A classification of coadjoint orbits carrying Gibbs ensembles

Abstract

A coadjoint orbit $O_{\hat{I}}»\subseteq {\mathfrak{g}}^{\ast }$ 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 $\hat{I}\pm \mapsto e^{-\hat{I}\pm (x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $\hat{I}{\text{c◯}}_{\hat{I}}»$. 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 $\hat{I}{\text{c◯}}_{\hat{I}}»$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $\hat{I}{\text{c◯}}_{\hat{I}}»/\mathfrak{z}(\mathfrak{g})$ diffeomorphically onto the interior of the convex hull of the coadjoint orbit $O_{\hat{I}}»$. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of $e^{-\hat{I}\pm (x)}$ for one $x\in \mathfrak{g}$ implies that $\hat{I}{\text{c◯}}_{\hat{I}}»\ne \varnothing$ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits $O_{\hat{I}}»$ as above.

50pp