# The Nos\'e-Hoover, Dettmann, and Hoover-Holian Oscillators

## TL;DR

This paper explores the relationships among Nosé-Hoover, Dettmann, and Hoover-Holian oscillators, providing new proofs of their long-term energy behavior and linking mechanical models to Gibbsian statistical mechanics.

## Contribution

It introduces new proofs for the long-time energy behavior of these oscillators and connects their dynamics to Hamiltonian mechanics and Gibbs' statistical mechanics.

## Key findings

- Yang's trajectories cross the energy plane repeatedly
- New proofs of long-time energy behavior
- Stationary Gibbsian distribution for the Hoover-Holian oscillator

## Abstract

To follow up recent work of Xiao-Song Yang on the Nos\'e-Hoover oscillator we consider Dettmann's harmonic oscillator, which relates Yang's ideas directly to Hamiltonian mechanics. We also use the Hoover-Holian oscillator 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 $(\zeta, \xi)$ vary the energy. Dettmann's description includes a time-scaling variable $s$, as does Nos\'e's original work. Time scaling controls the rates at which the $(q,p,\zeta)$ variables change. The ergodic Hoover-Holian oscillator provides the stationary Gibbsian probability density for the time-scaling variable $s$. Yang considered {\it qualitative} features of Nos\'e-Hoover dynamics. He showed that longtime Nos\'e-Hoover trajectories change energy, repeatedly crossing the $\zeta = 0$ plane. We use moments of the motion equations to give two new, different, and brief proofs of Yang's long-time limiting result.