Onsager-Kraichnan Condensation in Decaying Two-Dimensional Quantum Turbulence

Despite the prominence of Onsager’s point-vortex model as a statistical description of 2D classical turbulence, a first-principles development of the model for a realistic superfluid has remained an open problem. Here we develop a mapping of a system of quantum vortices described by the homogeneous 2D Gross-Pitaevskii equation (GPE) to the point-vortex model, enabling Monte Carlo sampling of the vortex microcanonical ensemble. We use this approach to survey the full range of vortex states in a 2D superfluid, from the vortex-dipole gas at positive temperature to negative-temperature states exhibiting both macroscopic vortex clustering and kinetic energy condensation, which we term an Onsager-Kraichnan condensate (OKC). Damped GPE simulations reveal that such OKC states can emerge dynamically, via aggregation of small-scale clusters into giant OKC clusters, as the end states of decaying 2D quantum turbulence in a compressible, finite-temperature superfluid. These statistical equilibrium states should be accessible in atomic Bose-Einstein condensate experiments.

Figure 1

Properties of neutral $N$-vortex states in statistical equilibrium, corresponding to the end states of decaying 2DQT. (a) Clustering measures $f_c$ and $c_B$ and structure function $w(ϵ)$; vertical gray lines indicate the expected energy ($⟨ϵ⟩$) and the boundary between positive- and negative-temperature states (${ϵ}_{\infty }$). (b) Average and maximum cluster charges ($|{\kappa }_c|$) and radii ($r_c$) obtained using the RCA (see text). (c) IKE spectrum in the point-vortex-like approximation [Eq. (3)] (solid black line, gray shaded area), and obtained from the GPE wave functions [13, 19] (magenta circles). Straight lines show analytical $k^{-3}$ and $k^{-1}$ power laws (see text). Inset: typical vortex configurations (field of view $L\times L$): dots indicate vortices, which have been sorted into dipoles, free vortices and clusters by the RCA (see legend). Lines show the minimal spanning tree of clusters and identify dipoles. (d) Distribution of $N_{{\kappa }_c}$. Shaded areas behind curves in (a),(b),(c) indicate the width of the equilibrium distribution ($\pm 1$ standard deviations). Ensemble sizes are given in [58].

Figure 2

Dynamical dGPE evolution of nonequilibrium neutral $N$-vortex states towards statistical equilibrium (see also animations in [40]). (a) RCA decomposition after equilibration ($t=9500\hslash /\mu$), with initial conditions inset. Streamlines show the incompressible velocity field. Field of view is $L\times L$. (b) IKE spectrum, compared to the statistical equilibrium distribution from Fig. 1 (gray solid line, thickness indicates $\pm 1$ standard deviations). Other symbols in (a),(b) as in Fig. 1. (c) Clustered fraction $f_c$. (d) Correlation function $c_2$. (e) Absolute charge $|{\kappa }_c|$ of the largest vortex cluster. Spectra and measures are shown as a moving average from $t$ to $t+500\hslash /\mu$, and $E^i(k)$ is normalized by $N(t)$; decay of $N(t)$ for $ϵ(0)>-3$ is negligible ($\lesssim 5\%$). Horizontal shading in (c)—(e) shows the statistical equilibrium distributions from Figs. 1 and 1 (shaded areas indicate $\pm 1$ standard deviations). Note that we compare the $ϵ(0)=-3$ evolution to the statistical equilibrium at $ϵ=-2.5$, since dipole annihilation [for $ϵ=-3$, $N({10}^4)=178$] leads to $ϵ({10}^4)\approx -2.5$.