Infinitesimal Legendre symmetry in the Geometrothermodynamics programme

García- Peláez, Daniel; López-Monsalvo, C. S.

doi:10.1063/1.4891921

Infinitesimal Legendre symmetry in the Geometrothermodynamics programme

Journal

Journal of Mathematical Physics

ISSN

0022-2488

1089-7658

Date Issued

2014

Author(s)

García- Peláez, Daniel

Facultad de Ingeniería - CampCM

López-Monsalvo, C. S.

Type

text::journal::journal article

DOI

10.1063/1.4891921

URL

https://scripta.up.edu.mx/handle/20.500.12552/4479

Abstract

The work within the Geometrothermodynamics programme rests upon the metric structure for the thermodynamic phase-space. Such structure exhibits discrete Legendre symmetry. In this work, we study the class of metrics which are invariant along the infinitesimal generators of Legendre transformations. We solve the Legendre-Killing equation for a K-contact general metric. We consider the case with two thermodynamic degrees of freedom, i.e., when the dimension of the thermodynamic phase-space is five. For the generic form of contact metrics, the solution of the Legendre-Killing system is unique, with the sole restriction that the only independent metric function – Ω – should be dragged along the orbits of the Legendre generator. We revisit the ideal gas in the light of this class of metrics. Imposing the vanishing of the scalar curvature for this system results in a further differential equation for the metric function Ω which is not compatible with the Legendre invariance constraint. This result does not allow us to use Quevedo's interpretation of the curvature scalar as a measure of thermodynamic interaction for this particular class.