé-Hoover trajectories only on specific energy shells.

The proposed Hamiltonian thermostat is experimentally compatible.

Abstract

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's Hamiltonian. Dettmann's Hamiltonian produces the same trajectories as the Nos\'e-Hoover equations only on a specific energy shell, but not on the neighboring ones. This fact explains why the Nos\'e-Hoover equations are not volume preserving. The Hamiltonian that we put forward with [arXiv:1203.5968] instead produces thermostated dynamics irrespective of the energy value. The main advantage of our Hamiltonian thermostat over previous ones is that it contains kinetic energy terms that are of standard form with coordinate-independent masses and consequently is readily matched in laboratory experiments.