The geometry of thermodynamics

TL;DR

This paper reviews geometrothermodynamics, a geometric approach to classical thermodynamics, analyzing its effectiveness for different systems like gases and black holes, and highlighting its limitations.

Contribution

It introduces a Legendre invariant metric framework for thermodynamics and evaluates its geometric properties across various physical systems.

Findings

Reproduces thermodynamic behavior of van der Waals gas

Captures properties of Reissner-Nordström black holes

Fails to adequately describe Kerr black holes

Abstract

We present a review of the main aspects of geometrothermodynamics, an approach which allows us to associate a specific Riemannian structure to any classical thermodynamic system. In the space of equilibrium states, we consider a Legendre invariant metric, which is given in terms of the fundamental equation of the corresponding thermodynamic system, and analyze its geometric properties in the case of the van der Waals gas, and black holes. We conclude that the geometry of this particular metric reproduces the thermodynamic behavior of the van der Waals gas, and the Reissner-Nordstr\"om black hole, but it is not adequate for the thermodynamic description of Kerr black holes.