2026/01/08 by Karl-Hermann Neeb · 1 voice
Mathematics · Physics and Astronomy · #math.SG #math-ph
A coadjoint orbit Oλ⊆ \mathfrak g^* of a Lie group G is said to carry a Gibbs ensemble if the set of all x ∈ \mathfrak g, for which the function α↦ e-α(x) on the orbit is integrable with respect to the Liouville measure, has non-empty interior Ωλ. 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 Ωλ is the geometric temperature, a parameter space for a family of Gibbs measures on the coadjoint orbit. The corresponding Fenchel--Legendre transform maps Ωλ/\mathfrak z(\mathfrak g) diffeomorphically onto the interior of the convex hull of the coadjoint orbit Oλ. This provides an interesting perspective on the underlying information geometry. We also show that already the integrability of e-α(x) for one x ∈ \mathfrak g implies that Ωλ\not=∅ and that, for general Hamiltonian actions, the existence of Gibbs measures implies that the range of the momentum maps consists of coadjoint orbits Oλ as above.