Nonequilibrium thermodynamics as a symplecto-contact reduction

Both statistical phase space (SPS), T*R^{3N} of N -body particle system and kinetic theory phase space (KTPS), and the cotangent bundle T*P(Γ) of the probability space P(Γ) thereon, carry canonical symplectic structures. Starting from this first principle, we provide a canonical derivation of thermodynamic phase space (TPS) of nonequilibrium thermodynamics as a contact manifold. Regarding the observation of observables as a moment map, we apply the Marsden-Weinstein reduction and obtain a mesoscopic phase space in between KTPS and TPS as an (infinite dimensional) symplectic fibration. We then show that the relative information entropy (aka Kullback-Leibler divergence) ρ → D_{KL}(ρ∥ν_0) defines a generating function that provides a covariant construction of thermodynamic equilibrium as a Legendrian submanifold. This Legendrian submanifold is not necessarily graph-like. We interpret Maxwell’s construction via the equal area law as the procedure of finding a continuous, not necessarily differentiable, Gibbs potential and explain the associated phase transition. Our derivation complements the previously proposed contact geometric description of thermodynamic equilibria and explains the origin of phase transition and the Maxwell construction in this framework. This is a joint work with Jinwook Lim.