Thermodynamics based on the principle of least abbreviated action: entropy production in a network of coupled oscillators

TL;DR

This paper introduces a thermodynamic framework based on the Maupertuis principle, using action-angle variables to define temperature and entropy, and applies it to coupled oscillators to analyze entropy production and synchronization.

Contribution

It develops a novel thermodynamic formalism using Hamiltonian action-angle variables and applies it to nonequilibrium systems like coupled oscillators, linking entropy production to dissipation.

Findings

Entropy increases during relaxation to equilibrium.

The formalism reproduces microcanonical quantities.

Entropy production relates to dissipation in synchronization.

Abstract

We present some novel thermodynamic ideas based on the Maupertuis principle. By considering Hamiltonians written in terms of appropriate action-angle variables we show that thermal states can be characterized by the action variables and by their evolution in time when the system is nonintegrable. We propose dynamical definitions for the equilibrium temperature and entropy as well as an expression for the nonequilibrium entropy valid for isolated systems with many degrees of freedom. This entropy is shown to increase in the relaxation to equilibrium of macroscopic systems with short-range interactions, which constitutes a dynamical justification of the Second Law of Thermodynamics. Several examples are worked out to show that this formalism yields the right microcanonical (equilibrium) quantities. The relevance of this approach to nonequilibrium situations is illustrated with an application to a network of coupled oscillators (Kuramoto model). We provide an expression for the entropy production in this system finding that its positive value is directly related to dissipation at the steady state in attaining order through synchronization.