A classification of coadjoint orbits carrying Gibbs ensembles
Karl-Hermann Neeb
TL;DR
This work classifies coadjoint orbits O_λ of finite-dimensional Lie algebras that carry Gibbs ensembles by introducing the geometric temperature Ω_λ and analyzing the finiteness domain of the Laplace transform of the Liouville measure μ_λ. The authors establish that Ω_λ ≠ ∅ occurs exactly when the ambient Lie algebra g is admissible and λ lies in the dual cone W_min^*, with an adapted, aligned root system yielding a diffeomorphism from Ω_λ/{\frak z}(g) onto the interior of conv(O_λ). They develop a comprehensive reduction framework (including affine actions on symplectic spaces and semidirect sums) to handle nilpotent, mixed, and reductive cases, prove a Compactness/Temperedness backbone (Compactness Theorem and tempering of μ_λ), and show measure-disintegration over cross-sections to express invariant measures as integrals of Liouville measures on orbits. The results connect Lie-group thermodynamics, information geometry, and convex analysis, offering a structural view of equilibrium states in finite-dimensional settings and laying groundwork for noncommutative, infinite-dimensional extensions. Overall, the paper provides a precise geometric characterization of when Gibbs ensembles exist on coadjoint orbits and reveals a deep link between domain properties of Laplace transforms and the convex-geometric structure of orbit convex hulls.
Abstract
A coadjoint orbit $O_λ\subseteq {\mathfrak g}^*$ 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 $α\mapsto e^{-α(x)}$ on the orbit is integrable with respect to the Liouville measure, has non-empty interior $Ω_λ$. 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 $Ω_λ$ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps $Ω_λ/{\mathfrak z}({\mathfrak g})$ diffeomorphically onto the interior of the convex hull of the coadjoint orbit $O_λ$. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of $e^{-α(x)}$ for one $x \in {\mathfrak g}$ implies that $Ω_λ\not=\emptyset$ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits $O_λ$ as above.