Liouville geometry of classical thermodynamics

TL;DR

This paper develops a geometric framework for classical thermodynamics using contact geometry and homogeneous coordinates, unifying energy and entropy representations and incorporating the Gibbs-Duhem relation.

Contribution

It introduces a homogeneous geometric formulation on the cotangent bundle of extensive variables, resolving representation distinctions and integrating classical thermodynamic relations.

Findings

Geometric formulation unifies energy and entropy representations.

Homogeneity in coordinates captures Gibbs-Duhem relation.

In-depth study of thermodynamic geometry on cotangent bundles.

Abstract

In the contact-geometric formulation of classical thermodynamics distinction is made between the energy and entropy representation, which can be resolved by taking homogeneous coordinates for the intensive variables. This results in a geometric formulation on the cotangent bundle of the manifold of extensive variables, where all geometric objects are homogeneous in the cotangent variables. The resulting geometry is studied in-depth. Additional homogeneity with respect to the extensive variables, corresponding to the classical Gibbs-Duhem relation, is treated within the same geometric framework.