Conformal Gauge Transformations in Thermodynamics

In this work we consider conformal gauge transformations of the geometric structure of thermodynamic fluctuation theory. In particular, we show that the Thermodynamic Phase Space is naturally endowed with a non-integrable connection, defined by all those processes that annihilate the Gibbs 1-form, i.e. reversible processes. Therefore the geometry of reversible processes is invariant under re-scalings, that is, it has a conformal gauge freedom. Interestingly, as a consequence of the non-integrability of the connection, its curvature is not invariant under conformal gauge transformations and, therefore, neither is the associated pseudo-Riemannian geometry. We argue that this is not surprising, since these two objects are associated with irreversible processes. Moreover, we provide the explicit form in which all the elements of the geometric structure of the Thermodynamic Phase Space change under 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