A classification of coadjoint orbits carrying Gibbs ensembles

TL;DR

This paper classifies coadjoint orbits of finite-dimensional Lie algebras that support Gibbs ensembles, linking geometric temperature, convex hulls, and information geometry within Lie group thermodynamics.

Contribution

It provides a complete classification of coadjoint orbits supporting Gibbs measures and explores their geometric and thermodynamic properties.

Findings

Classification of all coadjoint orbits with Gibbs ensembles.

The geometric temperature parameter space is diffeomorphic to the interior of the convex hull.

Existence of Gibbs measures implies certain properties of the momentum map range.

Abstract

A coadjoint orbit $O_\lambda \subseteq {\mathfrak g}^*$ of a Lie group $G$ is said to carry a Gibbs ensemble if the set of all $x \in {\mathfrak g}$, for which the function $\alpha \mapsto e^{-\alpha(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $\Omega_\lambda$. We describe a classification of all coadjoint orbits of finite-dimensional Lie algebras with this property. In the context of Souriau's Lie group thermodynamics, the subset $\Omega_\lambda$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $\Omega_\lambda/{\mathfrak z}({\mathfrak g})$ diffeomorphically onto the interior of the convex hull of the coadjoint orbit $O_\lambda$. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of $e^{-\alpha(x)}$ for one $x \in {\mathfrak g}$ implies that $\Omega_\lambda \not=\emptyset$ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits $O_\lambda$ as above.