Symplectic theory of heat and information geometry

Abstract

We present in this chapter a new formulation of heat theory and Information Geometry through symplectic and Poisson structures based on Jean-Marie Souriau's symplectic model of statistical mechanics, called “Lie Groups Thermodynamics.” Souriau model was initially described in chapter IV “Statistical Mechanics” of his book “Structure of dynamical systems” published in 1969. This model gives an archetypal, and purely geometric, characterization of Entropy, which appears as an invariant Casimir function in coadjoint representation, from which we will deduce a geometric heat equation as Euler-Poincaré equation. The approach also allows generalizing the Fisher metric of information geometry thanks to the KKS (Kirillov, Kostant, Souriau) 2-form in the affine case via the Souriau's cocycle. In this model, the Souriau's moment map and the coadjoint orbits play a central role. Ontologically, this model provides joint geometric structures for Statistical Mechanics, Information Geometry and Probability. Entropy acquires a geometric foundation as a function parameterized by mean of moment map in dual Lie algebra, and in term of foliations. Souriau established the generalized Gibbs laws when the manifold has a symplectic form and a connected Lie group G operates on this manifold by symplectomorphisms. Souriau Entropy is invariant under the action of the group acting on the homogeneous symplectic manifold. As quoted by Souriau, these equations are universal and could be also of great interest in Mathematics. The dual space of the Lie algebra foliates into coadjoint orbits that are also the level sets on the entropy that could be interpreted in the framework of Thermodynamics by the fact that motion remaining on these surfaces is non-dissipative, whereas motion transversal to these surfaces is dissipative. We will also explain the second Principle in thermodynamics by definite positiveness of Souriau tensor extending the Koszul-Fisher metric from Information Geometry. Entropy as Casimir function is characterized by Koszul Poisson Cohomology. We will finally introduce Gaussian distribution on the space of Symmetric Positive Definite (SPD) matrices, through Souriau's covariant Gibbs density by considering this space as the pure imaginary axis of the homogeneous Siegel upper half space where Sp(2n,R)/U(n) acts transitively. Gauss density of SPD matrices is computed through Souriau's moment map and coadjoint orbits. We will illustrate the model first for Poincaré unit disk, then Siegel unit disk and finally upper half space. For this example, we deduce Gauss density for SPD matrices.