Deterministic Time-Reversible Thermostats : Chaos, Ergodicity, and the Zeroth Law of Thermodynamics

TL;DR

This paper evaluates the stability and ergodicity of deterministic, time-reversible thermostats, identifying three types that reliably generate canonical distributions and obey the Zeroth Law, making them effective for statistical mechanics simulations.

Contribution

It demonstrates that Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman thermostats are ergodic and suitable as deterministic heat baths, with consistent Lyapunov spectra and canonical distribution generation.

Findings

Three thermostat types are ergodic and generate Gibbs distributions.

These thermostats obey the Zeroth Law of Thermodynamics.

They provide deterministic alternatives to stochastic Langevin dynamics.

Abstract

The relative stability and ergodicity of deterministic time-reversible thermostats, both singly and in coupled pairs, are assessed through their Lyapunov spectra. Five types of thermostat are coupled to one another through a single Hooke's-Law harmonic spring. The resulting dynamics shows that three specific thermostat types, Hoover-Holian, Ju-Bulgac, and Martyna-Klein-Tuckerman, have very similar Lyapunov spectra in their equilibrium four-dimensional phase spaces and when coupled in equilibrium or nonequilibrium pairs. All three of these oscillator-based thermostats are shown to be ergodic, with smooth analytic Gaussian distributions in their extended phase spaces ( coordinate, momentum, and two control variables ). Evidently these three ergodic and time-reversible thermostat types are particularly useful as statistical-mechanical thermometers and thermostats. Each of them generates Gibbs' universal canonical distribution internally as well as for systems to which they are coupled. Thus they obey the Zeroth Law of Thermodynamics, as a good heat bath should. They also provide dissipative heat flow with relatively small nonlinearity when two or more such bath temperatures interact and provide useful deterministic replacements for the stochastic Langevin equation.