The Nosé-Hoover, Dettmann, and Hoover-Holian Oscillators
Hoover William G. 1*, Sprott Julien C. 2, Hoover Carol G. 1
1 Ruby Valley Research Institute
601 Highway Contract 60
Ruby Valley, Nevada 89833
*E-mail: hooverwilliam@yahoo.com2 University of Wisconsin-Madison
Department of Physics
Madison, Wisconsin 53706
Received:
Received: 10 July 2019; accepted: 27 July 2019; published online: 13 August 2019
DOI: 10.12921/cmst.2019.0000031
Abstract:
To follow up recent work of Xiao-Song Yang [1] on the Nosé-Hoover oscillator [2–5] we consider Dettmann’s harmonic oscillator [6, 7], which relates Yang’s ideas directly to Hamiltonian mechanics.We also use the Hoover-Holian oscillator [8] to relate our mechanical studies to Gibbs’ statistical mechanics. All three oscillators are described by a coordinate q and a momentum p. Additional control variables (ζ; ξ) vary the energy. Dettmann’s description includes a time-scaling variable s, as does Nosé’s original work [2, 3]. Time scaling controls the rates at which the (q; p; ζ) variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable s. Yang considered qualitative features of Nosé-Hoover dynamics. He showed that longtime Nosé-Hoover trajectories change energy, repeatedly crossing the ζ = 0 plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang’s long-time limiting result.
Key words:
Dettmann oscillator, Hoover-Holian oscillator, nonlinear dynamics, Nosé-Hoover oscillator
References:
[1] X.S. Yang, Qualitative Analysis of the Nosé-Hoover Oscillator, submitted to Qualitative Theory of Dynamical Systems, 2019.
[2] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, The Journal of Chemical Physics 81, 511–519 (1984).
[3] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[4] Wm.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[5] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[6] W.G. Hoover, Mécanique de Nonéquilibre à la Californienne, Physica A 240, 1–11 (1997).
[7] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1997).
[8] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[9] L. Wang, X.S. Yang, The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nosé-Hoover Oscillator, International Journal of Bifurcation and Chaos 27, 1750111 (2017).
[10] L.Wang, X.S. Yang, Global Analysis of a Generalized Nosé-Hoover Oscillator, Journal of Mathematical Analysis and Applications 464, 370–379 (2018).
[11] W.G. Hoover, J.C. Sprott, C.G. Hoover, A Tutorial. Adaptive Runge-Kutta Integration for Stiff Systems: Comparing the Nosé and Nosé-Hoover Oscillator Dynamics, American Journal of Physics 84, 786–794 (2016).
[12] W.G. Hoover, Computational Statistical Mechanics, Elsevier, New York (1991). Available free online at williamhoover.info.
[13] P.K. Patra, W.G. Hoover, C.G. Hoover, 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, 1–11 (2016).
[14] R. French, The Banach-Tarski Theorem, The Mathematical Intelligencer 10, 21–28 (1988).
To follow up recent work of Xiao-Song Yang [1] on the Nosé-Hoover oscillator [2–5] we consider Dettmann’s harmonic oscillator [6, 7], which relates Yang’s ideas directly to Hamiltonian mechanics.We also use the Hoover-Holian oscillator [8] to relate our mechanical studies to Gibbs’ statistical mechanics. All three oscillators are described by a coordinate q and a momentum p. Additional control variables (ζ; ξ) vary the energy. Dettmann’s description includes a time-scaling variable s, as does Nosé’s original work [2, 3]. Time scaling controls the rates at which the (q; p; ζ) variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable s. Yang considered qualitative features of Nosé-Hoover dynamics. He showed that longtime Nosé-Hoover trajectories change energy, repeatedly crossing the ζ = 0 plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang’s long-time limiting result.
Key words:
Dettmann oscillator, Hoover-Holian oscillator, nonlinear dynamics, Nosé-Hoover oscillator
References:
[1] X.S. Yang, Qualitative Analysis of the Nosé-Hoover Oscillator, submitted to Qualitative Theory of Dynamical Systems, 2019.
[2] S. Nosé, A Unified Formulation of the Constant Temperature Molecular Dynamics Methods, The Journal of Chemical Physics 81, 511–519 (1984).
[3] S. Nosé, A Molecular Dynamics Method for Simulations in the Canonical Ensemble, Molecular Physics 52, 255–268 (1984).
[4] Wm.G. Hoover, Canonical Dynamics. Equilibrium Phase-Space Distributions, Physical Review A 31, 1695–1697 (1985).
[5] H.A. Posch, W.G. Hoover, F.J. Vesely, Canonical Dynamics of the Nosé Oscillator: Stability, Order, and Chaos, Physical Review A 33, 4253–4265 (1986).
[6] W.G. Hoover, Mécanique de Nonéquilibre à la Californienne, Physica A 240, 1–11 (1997).
[7] C.P. Dettmann, G.P. Morriss, Hamiltonian Reformulation and Pairing of Lyapunov Exponents for Nosé-Hoover Dynamics, Physical Review E 55, 3693–3696 (1997).
[8] W.G. Hoover, B.L. Holian, Kinetic Moments Method for the Canonical Ensemble Distribution, Physics Letters A 211, 253–257 (1996).
[9] L. Wang, X.S. Yang, The Coexistence of Invariant Tori and Topological Horseshoes in a Generalized Nosé-Hoover Oscillator, International Journal of Bifurcation and Chaos 27, 1750111 (2017).
[10] L.Wang, X.S. Yang, Global Analysis of a Generalized Nosé-Hoover Oscillator, Journal of Mathematical Analysis and Applications 464, 370–379 (2018).
[11] W.G. Hoover, J.C. Sprott, C.G. Hoover, A Tutorial. Adaptive Runge-Kutta Integration for Stiff Systems: Comparing the Nosé and Nosé-Hoover Oscillator Dynamics, American Journal of Physics 84, 786–794 (2016).
[12] W.G. Hoover, Computational Statistical Mechanics, Elsevier, New York (1991). Available free online at williamhoover.info.
[13] P.K. Patra, W.G. Hoover, C.G. Hoover, 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, 1–11 (2016).
[14] R. French, The Banach-Tarski Theorem, The Mathematical Intelligencer 10, 21–28 (1988).
