有界域中点涡的轴对称 Onsager 团簇态
Axis-symmetric Onsager Clustered States of Point Vortices in a Bounded Domain
AI总结：
本文基于微正则平均场研究二维圆盘内中性点涡的轴对称 Onsager 团簇态，刻画其对称性、无旋转态条件，并给出高能渐近密度及手性系统的新精确分布。
摘要：
我们研究了限制在二维圆盘内的中性点涡系统的轴对称 Onsager 团簇态。分析基于微正则系综中有界点涡的平均场。团簇涡态由逆温度 β 和旋转频率 ω 刻画，二者分别是能量 E 和角动量 L 的共轭变量。轴对称团簇涡态（与方位角无关）的形成涉及相反环量涡的分离，以及相同环量涡在原点和边缘附近的聚集。该态保持 SO(2) 对称性，同时破缺 Z2 对称性。我们发现，在均匀态（E=0）附近，无旋转态（ω=0）出现在 L²/E 和 β 的特定取值处。在高能情形下，我们得到渐近精确的涡密度分布，其有效性条件给出了无旋转态 β 的下界。值得注意的是，高能下所得边缘附近的涡密度分布，为相应的手性涡系统提供了一种新的精确涡密度分布。
英文摘要：
We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to a two-dimensional disc. Our analysis is based on the mean field of bounded point vortices in the microcanonical ensemble. The clustered vortex states are specified by the inverse temperature $β$ and the rotation frequency $ω$, which are the conjugate variables of energy $E$ and angular momentum $L$, respectively. The formation of the axis-symmetric clustered vortex states (azimuthal angle independent) involves the separating of vortices with opposite circulation and the clustering of vortices with same circulation around origin and edge. The state preserves $\rm SO(2)$ symmetry while breaks $\mathbb Z_2$ symmetry. We find that, near the uniform state ($E=0$), the rotation free state ($ω=0$) emerges at particular values of $L^2/E$ and $β$. At large energies, we obtain asymptotically exact vortex density distributions, whose validity condition gives rise the lower bound of $β$ for the rotation free states. Noticeably, the obtained vortex density distribution near the edge at large energies provides a novel exact vortex density distribution for the corresponding chiral vortex system.
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