Thermodynamics a la Souriau on Kähler Non Compact Symmetric Spaces for Cartan Neural Networks

TL;DR

The paper develops a unified geometric framework for thermodynamics on non-compact Kahler symmetric spaces U/H, focusing on Cartan Neural Networks where hidden layers are modeled by these manifolds. It distinguishes Gibbs states arising from integrable geodesic dynamics from Souriau-type non-abelian Gibbs states defined via moment maps on Kahler manifolds, showing that convergence constraints select the admissible generalized temperatures as an adjoint orbit of a positivity domain in the Cartan subalgebra. Through master examples (Poincaré plane H2 and Siegel plane SH2) and the SL(3,R)/SO(3) model, it derives partition functions, Gibbs distributions, and the associated geothermodynamic metrics and curvatures, highlighting covariance under the full symmetry group and the role of Paint Group symmetry for Calabi–Vesentini manifolds. The work connects Rao–Chentsov–Amari information geometry to Souriau thermodynamics and outlines a program to deploy these distributions in data-encoding layers of CaNNs, with explicit instructions for extending to broader Tsits Satake universality classes. Overall, it lays foundational tools for geometry-aware probabilistic modeling on non-flat manifolds, with potential applications to advanced ML architectures that leverage symmetry and solvable coordinates.

Abstract

In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non compact symmetric spaces $\mathrm{U/H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a distinction between the generalized thermodynamics associated with Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U/H}$ provided by generalized thermodynamics {à} la Souriau. Main result is the proof that $\mathrm{U/H}$.s supporting Gibbs distributions are only the Kähler ones. For the latter, we solve the problem of determining the space of temperatures, namely of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset\mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset\mathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore we claim that Rao's, Chentsov's, Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. The most important property of the Gibbs probability distributions provided by the here introduced setup is their covariance with respect to the action of the full group of symmetries $\mathrm{U}$. The partition function is invariant against $\mathrm{U}$ transformations and the set of its arguments, namely the generalized temperatures, can be always reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$.

Figures (8)

• Figure 1: The cone $\Omega$ of Souriau allowed temperature vectors in the $\mathfrak{sl}(2,\mathbb{R})$ Lie algebra space as defined in eq.(refkunegonda).
• Figure 2: Examples of plots of the Gibbs probability distributions (\ref{['perigord']}) over the Poincaré disk, labeled by different set of temperatures. The exponential gaussian decay toward infinity is visually evident, as much as the deformed bell shape. In the first image we compare two distribution with the same values of $\delta\mu$ but with a different angle $\theta$. In the second image we compare two distributions that differ in all parameters.
• Figure 3: In this figure we compare two Gibbs distributions on the Poincaré plane corresponding to a lower and higher value of the norm (\ref{['normaltale']}). As one sees for a high value of the norm the distribution is very sharply shaped around its maximal value, while for lower norm it is much broader. For high norm we know with much more precision the actual location of the stochastic variable in the plane.
• Figure 4: In the first and second picture of this figure we display the behavior of the four intrinsic curvature components $\mathcal{F},\mathcal{G},\mathcal{Q},\mathcal{P}$ as respectively seen from the two sides of the vertical plane that has the the line $\delta=\mu$ as base. Such line corresponds to the boundary of the cone in fig. \ref{['balzano1']}. All the curvature components become singular on such line.
• Figure 5: In this figure we show for a few pairs of values of $\lambda$ and $\mu$ the integrand $\mathfrak{F}$ in the integration variables $w_{1,2}\, = \, \log{\rho_{1,2}}$. The bell shape and the uniform exponential decay to zero at infinity in all directions guarantees the convergence of the two remaining integrals on $w_{1,2}$.

Theorems & Definitions (28)

• Definition 3.1
• Theorem 3.1
• Proof 3.1.1
• Definition A.1
• Definition A.2
• Definition A.3
• Definition A.4
• Theorem A.1
• Definition A.5
• Definition A.6