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arXiv:2503.11286 · Physical Review A

Out-of-equilibrium behavior of quantum vortices: A comparison of point vortex dynamics and Fokker-Planck evolution

R. J. Tattersall, Andrew William Baggaley, T. P. Billam

Published Jun 9, 2025• 0 citations
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Abstract

An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, E 0 / N v ≲ − 4 , when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent z in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for E 0 / N v ≳ − 4 , to a value which depends explicitly on the initial conditions for E 0 / N v ≲ − 5 .

DOI
10.1103/l8yk-kk5c

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An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, E 0 / N v ≲ − 4 , when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent z in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for E 0 / N v ≳ − 4 , to a value which depends explicitly on the initial conditions for E 0 / N v ≲ − 5 .
© (2025) (American Physical Society) All rights reserved.An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, E0/Nv −4, when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent z in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for E0/Nv −4, to a value which depends explicitly on the initial conditions for E0/Nv −5.
© (2025) (American Physical Society) All rights reserved.An out-of-equilibrium two-dimensional superfluid relaxes towards equilibrium via a process of coarsening, driven by the annihilation of vortices with antivortices. Here we present a comparison of two different numerical models of this process, a dissipative point vortex model and Fokker-Planck evolution, across a wide range of initial configurations and levels of dissipation. We find that our dissipative point vortex model is well approximated by Fokker-Planck evolution only for very low initial energies per vortex, E0/Nv −4, when almost all vortices and antivortices are closely bound into dipoles. We observe that the dynamical critical exponent z in the dissipative point vortex model undergoes a crossover, from a roughly constant value close to 2 for E0/Nv −4, to a value which depends explicitly on the initial conditions for E0/Nv −5.
Understanding the elementary mechanism for the dissipation of vortex energy in quantum liquids is one central issue in quantum hydrodynamics, such as quantum turbulence in systems ranging from neutron stars to atomic condensates. In a two-dimensional (2D) Bose-Einstein condensate (BEC) at zero temperature, in addition to the vortex drift-out process from the boundary, a vortex-antivortex pair can annihilate in bulk, but controversy remains on the number of vortices involved in the annihilation process. We find that a dynamic crossover exists from four-body to three-body vortex annihilation processes with time evolution in decaying 2D quantum turbulence of a boundary-less uniform quasi-2D BEC. Such a dynamical crossover depends on the initial vortex pair density and occurs when the sound waves generated in the vortex annihilation process surpass a threshold energy. As the confinement along the third direction is relaxed in a quasi-2D BEC, the threshold sound wave energy decreases due to the three-dimensional (3D) vortex line curve and reconnection, shifting the dynamical crossover to the early time. Our work reveals an elementary mechanism for the dissipation of vortex energy that may help understand exotic matter and dynamics in quantum liquids.
A large ensemble of quantum vortices in a superfluid may itself be treated as a novel kind of fluid that exhibits anomalous hydrodynamics. Here we consider the dynamics of vortex clusters under thermal friction and present an analytic solution that uncovers a new universality class in the out-of-equilibrium dynamics of dissipative superfluids. We find that the long-time dynamics of the vorticity distribution is universal in the form of an expanding Rankine vortex (i.e., top-hat distribution) independent of initial conditions. This highlights a fundamentally different decay process to classical fluids, where the Rankine vortex is forbidden by viscous diffusion. Numerical simulations of large ensembles of point vortices confirm the universal expansion dynamics and further reveal the emergence of a frustrated lattice structure marked by strong correlations. We present experimental results of expanding vortex clusters in a quasi-two-dimensional Bose-Einstein condensate that are in excellent agreement with the vortex fluid theory predictions, demonstrating that the signatures of vortex fluid theory can be observed with as few as N∼11 vortices. Our theoretical, numerical, and experimental results establish the validity of the vortex fluid theory for superfluid systems.
Understanding the elementary mechanism for the dissipation of vortex energy in quantum liquids is one central issue in quantum hydrodynamics, such as quantum turbulence in systems ranging from neutron stars to atomic condensates. In a two-dimensional (2D) Bose-Einstein condensate (BEC) at zero temperature, besides the vortex drift-out process from the boundary, vortex-antivortex pair can annihilate in the bulk, but controversy remains on the number of vortices involved in the annihilation process. We find there exists a dynamical transition from four-body to three-body vortex annihilation processes with the time evolution in a boundary-less uniform quasi-2D BEC. Such dynamical transition depends on the initial vortex pair density, and occurs when the sound waves generated in the vortex annihilation process surpass a critical energy. With the confinement along the third direction is relaxed in a quasi-2D BEC, the critical sound wave energy decreases due to the 3D vortex line curve and reconnection, shifting the dynamical transition to the early time. Our work reveals an elementary mechanism for the dissipation of vortex energy that may help understand exotic matter and dynamics in quantum liquids.
A critical point is associated with a diverging relaxation time which always makes the dynamics across a critical point fascinating. For example, when the temperature of a classical system undergoing a classical phase transition (CPT) at a finite temperature T c is suddenly changed from a value higher than the critical temperature to a lower value across the critical point, the system does not equilibrate instantaneously. Domains of ordered regions are formed which grow following a phase ordering dynamics which leads to a dynamical scaling [104]. In this section, we will discuss the recent studies of non-equilibrium dynamics of quantum systems driven across QCPs where the dynamics is unitary unlike the dynamics across a finite temperature critical point. The non-equilibrium dynamics of a transverse XY spin chain was first investigated in a series of papers [48, 46, 47] where the time evolution of the model was studied in the presence of various time-dependent magnetic fields and the nonergodic behavior of the magnetization was pointed out. A similar result was also obtained in [505]. There is a recent upsurge in studies of non-equilibrium dynamics of a quantum system swept across a QCP. These studies are important for exploring the universality associated with quantum critical dynamics. Moreover, recent experiments with ultracold atomic gases [343, 663, 478, 87] have stimulated numerous theoretical studies. The main properties of these atomic gases are low dissipation rates and phase coherence over a long time so that the dynamics is well described by the usual quantum evolution of a closed system. In the subsequent sections, we shall discuss that when a quantum system initially prepared in its ground state is driven across a QCP, the dynamics fails to be adiabatic however slow the rate of change in the parameters of the Hamiltonian may be. This is due to the divergence of the characteristic time scale of the quantum system, namely, the relaxation time close to the QCP. This non-adiabaticity results in the occurrence of defects in the final state of the quantum Hamiltonian.