Comment on “Preserving the Boltzmann ensemble in replica-exchange molecular dynamics” [J. Chem. Phys. 129, 164112 (2008)

Abstract

A brief discussion of the ergodic description of constant temperature molecular dynamics (MD) is provided; the discussion is based on the analysis of criticisms raised in a recent paper [B. Cooke and S. C. Schmidler, J. Chem. Phys.129, 164112 (2008)]. In the paper, the following criticisms relating to the basic concepts of constant temperature MD are made in mathematical manners: (I) the Nosé–Hoover (NH) equation is not measure-preserving; (II) NH system and NH chain system are not ergodic under the Boltzmann measure; and (III) the Nosé Hamiltonian system as well as the Nosé–Poincaré Hamiltonian system is not ergodic. In this comment, I show the necessity for the reconsideration of these criticisms. The NH equation is measure-preserving, where the measure carries the Boltzmann–Gibbs density; this fact provides the compatibility between MD equation and the Boltzmann–Gibbs distribution. The arguments advanced in support of the above criticisms are unsound; ergodicities of those systems are still not theoretically judged. I discuss exact ergodic-theoretical expressions appropriate for constant temperature MD, and explain the reason behind the incorrect recognitions.