Vortex configuration dependent equilibrium and non-equilibrium states in two-dimensional quantum turbulence

TL;DR

This study uses the two-dimensional Gross-Pitaevskii equation to probe how four vortex configurations in quantum turbulence—dipole, plasma, cluster, and lattice—relax toward equilibrium or settle into non-thermal fixed points. By analyzing incompressible, compressible, and quantum kinetic-energy spectra, particle-number spectra, and the instantaneous particle transfer function, the authors reveal that the cluster configuration relaxes fastest, displaying Kolmogorov-like $\varepsilon^{i}_{kin}(k)\sim k^{-5/3}$ in the inertial range and full mode thermalization in the compressible sector, alongside a Gaussian transfer-function indicative of equilibrium. In contrast, the dipole, plasma, and lattice configurations exhibit persistent non-thermal behavior with Vinen-like incompressible scaling $\varepsilon^{i}_{kin}(k)\sim k^{-1}$ and limited thermalization above a critical wavenumber $k_c^{th}$, whose time evolution differs across configurations. The work demonstrates clear, configuration-dependent routes to thermalization in 2D quantum turbulence, illustrating how initial topology shapes energy cascades, self-similar scaling, and energy transfer statistics, with implications for controlling quantum turbulence in Bose-Einstein condensates.

Abstract

In this work, we analyze the evolution of four vortex configurations, namely, dipole, plasma, cluster, and lattice, using the two-dimensional mean-field Gross-Pitaevskii equation, focusing on their dynamical decay and approach to the equilibrium. Our analysis reveals that the cluster vortex configuration reaches equilibrium more rapidly than the others, while the dipole, plasma, and lattice configurations exhibit persistent non-equilibrium behavior, tending toward non-thermal fixed points. Specifically, the cluster configuration follows Kolmogorov-like scaling ($\varepsilon^{i}(k)\sim k^{-5/3}$) in the incompressible spectrum, while the other configurations follow Vinen-like scaling ($\varepsilon^{i}(k)\sim k^{-1}$). In the compressible spectrum, the cluster case exhibits a $k$ scaling, indicating full mode equilibration, while for the other configurations, the modes thermalize only above a critical wave number. Additionally, the transfer function for the cluster configuration displays a Gaussian distribution, typical of equilibrium states, while the other configurations exhibit skewed Gaussian or exponential distributions, indicative of their non-equilibrium nature. Finally, the particle number spectra show that the cluster case follows dynamical scaling closer to equilibrium, while the dipole, plasma, and lattice configurations evolve towards non-thermal fixed points. Our findings provide new insights into the dynamics of vortex configurations and their approach to equilibrium or non-equilibrium states, offering guidance for future studies on quantum turbulence and its control.

Figures (12)

• Figure 1: The density of the wavefunctions for four different vortex configurations: (a) Dipole gas (b) Plasma (c) Cluster (d) Lattice. The blue markers are placed around vortices and the red markers around antivortices. Subsequent rows show the density of the wavefunction at times $t=0$, $10000$, $50000$ and $100000$ for the four distributions respectively.
• Figure 2: The time evolution of the total number of vortices $N_{v}(t)$ present in the system for the four different initial vortex-antivortex configurations: (a) Dipole, (b) Plasma, (c) Cluster, and (d) Lattice. In the dipole configuration, the vortex number decays with a scaling exponent of $-1/3$, followed by a slower decay with the smaller exponent of $-1/4$ at later times. The Plasma configuration undergoes a relatively steady decay with a scaling exponent $-1/3$. The cluster configuration undergoes a multi-stage decay, where the exponents decay from $-1/3$ to $-1/2$, and eventually approaches $-1$ at long times. In the lattice case, once the lattice structure melts, the subsequent decay of vortex number follows a $-1/3$ scaling, similar to the dipole case.
• Figure 3: Time evolution of the second-order vortex sign correlation function $C_2$ for four different vortex configurations (see legend for configuration types). The Dipole and Lattice configurations converge to a similar vortex configuration with $C_2 \sim 0.38$, while the Plasma configuration settles to $C_2 \sim 0.5$. The Cluster configuration stabilizes at $C_2 \gtrsim 0.5$, indicating different vortex dynamics for each configuration.
• Figure 4: The time evolution of kinetic energy and its components for the four different vortex configurations: (a) dipole, (b) plasma, (c) cluster, and (d) lattice. The blue line represents the total kinetic energy ($E_{kin}$), the red line represents incompressible kinetic energy ($E^{i}_{kin}$), the green line represents compressible kinetic energy ($E^{c}_{kin}$), and the brown line represents quantum kinetic energy ($E^{q}_{kin}$). The black line represents the total energy of the system, rescaled by a factor $\alpha$, with values of 0.03 and 0.09 for the top and bottom rows, respectively. The depletion in incompressible kinetic energy reflects the vortex-antivortex annihilations. The energy extracted from the annihilation processes are transferred to other components: compressible KE, quantum KE and interaction energy. Differences in the rate and magnitude of this redistribution across the four cases arise from the distinct initial vortex configurations, which govern how efficiently dipoles form, interact, and annihilate.
• Figure 5: A heatmap showing the time evolution of the incompressible kinetic energy ($\varepsilon^i_{kin}$) spectra for four different vortex configurations: (a) dipole, (b) plasma, (c) cluster, and (d) lattice. In all cases, the spectral intensity shifts toward lower wavenumbers over time, indicating an inverse cascade. The cluster case shows additional depletion at low wavenumbers and energy transfer from high $k$ modes to intermediate scales, due to the dynamic instability of the initially perfect clusters.