The Nosé-Hoover, Dettmann, and Hoover-Holian Oscillators

• Autorzy: Hoover W. G. , Sprott J. C. , Hoover C. G.

Identyfikatory

DOI

10.12921/cmst.2019.0000031

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Nosé-Hoover oscillator; Dettmann oscillator; Hoover-Holian oscillator; nonlinear dynamics

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2019
• Tom: Vol. 25, No. 3
• Strony: 121--124
• Opis fizyczny: Bibliogr. 14 poz., rys.

Twórcy

autor

Hoover W. G.

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833

autor

Sprott J. C.

• University of Wisconsin-Madison Department of Physics Madison, Wisconsin 53706

autor

Hoover C. G.

• Ruby Valley Research Institute 601 Highway Contract 60 Ruby Valley, Nevada 89833

Bibliografia

• [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 PhaseSpace 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 NoséHoover 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 williamhoov er.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).

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).