Instituto de Física, UFRGS, Porto Alegre, Rio Grande do Sul 91501-970, Brazil.
Phys Rev E. 2023 Feb;107(2-1):024115. doi: 10.1103/PhysRevE.107.024115.
We use Kirchhoff's vortex formulation of 2D Euler fluid equations to explore the equilibrium state to which a 2D incompressible fluid relaxes from an arbitrary initial flow. The vortex dynamics obeys Hamilton's equations of motion with x and y coordinates of the vortex position forming a conjugate pair. A state of fluid can, therefore, be expressed in terms of an infinite number of infinitesimal vortices. If the vortex dynamics is mixing, the final equilibrium state of the fluid should correspond to the maximum of Boltzmann entropy, with the constraint that all the Casimir invariants of the fluid must be preserved. This is the fundamental assumption of Lynden-Bell's theory of collisionless relaxation. In this paper, we will present a Monte Carlo method which allows us to find the maximum entropy state of the fluid starting from an arbitrary initial condition. We will then compare this prediction with the results of molecular dynamics simulation and demonstrate that the final state to which the fluid evolves is, actually, very different from that corresponding to the maximum of entropy. This indicates that the mixing assumption is not correct. We will then present a different approach based on core-halo distribution which allows us to accurately predict the final state to which the fluid will relax, starting from an arbitrary initial condition.
我们使用二维 Euler 流体方程的 Kirchhoff 涡旋公式来探索二维不可压缩流体从任意初始流态松弛到平衡态的过程。涡旋动力学遵循 Hamilton 运动方程，涡旋位置的 x 和 y 坐标形成共轭对。因此，流体的状态可以用无数个无穷小的涡旋来表示。如果涡旋动力学是混合的，那么流体的最终平衡态应该对应于 Boltzmann 熵的最大值，同时必须保持流体的所有 Casimir 不变量。这是 Lynden-Bell 无碰撞松弛理论的基本假设。在本文中，我们将提出一种蒙特卡罗方法，该方法允许我们从任意初始条件出发找到流体的最大熵状态。然后，我们将把这个预测与分子动力学模拟的结果进行比较，并证明流体演化的最终状态实际上与熵最大值的状态非常不同。这表明混合假设是不正确的。然后，我们将提出一种基于核-晕分布的不同方法，该方法允许我们从任意初始条件出发，准确预测流体的最终松弛状态。
