Ergodicity of one-dimensional systems coupled to the logistic thermostat

TL;DR

This study investigates the ergodic behavior of three one-dimensional Hamiltonian systems with different potentials when coupled to a logistic thermostat, using multiple criteria to confirm ergodicity and demonstrating its robustness across various initial conditions.

Contribution

It provides the first comprehensive numerical analysis confirming the ergodicity of the logistic thermostat for different one-dimensional systems with multiple validation criteria.

Findings

No regular trajectories observed at certain parameters.

Time and ensemble distributions converge for all systems.

Logistic thermostat effectively induces ergodicity in stiff systems.

Abstract

We analyze the ergodicity of three one-dimensional Hamiltonian systems, with harmonic, quartic and Mexican-hat potentials, coupled to the logistic thermostat. As criteria for ergodicity we employ: the independence of the Lyapunov spectrum with respect to initial conditions; the absence of visual "holes" in two-dimensional Poincar\'e sections; the agreement between the histograms in each variable and the theoretical marginal distributions; and the convergence of the global joint distribution to the theoretical one, as measured by the Hellinger distance. Taking a large number of random initial conditions, for certain parameter values of the thermostat we find no indication of regular trajectories and show that the time distribution converges to the ensemble one for an arbitrarily long trajectory for all the systems considered. Our results thus provide a robust numerical indication that the logistic thermostat can serve as a single one-parameter thermostat for stiff one-dimensional systems.