  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 the
cotangent bundle $T^*\mathcal P(\Gamma)$ of the probability space $\mathcal
P(\Gamma)$ 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 Legendrian submanifold. This Legendrian
submanifold is not necessarily graph-like. We interpret the Maxwell
construction of \emph{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.
