The macroscopic Kaehler metric of Geometric Thermodynamics versus the microscopic one on the Event Manifold: Exact Partition Functions on CV manifolds. Extended Souriau temperatures and spontaneous magnetizations

Abstract

In this paper we clarify the relation between Geometric Thermodynamics and Information Geometry based on the Fisher matrix. On the macroscopic odd-dimensional contact manifold of thermodynamic variables, we introduce for the first time a metric, whose pull-back on the isoentropic symplectic submanifolds transverse to the Reeb field is K\"ahlerian. The pull-back of such metric on equilibrium states, that are lagrangian submanifolds, is the Fisher Hessian. Then we consider the Souriau-like Thermodynamics that uses Calabi-Vesentini (CV) manifolds as Kaehlerian microscopic event manifolds and the Killing moment maps as observable functions. A systematic use of the theory of compact abelian structures and the setup of Special K\"ahler Geometry in which CV manifolds are encoded allows us to perform the explicit integration defining the partition function for any entry in the CV Tits Satake universality class. The additional actions completing the abelian structure are non linear Casimir functions of the Killing moment-maps and suggest a generalization of Souriau thermodynamics that partially breaks the isometry group symmetry by means of the non vanishing mean values of the Casimir functions in a manner similar to the spontaneous magnetization in ferromagnetism. Our new exact Gibbs distributions provide the analogue for Cartan Neural Networks of the Gaussian probability distributions in flat space used in conventional Machine Learning.