Hessian matrix, specific heats, Nambu brackets, and thermodynamic geometry

TL;DR

This paper uses Nambu brackets to connect divergences in heat capacities with thermodynamic geometry, providing new relations between Hessian matrices and specific heats, and applies this to Meyers-Perry black holes with three spins.

Contribution

It introduces a bracket-based method to relate thermodynamic metrics, Hessian matrices, and heat capacities, extending previous work and analyzing black hole thermodynamics.

Findings

Divergences in heat capacities correspond to geometric divergences.

Derived relations between Hessian matrices and specific heats.

Analyzed thermodynamics of Meyers-Perry black holes with three spins.

Abstract

As an extension to our earlier work \cite{Mirza2}, we employ the Nambu brackets to prove that the divergences of heat capacities correspond to their counterparts in thermodynamic geometry. We also obtain a simple representation for the conformal transformations that connect different thermodynamics metrics to each other. Using our bracket approach, we obtain interesting exact relations between the Hessian matrix with any number of parameters and specific heat capacities. Finally, we employ this approach to investigate some thermodynamic properties of the Meyers-Perry black holes with three spins.