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1 2019

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

Hoover W. G., Sprott J. C., Hoover C. G.

Computational Methods in Science and Technology

2019

Vol. 25, No. 3

121--124

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.