Both statistical phase space (SPS), $\Gamma = T^*\mathbb R^{3N}$ of $N$-body
particle system $\mathcal F$, and kinetic theory phase space (KTPS), 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. Regarding the
collective observation of observables 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 an (infinite dimensional) symplectic fibration. We then show
that the $\mathcal F$-reduction of the relative information entropy (aka
Kullback-Leibler divergence) defines a generating function that provides a
covariant construction of thermodynamic equilibrium as a Legendrian
submanifold. This Legendrian submanifold is not necessarily holonomic. We
interpret the Maxwell construction as the procedure of finding a continuous,
not necessarily differentiable, thermodynamic potential and explain the
associated phase transition. We do this by identifying the procedure with that
of finding a graph selector in symplecto-contact geometry and in the
Aubry-Mather theory of dynamical system. 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.



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