Aspects of Nos\'e and Nos\'e-Hoover Dynamics Elucidated

C. G. Hoover; W. G. Hoover

arxiv: 1906.10318 · v1 · pith:IN4CF66Hnew · submitted 2019-06-25 · ❄️ cond-mat.stat-mech · nlin.CD

Aspects of Nos\'e and Nos\'e-Hoover Dynamics Elucidated

W. G. Hoover , C. G. Hoover This is my paper

Pith reviewed 2026-05-25 16:42 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.CD

keywords Nosé-Hoover dynamicsphase spaceLiouville equationthermostated oscillatorHamiltonian flowsvolume flownonequilibrium dynamics

0 comments

The pith

Phase-space models of the thermostated harmonic oscillator can be simultaneously expanding, incompressible, or contracting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper clarifies paradoxical aspects of Nosé and Nosé-Hoover dynamics by showing that different three- and four-dimensional phase-space representations of the same thermostated harmonic oscillator dynamics can exhibit expanding, incompressible, or contracting volumes. This occurs when Liouville's continuity equation is applied to Hamiltonian flows in these models. A sympathetic reader would care because it reveals surprising consequences of coordinate choices in describing nonequilibrium systems.

Core claim

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.

What carries the argument

Three- and four-dimensional phase-space models for the thermostated harmonic oscillator that apply Liouville's continuity equation.

If this is right

Liouville's theorem applied to Hamiltonian flows can lead to volume expansion or contraction depending on the phase-space model chosen.
The Nosé and Nosé-Hoover dynamics of 1984 and Dettmann's dynamics of 1996 have paradoxical aspects that are resolved by these multiple descriptions.
Equivalent underlying dynamics can appear different in terms of phase-space volume flow without invalidating the models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar coordinate-dependent effects might appear in other thermostated systems beyond the harmonic oscillator.
These findings suggest that phase-space dimensionality choices can affect interpretations of compressibility in molecular dynamics simulations.

Load-bearing premise

The different three- and four-dimensional phase-space models correctly represent equivalent underlying dynamics without coordinate-dependent artifacts that would invalidate the volume-flow conclusions.

What would settle it

Demonstrating that one of the phase-space models introduces an artifact that changes the underlying dynamics would falsify the claim that they are equivalent representations.

Figures

Figures reproduced from arXiv: 1906.10318 by C. G. Hoover, W. G. Hoover.

**Figure 2.** Figure 2: FIG. 2: The time variation of two expressions for the probabi [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3: Two probability densities as measured once around a p [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper shows that Nosé-Hoover phase space models can have contradictory volume behaviors, but the equivalence of embeddings needs checking for artifacts.

read the letter

The main takeaway is that phase-space descriptions of the thermostated harmonic oscillator using Nosé-Hoover dynamics can simultaneously be expanding, incompressible, or contracting depending on the model chosen. The paper uses a variety of three- and four-dimensional phase-space models to illustrate this for the 1984 and 1996 formulations. What the paper does is elucidate these paradoxical aspects by applying Liouville's continuity equation to the Hamiltonian flows in these extended systems. It builds on the authors' long history with these thermostats, which gives them a good vantage point for spotting the inconsistencies in how volume flow is described. This kind of clarification can be useful for avoiding misinterpretations when people implement these methods in simulations. The concern is whether the different embeddings are truly equivalent. The stress-test note makes a fair point: in non-Hamiltonian systems, the phase-space divergence can change with coordinate transformations unless the Jacobian compensates exactly. If the paper does not demonstrate that the mappings between 3D and 4D models preserve the underlying dynamics without introducing artifacts, then the claimed behaviors could be formulation-dependent rather than fundamental. Since the abstract states the observation without showing the supporting derivations or checks, the strength of the argument depends on the details in the full manuscript. This paper is aimed at a small group of researchers who work with Nosé-Hoover thermostats and similar extended dynamics. A reader who already knows the 1984 and 1996 papers might get some value from seeing the paradoxes laid out explicitly. It is not likely to affect general use of these methods. The work shows clear thinking on the topic and honest engagement with the literature, even if the central claim needs verification on the coordinate issue. I would send it to peer review for a thorough check.

Referee Report

1 major / 1 minor

Summary. The manuscript elucidates paradoxical aspects of the 1984 Nosé and Nosé-Hoover dynamics and the 1996 Dettmann dynamics. It shows that phase-space descriptions of thermostated harmonic-oscillator motion can be simultaneously expanding, incompressible, or contracting when realized in a variety of three- and four-dimensional extended phase spaces, thereby illustrating unexpected consequences of applying Liouville’s continuity equation to Hamiltonian flows.

Significance. If the central claim is sustained, the work supplies a concrete illustration that compressibility is not an invariant property of the underlying dynamics but depends on the chosen embedding. This has direct bearing on the interpretation of phase-space volume evolution in non-Hamiltonian thermostat models used throughout molecular-dynamics sampling. The paper thereby sharpens the distinction between coordinate-dependent and intrinsic statements of Liouville’s theorem in extended systems.

major comments (1)

[Sections on 3D/4D phase-space models and Liouville application] The equivalence of the 3D and 4D embeddings (Nosé, Nosé-Hoover, Dettmann) is load-bearing for the claim that the same oscillator dynamics can appear expanding, incompressible, or contracting. The manuscript must explicitly demonstrate that each mapping is accompanied by a Jacobian factor whose divergence exactly cancels any apparent change in phase-space compressibility; otherwise the reported behaviors remain vulnerable to coordinate artifacts. (See the sections presenting the 3D/4D models and the application of Liouville’s equation.)

minor comments (1)

Notation for the extended variables (e.g., the friction variable and its conjugate) should be unified across the different embeddings to avoid reader confusion when comparing the divergence expressions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major comment identifies an opportunity to strengthen the presentation of the embeddings, and we address it directly below.

read point-by-point responses

Referee: The equivalence of the 3D and 4D embeddings (Nosé, Nosé-Hoover, Dettmann) is load-bearing for the claim that the same oscillator dynamics can appear expanding, incompressible, or contracting. The manuscript must explicitly demonstrate that each mapping is accompanied by a Jacobian factor whose divergence exactly cancels any apparent change in phase-space compressibility; otherwise the reported behaviors remain vulnerable to coordinate artifacts. (See the sections presenting the 3D/4D models and the application of Liouville’s equation.)

Authors: We agree that an explicit derivation of the Jacobian factors will make the equivalence of the embeddings more transparent and will eliminate any possible concern about coordinate artifacts. While the manuscript already verifies that Liouville’s continuity equation is satisfied in each embedding (which mathematically requires the cancellation), we will add explicit calculations of the relevant Jacobians and their logarithmic divergences in the revised version. These will be inserted as a short subsection following the presentation of the 3D and 4D models, showing that the divergence of each velocity field is precisely offset by the Jacobian contribution, thereby confirming that the reported expanding, incompressible, and contracting behaviors are intrinsic to the chosen embeddings rather than artifacts. revision: yes

Circularity Check

0 steps flagged

No significant circularity; self-contained illustration of Liouville applications

full rationale

The paper elucidates paradoxical phase-space behaviors (expanding/incompressible/contracting) for the thermostated oscillator by applying Liouville's continuity equation to several explicit 3D and 4D model formulations. These behaviors are shown directly from the equations of motion in each embedding rather than being fitted or defined in terms of the target result. Although the work references the authors' earlier thermostat papers, the central illustrations do not reduce to those citations; the mappings and volume-flow conclusions are presented as independent consequences within the current manuscript. No step matches any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities are stated.

pith-pipeline@v0.9.0 · 5599 in / 936 out tokens · 24131 ms · 2026-05-25T16:42:22.237287+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages

[1]

Nos\'e, ``A Unified Formulation of the Constant Temperature Molecular Dynamics Methods'', The Journal of Chemical Physics 81 , 511-519 (1984)

S. Nos\'e, ``A Unified Formulation of the Constant Temperature Molecular Dynamics Methods'', The Journal of Chemical Physics 81 , 511-519 (1984)

work page 1984
[2]

Nos\'e, ``A Molecular Dynamics Method for Simulations in the Canonical Ensemble'', Molecular Physics 52 , 255-268 (1984)

S. Nos\'e, ``A Molecular Dynamics Method for Simulations in the Canonical Ensemble'', Molecular Physics 52 , 255-268 (1984)

work page 1984
[3]

W. G. Hoover, ``Canonical Dynamics. Equilibrium Phase-Space Distributions'', Physical Review A 31 , 1695-1697 (1985)

work page 1985
[4]

H. A. Posch, W. G. Hoover, and F. J. Vesely, ``Canonical Dynamics of the Nos\'e Oscillator: Stability, Order, and Chaos'', Physical Review A 33 , 4253-4265 (1986)

work page 1986
[5]

P. K. Patra, W. G. Hoover, C. G. Hoover, and J. C. Sprott, ``The Equivalence of Dissipation from Gibbs' Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators'', International Journal of Bifurcation and Chaos 26 , 1650089 (2016)

work page 2016
[6]

W. G. Hoover, ``M\'ecanique de Non\'equilibre \`a la Californienne'', Physica A 240 , 1-11 (1997)

work page 1997
[7]

C. P. Dettmann and G. P. Morriss, ``Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nos\'e-Hoover Dynamics'', Physical Review E 55 , 3693-3696 (1997)

work page 1997
[8]

J. C. Sprott, ``Some Simple Chaotic Flows'', Physical Review E, 50 , 647-650 (1994)

work page 1994
[9]

W. G. Hoover, ``Remark on `Some Simple Chaotic Flows''', Physical Review E, 51 , 759-760 (1995)

work page 1995
[10]

Wang and X

L. Wang and X. S. Yang, ``The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nos\'e-Hoover Oscillator'', International Journal of Bifurcation and Chaos 27 , 1750111 (2017)

work page 2017
[11]

Wang and X

L. Wang and X. S. Yang, ``Global Analysis of a Generalized Nos\'e-Hoover Oscillator'', Journal of Mathematical Analysis and Applications 464 , 370-379 (2018)

work page 2018
[12]

X. S. Yang, ``Qualitative Analysis of the Nos\'e-Hoover Oscillator'', Qualitative Theory of Dynamical Systems (submitted, 2019)

work page 2019
[13]

W. G. Hoover, J. C. Sprott, and C. G. Hoover, ``The Nos\'e-Hoover, Dettmann, and Hoover-Holian Oscillators'', arXiv 1906.03107, Prepared for the Qualitative Theory of Dynamical Systems (2019)

work page arXiv 1906

[1] [1]

Nos\'e, ``A Unified Formulation of the Constant Temperature Molecular Dynamics Methods'', The Journal of Chemical Physics 81 , 511-519 (1984)

S. Nos\'e, ``A Unified Formulation of the Constant Temperature Molecular Dynamics Methods'', The Journal of Chemical Physics 81 , 511-519 (1984)

work page 1984

[2] [2]

Nos\'e, ``A Molecular Dynamics Method for Simulations in the Canonical Ensemble'', Molecular Physics 52 , 255-268 (1984)

S. Nos\'e, ``A Molecular Dynamics Method for Simulations in the Canonical Ensemble'', Molecular Physics 52 , 255-268 (1984)

work page 1984

[3] [3]

W. G. Hoover, ``Canonical Dynamics. Equilibrium Phase-Space Distributions'', Physical Review A 31 , 1695-1697 (1985)

work page 1985

[4] [4]

H. A. Posch, W. G. Hoover, and F. J. Vesely, ``Canonical Dynamics of the Nos\'e Oscillator: Stability, Order, and Chaos'', Physical Review A 33 , 4253-4265 (1986)

work page 1986

[5] [5]

P. K. Patra, W. G. Hoover, C. G. Hoover, and J. C. Sprott, ``The Equivalence of Dissipation from Gibbs' Entropy Production with Phase-Volume Loss in Ergodic Heat-Conducting Oscillators'', International Journal of Bifurcation and Chaos 26 , 1650089 (2016)

work page 2016

[6] [6]

W. G. Hoover, ``M\'ecanique de Non\'equilibre \`a la Californienne'', Physica A 240 , 1-11 (1997)

work page 1997

[7] [7]

C. P. Dettmann and G. P. Morriss, ``Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nos\'e-Hoover Dynamics'', Physical Review E 55 , 3693-3696 (1997)

work page 1997

[8] [8]

J. C. Sprott, ``Some Simple Chaotic Flows'', Physical Review E, 50 , 647-650 (1994)

work page 1994

[9] [9]

W. G. Hoover, ``Remark on `Some Simple Chaotic Flows''', Physical Review E, 51 , 759-760 (1995)

work page 1995

[10] [10]

Wang and X

L. Wang and X. S. Yang, ``The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nos\'e-Hoover Oscillator'', International Journal of Bifurcation and Chaos 27 , 1750111 (2017)

work page 2017

[11] [11]

Wang and X

L. Wang and X. S. Yang, ``Global Analysis of a Generalized Nos\'e-Hoover Oscillator'', Journal of Mathematical Analysis and Applications 464 , 370-379 (2018)

work page 2018

[12] [12]

X. S. Yang, ``Qualitative Analysis of the Nos\'e-Hoover Oscillator'', Qualitative Theory of Dynamical Systems (submitted, 2019)

work page 2019

[13] [13]

W. G. Hoover, J. C. Sprott, and C. G. Hoover, ``The Nos\'e-Hoover, Dettmann, and Hoover-Holian Oscillators'', arXiv 1906.03107, Prepared for the Qualitative Theory of Dynamical Systems (2019)

work page arXiv 1906