Nonequilibrium thermodynamics as a symplecto-contact reduction and relative information entropy

Abstract

Both statistical phase space (SPS), which is Γ = T* $\mathbb{R}$^3N of N-body particle system, and kinetic theory phase space (KTPS), which is the cotangent bundle T* $\mathbb{P}$(Γ) of the probability space $\mathbb{P}$(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 in two steps. First, regarding the collective observation of observables in SPS as a moment map defined on KTPS, we apply the Marsden–Weinstein reduction and obtain a mesoscopic phase space in between KTPS and TPS as a (infinite-dimensional) symplectic fibration. Then we show that the reduced relative information entropy defines a generating function that provides a covariant construction of a thermodynamic equilibrium as a Legen-drian submanifold. This Legendrian submanifold is not necessarily graph-like. We interpret the Maxwell construction of equal-area law as the procedure of finding a continuous, not necessarily differentiable, thermodynamic potential and explain the associated phase transition by identifying the procedure with that of finding a graph selector in symplecto-contact geometry and in the Aubry-Mather theory of dynamical system.