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.

两者都是统计相空间（SPS），即 Γ = T *$\mathbb{右}$^{3 N}的N体粒子系统，以及动力学理论相空间（KTPS），即余切丛T *$\mathbb{磷}$概率空间的 (Γ)$\mathbb{磷}$(Γ) 其上带有规范辛结构。从这个第一原理出发，我们分两步提供了非平衡热力学的热力学相空间（TPS）作为接触流形的规范推导。首先，将 SPS 中可观测量的集体观测视为 KTPS 上定义的矩图，我们应用 Marsden-Weinstein 约简并获得 KTPS 和 TPS 之间的介观相空间作为（无限维）辛纤维。然后我们证明，减少的相对信息熵定义了一个生成函数，该函数提供了作为勒根德里安子流形的热力学平衡的协变结构。这个勒让德子流形不一定是图形的。我们将等面积定律的麦克斯韦构造解释为寻找连续的、不一定可微的热力学势的过程，并通过将该过程与在辛接触几何和奥布里中寻找图选择器的过程进行识别来解释相关的相变。 -马瑟动力系统理论。