On the dispute between Boltzmann and Gibbs entropy

TL;DR

This paper defends Boltzmann's entropy as the correct microcanonical entropy, demonstrating its validity for systems with ensemble equivalence and supporting the existence of negative temperature states through analytical and numerical evidence.

Contribution

It proves Boltzmann entropy's correctness for systems with ensemble equivalence and clarifies the debate with analytical and simulation support.

Findings

Boltzmann entropy is valid for systems with ensemble equivalence.

Negative temperature states are physically accessible and consistent with statistical mechanics.

Analytical and numerical results support Boltzmann's entropy over Gibbs' in certain systems.

Abstract

Very recently, the validity of the concept of negative temperature has been challenged by several authors since they consider Boltzmann's entropy (that allows negative temperatures) inconsistent from a mathematical and statistical point of view, whereas they consider Gibbs' entropy (that does not admit negative temperatures) the correct definition for microcanonical entropy. In the present paper we prove that for systems with equivalence of the statistical ensembles Boltzmann entropy is the correct microcanonical entropy. Analytical results on two systems supporting negative temperatures, confirm the scenario we propose. In addition, we corroborate our proof by numeric simulations on an explicit lattice system showing that negative temperature equilibrium states are accessible and obey standard statistical mechanics prevision.