Boltzmann, Gibbs and the Concept of Equilibrium

TL;DR

This paper explores the differences between Boltzmann and Gibbs definitions of equilibrium and entropy, proposing a unified approach that treats equilibrium as a continuous property and tests it using the Kac ring model.

Contribution

It introduces a unified framework for statistical mechanics that redefines equilibrium as a continuous measure and incorporates both Boltzmann and Gibbs approaches.

Findings

Equilibrium can be modeled as a degree rather than a binary state.

The proposed framework aligns Boltzmann and Gibbs concepts of entropy.

The Kac ring model supports the validity of the continuous equilibrium approach.

Abstract

The Boltzmann and Gibbs approaches to statistical mechanics have very different definitions of equilibrium and entropy. The problems associated with this are discussed and it is suggested that they can be resolved, to produce a version of statistical mechanics incorporating both approaches, by redefining equilibrium not as a binary property (being/not being in equilibrium) but as a continuous property (degrees of equilibrium) measured by the Boltzmann entropy and by introducing the idea of thermodynamic-like behaviour for the Boltzmann entropy. The Kac ring model is used as an example to test the proposals.