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Entropy 2015, 17(9), 6150-6168; doi:10.3390/e17096150

Conformal Gauge Transformations in Thermodynamics

1
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 México D.F., Mexico
2
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, A.P. 70-543, 04510 México D.F., Mexico
3
Dipartimento di Fisica, Università di Roma La Sapienza, P.le Aldo Moro 5, I-00185 Rome, Italy
These authors contributed equally to this work.
*
Author to whom correspondence should be addressed.
Academic Editor: George Ruppeiner
Received: 27 June 2015 / Revised: 25 August 2015 / Accepted: 28 August 2015 / Published: 2 September 2015
(This article belongs to the Special Issue Geometry in Thermodynamics)
View Full-Text   |   Download PDF [237 KB, uploaded 2 September 2015]

Abstract

In this work, we show that the thermodynamic phase space is naturally endowed with a non-integrable connection, defined by all of those processes that annihilate the Gibbs one-form, i.e., reversible processes. We argue that such a connection is invariant under re-scalings of the connection one-form, whilst, as a consequence of the non-integrability of the connection, its curvature is not and, therefore, neither is the associated pseudo-Riemannian geometry. We claim that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all of the elements of the geometric structure of the thermodynamic phase space change under a re-scaling of the connection one-form. We call this transformation of the geometric structure a conformal gauge transformation. As an example, we revisit the change of the thermodynamic representation and consider the resulting change between the two metrics on the thermodynamic phase space, which induce Weinhold’s energy metric and Ruppeiner’s entropy metric. As a by-product, we obtain a proof of the well-known conformal relation between Weinhold’s and Ruppeiner’s metrics along the equilibrium directions. Finally, we find interesting properties of the almost para-contact structure and of its eigenvectors, which may be of physical interest. View Full-Text
Keywords: thermodynamic geometry; contact geometry; gauge transformations thermodynamic geometry; contact geometry; gauge transformations
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Bravetti, A.; Lopez-Monsalvo, C.S.; Nettel, F. Conformal Gauge Transformations in Thermodynamics. Entropy 2015, 17, 6150-6168.

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