Nonequilibrium Statistical Mechanics of Two-Dimensional Vortices

It has been observed empirically that two-dimensional vortices tend to cluster, forming a giant vortex. To account for this observation, Onsager introduced the concept of negative absolute temperature in equilibrium statistical mechanics. In this Letter, we show that in the thermodynamic limit a system of interacting vortices does not relax to the thermodynamic equilibrium but becomes trapped in a nonequilibrium stationary state. We show that the vortex distribution in this nonequilibrium stationary state has a characteristic core-halo structure, which can be predicted a priori. All the theoretical results are compared with explicit molecular dynamics simulations.

Figure 1

The equilibrium vortex density distribution starting from an initial state in which vortices are uniformly distributed inside an ellipse with $a=1.0$ and $b=0.5$, shown by the dashed curve, calculated by solving numerically the nonlinear PB equation (4). The equilibrium distribution has a negative temperature corresponding to $\beta =-1.19$.

Figure 2

Snapshots of the molecular dynamics simulation for elliptical vortex distribution with $a=1.0$, $b=0.5$, and $N={10}^6$ at (a) $t=0$, (b) $t=T/8$, and (c) $t=T/4$, where $T=2\pi /\omega$ and $\omega$ is given by Eq. (13). In (d), we show the time evolution of the angle between the semimajor axis of ellipse and the $x$ axis, $\theta$. The circles correspond to the results obtained from MDS, and the line is the theoretical prediction $\theta =\omega t$.

Figure 3

(a) Snapshot of the phase space obtained using a molecular dynamics simulation. (b) The theoretical prediction obtained using Eq. (14), with no adjustable parameters. The black region corresponds to the high-density core, whereas the gray region corresponds to the low-density halo. The core-halo structure rotates in the lab frame. The core has population inversion—the high-energy states are occupied up to the maximum density $\eta$ permitted by the Vlasov dynamics.

Figure 4

Level curves—in the rotating reference frame—of the effective potential [Eq. (12)], generated by a uniform ellipse rotating with angular velocity $\omega$. The dashed curve shows the ellipse boundary with $a=1.0$ and $b=0.5$. The thick curve corresponds to the separatrix that contains two hyperbolic fixed points located at $\tilde{x}=\pm {\tilde{x}}_{\mathrm{fix}}$ and $\tilde{y}=0$.