论文标题
关于几何学在统计力学和热力学中的作用
On the role of geometry in statistical mechanics and thermodynamics I: Geometric perspective
论文作者
论文摘要
本文包含用于非平衡可逆 - 反转耦合（通用）的通用方程式的完全几何公式。尽管通用是汉密尔顿力学和梯度动力学的总和，但由于汉密尔顿力学的几何起源和渐变动力学的多样化，它具有不平衡热力学的各种模型的框架，但它具有不清楚的几何结构。差异可以通过动力学的升力来克服，例如，这导致了梯度动力学形式。此外，可以将抬高的向量场分为其全体和垂直代表，这提供了动态还原的几何方法。提起的动力学也可以给出物理含义，这里称为速率生成。最后，可以在接触几何形状中配制升力，其中第二个热力学定律明确包含在进化方程中。
This paper contains a fully geometric formulation of the General Equation for Non-Equilibrium Reversible-Irreversible Coupling (GENERIC). Although GENERIC, which is the sum of Hamiltonian mechanics and gradient dynamics, is a framework unifying a vast range of models in non-equilibrium thermodynamics, it has unclear geometric structure, due to the diverse geometric origins of Hamiltonian mechanics and gradient dynamics. The difference can be overcome by cotangent lifts of the dynamics, which leads, for instance, to a Hamiltonian form of gradient dynamics. Moreover, the lifted vector fields can be split into their holonomic and vertical representatives, which provides a geometric method of dynamic reduction. The lifted dynamics can be also given physical meaning, here called the rate-GENERIC. Finally, the lifts can be formulated within contact geometry, where the second law of thermodynamics is explicitly contained within the evolution equations.
