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Both statistical phase space (SPS), which is $\Gamma = T^*\mathbb R^{3N}$ of$N$-body particle system, and kinetic theory phase space (KTPS), which is thecotangent bundle $T^*\mathcal P(\Gamma)$ of the probability space $\mathcalP(\Gamma)$ thereon, carry canonical symplectic structures. Starting from thisfirst 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 momentmap defined on KTPS, we apply the Marsden-Weinstein reduction and obtain amesoscopic phase space in between KTPS and TPS as a (infinite dimensional)symplectic fibration. Then we show that the reduced relative informationentropy defines a generating function that provides a covariant construction ofa thermodynamic equilibrium as a Legendrian submanifold. This Legendriansubmanifold is not necessarily graph-like. We interpret the Maxwellconstruction of \emph{equal-area law} as the procedure of finding a continuous,not necessarily differentiable, thermodynamic potential and explain theassociated phase transition by identifying the procedure with that of finding agraph selector in symplecto-contact geometry and in the Aubry-Mather theory ofdynamical system.

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