Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
Statistical mechanics of two-dimensional Euler flows and minimum enstrophy states
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二维欧拉流和最小熵状态的统计力学
DOI：
10.1140/epjb/e2010-00269-0
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发表时间：
2009
期刊：
影响因子：
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通讯作者：
Bérengère Dubrulle
中科院分区：
文献类型：
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作者：
A. Naso;A. Naso;P. Chavanis;Bérengère Dubrulle
基于能量守恒、环流和微观涡度拟能，考虑了二维不可压欧拉方程的简化热力学方法。统计平衡态是通过在这些唯一的约束下最大化Miller-Robert-Sommeria（MRS）熵来获得的。我们假设这些约束是由强迫和耗散的性质选择的。我们发现，涡度起伏是高斯的，而平均流量的特点是由一个线性关系。进一步证明了在能量、环流和微观拟能固定的情况下熵的最大化等价于在能量和环流固定的情况下宏观拟能的最小化。这提供了一个理由的最小拟能原理从统计力学时，只有微观拟能是保守的无限类的Casimir约束。导出了趋于统计平衡态的弛豫方程。这些方程可以作为确定最大熵或最小涡度拟能状态的数值算法。我们使用这些弛豫方程来研究矩形域中几何诱导的相变。特别地，我们用弛豫方程说明了Chavanis和Sommeria [J.Fluid Mech.314，267（1996）]预测的单极子和偶极子之间的转变。我们考虑到稳定以及亚稳态，并表明亚稳态是强大的，并具有负比热。这是第一个在这种背景下负比热的证据。我们还认为，鞍点的熵可以是长寿的，并发挥作用的动力学，因为系统可能不会自发地产生扰动，使他们不稳定。
A simplified thermodynamic approach of the incompressible 2D Euler equation is considered based on the conservation of energy, circulation and microscopic enstrophy. Statistical equilibrium states are obtained by maximizing the Miller-Robert-Sommeria (MRS) entropy under these sole constraints. We assume that these constraints are selected by properties of forcing and dissipation. We find that the vorticity fluctuations are Gaussian while the mean flow is characterized by a linearrelationship. Furthermore, we prove that the maximization of entropy at fixed energy, circulation and microscopic enstrophy is equivalent to the minimization of macroscopic enstrophy at fixed energy and circulation. This provides a justification of the minimum enstrophy principle from statistical mechanics when only the microscopic enstrophy is conserved among the infinite class of Casimir constraints. Relaxation equations towards the statistical equilibrium state are derived. These equations can serve as numerical algorithms to determine maximum entropy or minimum enstrophy states. We use these relaxation equations to study geometry induced phase transitions in rectangular domains. In particular, we illustrate with the relaxation equations the transition between monopoles and dipoles predicted by Chavanis and Sommeria [J. Fluid Mech.314, 267 (1996)]. We take into account stable as well as metastable states and show that metastable states are robust and have negative specific heats. This is the first evidence of negative specific heats in that context. We also argue that saddle points of entropy can be long-lived and play a role in the dynamics because the system may not spontaneously generate the perturbations that destabilize them.