论文标题
非平衡热力学作为符号接触的还原和相对信息熵
Nonequilibrium thermodynamics as a symplecto-contact reduction and relative information entropy
论文作者
论文摘要
这两个统计相空间（SPS），即$γ= T^*\ Mathbb r^{3n} $的$ n $体粒子系统和动力学理论阶段空间（KTPS），这是cotangent $ t^*\ MATHCAL P（MATHCAL P（γ）$ \ Mathcal P（γ）$ canical can can can can can canon，从第一个原理开始，我们提供了非平衡热力学的热力学相空间（TPS）的规范推导，作为两个步骤的接触歧管。首先，关于在KTPS上定义的SPS中观察物的集体观察，我们应用了Marsden-Weinstein还原，并在KTPS和TPS之间获得介镜相位空间作为（无限尺寸）符号纤维纤维。然后，我们表明，减少的相对信息熵定义了生成函数，该函数可提供热力学平衡作为Legendrian Submanifold的协变构建。这个Legendrian Submanifold不一定类似于图形。我们将\ emph {等于区域定律}的麦克斯韦构造解释为找到连续的，不一定可区分的，热力学潜力的过程，并通过与在somplecto-contact几何形状中找到图形选择器的过程来解释相关的相变。
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 $T^*\mathcal P(Γ)$ of the probability space $\mathcal P(Γ)$ 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 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.
