arXiv subjects
Publications and source records attributed to Jérémie Pierard de Maujouy.
Gibbs states are probability distributions defined on Hamiltonian G-manifolds. They are parametrized by an open set of the Lie algebra g and arise as natural equilibrium states with regard to the action of G on the system. In this paper, we focus on nilpotent orbits of SL(n,R). We show that for n>2, those orbits do not admit any Gibbs state.
Gibbs states are probability distributions defined on Hamiltonian G-manifolds that are naturally parametrized by elements of the Lie algebra g. In this paper, we focus on a specific case of the simplest Hamiltonian G-manifolds, the coadjoint orbits of Lie algebras. We look at the nilpotent coadjoint orbits of the classical Lie algebras, or equivalently the nilpotent adjoint orbits. We show that Gibbs states do not exist on nilpotent orbits that are stable under multiplication by -1, and proceed to classify those for all classical Lie algebras.
Frame bundles equipped with a principal connection have their local structure characterised by a 1-form, called the Cartan connection 1-form, which gathers the principal connection form and the soldering form. We introduce generalised frame bundles as a smooth manifold equipped with a coframe with value in a suitable Lie algebra and which furthermore satisfies a weakened version of the Maurer-Cartan equation. From this structure it is possible to construct a Lie algebra action on the manifold. We study the question of whether it is possible to construct a Lie group action and build a base manifold over which the initial manifold is a frame bundle. We find that generalised frame bundles can have singular underlying manifolds.
We study the Hessian geometry associated with an ideal gas in a spherical centrifuge. According to Souriau, a spherically confined ideal gas admit states of thermal and rotational equilibrium. These states, called Gibbs states, form an exponential family with an action of the Euclidean rotation group. We investigate its Hessian (Fisher-Rao) geometry and show that in the high angular velocity limit, the geometry can be compared to the Hessian geometry of a spherical rigid body which we show to be isometric to a hyperbolic space.
We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle starting from a structure-less differentiable 10-manifold. The obtained structure is locally equivalent to a frame bundle which we term "generalised frame bundle". In the same time, we enrich the model with a Dirac spinor coupled to the Einstein-Cartan spacetime. The obtained variational equations generalise the usual Einstein-Cartan-Dirac field equations in that they are shown to imply the usualy field equations when the generalised frame bundle is a standard frame bundle.
We clarify the structure obtained in Hélein and Vey's proposition for a variational principle for the Einstein-Cartan gravitation formulated on a frame bundle starting from a structure-less differentiable 10-manifold (arXiv:1508.07765v2 [math-ph]). The obtained structure is locally equivalent to a frame bundle. We show that under global hypotheses the formalism allows to produce variational principles which induce a frame bundle structure with a fibration over a base manifold and a variational principle on the base manifold. Our perspective on the model allows us to extend the Einstein-Cartan model to the Einstein-Cartan-Dirac theory which includes a Dirac spinor field coupled to the torsion field.