On the Actual Difference between the Nose and the Nose-Hoover Thermostats: A Critical Review of Canonical Temperature Control by Molecular Dynamics

This article has to be intended both as a review and as an historical tribute to the ideas developed almost 40 years ago by S. Nose, establishing the theoretical foundations of the implementation and use of thermostats in molecular dynamics (MDs). The original motivation of this work is enriched by an observation related to the connection between the Nose seminal expression of temperature control and the extension proposed in 1985 by W. G. Hoover, known as the Nose-Hoover thermostat. By carefully rederiving the equations of motion in both formalisms, it appears that all features of Nose-Hoover framework (replacement of the Nose variables by a single friction coefficient in the equations of motion) are already built in the Nose approach. Therefore, one is able to work directly within the Nose formalism with the addition of a single variable only, by greatly extending its general impact and simplicity and somewhat making redundant the Nose-Hoover extension. Having been implicitly (and somewhat inadvertently) put to good use by a multitude of users over the past 40 years (in the context of classical and first-principles MDs), this finding does not need any specific application to be assessed. This paper is both a review and historical tribute to the ideas developed almost 40 years ago by S. Nose establishing the theoretical foundations of implementation and use of thermostats in molecular dynamics. By carefully rederiving the equations of motion in both formalisms, it appears that all features of Nose-Hoover framework (replacement of the Nose variables by a single-friction coefficient in the equations of motion) are already built in the Nose approach.image (c) 2023 WILEY-VCH GmbH