Ensemble equivalence for mean field models and plurisubharmonicity

TL;DR

This paper proves that entropy is globally concave for a broad class of mean field models, linking microscopic properties to geometric structures, and demonstrates thermodynamic ensemble equivalence under these conditions.

Contribution

It establishes a novel connection between entropy concavity in mean field models and Kahler geometry, extending thermodynamic equivalence results to non-continuous interactions.

Findings

Entropy is globally concave for various mean field models.

Thermodynamic ensemble equivalence holds for this class of models.

Critical inverse temperatures match the asymptotic slopes of microcanonical entropies.

Abstract

We show that entropy is globally concave with respect to energy for a rich class of mean field interactions, including regularizations of the the point-vortex model in the plane, plasmas and self-gravitating matter in 2D, as well as the higher dimensional logarithmic interactions appearing in conformal geometry and power laws. The proofs are based on a corresponding "microscopic" concavity result at finite N, shown by leveraging an unexpected link to Kahler geometry and plurisubharmonic functions. Under more restrictive homogeneity assumptions strict concavity is obtained using a uniqueness result for free energy minimizers, established in a companion paper. The results imply that thermodynamic equivalence of ensembles holds for this class of mean field models. As an application it is shown that the critical inverse negative temperatures - in the macroscopic as well as the microscopic setting - coincide with the asymptotic slope of the corresponding microcanonical entropies. Along the way we also extend previous results on the thermodynamic equivalence of ensembles for continuous weakly positive definite interactions, concerning positive temperature states, to the general non-continuous case. In particular, singular situations are exhibited where, somewhat surprisingly, thermodynamic equivalence of ensembles fails at energy levels sufficiently close to the minimum energy level.