Thermodynamic curvature and ensemble nonequivalence

TL;DR

This paper explores the use of thermodynamic geometric methods to understand ensemble nonequivalence in systems with two degrees of freedom, revealing how curvature divergences relate to stability changes, especially in black hole thermodynamics.

Contribution

It derives simple formulas for curvature scalars of Hessian metrics and proves a conjecture linking different thermodynamic metrics to ensemble behaviors.

Findings

Curvature scalars diverge at specific heat divergence lines.

Different Hessian metrics correspond to different ensemble behaviors.

Curvature divergence matches stability changes in ensembles.

Abstract

In this work we consider thermodynamic geometries defined as Hessians of different potentials and derive some useful formulae that show their complementary role in the description of thermodynamic systems with two degrees of freedom that show ensemble nonequivalence. From the expressions derived for the metrics, we can obtain the curvature scalars in a very simple and compact form. We explain here the reason why each curvature scalar diverges over the line of divergence of one of the specific heats. This application is of special interest in the study of changes of stability in black holes as defined by Davies. From these results we are able to prove on a general footing a conjecture first formulated by Liu, L\"u, Luo and Shao stating that different Hessian metrics can correspond to different behaviors in the various ensembles. We study the case of two thermodynamic dimensions. Moreover, comparing our result with the more standard turning point method developed by Poincar\'e, we obtain that the divergence of the scalar curvature of the Hessian metric of one potential exactly matches the change of stability in the corresponding ensemble.