Abstract

We construct a Hamiltonian field theory model that combines hydrodynamics and thermodynamics. In this model, the continuity equation and the Euler equation for a potential flow are used as the Hamilton equations; and equations of state (or the Gibbs equations, which are equivalent to them ), as equations of second-class constraints. The Hamiltonian function density reduced to the surface of second-class constraints is the sum of densities of the kinetic and potential energies; free energy then plays the role of potential energy. The surface of second-class constraints is endowed with a natural symplectic structure. Canonical variables are also defined on the second-class constraint surface such that all physical variables can be expressed as functions of these variables. In particular, from the standpoint of the Hamiltonian formalism, entropy is interpreted as a generalized velocity — a Lagrange multiplier for the corresponding second-class constraint expressing the temperature as a function of the canonical variables at the last reduction stage. This multiplier is expressed in terms of the canonical variables at the last stage, yielding a nontrivial equation of motion for entropy. The model must be made more concrete by fixing the dependence of the specific free energy on its arguments. We choose the simplest nontrivial variant, the monatomic van der Waals gas whose atoms are in the ground state. The canonical Hamilton equations allow calculating the rate of change in the entropy of this dynamic system. For the physically interesting case where the system evolution leads to equilibrium, the entropy and its rate of change are functionals of the solution of dynamic equations for the density. A numerical solution of these equations gives a monotonic growth of the entropy (for a finite evolution time). The equation can be linearized to find the time asymptotics; an elliptic equation with the “wrong” sign of the analogue of the squared speed of sound, rather than a hyperbolic equation, is obtained for the asymptotic evolution of the density deviation from the equilibrium value. Thus, the time reversibility of the solution is lost.

中文翻译：

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哈密​​顿动力学中熵和时间不可逆性的增加

摘要

我们构建了一个结合了流体力学和热力学的哈密顿场理论模型。在该模型中，势流的连续性方程和欧拉方程被用作哈密顿方程；和状态方程（或与它们等价的 Gibbs 方程），作为二类约束方程。减少到第二类约束表面的哈密顿函数密度是动能和势能的密度之和；自由能则起到势能的作用。第二类约束的表面被赋予了自然的辛结构。在第二类约束面上也定义了典型变量，使得所有物理变量都可以表示为这些变量的函数。特别是，从哈密顿形式主义的角度来看，熵被解释为广义速度——对应的二类约束的拉格朗日乘数，将温度表示为最后一个还原阶段的规范变量的函数。这个乘数用最后阶段的规范变量表示，产生一个非平凡的熵运动方程。必须通过确定特定自由能对其参数的依赖性来使模型更加具体。我们选择最简单的非平凡变体，即原子处于基态的单原子范德华气体。典型的哈密顿方程允许计算这个动态系统的熵的变化率。对于系统演化导致平衡的物理上有趣的情况，熵及其变化率是密度动态方程解的函数。这些方程的数值解给出了熵的单调增长（对于有限的演化时间）。可以将方程线性化以找到时间渐近线；对于密度偏离平衡值的渐近演化，获得了一个带有声速平方类似物“错误”符号的椭圆方程，而不是双曲线方程。因此，解决方案的时间可逆性丢失。不是双曲线方程，而是从平衡值的密度偏差的渐近演化中获得的。因此，解决方案的时间可逆性丢失。不是双曲线方程，而是从平衡值的密度偏差的渐近演化中获得的。因此，解决方案的时间可逆性丢失。