The French mathematician and physicist Jean-Marie Souriau studied Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold and considered their possible applications in Physics and Cosmology. These Gibbs states are presented here with detailed proofs of all the stated results. A companion paper to appear will present examples of Gibbs states on various symplectic manifolds on which a Lie group of symmetries acts by a Hamiltonian action, including the Poincaré disk and the Poincaré half-plane.
"On Gibbs States of Mechanical Systems with Symmetries." J. Geom. Symmetry Phys. 57 45 - 85, 2020. https://doi.org/10.7546/jgsp-57-2020-45-85