Souriau-Casimir Lie Groups Thermodynamics and Machine Learning

Abstract

In 1969, Jean-Marie Souriau introduced a “Lie Groups Thermodynamics” in the framework of Symplectic model of Statistical Mechanics. This Souriau’s model considers the statistical mechanics of dynamic systems in their “space of evolution” associated to a homogeneous symplectic manifold by a Lagrange 2-form, and defines in case of non-null cohomology (non equivariance of the coadjoint action on the moment map with appearance of an additional cocyle) a Gibbs density (of maximum Entropy) that is covariant under the action of dynamic groups of physics (eg., Galileo’s group in classical physics). Souriau method could then be applied on Lie Groups to define a covariant maximum Entropy density by Kirillov representation theory. Based on this model, we will introduce a geometric characterization of Entropy as a generalized Casimir invariant function in coadjoint representation, and Massieu characteristic function, dual of Entropy by Legendre transform, as a generalized Casimir invariant function in adjoint representation, where Souriau cocycle is a measure of the lack of equivariance of the moment mapping. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the Entropy. The information manifold foliates into level sets of the Entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on this complex surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will explain the 2nd Principle in thermodynamics by definite positiveness of Souriau tensor extending the (Koszul-)Fisher metric from Information Geometry.