The conformal metric structure of Geometrothermodynamics

TL;DR

This paper investigates the invariance properties of metrics in Geometrothermodynamics, demonstrating that only total Legendre transformations preserve curvature invariance and providing an explicit invariant metric form.

Contribution

It introduces a new metric invariant under total Legendre transformations, ensuring curvature independence from the fundamental representation in Geometrothermodynamics.

Findings

Invariance of curvature under total Legendre transformations is established.

An explicit form of an invariant metric is derived.

Analysis of a two-degree-of-freedom system confirms theoretical results.

Abstract

We present a thorough analysis on the invariance of the most widely used metrics in the Geometrothermodynamics (GTD) programme. We centre our attention in the invariance of the curvature of the space of equilibrium states under a change of fundamental representation. Assuming that the systems under consideration can be described by a fundamental relation which is a homogeneous function of a definite order, we demonstrate that such invariance is only compatible with total Legendre transformations in the present form of the programme. We give the explicit form of a metric which is invariant under total Legendre transformations and whose induced metric produces a curvature which is independent of the fundamental representation. Finally, we study a generic system with two degrees of freedom and whose fundamental relation is homogeneous of order one.