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

## TL;DR

This paper clarifies complex and paradoxical behaviors in Nosé and Nosé-Hoover thermostated dynamics, revealing how phase-space volume can expand, contract, or remain incompressible, challenging traditional Liouville's theorem interpretations.

## Contribution

It provides new phase-space models that explain the paradoxical aspects of thermostated Hamiltonian dynamics, enhancing understanding of their geometric properties.

## Key findings

- Phase-space can be expanding, incompressible, or contracting.
- Liouville's equation has surprising implications in thermostated systems.
- Models illustrate complex volume behavior in thermostated oscillators.

## 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.