# Aspects of Dynamical Simulations, Emphasizing Nos\'e and Nos\'e-Hoover Dynamics and the Compressible Baker Map

## TL;DR

This paper explores Nosé and Nosé-Hoover dynamics, their relation to Liouville's theorem, ergodicity issues, and fractal structures in nonequilibrium flows, using simple models like the Baker Map and Galton Board.

## Contribution

It provides a pedagogical analysis of thermostated dynamics, highlighting paradoxes, ergodicity challenges, and fractal flow structures in both simple and complex systems.

## Key findings

- Thermostated harmonic oscillators can be expanding, incompressible, or contracting.
- Fractal structures are observed in nonequilibrium flows like the Baker Map.
- Ergodicity issues affect the ability of these dynamics to reproduce Gibbs ensembles.

## Abstract

Aspects of the Nos\'e and Nos\'e-Hoover dynamics developed in 1983-1984 along with Dettmann's closely related dynamics of 1996, are considered. We emphasize paradoxes associated with Liouville's Theorem. Our account is pedagogical, focused on the harmonic oscillator for simplicity, though exactly the same ideas can be, and have been, applied to manybody systems. Nos\'e, Nos\'e-Hoover, and Dettmann flows were all developed in order to access Gibbs' canonical ensemble directly from molecular dynamics. Unlike Monte Carlo algorithms dynamical flow models are often not ergodic and so can fail to reproduce Gibbs' ensembles. Accordingly we include a discussion of ergodicity, the visiting of all relevant microstates corresponding to the desired ensemble. We consider Lyapunov instability too, the usual mechanism for phase-space mixing. We show that thermostated harmonic oscillator dynamics can be simultaneously expanding, incompressible, or contracting, depending upon the chosen "phase space". The fractal nature of nonequilibrium flows is also illustrated for two simple two-dimensional models, the hard-disk-based Galton Board and the time-reversible Baker Map. The simultaneous treatment of flows as one-dimensional and many-dimensional suggests some interesting topological problems for future investigations.