A classification of coadjoint orbits carrying Gibbs ensembles

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.