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

2026-06-08

AI Summary

This work proposes a unified framework bridging geometric thermodynamics and information geometry by constructing a novel Kähler metric on the macroscopic thermodynamic contact manifold, whose pullback to isentropic symplectic submanifolds yields a Kähler structure and which reduces at equilibrium to the Fisher–Hessian metric. On the microscopic level, the Calabi–Vesentini manifold serves as a Kähler event space, enabling—through the interplay of Killing moment maps, compact Abelian group actions, and special Kähler geometry—the first explicit analytic expression of partition functions for arbitrary Tits–Satake universality classes. The study innovatively extends Souriau’s thermodynamic formalism by introducing nonlinear Casimir functions to model spontaneous symmetry breaking and constructs a new class of exact Gaussian-like Gibbs distributions, thereby establishing a rigorous theoretical foundation for geometric machine learning and thermodynamic modeling.

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 Kählerian. 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 Kähler 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.

Problem

• Geometric Thermodynamics
• Information Geometry
• Kähler manifold
• Partition Function
• Spontaneous Magnetization

Innovation

• Geometric Thermodynamics
• Kähler manifold
• Information Geometry
• Souriau thermodynamics
• Partition function