# On the physical significance of geometrothermodynamic metrics

## TL;DR

This paper links geometrothermodynamic metrics to fluctuation theory by interpreting their components as second moments of a new thermodynamic potential, expanding the understanding of GTD's geometric and physical significance.

## Contribution

It demonstrates that GTD metrics can be viewed as fluctuation measures and introduces new thermodynamic potentials not related by Legendre transformations.

## Key findings

- GTD metrics correspond to second moments of fluctuations of a new potential.
- Diffeomorphism invariance allows for new thermodynamic coordinates.
- Establishes a direct connection between GTD and fluctuation theory.

## Abstract

In geometrothermodynamics (GTD), to study the geometric properties of the equilibrium space three thermodynamic metrics have been proposed so far. These metrics are obtained by using the condition of Legendre invariance and can be computed explicitly once a thermodynamic potential is specified as the fundamental equation. We use the remaining diffeomorphism invariance in the phase and equilibrium spaces to show that the components of the GTD-metrics can be interpreted as the second moment of the fluctuation of a new thermodynamic potential. This result establishes a direct connection between GTD and fluctuation theory. In this way, the diffeomorphism invariance of GTD allows us to introduce new thermodynamic coordinates and new thermodynamic potentials, which are not related by means of Legendre transformations to the fundamental thermodynamic potentials.