Para-Sasakian geometry in thermodynamic fluctuation theory

In this work we tie concepts derived from statistical mechanics, information theory and contact Riemannian geometry within a single consistent formalism for thermodynamic fluctuation theory. We derive the concrete relations characterizing the geometry of the thermodynamic phase space stemming from the relative entropy and the Fisher-Rao information matrix. In particular, we show that the thermodynamic phase space is endowed with a natural para-contact pseudo-Riemannian structure derived from a statistical moment expansion which is para-Sasaki and eta-Einstein. Moreover, we prove that such manifold is locally isomorphic to the hyperbolic Heisenberg group. In this way we show that the hyperbolic geometry and the Heisenberg commutation relations on the phase space naturally emerge from classical statistical mechanics. Finally, we argue on the possible implications of our results.