Axis-symmetric Onsager clustered states of point vortices in a bounded domain

Yanqi Xiong,Jiawen Chen and Xiaoquan Yu,2

1 Graduate School of China Academy of Engineering Physics,Beijing 100193,China

2 Department of Physics,Centre for Quantum Science,and Dodd-Walls Centre for Photonic and Quantum Technologies,University of Otago,Dunedin,New Zealand

Abstract We study axis-symmetric Onsager clustered states of a neutral point vortex system confined to a twodimensional disc.Our analysis is based on the mean field of bounded point vortices in the microcanonical ensemble.The clustered vortex states are specified by the inverse temperature β and the rotation frequency ω,which are the conjugate variables of energy E and angular momentum L,respectively.The formation of the axis-symmetric clustered vortex states (azimuthal angle independent) involves the separating of vortices with opposite circulation and the clustering of vortices with the same circulation around the origin and edge.The state preserves SO(2) symmetry while breakingZ2 symmetry.We find that,near the uniform state,the rotation-free clustered state(ω=0)emerges at particular values of L2/E and β.At large energies,we obtain asymptotically exact vortex density distributions,whose validity condition gives rise to the lower bound of β for the rotation-free states.Noticeably,the obtained vortex density distribution near the edge at large energies provides a novel exact vortex density distribution for the corresponding chiral vortex system.

Keywords: vortex clusters,negative temperature,exact solutions,quantum vortices

1.Introduction

In two-dimensional (2D) fluid turbulence,energy at small scales can transport to large scales known as inverse energy cascade[1–4].This process involves formations of large scale vortex patterns.Onsager explained the formation of large scale structures by studying equilibrium statistical mechanics of point vortices in a bounded domain.The macroscopic vortex structure is associated with the clustering of like-sign point vortices at negative temperatures [1,5].These coherent large structures occur in various systems.Examples are the Great Red Spot in Jupiter’s atmosphere [6],giant vortex clusters in atomic Bose–Einstein condensates (BECs) [7–9],and vortex clustering in quantum fluids of exciton–polaritons [10].

The clustering phenomena of vortices have attracted much attention[11–38].For a given 2D domain,searching for the maximum entropy clustered vortex state is at the center of investigations.For circularly symmetric domains,previous studies on neutral vortex systems focus on zero angular momentum cases [18,25,27,32,33].The role of finite angular momentum in the formation of clustered states in a neutral vortex system remains less well-explored.

In this paper,we study axis-symmetric clustered vortex states through the mean field approach.The mean field theory to describe the formation of negative temperature clustered vortex states was formulated systematically by Joyce and Montgomery [39].The mean field equations,which were obtained by maximizing the entropy of the vortex system,are essential for analyzing possible clustered states.We consider a neutral vortex system consisting of an equal number of positive and negative vortices confined to a disc.For a given positive vortex number N+and a negative vortex number N-,clustered vortex states are specified by energy E and angular momentum L or their conjugate variables inverse temperature β(E,L) and rotation frequency ω(E,L).We find that in the limit β →0,ω →∞while keeping βω finite,positive and negative vortex density distributions are Gaussian distributions centered at the origin and edge,respectively.For rotation-free states (ω=0),we find asymptotically exact positive and negative vortex density distributions at large energies.In particular,the one maximized on the edge provides a new exact solution to the mean field equations for the corresponding chiral vortex system.The lower bound of β is obtained from the validity condition of the asymptotically exact solutions at high energies,above which rotation-free clustered states exist.To analyze clustered states closed to the uniform state at low energies,we generalized the perturbation theory,which was initially developed for chiral systems[17],to the neutral case.Using this perturbation theory we find the critical value of β for the onset of the rotation-free clustered vortex state,providing an upper bound of β.

2.Model

The point-vortex model describes the dynamics of wellseparated quantum vortices in a superfluid at low temperature[40],2D classical inviscid,incompressible fluids[21,41]and guiding-center plasma[39].Negative temperature states occur due to the bounded phase space of a 2D confined point vortex system.Above a certain energy,the number of available states decreases as the function of energy and consequently,the system becomes more ordered as energy increases [1].

We consider a system consisting of a large number of point vortices confined to a uniform disc of radius R.The system is neutral and contains N+positive vortices and N-=N+negative vortices.The Hamiltonian is [42]

In a BEC,the Hamiltonian equation (1) is measured 8n unit E0=ρmaκ2/4π,where ρ is the superfluid density,κ=h/mais the circulation quantum and mais the atomic mass.In this unit,κi=±1/N±and the 1/N±scaling gives a well-defined mean field limit[43,44].For a vortex at position rj,its image locates atto ensure that the fluid velocity normal to the boundary vanishes.The Hamiltonian (1) has rotational SO(2) symmetry due to the disc geometry andZ2symmetry(invariant under κi→-κi).Hereafter we set R=1 without losing generality.

To investigate formations of large-scale clustered patterns,it is necessary to consider the continuous effective Hamiltonian in the large N±limit [39]:

Here σ(r)≡n+(r)-n-(r) is the vorticity field,

is the local density of positive (negative) vortices,andis the position of the vortex i with circulation ±1/N±.The vortex densities n±satisfy the normalization condition

The Green’s functionφ(r-r′)satisfies∇2φ(r-r′)=-4πδ(r-r′).Hereφ(r-r′)=0 on the boundary(|r|=1),andφ(r-r′)～-2 log∣r-r′∣as ∣r-r′∣→0 [45].The stream function

satisfies the Poisson equation

with the boundary condition ψ(r=1,θ)=C.Here C is a constant.Recall that the radial velocity

This boundary condition ensures that there is no flow across the boundary of the domain.Without losing generality,we choose C=0,which is equivalent to including image terms in equation (1).

For a rotationally symmetric domain,energy

and angular momentum

are conserved quantities.

The most probable density distribution is given by maximizing the entropy function

at fixed values of N+,N-,E and L[39].From the variational equation

where β,α and μ±are Lagrange multipliers and γ±=-μ±-1.The parameters β,ω ≡α/β and μ±have the interpretation of inverse temperature,rotation frequency and chemical potentials,respectively.

3.Onset of clustering

In this section,we analyze the possible stable large scale coherent structures described by equation (6) and equation(12)near the uniform state.Here we generalized the method which was developed for analyzing chiral vortex matter [17],to the neutral case.

Let us start at a solution n±of equation (12)at energy E and angular momentum L,and consider a nearby solution n±+δn±at E+δE and L+δL.The corresponding changes are

To leading order,we obtain

where δγ-,δγ+,δβ and δα are changes of Lagrange multipliers.Plugging equation (17) into equations (13–16), we have

and n=n++n-is the total density.Variation of equation(6)gives us

Our aim is to find stable clustered states which emerge from the homogeneous state n-=n+=n0=1/π.For the homogeneous state,σ=0,ψ=0,α=0,L=0 and E=0.We assume that δα is in the same order as δψ and from equation (18) we obtain

Let us introduce operatorL:

Then equation (20) becomes a zero mode equation of the operatorL.The onset of large scale vortex clusters occurs if equation (25) has non-zero solutions.The value of β is undefined in the homogeneous phase within our mean field approach and depends on the mode developing from the uniform state.Since the operatorL is defined on a disc with the Dirichlet boundary condition,it is natural to decompose equation(25) in azimuthal Fourier harmonics ψswhich is characterized by the mode number s and satisfies ∂2ψs/∂θ2=-s2ψs:

where ∈≪1 is a small amplitude and fsis the mode coefficient.Then each mode satisfies

where ψs(r,θ) satisfies the boundary condition ψs(r=1,θ)=0.We denote δL=L0∈,δE=E0∈2and δα=∈βω.

We find that

solves equation (27) with

Here Js(r)is the Bessel function of the first kind.Consistently,

For given L0and E0,the parameters cs,k,and bsare determined by equations(31)and(32)combined with the Dirichlet boundary condition

The single-valueless of the stream function requires that s has to be an integer,namely,s∈Z.

For s ≠0,L0=0,a=-ω=0,bs=0,

and k=js,m,where js,mis the mth zero of the Bessel function of the first kind Js(r).

For s=0,

For given E0and L0,c0and k are determined by

It is useful to introduce

as a control parameter.

The ratio Γ(k) reaches its maximum value at k=k*with j1,1＜k*＜j2,1(see figure 1).For a given Γ0＜Γ(k*),there are more than one value of kcsuch that Γ(kc)=Γ0.Guided by the maximum entropy principle,the minimal value of kccorresponds to the equilibrium state.For k →0,E0→0,L0→0,this mode describes the uniform state.

Fig.1.Γ(k) as a function of k.The maximum value of Γ(k) is reached at k=k* and j1,1 ＜k*＜j2,1.

The modes s ≠0 break SO(2) symmetry and the maximum entropy state for given energy is the clustered vortex dipole state which corresponds to the s=1 mode [27].This clustered vortex dipole state has been recently realized in BEC experiments[7].In this paper,we focus on states related to the s=0 mode.

4.Axis-symmetric clustered states

In this section,we present some(asymptomatically) exact results on axis-symmetric neutral vortex clusters.For axis-symmetric states,the boundary condition equation(7)which is imposed by the most relevant physical condition is fulfilled automatically.

4.1.Gaussian vortex states

Let us first consider β →0.For finite ω,vortex distributions n±must be uniform.However,when ω →∞simultaneously such that α=ωβ is finite,non-trivial distributions can occur.In this special limit,the vortex densities have the profile of Gaussian distribution:

where α ∈(-∞,∞).

The corresponding stream function reads

is the exponential integral function.The stream function satisfies ψ(r=1)=0 and dψ/dr|r=1=0.

The angular momentum is

It is easy to see that L ≤1.Figures 2(a)–(b) show typical vortex densities for different values of α.Figures 2(c)–(d) show angular momentum and energy as functions of α.Note that the Gaussian state is available in the chiral vortex system as well [17].

4.2.Rotation-free vortex states

In this subsection,we consider clustered vortex states for ω=0 and finite β ＜0.

4.2.1.Onset of axis-symmetric clustered states.Closed to the uniform state,the clustered states can be analyzed using the formalism developed in section 3.The polar angle θ-independent zero modes s=0 carry non-zero angular momentum.For s=0 modes,the rotation-free condition a=-ω=c0kJ1(k)/2=0[see equation(35)]requires that k=j1,m,where j1,mis the mth zero of the Bessel function of the first kind J1(r).These modes occur atand breakZ2symmetry.The m=1 mode starts to emerge at β=βt=β1,1⋍-1.835 and has the highest statistical weight among the rotation-free modes(ω=0):

4.2.2.High energy configuration.All the rotation-free and axis-symmetry states satisfy

The most relevant solution of equation(46)should be the nonlinear continuation of the zero mode ψ0and describes the axis-symmetry equilibrium state with zero rotation frequency.

At large energies,vortices with opposite signs are wellseparated and the overlap between n+and n-can be neglected.In this limit,exact results are available.Let us assume that positive vortices are concentrated in the center of the disc and negative vortices are distributed along the edge of the disc.The density distribution of positive vortices near r=0 can be obtained analytically by neglecting the influence of negative vortices:

Fig.2.Vortex densities for α=1 (a) and α=6 (b).The angular momentum and the energy as functions of α are shown in (c) and(d),respectively.

with the boundary conditions ψ(0)=0 andψ′(0)=0 [17].HereA=is fixed by the normalization condition of n+and β*=-2.The supercondensation occurs at β=β*,involving point-like concentration of the positive vortices and the divergence of energy [17,46].

Near r=1,we can neglect the influence of positive vortices and find the density distribution of negative vortices

where the boundary conditions are ψ(1)=0 andψ′(1)=0.

Note that ψ(0) and ψ(1) can be chosen as arbitrary constants and here we choose them to be zero for convenience.The boundary conditionψ′(0)=0 ensures thatn′+(0)=0 and n+has no singular behavior near r=0.Similarly,the boundary conditionψ′(1)=0 implies thatn′-(1)=0 and the absence of singular behavior of n-near r=1.As approximations of vortex densities at large energies,equations (47) and (48) should be evaluated for β*＜β.Combining the critical value of β at which the onset of clustering occurs,we obtain the parameter regime for the rotation-free clustered vortex state:

Figure 3 shows the vortex density distributions at high energies.

In the deep clustered state,positive vortices are concentrated in a small region and the total energy is contributed dominantly from positive vortices.So as β →β*,

At large energies,the angular momentum is

and as β →β*,L→Lmaxwith

5.Exact results for chiral vortex clusters

As stated in the previous section,equation (48) is the exact solution to equation (46),provided that n+is neglected.Hence equation (48) provides an exact vortex density distribution for a chiral system,which is distinct from the wellknown exact distribution.In this section,we make a summary of relevant exact results and make a comparison between our findings and the known distribution.

Fig.3.Vortex density distributions at high energies.The densities of positive vortices (a) and negative vortices (b) are evaluated via equations (47) and (48),respectively.

For a rotation-free (ω=0) and axis-symmetric chiral system,equation (46) becomes

There is a known exact solution to equations (53) and(54),which is equation (47):

This solution is valid for β ＞-2.The corresponding stream function could be different depending on the boundary conditions.The vortex density equation (55) exhibits distinct behaviors in different parameter regimes.The vortices accumulate around the edge for 0 ＜β while for-2 ＜β ＜0 the vortices are center-concentrated(see figure 4).Note that in some literature,equation (53) does not have the prefactor 4π and hence the solution looks slightly different [35,47].

Fig.4.Typical profiles of the vortex density distribution described by equation (55) in two distinct parameter regimes: 0 ＜β and -2 ＜β ＜0.

Distinct from equation(55),our finding is equation(48):

with boundary conditions

The solution equation (58) holds for β ＜1 and β ≠0.If requiring thatn′(r=0) is finite,β ＜-1/2.For 0 ＜β ＜1,equation (58) shows center-concentrated distribution and n(r →0)→∞.For -1/2 ＜β ＜0,the vortex density is singular at origin,namelyn′(r→ 0)→∞.Vortices distribute around the edge for β ＜-1/2.In contrast to the known exact solution equation (55),the distribution equation (58) is peaked on the boundary at a negative temperature and is maximized at the origin at a positive temperature.Figure 5 shows typical behaviors of the vortex density in these parameter regimes.

6.Conclusions

Axis-symmetric clustered vortex states for a neutral vortex system confined to a disc are investigated.Combining the perturbation theory near the uniform state and asymptotic analysis at high energies,we find the parameter regime for which the rotation-free states are supported.At large energies,the distributions of positive vortices and negative vortices are well-separated and the edge-concentrated part provides a new exact vortex density distribution for the corresponding chiral vortex system.

Fig.5.Typical profiles of the vortex density distribution described by equation (58) in three distinct parameter regimes: 0 ＜β ＜1,-1/2 ＜β ＜0 and β ＜-1/2.

The onset of a non-axisymmetric vortex cluster in chiral vortex systems appears to proceed via a second-order phase transition [16,17].It would be interesting to investigate possible non-axisymmetric states for neutral systems carrying finite angular momentum.Thanks to the recent experimental advances [7–9],our work would motivate experimentally investigating axis-symmetric clustered phases in a homogeneous Bose–Einstein condensate trapped in cylindrically symmetric potentials.Due to the presence of conservation of angular momentum,axis-symmetric clustered phases are expected to have a longer lifetime than the giant vortex dipole state [7].

Acknowledgments

We acknowledge J Nian,T P Billam,M T Reeves and A S Bradley for useful discussions.X.Y.acknowledges support from the National Natural Science Foundation of China(Grant No.12175215),the National Key Research and Development Program of China (Grant No.2022YFA 1405300),and NSAF (Grant No.U1930403).