Axis-symmetric Onsager Clustered States of Point Vortices in a Bounded Domain

We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to two-dimensional disc. Our analysis is based on the mean field bounded vortices in microcanonical ensemble. The are specified by inverse temperature $\beta$ and rotation frequency $\omega$, which conjugate variables energy $E$ angular momentum $L$, respectively. formation (azimuthal angle independent) involves separating with opposite circulation clustering same around origin edge. state preserves $\rm SO(2)$ symmetry while breaks $\mathbb Z_2$ symmetry. find that, near uniform ($E=0$), free ($\omega=0$) emerges at particular values $L^2/E$ $\beta$. At large energies, we obtain asymptotically exact density distributions, whose validity condition gives rise lower bound for states. Noticeably, obtained distribution edge energies provides novel corresponding chiral system.