Jean-Marie Souriau’s Symplectic Foliation Model of Sadi Carnot’s Thermodynamics

Thermodynamics explanations by Geometric model were initiated by all precursors Carnot, Gibbs, Duhem, Reeb and Carathéodory. It is only recently that Symplectic Foliation Model introduced in the domain of Geometric statistical Mechanics has given a geometric definition of Entropy as invariant Casimir function on Symplectic leaves (coadjoint orbit of the Lie Group acting on the system, where orbits are interpreted as level sets of Entropy). We give a symplectic foliation interpretation of Thermodynamics based on Jean-Marie Souriau’s "Lie Groups Thermodynamics". This model gives a Lie algebra cohomological characterization of Entropy, as an invariant Casimir function in coadjoint representation. The dual space of the Lie algebra foliates into coadjoint orbits identified with the Entropy level sets. In the framework of Thermodynamics, a dynamics on symplectic leaves, described by Poisson bracket, is associated to non-dissipative phenomenon, and on transversal Riemannian foliation (level sets of energy), dynamics characterized by metric bracket induce entropy production from symplective leaf to leaf. Souriau’s model is interpreted by Libermann's foliations, clarified as dual to Poisson -structure of Haefliger.