# Homogeneity and thermodynamic identities in geometrothermodynamics

## TL;DR

This paper classifies thermodynamic systems based on the homogeneity of their fundamental equations, affecting key thermodynamic identities, and applies these ideas to black hole geometrothermodynamics to resolve metric arbitrariness.

## Contribution

It introduces a classification scheme for thermodynamic systems using homogeneity properties and applies it to fix metric arbitrariness in black hole geometrothermodynamics.

## Key findings

- Homogeneous functions correspond to ordinary thermodynamic systems.
- Generalized homogeneous functions describe non-ordinary systems.
- The formalism can fix arbitrariness in Legendre invariant metrics for black holes.

## Abstract

We propose a classification of thermodynamic systems in terms of the homogeneity properties of their fundamental equations. Ordinary systems correspond to homogeneous functions and non-ordinary systems are given by generalized homogeneous functions. This affects the explicit form of the Gibbs-Duhem relation and Euler's identity. We show that these generalized relations can be implemented in the formalism of black hole geometrothermodynamics in order to completely fix the arbitrariness present in Legendre invariant metrics.