Irreversible dynamics of vortex reconnections in quantum fluids
IIrreversible dynamics of vortex reconnections in quantum fluids
Alberto Villois
Department of Physics, University of Bath, Bath, BA2 7AY, UK
Davide Proment
School of Mathematics, University of East Anglia,Norwich Research Park, Norwich NR4 7TJ, United Kingdom
Giorgio Krstulovic
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS,Laboratoire Lagrange, Bd de l’Observatoire, CS 34229, 06304 Nice cedex 4, France.
We statistically study of vortex reconnection in quantum fluids by evolving different realizationsof vortex Hopf links using the Gross–Pitaevskii model. Despite the time-reversibility of the model,we report a clear evidence that the dynamics of the reconnection process is time-irreversible, asreconnecting vortices tend to separate faster than they approach. Thanks to a matching theorydevised concurrently in
Proment and Krstulovic (2020) [1], we quantitatively relate the origin ofthis asymmetry to the generation of a sound pulse after the reconnection event. Our results havethe prospect of being tested in several quantum fluid experiments and, theoretically, may shed newlight on the energy transfer mechanisms in both classical and quantum turbulent fluids.
Introduction . Irreversibility emerges naturally in mostinteracting systems characterized by a huge number ofdegrees of freedom. Its manifestation is associated toa time-symmetry breaking: the arrow of time appearsinherently defined in the dynamics and an experiencedobserver is able to distinguish what are before and after .In dissipative systems the arrow of time naturally re-flects the dynamics that minimizes the energy. Classicalviscous fluids present valuable examples. Where no ex-ternal forces are applied, an initial laminar flow decaysin time until its kinetic energy is totally converted intoheat. A less simpler example is the particle pair disper-sion in turbulent flows. Although two tracers separatefrom each other backward and forward in time with thesame Richardson scaling, their rates are different [2]: par-ticles separate slower forward in time than backward.Conservative (energy-preserving) systems are moresubtle. The arrow of time is defined only in a staticalsense by exploiting an entropy function that approachesits extremal as time progresses. The simplest exampleof this kind is the free expansion experiment of a gas:even if the gas particles interact microscopically throughconservative collisions, on average their macroscopic po-sition and velocity distribution obeys the Boltzmann ki-netic equation which is time-irreversible.Quantum fluids are exotic types of fluids characterizedby the total absence of viscosity, thus being conservative.Examples of such systems are superfluid liquid helium [3]and Bose–Einstein condensates (BECs) made of dilutegases of bosons [4], Cooper-paired fermions [5], or mas-sive photons [6]. As a consequence of the wave natureof their bosonic constituents, quantum fluids have twostriking properties: vortices arise as topological defects inthe order parameter and their circulation takes only dis-crete multiples of the quantum of circulation Γ = h/m , where h is the Planck constant and m is the boson’smass. These defects, referred in the following as vortexfilaments, arrange themselves into loops and present acomplicate dynamics whose still misses a general solu-tion. A key point in such dynamics is the occurrence ofreconnection events. A vortex reconnection is the processof interchange of two sections of different filaments, see asketch in Fig. 1(a). It happens at small spatial and fasttime scales [7], and allows the filament topology to vary.For the sake of simplicity, we consider in this Lettera quantum fluid described by a single scalar order pa-rameter. In the limit of zero temperature, this quantumfluid accommodates only two distinct excitation fami-lies: vortex-type excitations, in the form of filaments,and compressible density/phase excitations, that is soundwaves. While the full dynamics is energy-preserving, theenergy may continually flow between these two excitationfamilies. In this perspective, we provide a statistical anal-ysis over many realizations of vortex reconnections, un-veiling an inherent irreversible dynamics of the reconnec-tion process. Moreover, we show how the linear momen-tum and energy transfers, from vortex-type excitations tocompressible density/phase excitations, is related to thegeometrical parameters (macroscopic reconnection angleand concavity parameter) of the reconnecting filaments,explaining the origin of such the irreversibility. Main results . We choose an initial configuration charac-terized by a Hopf link vortex filament, see Fig.1(b), where(almost) all the superfluid kinetic energy is stored intothe vortex-type excitations. Similarly to vortex knots,the Hopf links naturally decay into topologically simplerconfigurations [8–10] by performing a set of vortex recon-nections. To study the evolution of the Hopf link, we useof the Gross–Pitaevskii (GP) model, a nonlinear partialdifferential equation formally derived to mimic the order a r X i v : . [ c ond - m a t . o t h e r] M a y FIG. 1. (Color online) (a) Sketch of a vortex reconnection event in quantum fluids: at the reconnection time t r the reconnectingfilaments are locally tangent to the plane xOy , here depicted in grey, and form the reconnecting angle φ + . The vorticity of thefilaments is depicted with grey arrows. (b) The Hopf link initial condition used to create the different realizations, with visualindication of the offset parameters ( d , d ). parameter ψ of a BEC made of dilute locally-interactingbosons, but qualitatively able to mimic a generic quan-tum fluid [11]. The GP equation, casted in terms of thehealing length ξ and the sound velocity c , reads i ∂ψ∂t = c √ ξ (cid:18) − ξ ∇ ψ + mρ | ψ | ψ (cid:19) , (1)where ρ is the bulk superfluid density and m the massof a boson. When the GP equation is linearized aboutthe uniform bulk value | ψ | = (cid:112) ρ /m , dispersive effectsarise at scales smaller than ξ and (large-scale) soundwaves effectively propagate at speed c . In this Letterlengths and times are expressed in units of ξ and τ = ξ/c ,respectively. Thanks to the Madelung transformation ψ ( x , t ) = (cid:112) ρ ( x , t ) /m exp[ iφ ( x , t ) / ( √ cξ )], eq. (1) canbe interpreted as a model for an irrotational inviscidbarotropic fluid of density ρ and velocity v = ∇ φ . Vor-tices arise as topological defects of circulation Γ = h/m =2 √ πcξ and vanishing density core size order of ξ [12].In the previous formula, h is the Planck constant.We integrate numerically the GP model using a stan-dard pseudo-spectral code evolved in time by a forth-order Runge–Kutta scheme. The computational box isperiodic with sides of length L = 128 ξ ; 256 colloca-tion points are used. The initial Hopf link is preparedby superimposing two rings of radius R = 18 ξ , each ofthem lying on a plane orthogonal to the other. The orderparameter of each ring is numerically obtained by usinga Newton–Raphson and biconjugate-gradient technique[13], allowing to minimize the initial sound excitations inthe system. A set of 49 different realizations are obtainedby changing the offsets ( d , d ) of one ring as sketched inFig.1(b), taking d i ∈ [ − ξ, ξ ] with unit step of 3 ξ . Dur-ing the evolution one or more reconnection events occur.It has been shown [14–17] that about the reconnectionevent, the distance between the two filaments behaves as δ ± ( t ) = A ± (Γ | t − t r | ) / , (2)where A ± are dimensionless pre-factors and t r is the re-connection time; the superscripts − and + label the cases before and after the reconnection, respectively. In eachHopf link realization, we carefully track [18] all recon- FIG. 2. (Color online) Values of approach and separationpre-factors A + and A − . Red points correspond to data ofthe present work. Gray left and right triangles correspond toreconnections of free and trapped vortices respectively, fromGalantucci et al.[16]; other symbols from Villois et al. [15]. necting events and measure A ± . Their values are plot-ted in red dots in Fig. 2. Remarkably, the reconnectingfilaments always separate faster (or at an almost equalrate) than they approach, that is A + ≥ A − . The clearasymmetry recorded in the distribution of the A ± s is thefingerprint of the irreversible dynamics characterizing thevortex reconnection process. For completeness, we alsoreport in the figure, using different symbols, the pre-factor measurements obtained in previous works [15, 16],which corroborate even further our results. In what fol-lows we quantitatively relate the asymmetry in the dis-tribution of the pre-factors with the irreversible energytransfer between the vortex-type and density/phase ex-citation families occurring during a reconnection event.Previous numerical studies of the GP model have indeedreported the clear emission of a sound pulse during re-connection events [19, 20]. A series of snapshots showingthe sound pulse emitted during the decay of the Hopflink in one of our realizations is reported in [1].The simple linear theory neglecting the nonlinear termof the GP model [14, 15], valid in the limit δ ± →
0, pro-vides an insight into the dynamics of reconnecting pa-rameters as the the order parameter can be found ana-lytically. It predicts that the filaments reconnect tangentto a plane, in our reference frame the z = 0, see Fig. 1(a), FIG. 3. (Color online) Spatiotemporal plot of density aboutthe reconnection event denoted by the blue central point. Thetwo dashed green lines are z = c ( t − t r ). and that the projections of the filaments onto it approachand separate following the branches of a hyperbola. The macroscopic (post) reconnection angle , formed by the hy-perbola asymptotes, results in φ + = 2 arccot( A r ) , where A r = A + /A − . (3)Moreover he projections of the filaments onto the orthog-onal plane y = 0 is a parabola (not shown in here, see[1] for more details). Without any loss of generality, weset Λ /ζ the concavity of such parabola, where Λ is the concavity parameter and ζ is an arbitrary length scale.In all the reconnection events detected, we observe aneat sound pulse generated after the reconnection andpropagating towards the positive z -axis. Figure 3 showsthe behavior of the superfluid density along the z direc-tion versus times t − t r in one of the many reconnectionevents. The reconnection point (0 ,
0) is represented bythe blue dot. A (depression) sound pulse is generatedsoon after the reconnection and propagates towards thepositive z -direction at a speed qualitatively close to thespeed of sound in the bulk, refer to the green dashed line z = c ( t − t r ). Note that the other low density regions, cor-responding to the density depletions of the vortex cores,which move much slower.To explain the generation and directionality of suchpulse we devise a novel theoretical approach, detailed in[1], and summarize in the following. Let us denote by R ± ( s, t ) and R ± ( s, t ) the reconnecting filaments, with s being their spatial parametrization variable. Far fromthe reconnection point (both before and after), the dy-namics of the vortex filaments are mostly driven by theBiot-Savart (BS) model, which describes the motion of δ -supported vorticity in an incompressible inviscid flow[21]; note that this limit can be formally derived fromGP [22]. In our realizations, BS is valid at distances δ ± ( t ) (cid:29) δ lin , whereas for δ ± ( t ) (cid:28) δ lin the dynamics isdetermined by the linear approximation, given δ lin is acrossover scale of order of the healing length. We assumeboth descriptions approximately valid when the filamentsare at the distance δ ± ( t ± ) ≈ δ lin . This hypothesis, val-idated by previous GP simulations [15, 16], allows us to perform an asymptotic matching.We can therefore compute the difference, before and af-ter the reconnection, of BS linear momentum ∆ P fil usingthe positions of the filaments R ± ( s, t ± ) and R ± ( s, t ± )coming from the linear approximation. As shown in [1],note that these depend only on the reconnection angle φ + (or equivalently A r ) and the concavity parameter Λ.Within BS, the linear momentum is given as the line in-tegral P fil ( t ) = ρ Γ (cid:72) R ( s, t ) × d R ( s, t ) [23]. As the totallinear momentum of the superfluid is conserved in GP[24], the linear momentum carried by the sound pulsecreated after the reconnection must compensate the lossof linear momentum accounted by ∆ P fil and reads [1] P pulse = − ∆ P fil ∝ (0 , , φ + ) , (4)independently on the δ lin chosen. This result is remark-able: the sound pulse linear momentum is (overall) non-zero only in the positive z -direction, as observed in allour reconnections events, its amplitude is independent ofΛ and minimal for φ − = π/ E Ckin , associated to sound excita-tions, and an incompressible component E Ikin , associatedto vortex-type excitations. In all our realizations, we ob-serve a sharp growth of E Ckin during each reconnectionevent. An example of its evolution, normalized by thetotal (constant) energy E tot , is shown in Fig. 4(a): herethe red dot indicates the reconnection time, and the greenregion indicates the times when δ ± ( t ) ≤ δ lin = 6 ξ . Theincrease of E Ckin during the reconnection event is relatedto the loss of incompressible kinetic energy E Ikin . Forall the reconnection events measured in our realizations,we compute the energy transferred to the sound pulse as E pulse = − ∆ E Ikin , where ∆ E Ikin = E Ikin ( t + ) − E Ikin ( t − ).Figure 4(b) shows the measured E pulse /E tot data versus A r : there is clear correlation between these two quanti-ties, with a best-fit scaling of E pulse /E tot ∝ ( A r − . .To the simplest approximation, called local inductionapproximation (LIA), the BS superfluid kinetic energy isproportional to the total length of the filaments. As therepresentations R ( s, t ) and R ( s, t ) have infinite lengths(as in the linear regime they do not close) we choose toaccount only for the length of finite sections of the fil-aments contained in a cylinder of radius R (cid:29) δ lin , cen-tered at the reconnection point and parallel to the z -axis.Evaluating E pulse reduces thus to the computation of thedifference ∆ L ( A r , | Λ | /ζ, δ lin , R/δ lin ) of the length of thesesections, see [1] for more details. As the total GP energyis conserved, we have that E pulse /E tot = − ∆ L / L , (5)given L is the initial length of the Hopf link filament. Forany given choices of δ lin and R , all the admissible values of FIG. 4. (Color online) (a) Relative increase of compressiblekinetic energy (solid blue) and relative vortex length change(dashed red) about a reconnection event (denoted by the reddots) for a typical realization. The green area correspondsto the interval defined by δ ± ( t ) ≤ δ lin = 6 ξ. (b) Relativeenergy transferred to waves during the reconnection process.The cyan zone denotes the allowed values from the matchingtheory. the theoretical estimation ∆ L , rendered in cyan color inFig. 4(b), are bounded between two lines obtained settingΛ = 0 (dashed line) and | Λ | → ∞ (solid line). The GPdata are all distributed within these admissible values,thus confirming the accuracy of the matching theory.Remarkably, the estimation of E pulse explains in astraightforward way the time asymmetry between therates of approach and separation reported in Fig. 2. In-dependently on the value of the concavity parameter Λ,the energy of the sound pulse is only non-negative when A + ≥ A − , meaning that unless energy is externally pro-vided to the reconnecting vortices, it is energetically im-possible to have a reconnection event where A + < A − ,or equivalently, where φ + > π/ Closing remarks . In this Letter we reported numer-ical evidence of the irreversible dynamics of vortex re-connections in a scalar quantum fluid, and explain itsorigin thanks to a matching theory developed concur-rently in [1]. Our results can be extended to more compli-cated quantum fluids where non-local interactions and/orhigher order nonlinearities are included, like BECs withdipolar interactions, cold Fermi gases, and superfluid liq-uid He.In quantum fluid experiments, the detailed study ofvortex reconnections is still in its infancy. In currentBECs made of dilute gases, reconnecting vortices are cre-ated only in a non-reproducible way using fast temper- ature quenches [26]; however new protocols have beenproposed to create vortices in a reproducible manner [27].In such setups, once the reconnection plane is identified,it should be feasible to measure the rates of approachand separation and detecting directionality of the soundpulse, using for instance Bragg spectroscopy [28]. Insuperfluid liquid He experiments, vortex reconnectionshave been detected so far only at relatively hight tem-perature where the normal component is non-negligible[29]. This latter may provide energy but also dissipate itthrough mutual friction, hence measuring experimentallythe distribution of the rates of approach and separationat different temperatures would be particularly desirable.Finally, let us come back to the concept of irreversibil-ity. In the realizations presented in this Letter, almostall of the superfluid kinetic energy is initially stored inthe vortex-type excitations. This is likely to cause theobserved statistical asymmetry in the distribution of therates of approach and separation to be maximized. Atfinite temperatures or in a turbulent tangle, fluctuationscan provide extra energy to reduce this asymmetry, per-haps allowing also for φ + > π/
2, but the time-asymmetryshould in principle remains as an inherent mechanism al-lowing the system to reach the equilibrium. From a fluiddynamical point of view, let us to remark that vortexreconnections are allowed and regular, in classical fluids,due to the presence of viscosity, while in quantum fluids,thanks to a dispersive term. Showing whether the result-ing dynamics of these two different fluids are equivalentor not, in the limit where their respective regularizationscale tends to zero, is an appealing open problem. Com-paring the results presented in this Letter with a similarstudy in Navier–Stokes or a carefully regularized Biot-Savart model might provide some insights on the sponta-neous stochasticity and the dissipative anomaly of turbu-lent flows, two concepts closely related to irreversibility.The authors acknowledge L. Galantucci for providingsome of the data displayed in Fig. 2. G.K., D.P. and A.V.were supported by the cost-share Royal Society Interna-tional Exchanges Scheme (IE150527) in conjunction withCNRS. A.V. and D.P. were supported by the EPSRCFirst Grant scheme (EP/P023770/1). G.K. and D.P. ac-knowledge the Federation Doeblin for supporting D.P.during his sojourn in Nice. G.K. was also supported bythe ANR JCJC GIANTE ANR-18-CE30-0020-0.1 and bythe EU Horizon 2020 research and innovation programmeunder the grant agreement No 823937 in the frameworkof Marie Skodowska-Curie HALT project. Computationswere carried out at M´esocentre SIGAMM hosted at theObservatoire de la Cˆote d’Azur and on the High Per-formance Computing Cluster supported by the Researchand Specialist Computing Support service at the Univer-sity of East Anglia. Part of this work has been presentedat the workshop “Irreversibility and Turbulence” hostedby Fondation Les Treilles in September 2017. G.K. andD.P. acknowledge Fondation Les Treilles and all partici-pants of the workshop for the frightful scientific discus-sions and support. [1] D. Proment and G. Krstulovic, Article availableat https://gkrstulovic.gitlab.io/publication/promentkrstulovicmatching2020/ (2020).[2] J. P. Salazar and L. R. Collins, Annual Re-view of Fluid Mechanics , 405 (2009),https://doi.org/10.1146/annurev.fluid.40.111406.102224.[3] R. J. Donnelly, Quantized vortices in helium II , Vol. 2(Cambridge University Press, 1991).[4] L. Pitaevskii and S. Stringari,
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