# Onsager vortex clusters on a sphere

Jiawen Chen  Graduate School of China Academy of Engineering Physics, Beijing
100193, China  Xiaoquan Yu<EMAIL_ADDRESS>Graduate School of China Academy
of Engineering Physics, Beijing 100193, China  Department of Physics, and
Centre for Quantum Science, University of Otago, Dunedin 9016, New Zealand

###### Abstract

We study Onsager vortex clustered states in a shell-shaped superfluid
containing a large number of quantum vortices. In the incompressible limit and
at low temperatures, the relevant problem can be boiled down to the
statistical mechanics of neutral point vortices confined on a sphere. We
analyze rotation free vortex clustered states within the mean field theory in
the microcanonical ensemble. We find that the sandwich state, which involves
the separating of vortices with opposite circulation and the clustering of
vortices with same circulation around the poles and the equator, is the
maximum entropy vortex distribution, subject to zero angular momentum
constraint. The dipole momentum vanishes for the sandwich state and the
quadrupole tensor serves as an order parameter to characterize the vortex
cluster structure. For given finite angular momentum, the equilibrium vortex
distribution forms a dipole structure, i.e., vortices with opposite sign are
separated and are accumulated around the south and north pole, respectively.
The conditions for the onset of clustering, and the exponents associated with
the quadrupole moment and the dipole moment as functions of energy, are
obtained within the mean field theory. At large energies, we obtain
asymptotically exact vortex density distributions using the stereographic
projection method, which give rise the parameter bounds for the vortex
clustered states. The analytical predictions are in excellent agreement with
microcanonical Monte Carlo simulations.

##  I Introduction

Coherent large vortex structures can occur in various bounded flows, examples
range from Great Red Spot in Jupiter’s atmosphere Vasavada  _et al._ (1998),
to giant vortex clusters in atomic Bose-Einstein condensates (BECs) Gauthier
_et al._ (2019); Johnstone  _et al._ (2019), and in dissipative quantum fluids
of exciton-polaritons Panico  _et al._ (2023). For a vortex system in a
bounded domain, the phase space per vortex is also bounded. The formation of
vortex clusters is due to that the system favors the clustering of like-sign
vortices at high energies Onsager (1949); Eyink and Sreenivasan (2006).
Onsager formulates the statistical mechanics of a bounded point vortex system
in the microscopic ensemble and demonstrates that the clustered state is at
negative absolute temperature. Since then clustering of vortices on bounded
domains has been investigated extensively Joyce and Montgomery (1973); Edwards
and Taylor (1974); Williamson (1977); Pointin and Lundgren (1976); Campbell
and O’Neil (1991); Smith (1989); Smith and O’Neil (1990); Kiessling (1995);
Yatsuyanagi  _et al._ (2005); Esler and Ashbee (2015); Qi and Marston (2014);
Siggia and Aref (1981); Reeves  _et al._ (2012); Billam  _et al._ (2015);
Simula  _et al._ (2014); Gurarie and Chow (2004); Kiessling and Wang (2012);
Yu  _et al._ (2016); Salman and Maestrini (2016); Valani  _et al._ (2018); Han
and Tsubota (2018, 2019); Skaugen and Angheluta (2016, 2017); Kanai and Guo
(2021); Patil and Dunkel (2021); Pandit  _et al._ (2017); Chavanis and
Sommeria (1996); Chavanis (2002); Xiong  _et al._ (2023). The clustering of
vortices is also closely related to the end state of inverse energy cascades
Billam  _et al._ (2014); Skaugen and Angheluta (2016); Reeves  _et al._
(2013), which involves energy transport from small to large scales in two-
dimensional (2D) turbulence.

Previous studies on Onsager vortices mainly focus on systems confined on a
flat region. Recent experimental realizations of BECs Aveline  _et al._ (2020)
and ultracold atomic bubbles Carollo  _et al._ (2022) in International Space
Station open a possibility to investigate a bubble trapped superfluid and
vortex physics on curved surfaces experimentally. Curvature and topology of a
surface have non-trivial effects on BECs Tononi and Salasnich (2019); Tononi
_et al._ (2020); Tononi and Salasnich (2023), vortex dynamics Bogomolov
(1978); CROWDY (2004); Hally (1980); Polvani and Dritschel (1993); Kimura
(1999); Newton (2001); Alobaidi  _et al._ (2006); Aref (2007); Vitelli and
Turner (2004); Turner  _et al._ (2010); Dritschel and Boatto (2015); Dritschel
_et al._ (2015); Doshi and Gromov (2021); Bereta  _et al._ (2021); Caracanhas
_et al._ (2022); Padavić  _et al._ (2020); Saito and Hayashi (2023); Li and
Efimkin (2023); Xiong and Yu (2024), and the Berezinskii-Kosterlitz-Thouless
(BKT) transition Tononi  _et al._ (2022). The formation and structure of
Onsager vortices also strongly depend on the surface geometry. Numerical
simulations of the Gross-Pitaevskii equation show that, different from the
giant dipole configuration for vortices confined to a discs Simula  _et al._
(2014); Yu  _et al._ (2016), in a spherical shell trap uniformly distributed
neutral vortices form a quadruple cluster through the evaporative heating
process Kanai and Guo (2021). However, it is not clear whether the quadruple
clustered state is the statistically most favored state for given energy,
angular momentum and vortex number.

In this paper, we systematically investigate rotation-free clustered states of
a neutral vortex system confined on a sphere. We extend the mean field theory
of vortex statistical mechanics developed by Joyce and Montgomery Joyce and
Montgomery (1973) to a closed surface. In particular, we analyze the self-
consistent equation of the stream function on a sphere near the uniform state
and in the high energy limit. We find that, under constraint of zero angular
momentum, the sandwich state where vortices with opposite sign are distributed
around the poles and the equator respectively, is the maximum entropy state
and has higher statistical weight than the quarupole state. This theoretical
prediction is conformed by Monte Carlo simulations.

##  II Point vortices on a closed surface

For a superfluid on a plane, the circulation of a quanutm vortex is quantized
in units of circulation quantum $\kappa\equiv 2\pi\hbar/m$ Pitaevskii and
Stringari (2016), and the vorticity has a singularity at the vortex core
$\mathbf{r}_{i}$:
$\omega(\mathbf{r})=\nabla\times\mathbf{u}=\kappa_{i}\delta(\mathbf{r}-\mathbf{r}_{i})=\kappa\sigma_{i}\delta(\mathbf{r}-\mathbf{r}_{i})$
with sign $\sigma_{i}=\pm 1$ for singly charged vortices. Here $m$ is the
atomic mass and $\mathbf{u}$ is the superfluid velocity. The quantization
ensures that the vorticity of a quantum vortex concentrates around the core
region Eyink and Sreenivasan (2006). Hence when the mean separation between
quantum vortices $\ell$ is much larger than the vortex core size $\xi$ Fetter
(1998); Barenghi and Parker (2016), the point vortex model (PVM) describes
well the dynamics of quantum vortices in the incompressible limit Onsager
(1949); Eyink and Sreenivasan (2006); Novikov (1975); Aref  _et al._ (1999),
provided vortex annihilation can be neglected. The PVM also describes 2D
superfluid transition at finite temperature, which is the celebrated BKT
transition Kosterlitz and Thouless (1973); Kosterlitz (1974).

On curved surfaces, a covariant PVM has been well formulated Hally (1980);
Kimura (1999); Dritschel and Boatto (2015). Here we summarize the main results
of the PVM on a closed surface. In isothermal coordinates, the metric on the
surface is

$\displaystyle
ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=h(x^{1},x^{2})[(dx^{1})^{2}+(dx^{2})^{2}],$
(1)

where $g_{12}=g_{21}=0$, $g_{11}=g_{22}=h(x^{1},x^{2})$, and $h(x^{1},x^{2})$
is a positive function and exists locally for 2D surfaces Chern (1955). For
incompressible fluids, the component fluid velocity
$u_{a}=\epsilon_{a}^{\,\,\,b}\nabla_{b}\psi$, where $\psi$ is the stream
function, $\nabla_{b}$ is the Levi-Civita covariant derivative,
$\epsilon^{\beta}_{\,\,\,\,\alpha}=g^{\beta\gamma}\epsilon_{\gamma\alpha}$,
$\epsilon_{\gamma\alpha}=\sqrt{\det
g_{\mu\nu}}\,\tilde{\epsilon}_{\gamma\alpha}$ is the Levi-Civita tensor and
$\tilde{\epsilon}_{\gamma\alpha}$ is the Levi-Civita symbol. The vorticity
$\omega=\epsilon^{\mu}_{\,\,\,\,\nu}\nabla_{\mu}u^{\nu}=-\Delta\psi$, where
$\Delta\equiv\nabla^{\mu}\nabla_{\mu}=(1/\sqrt{g})\partial_{i}\left(\sqrt{g}g^{ij}\partial_{j}\right)=h^{-1}(x^{1},x^{2})(\partial^{2}_{x^{1}}+\partial^{2}_{x^{2}})$
is the Laplace-Bertrami operator and $g=\det g_{ij}$.

Let us also introduce complex coordinates $z=x^{1}+ix^{2}$,
$\partial_{z}\equiv(\partial_{1}-i\partial_{2})/2$,
$\partial_{\bar{z}}\equiv(\partial_{1}+i\partial_{2})/2$ and complex velocity
$u\equiv u^{1}-iu^{2}$. In complex notations, $u=2ih^{-1}\partial_{z}\psi$ and
the stream function

$\displaystyle\psi(z)=\kappa\sum_{i}\sigma_{i}G(z,z_{i}),$ (2)

where $z_{i}$ is the position of the $i$th vortex, $G(z,z_{i})$ is the Green’s
function satisfying Dritschel and Boatto (2015)

$\displaystyle\Delta G(z,z_{i})=-\delta_{z,z_{i}}+1/\Omega,$ (3)

$\Omega$ is the area of the surface, and $\delta_{z,z_{k}}\equiv
h^{-1}\delta(z-z_{k})$.

The Hamiltonian of point vortices reads

$\displaystyle H=-\frac{\rho_{s}}{2}\sum_{i\neq
j}\kappa_{i}\kappa_{j}G(z_{i},z_{j})-\frac{\rho_{s}}{2}\sum_{i}\kappa^{2}_{i}R_{\rm
robin}(z_{i}),$ (4)

where $\rho_{s}$ is the superfluid (mass) density,

$\displaystyle R_{\rm robin}(z)=\lim_{w\rightarrow
z}\left[G(z,w)+\frac{1}{2\pi}\log d(z,w)\right]$ (5)

is the celebrated Robin function and $d(z,w)$ is the geodesic distance between
points $w$ and $z$. When $w\rightarrow z$,
$d(z,w)=ds+o(|w-z|)=h|w-z|+o(|w-z|)$ and only the leading order term
contributes the Robin function.

The kinetic energy of the fluid is

$\displaystyle E=\frac{\rho_{s}}{2}\int
d\Omega\,|u|^{2}=\frac{\rho_{s}}{2}\int
d\Omega\,\omega\,\psi=H+H_{\text{singular}},$ (6)

where $H_{\text{singular}}=(\rho_{s}/4\pi)\sum_{i}\kappa^{2}_{i}\log
d(z_{i},z_{i})$. For the point vortex model, $H_{\text{singular}}=\infty$. In
practice, this term is regularized by the vortex core size $\xi$ which serves
the ultraviolet cut-off of the PVM, and
$H_{\text{singular}}=(\rho_{s}/4\pi)\sum_{i}\kappa^{2}_{i}\log\xi$. Hereafter,
we set $\rho_{s}=\kappa=1$ for convenience.

##  III Statistical mechanics of point vortices on a closed surface

On a closed surface, the total vorticity must vanish, namely,
$\sum_{i}\kappa_{i}=0$. We consider a system consisting of $N_{+}$ positive
vortices and $N_{-}=N_{+}$ negative vortices. The total vortex number is
$N=N_{+}+N_{-}$. The model Eq. (4) exhibits Onsager clustered states at high
energies in a bounded domain. In clustered phases, the energy $H\sim N^{2}$,
hence in order to describe clustered phases in the proper thermodynamics
limit, we need to make the rescaling $\sigma_{i}\rightarrow\sigma_{i}/N_{\pm}$
and define the vortex number density and vortex charge density (vorticity) on
a curved surface as following:

$\displaystyle n_{\pm}(x^{\mu})$ $\displaystyle=\frac{1}{N_{\pm}\sqrt{\det
g_{\mu\nu}}}\sum^{N_{\pm}}_{i}\delta(x^{\mu}-x^{\mu}_{i}),$ (7)
$\displaystyle\sigma(x^{\mu})$ $\displaystyle=\frac{1}{N_{\pm}\sqrt{\det
g_{\mu\nu}}}\sum_{i}\sigma_{i}\delta(x^{\mu}-x^{\mu}_{i}).$ (8)

Here $n_{\pm}$ satisfy the normalization condition $\int{\rm
d}\Omega\,n_{\pm}=1$ and ${\sigma}(x^{\mu})=n_{+}(x^{\mu})-n_{-}(x^{\mu})$. In
the following we consider vortex distributions at large scales and treat
$\sigma$, $n_{\pm}$ and $\psi$ as smooth functions at scales much larger than
the mean seperation $\ell$.

For given vortex distributions $n_{\pm}$, the von Neumann entropy reads Joyce
and Montgomery (1973)

$\displaystyle S=-\int{\rm d}\Omega\left(n_{+}\log n_{+}+n_{-}\log
n_{-}\right),$ (9)

and the energy

$\displaystyle E=\frac{1}{2}\int{\rm d}\Omega\
|\nabla\psi(\textbf{r})|^{2}=\frac{1}{2}\int{\rm d}\Omega\
\psi(\textbf{r}){\sigma}(\textbf{r}),$ (10)

where we have used the relation

$\displaystyle\nabla^{2}\psi(\mathbf{r})$
$\displaystyle=-{\sigma}(\mathbf{r}).$ (11)

The most probable vortex density distribution is given by maximizing the
entropy Eq. (9) subject to given energy $E$ and vortex number $N_{\pm}$:

$\displaystyle\delta S-\beta\delta E-\mu_{+}\delta N_{+}/N_{+}-\mu_{-}\delta
N_{-}/N_{-}=0,$ (12)

where $\beta,\mu_{\pm}$ are Lagrange multipliers and have the interpretation
of inverse temperature and chemical potentials in microcanonical ensemble,
respectively. The variation equation (12) gives rise

$\displaystyle n_{\pm}$ $\displaystyle=\exp[\mp\beta\psi(r)+\gamma_{\pm}],$
(13)

where$\gamma_{\pm}=-\mu_{\pm}-1$. Combining Eq. (11) we obtain the self-
consistent equation for determining the structure of coherent Onsager vortex
clusters:

$\displaystyle\nabla^{2}\psi(\mathbf{r})=\exp[\beta\psi(r)+\gamma_{-}]-\exp[-\beta\psi(r)+\gamma_{+}].$
(14)

##  IV Onsager vortex clusters on a sphere

In this work, we focus on Onsager vortex clusters on a sphere.

###  IV.1 Stereographic coordinates

The Riemannian metric on a sphere is

$\displaystyle ds^{2}=R^{2}d\theta^{2}+R^{2}\sin^{2}\theta d\phi^{2},$ (15)

where $R$ is the radius, $\theta$ is the polar angle and $\phi$ is the
azimuthal angle. In spherical coordinates, the fluid velocity
$\textbf{u}=\nabla\psi\times\textbf{e}_{r}$, where $\textbf{e}_{r}$ is the
radial unit vector.

The Cartesian coordinates relate to the spherical coordinates via

$\displaystyle\xi=R\sin\theta\cos\phi,\quad\eta=R\sin\theta\sin\phi,\quad\zeta=R\cos\theta.$
(16)

The projection from the north pole to the $\zeta=0$ plane induces
stereographic coordinates $z=x^{1}+ix^{2}$, where $x^{1}$ and $x^{2}$ axes are
identified as the $\xi$ and $\eta$ axes, respectively (Fig. 1). For a sphere,
stereographic coordinates are isothermal coordinates and are related to the
spherical coordinates by

$\displaystyle z=R\cot\left(\frac{\theta}{2}\right)e^{i\phi}.$ (17)

In terms of stereographic coordinates, the metric reads

$\displaystyle h=\frac{4R^{4}}{(R^{2}+|z|^{2})^{2}}.$ (18)

It is sometimes convenient to introduce polar coordinates $z=\rho e^{i\phi}$,
where $\rho=\cot(\theta/2)$ and the polar angle $\phi$ is chosen to coincide
with the azimuthal angle. The relations between the three coordinate systems
are

$\displaystyle\xi$ $\displaystyle=$
$\displaystyle\frac{2x^{1}R^{2}}{R^{2}+|z|^{2}}=\sqrt{h}x^{1}=R\sin\theta\cos\phi,$
(19) $\displaystyle\eta$ $\displaystyle=$
$\displaystyle\frac{2x^{2}R^{2}}{R^{2}+|z|^{2}}=\sqrt{h}x^{2}=R\sin\theta\sin\phi,$
(20) $\displaystyle\zeta$ $\displaystyle=$ $\displaystyle
R\frac{R^{2}-|z|^{2}}{R^{2}+|z|^{2}}=R\cos\theta,$ (21)

and we use Cartesian, spherical or stereographic coordinates wherever
convenient.

Figure 1: Stereographic projection from the north pole. The point ${\rm
P}(\theta,\phi)$ on the southern hemisphere is projected onto the $\zeta=0$
plane and is denoted as ${\rm P}^{\prime}$.

In stereographic coordinates, the Green’s function reads Kimura (1999)

$\displaystyle G(z,z_{i})$
$\displaystyle=-\frac{1}{2\pi}\log\frac{|z-z_{i}|}{\sqrt{|z-z_{i}|^{2}+|z\bar{z}_{i}+R^{2}|}}$
(22)

and $R_{\rm robin}=(1/2\pi)\log 2$. In spherical coordinates,

$\displaystyle G(\theta,\phi;\theta_{i},\phi_{i})$
$\displaystyle=-\frac{1}{2\pi}\log\sqrt{2R^{2}-2\mathbf{r}\cdot\mathbf{r}_{i}}$
(23)
$\displaystyle=-\frac{1}{2\pi}\log\left(\sqrt{2}R\sqrt{1-\cos\Theta}\right)$
$\displaystyle=-\frac{1}{2\pi}\log\left(2R\sin\frac{\Theta}{2}\right)$

where $\Theta$ is the angle between $\mathbf{r}$ and $\mathbf{r}_{i}$, and
$\cos\Theta=\cos\theta\cos\theta_{i}+\sin\theta\sin\theta_{i}\cos(\phi-\phi_{i})$.
It is interesting to notice that
$\sqrt{2R^{2}-2\mathbf{r}\cdot\mathbf{r}_{i}}$ is the Euclidean distance, but
not the geodesic distance, between two points on the sphere.

###  IV.2 Onset of clustering

The onset of clustering involves global eigen-modes of the Laplace-Bertrami
operator. It is then convenient to use spherical coordinates. Hereafter we set
$R=1$ for convenience. On a sphere, besides the energy and the number of
vortices, the angular momentum $\mathbf{L}=\int\mathbf{r}\times\mathbf{u}$ is
also a conserved quantity. In terms of collective variables of vortices, the
components of the angular momentum read

$\displaystyle L_{\xi}$ $\displaystyle=\int{\rm
d}\Omega\,\sigma\sin\theta\cos\phi,$ (24) $\displaystyle L_{\eta}$
$\displaystyle=\int{\rm d}\Omega\,\sigma\sin\theta\sin\phi,$ (25)
$\displaystyle L_{\zeta}$ $\displaystyle=\int{\rm d}\Omega\,\sigma\cos\theta.$
(26)

Hereafter we choose the $\zeta$-axis in the direction of total angular
momentum and hence $\mathbf{L}=L_{\zeta}$.

Taking into account the conversation of the angular momentum, Eq.(12) becomes

$\displaystyle\delta S-\beta\delta E-\alpha\delta L_{\zeta}-\mu_{+}\delta
N_{+}/N_{+}-\mu_{-}\delta N_{-}/N_{-}=0,$ (27)

where $\alpha$ is the Lagrange multiplier associated with the angular momentum
and $\omega\equiv\alpha/\beta$ has the meaning of rotation frequency.
Accordingly, the vortex number density turns to be

$\displaystyle
n_{\pm}=\exp\left[\mp\beta\psi(\textbf{r})\mp\alpha\cos\theta+\gamma_{\pm}\right].$
(28)

Non-uniform solutions at negative temperatures to Eq. (28) describe Onsager
vortex states.

Let us consider a solution to Eq. (28) for the given values of $E$ and
$L_{\zeta}$, and a nearby solution $n_{\pm}+\delta n_{\pm}$ at $E+\delta E$
and $L_{\zeta}+\delta L_{\zeta}$. The corresponding changes in the constraints
are

$\displaystyle 0$ $\displaystyle=\int{\rm d}\Omega\ \delta n_{\pm},$ (29)
$\displaystyle\delta E$ $\displaystyle=\int{\rm d}\Omega\
\psi\delta\sigma+\frac{1}{2}\int{\rm d}\Omega\,\delta{\sigma}\delta\psi,$ (30)
$\displaystyle\delta L_{\zeta}$ $\displaystyle=\int{\rm d}\Omega\
\delta\sigma\cos\theta\,.$ (31)

Combining the variations of Eqs. (11) and (28) in leading order, we obtain

$\displaystyle\nabla^{2}\delta\psi$
$\displaystyle=-\delta{\sigma}(\textbf{r})$
$\displaystyle=n\psi\delta\beta+n\beta\delta\psi-
n_{+}\delta\gamma_{+}+n_{-}\delta\gamma_{-}+n\cos\theta\delta\alpha,$

where $n=n_{-}+n_{+}$ is the total density of vortices, and

$\displaystyle\nabla^{2}=\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\phi^{2}}$
(33)

is the Laplace-Bertrami operator in spherical coordinates.

We consider vortex clustered states near the homogeneous state
$n_{\pm}=n_{0}=1/4\pi$. For the homogeneous state $\alpha=0$ and
$\sigma=\psi=0$, then Eqs. (29),(30) and (31) become

$\displaystyle 0=$ $\displaystyle\delta\gamma_{+}+\delta\gamma_{-},$ (34)
$\displaystyle 0=$ $\displaystyle\delta\gamma_{+}-\delta\gamma_{-}-2\beta
n_{0}\int{\rm d}\Omega\,\delta\psi,$ (35) $\displaystyle\delta E=$
$\displaystyle\frac{1}{2}\int{\rm d}\Omega\,\delta{\sigma}\,\delta\psi,$ (36)
$\displaystyle\delta L_{\zeta}=$ $\displaystyle-2n_{0}\beta\int{\rm
d}\Omega\cos\theta\,\delta\psi-\frac{2}{3}\delta\alpha,$ (37)

where $\delta\alpha$ is assumed to be the same order as $\delta\psi$. Let us
introduce operator $\mathcal{P}$:

$\displaystyle\mathcal{P}\delta\psi$
$\displaystyle\equiv\nabla^{2}\delta\psi-2n_{0}\bigg{[}\beta\delta\psi-
n_{0}\beta\int d\Omega\,\delta\psi$ (38)
$\displaystyle-3n_{0}\beta\cos\theta\int
d\Omega\cos\theta\delta\psi-\frac{3}{2}\cos\theta\delta L_{\zeta}\bigg{]}.$

Then Eq. (IV.2) becomes a zero mode equation of the operator $\mathcal{P}$,
i.e.,

$\displaystyle\mathcal{P}\delta\psi=0.$ (39)

In order to solve the zero mode equation, we consider an expansion

$\displaystyle\delta\psi$
$\displaystyle=\sum^{\infty}_{\ell=0}\sum^{\ell}_{m=-\ell}\epsilon f_{\ell
m}\psi_{\ell m}(\theta,\phi),$ (40)

where $\psi_{\ell m}$ satisfies $\mathcal{P}\psi_{\ell m}=0$, $f_{\ell m}$ is
the mode coefficient, and $\epsilon\ll 1$ is a small amplitude. We denote
$\delta L=\epsilon L_{0}$, $\delta\alpha=\epsilon\beta\omega$, and $\delta
E=\epsilon^{2}E_{0}$.

We find that

$\displaystyle\psi_{\ell m}(\theta,\phi)=c_{\ell m}{\rm Re}({\rm Y}_{\ell
m})+b\cos\theta$ (41)

solves Eq. (39), if

$\displaystyle\beta=-\frac{\ell(\ell+1)}{2n_{0}},$ (42)

and

$\displaystyle b=-\frac{n_{0}\beta\omega}{1+n_{0}\beta}$ (43)

for $\ell\neq 1$. While for $\ell=1$, $b$ can be arbitrarily and we choose
that $b=0$ without losing generality. Here ${\rm Y}_{\ell m}$ is the spherical
harmonic function. For weekly clustered states, the vorticity field
$\sigma=\sigma_{\ell m}\equiv\delta\sigma=\epsilon\ell(\ell+1)\psi_{\ell m}$.

Given the values of $E_{0}$ and $L_{0}$, $c_{\ell m}$ and $\omega$ are
determined by

$\displaystyle E_{0}$ $\displaystyle=-\frac{1}{2}\int{\rm d}\Omega\,\psi_{\ell
m}(\theta,\phi)\nabla^{2}\psi_{\ell m}(\theta,\phi),$ (44) $\displaystyle
L_{0}$ $\displaystyle=-2n_{0}\beta\int{\rm d}\Omega\cos\theta\ \psi_{\ell
m}-\frac{2}{3}\beta\omega.$ (45)

For $\ell=m=0$, $E_{0}=0$ and $L_{0}=0$, hence only modes with $\ell>0$ are
relevant. In the following we focus on rotation free states, namely
$\omega=0$, and in this case $\psi_{\ell m}=c_{\ell m}\rm{Re}({\rm Y}_{\ell
m})$.

####  IV.2.1 Sandwich state: clusters with zero angular momentum

In this subsection, we consider clustered states with zero angular momentum
$L_{0}=0$. Clearly modes with $\ell=2$ have higher statistical weight (larger
entropy) than the modes with $\ell>2$.

For $m=0$,

$\displaystyle\psi_{20}=c_{20}{\rm
Y}_{20}=\frac{c_{20}}{4}\sqrt{\frac{5}{\pi}}(3\cos^{2}\theta-1),$ (46)

where $c_{20}=\sqrt{E_{0}/3}$. This mode describes the clustering of positive
(negative) vortices around the poles and negative (positive) vortices around
the equator [Fig.2 (a)]. We refer to this state as sandwich state. The onset
of the sandwich state occurs at $\beta=\beta_{c}=-12\pi$.

For $m=1,2$,

$\displaystyle\psi_{22}(\theta,\phi)$ $\displaystyle=c_{22}{\rm Re}({\rm
Y}_{22})=\frac{c_{22}}{4}\sqrt{\frac{15}{2\pi}}\cos(2\phi)\sin^{2}\theta,$
(47) $\displaystyle\psi_{21}(\theta,\phi)$ $\displaystyle=c_{21}{\rm Re}({\rm
Y}_{21})=-\frac{c_{21}}{4}\sqrt{\frac{15}{2\pi}}\cos\phi\sin 2\theta,$ (48)

where $c_{22}=c_{21}=\sqrt{2E_{0}/3}$. It happens that these two modes are
connected by a $\textrm{SO}(3)$ rotation, i.e., ${\rm
Y_{21}}(\mathbf{r}^{\prime})={\rm Y_{22}}(\mathbf{r})$, where
$\mathbf{r}^{\prime}=R(-\pi/4,\pi/2,\pi/2)\,\mathbf{r}$ and
$R(\theta_{1},\theta_{2},\theta_{3})=R_{\zeta}(\theta_{3})R_{\eta}(\theta_{2})R_{\zeta}(\theta_{1})$
is the $\textrm{SO}(3)$ rotation operator and $\\{\theta_{i=1,2,3}\\}$ are the
Euler angles. Therefore, modes $\psi_{22}$ and $\psi_{21}$ are degenerate and
describe the same state. We refer to this state as quadrupole state
[Fig.2(b)].

Figure 2: Vorticity $\sigma$ for the sandwich (a), the quadrupole (b) and the
dipole (c) state. The extreme values of $\sigma$ are scaled to unity for
convenience.

For states with $\ell=2$, the dipole moment defined as

$\displaystyle\mathbf{D}\equiv\int d\Omega\ \sigma(\mathbf{r})\,\mathbf{r}$
(49)

vanishes. In order to quantify the clustered states with zero angular
momentum, it is necessary to introduce the quadrupole tensor

$\displaystyle\mathcal{Q}_{ij}\left[\sigma(\textbf{r})\right]\equiv\frac{1}{2}\int
d\Omega\,\Big{[}3(\textbf{r}\cdot\textbf{e}_{i})(\textbf{r}\cdot\textbf{e}_{j})-\delta_{ij}\Big{]}{\sigma}(\textbf{r}),$
(50)

where $\\{\textbf{e}_{i=\xi,\eta,\zeta}\\}$ are unit vectors in Cartesian
coordinates. The quadrupole moment is defined as

$\displaystyle Q[\sigma]\equiv\sqrt{q^{2}_{1}+q^{2}_{2}+q^{2}_{3}},$ (51)

where $\\{q_{1},q_{2},q_{3}\\}$ are eigenvalues of $\mathcal{Q}_{ij}$. We
denote $Q_{s}$ and $Q_{q}$ as the quadrupole moment for the sandwich and the
quadrupole state, respectively.

For the sandwich state, $\ell=2,m=0$,

$\displaystyle\mathcal{Q}[\sigma_{20}]=\epsilon
c_{20}\sqrt{\frac{36\pi}{5}}\left(\begin{array}[]{ccc}-1&0&0\\\ 0&-1&0\\\
0&0&2\\\ \end{array}\right)$ (55)

and its eigenvalues are $\epsilon c_{20}\sqrt{36\pi/5}\,\\{-1,-1,2\\}$. The
quadrupole momentum for the sandwich state is

$\displaystyle Q_{s}=Q[\sigma_{20}]=Q_{0}|E-E_{c}|^{\nu},$ (56)

where $\nu=1/2$, $Q_{0}=\sqrt{216\pi/15}$ and $E_{c}=0$. For quadrupole
states, the quadrupole tensors are

$\displaystyle\mathcal{Q}[\sigma_{21}]$
$\displaystyle=-\epsilon{c_{21}}\sqrt{\frac{54\pi}{5}}\left(\begin{array}[]{ccc}0&0&1\\\
0&0&0\\\ 1&0&0\\\ \end{array}\right){,}\ $ (60)

and

$\displaystyle\mathcal{Q}[\sigma_{22}]$
$\displaystyle=\epsilon{c_{22}}\sqrt{\frac{54\pi}{5}}\left(\begin{array}[]{ccc}1&0&0\\\
0&-1&0\\\ 0&0&0\\\ \end{array}\right).$ (64)

Since $\sigma_{21}(R\mathbf{r})=\sigma_{22}(\mathbf{r})$, it is easy to see
that $\mathcal{Q}[\sigma_{21}]=R\mathcal{Q}[\sigma_{22}]R^{T}$, where
$R=R(-\pi/4,\pi/2,\pi/2)$. Their common eigenvalues are
$-{c_{21}}\epsilon\sqrt{54\pi/5}\,\\{-1,0,1\\}$. Moreover the quadrupole
momentum for the quadrupole state takes the same value as for the sandwich
state, namely, $Q_{q}=Q[\sigma_{21}]=Q[\sigma_{22}]=Q[\sigma_{20}]=Q_{s}$.

Using Eq. (9), we find that the entropy of the sandwich state is larger than
the quadrupole state,

$\displaystyle S[\sigma_{20}]>S[\sigma_{21}]=S[\sigma_{22}].$ (65)

Hence under condition of zero angular momentum $L_{0}=0$, _the sandwich state
has the highest statistical weight._ Although this conclusion is made based on
the perturbative analysis at low energies, it will be confirmed by later
analysis at high energies and Monte Carlo simulations at moderate energies.

####  IV.2.2 dipole states: vortex clustered states with finite angular
momentum

We now consider states with finite angular momentum $L_{0}\neq 0$. For
$\ell=1,m=0$, the corresponding stream function is

$\displaystyle\psi_{10}=c_{10}{\rm
Y}_{10}=c_{10}\sqrt{\frac{3}{4\pi}}\cos\theta,$ (66)

where $c_{10}=\sqrt{E_{0}}$.

The dipole moment for this state does not vanish and its magnitude reads

$\displaystyle D=|\mathbf{D}|=4\sqrt{\frac{\pi}{3}}\sqrt{E}.$ (67)

We refer this state as dipole state [Fig. 2 (c)] and the onset of the dipole
state occurs at $\beta=\beta^{d}_{c}=-4\pi$. For the dipole state, the angular
momentum is $L_{\zeta}=4\sqrt{\pi/3}\sqrt{E}$. Note that the vorticity
distribution for the dipole state at low energies is identical to the
vorticity distribution for a stationary vortex flow on a sphere Xiong and Yu
(2024).

###  IV.3 High energy configurations

In this subsection, we consider the sandwich state and the dipole state at
high energies. In the high energy limit, vortices and antivortices are well
separated and are concentrated in small regions of a sphere. It is then
convenient to use the stereographic projection method to find approximate
density distributions of vortices on a sphere.

In stereographic coordinates Eq. (14) becomes

$\displaystyle\
(1+z\bar{z})^{2}\partial_{z}\partial_{\bar{z}}\psi=e^{\beta\psi(z,\bar{z})+\gamma_{-}}-e^{-\beta\psi(z,\bar{z})+\gamma_{+}}$
(68)

for rotation free states ($\omega=0$). The stereographic projection maps the
southern hemisphere onto a disc on the $\zeta=0$ plane (Fig.1) and transforms
vortices in the region around southern pole and vortices in the region around
the equator to the region around the origin and the region near the boundary
of the disc, respectively.

####  IV.3.1 Sandwich states at high energies

In the high energy limit, the interaction between vortices near the poles and
antivortices near the equator can be neglected. Let us firstly consider the
region $|z|\ll 1$, in which Eq. (68) can be well approximated as

$\displaystyle\partial_{z}\partial_{\bar{z}}\psi=-e^{-\beta\psi(z,\bar{z})+\gamma_{+}}.$
(69)

For axisymmetric distributions, Eq. (69) becomes

$\displaystyle\frac{1}{4}\frac{1}{\rho}\frac{{\rm d}}{{\rm
d}\rho}\left(\rho\frac{{\rm d}}{{\rm
d}\rho}\psi\right)=-e^{-\beta\psi(\rho)+\gamma_{+}},$ (70)

and has the following exact solution

$\displaystyle\psi(\rho)$
$\displaystyle=-\frac{2}{\beta}\log\left(\frac{2}{\beta\rho^{2}e^{\gamma_{+}}-2}\right)$
(71)

with boundary condition $\psi(\rho=0)=\psi^{\prime}(\rho)|_{\rho=0}=0$ Xiong
_et al._ (2023). The vortex density reads

$\displaystyle
n_{+}(\rho)=\exp(-\beta\psi+\gamma_{+})=\frac{4e^{\gamma_{+}}}{(\beta\rho^{2}e^{\gamma_{+}}-2)^{2}}.$
(72)

For the sandwich state, since only half positive vortices are distributed
around the south pole (the other half positive vortices are distributed around
the north pole), the normalization condition is

$\displaystyle\frac{1}{2}=\int^{2\pi}_{0}{\rm
d}\phi\int_{0}^{1}\frac{4\rho}{(1+\rho^{2})^{2}}\,d\rho\,n_{+}(\rho),$ (73)

which gives rise the relation between $e^{\gamma_{+}}$ and $\beta$ [Fig. 10(a)
in Appendix]. When $\beta\rightarrow\beta_{s}=-16\pi$, vortices around the
pole collapse into a point, known as the supercondensation. Hence the
parameter range for the sandwich state is $\beta_{s}<\beta<\beta_{c}$. The
limit configuration at $\beta=\beta_{s}$ is
$n_{+}(\theta)=[\delta(\theta)+\delta(\theta-\pi)]/(4\pi\sin\theta)$. It is
worthwhile mentioning that the value of $\beta_{s}$ depends on the
normalization condition. For vortices confined to a disc on a plane, the
vortex density profile is the same as Eq. (72), however the vortex number is
normalized to unity, which gives rise that $\beta_{s}=-8\pi$ (in convention of
this paper) Smith and O’Neil (1990); Yu  _et al._ (2016); Kiessling (1995);
Patil and Dunkel (2021); Xiong  _et al._ (2023).

In spherical coordinates, the stream function and the vortices density on the
southern hemispheres [$\theta\in(\pi/2,\pi)$] are [see also Fig. 3(a)]

$\displaystyle\psi(\theta)$
$\displaystyle=-\frac{2}{\beta}\log\left[\frac{2}{\beta\cot^{2}\left(\frac{\theta}{2}\right)e^{\gamma_{+}}-2}\right],$
(74) $\displaystyle n_{+}(\theta)$
$\displaystyle=\frac{4e^{\gamma_{+}}}{\left[\beta\cot^{2}\left(\frac{\theta}{2}\right)e^{\gamma_{+}}-2\right]^{2}}.$
(75)

Figure 3: The vortex density distributions for the sandwich state at high
energies. (a) The distribution of positive vortices near poles [Eq. (75)] .
(b) The distribution of negative vortices near the equator at
$\beta=\beta_{s}$ [Eq. (82)].

Now we consider the distribution of negative vortices in the region near the
equator which is mapped to the region near the edge of the disc. For $|z|\sim
1$, Eq. (68) can be well approximated as

$\displaystyle 4\partial_{z}\partial_{\bar{z}}\psi$
$\displaystyle=e^{\beta\psi(z,\bar{z})+\gamma_{-}}.$ (76)

For axis-symmetric solutions, Eq. (76) becomes

$\displaystyle\frac{1}{\rho}\frac{{\rm d}}{{\rm d}\rho}\left(\rho\frac{{\rm
d}}{{\rm d}\rho}\psi\right)$ $\displaystyle=e^{\beta\psi(\rho)+\gamma_{-}},$
(77)

and admits an exact solution

$\displaystyle\psi(\rho)=\frac{2}{\beta}\log\left(\frac{2A\rho^{(-1+A/2)}}{2+A+(A-2)\rho^{A}}\right),$
(78)

with the boundary condition Xiong  _et al._ (2023)

$\displaystyle\psi(\rho=1)=\psi^{\prime}(\rho)|_{\rho=1}=0.$ (79)

Here $A=\pm\sqrt{4-2\beta e^{\gamma_{-}}}$. As it follows, the density is

$\displaystyle
n_{-}(\rho)=\frac{4A^{2}\rho^{(A-2)}e^{\gamma_{-}}}{\left[2+A+(A-2)\rho^{A}\right]^{2}}.$
(80)

The relation between $e^{\gamma_{-}}$ and $\beta$ is given by the
normalization condition Eq. (73) ($n_{+}\rightarrow n_{-}$). Moreover, the
smooth condition $n^{\prime}_{-}(\rho)|_{\rho=0}=0$ requires
$e^{\gamma_{-}}\beta<-5/2$ which gives rise to that $\beta<-7.11\pi$ [Fig. 10
(b) in Appendix]. Hence for $\beta$ within the range
$\beta_{s}<\beta<\beta_{c}$, the smooth condition is automatically satisfied.

In spherical coordinates, the stream function and the vortex density
distribution of negative vortices near the equator [$\theta\in(\pi/2,\pi)$]
read

$\displaystyle\psi(\theta)$
$\displaystyle=\frac{2}{\beta}\log\left[\frac{2A\cot(\theta/2)^{(-1+A/2)}}{2+A+(A-2)\cot(\theta/2)^{A}}\right],$
(81) $\displaystyle n_{-}(\theta)$
$\displaystyle=\frac{4A^{2}\cot^{(A-2)}(\theta/2)e^{\gamma_{-}}}{\left[2+A+(A-2)\cot^{A}(\theta/2)\right]^{2}}.$
(82)

Induced by the boundary condition Eq. (79), the fluid velocity
$u_{\phi}=-\partial\psi/\partial\theta$ on the equator vanishes, namely
$u_{\phi}(\theta=\pi/2)=\psi^{\prime}(\rho)\rho^{\prime}(\theta)|_{\theta=\pi/2}=0$,
which is consistent with the fluid velocity field on the equator at low
energies [$\psi_{20}^{\prime}(\theta)|_{\theta=\pi/2}=0$]. At
$\beta=\beta_{s}$, $e^{\gamma_{-}}\approx 0.167,A\approx 4.556$ [Fig. 10(b) in
Appendix], the vorticity for the sandwich state reads
$\sigma_{s}(\textbf{r})=\Big{[}\delta(\theta)+\delta(\theta-\pi)\Big{]}/(4\pi\sin\theta)-n_{-}(\theta),$
and the quadrupole moment reaches the maximum value $Q^{\rm max}_{s}\approx
1.59$. It is easy to see that the quadrupole state also reaches the high
energy limit at $\beta=\beta_{s}$, and the corresponding vorticity is
${\sigma}_{q}(\textbf{r})=\Big{[}\delta(\theta-\pi/4,\phi)+\delta(\theta-3\pi/4,\phi-\pi)-\delta(\theta-3\pi/4,\phi)-\delta(\theta-\pi/4,\phi-\pi)\Big{]}/(2\sin\theta)$
which gives rise the maximum quadrupole moment $Q^{\rm
max}_{q}=3\sqrt{2}/2>Q^{\rm max}_{s}$. Note that for the quadrupole state
locations of the clusters at supercondensation coincide with mechanical
equilibrium positions for $4$ point vortices on a sphere.

####  IV.3.2 Dipole states at high energies

At high energies, for the dipole state, vortices with opposite sign are
concentrated near the poles. The density distribution of positive vortices
near the southern pole is identical to Eq. (72) and the only difference is
that here the vortex density is normalized to unity but not 1/2. Hence for the
dipole state, the supercondensation occurs at $\beta=\beta^{d}_{s}=-8\pi$
which is the same as what for vortices confined to a disc on a plane Smith and
O’Neil (1990); Yu  _et al._ (2016); Xiong  _et al._ (2023). In spherical
coordinates, vortex densities are [see also Fig. 4]

$\displaystyle n_{-}(\theta)$
$\displaystyle=\frac{4e^{\gamma_{-}}}{\Big{[}\beta\cot^{2}(\theta/2)e^{\gamma_{-}}-2\Big{]}^{2}}\quad\theta\in(\pi/2,\pi),$
(83) $\displaystyle n_{+}(\theta)$
$\displaystyle=\frac{4e^{\gamma_{+}}}{\Big{[}\beta\tan^{2}(\theta/2)e^{\gamma_{+}}-2\Big{]}^{2}}\quad\theta\in(0,\pi/2),$
(84)

and the corresponding angular momentum is

$\displaystyle L_{\zeta}(\beta)=$ $\displaystyle\frac{8\pi
e^{\gamma_{-}}}{(2+\beta e^{\gamma_{-}})^{4}}\Big{[}\beta
e^{\gamma_{-}}\Big{(}\beta e^{\gamma_{-}}(-2\beta+5+32\log 2)$ (85)
$\displaystyle+20-32\log 2\Big{)}-16\beta e^{\gamma_{-}}(\beta
e^{\gamma_{-}}-1)\log(2-\beta e^{\gamma_{-}})+4\Big{]}.$

Figure 4: The total vortex density for the dipole state at high energies [Eqs.
(83) and (84)].

When $\beta\rightarrow\beta^{d}_{s}$,
$n_{-}=\delta(\theta-\pi)/(2\pi\sin\theta),n_{+}=\delta(\theta)/(2\pi\sin\theta)$,
and the angular momentum reaches the upper bound
$L_{\zeta}=L^{\rm{max}}_{\zeta}=2$.

For the dipole state at low energies, the fluid velocity $u_{\theta}=0$,
$u_{\phi}\propto\sin\theta$ and $u_{\phi}$ vanishes at poles (Fig. 5). At high
energies, the velocity field is

$\displaystyle
u_{\phi}(\theta)=-\frac{\partial\psi}{\partial\theta}=-\frac{2\cot(\theta/2)\gamma_{-}}{[-2+\beta\gamma_{-}\cot^{2}(\theta/2)]\sin^{2}(\theta/2)},$
(86)

where we have used Eq. (74). Consistently, it also vanishes at poles. At low
energies, fluid velocity field $u_{\phi}$ reaches the peak value on the
equator ($\theta=\pi/2$). While at high energies, two peaks of the velocity
field start to develop and move towards to poles as increasing the energy
(Fig. 5). This is because for tightly clustered vortices around the poles, the
fluid velocity is considerably large on the edges of the vortex clusters which
locate between the poles and the equator.

Figure 5: The velocity field $u_{\phi}$ for the dipole state at a low energy
(orange solid line) and at a high energy (blue solid line). The parameter
region for the dipole state is
$-8\pi=\beta^{d}_{s}<\beta<\beta^{d}_{c}=-4\pi$. When $\beta\to\beta^{d}_{s}$,
$u_{\phi}(\theta=\pi/2)\rightarrow 1/2\pi$. Note that here the fluid velocity
field on the northern hemisphere is obtained by shifting the coordinate of Eq.
(86) ($\theta\rightarrow\pi-\theta$). The two expressions work well only near
the poles and take different values on the equator. However the difference
decreases as increasing energy and for the $\beta$ value we choose here, the
discrepancy is invisible.

The quadrupole state at high energies can be easily constructed from Eqs. (83)
and (84). From the approximate high energy expressions of vortex densities, we
can calculate the entropy and find again that $S[\text{sandwich
state}]>S[\text{quadrupole state}]$.

####  IV.3.3 General solutions

A general solution to Eq. (69) is available Crowdy (1997); Henrici (1993) :

$\displaystyle\psi(z,\bar{z})=$
$\displaystyle-\frac{1}{\beta}\log\left[\frac{2f^{\prime}(z)\bar{f}^{\prime}(\bar{z})}{-c\beta(1+f(z)\bar{f}(\bar{z}))^{2}}\right],$
(87) $\displaystyle n(z,\bar{z})=$ $\displaystyle c\exp(-\beta\psi),$ (88)

where $f(z)$ is a meromorphic function of $z$ in a simply connected domain and
$\bar{f}(\bar{z})$ is the complex conjugate of $f(z)$. Furthermore , it is
required that $f(z)$ has at most simple poles and $f^{\prime}(z)\neq 0$ in the
consider domain Henrici (1993); CROWDY (2004).

Here we show that Eqs. (71) and (78) fall into this general solution by
specifying function $f(z)$ properly. Let us firstly consider the vortex
distribution around the pole. For an axisymmetric distribution, the stream
function $\psi(z,\bar{z})$ must be the function of $z\bar{z}$ or $\rho$ only.
Hence $f(z)$ has a unique form and $f(z)=az^{b}$, here
$a\in\mathbb{C},b\in\mathbb{Z}$. The boundary condition
$\psi(\rho=0)=\psi^{\prime}(\rho)|_{\rho=0}=0$ determines that
$a=\sqrt{-c\beta/2}$, $b=1$, and $c=e^{\gamma_{+}}$, which leads to Eq. (71)
precisely.

For the distribution near the edge (or equator in spherical coordinates), the
general solution Eq. (87) gives rise to Eq. (78) by choosing $f(z)=\
\sqrt{(A-2)/(A+2)}\ z^{A/2}$, where $A=\pm\sqrt{4-2\beta e^{\gamma_{-}}}$, and
$c=e^{\gamma_{-}}/4$. Here the boundary condition
$\psi(\rho=1)=\psi^{\prime}(\rho)|_{\rho=1}=0$ is fulfilled. However it is
important to notice that the exponent $A/2$ is _not_ an integer in general.
Such a choice of $f(z)$ may not be a meromorphic function and it does not meet
the requirements for Eq. (87). As shown previously, such a solution is indeed
the solution to Eq. (77), suggesting that the validity condition of Eq. (88)
can be extended.

###  IV.4 Gaussian clustered states

In this section, we discuss a limit distribution when $\beta\rightarrow 0$,
$\omega\rightarrow\infty$ simultaneously while $\alpha=\beta\omega$ is finite.
In this limit, Eq. (28) admits an exact solution:

$\displaystyle n_{\pm}(\theta)=$
$\displaystyle\frac{\alpha}{4\pi\sinh\alpha}\exp(\mp\alpha\cos\theta),$ (89)
$\displaystyle\psi(\theta)=$
$\displaystyle-\frac{1}{8\pi\sinh\alpha}\Big{[}e^{\alpha}{\rm
Ei}\Big{(}\alpha(\cos\theta-1)\Big{)}-e^{\alpha}{\rm
Ei}\Big{(}-\alpha(\cos\theta+1)\Big{)}$ $\displaystyle+e^{-\alpha}{\rm
Ei}\Big{(}\alpha(1-\cos\theta)\Big{)}-e^{-\alpha}{\rm
Ei}\Big{(}\alpha(\cos\theta+1)\Big{)}$ $\displaystyle-2\alpha C\
\rm{arctanh}(\cos\theta)\Big{]},$ (90)

where $\alpha\in(-\infty,\infty)$, ${\rm Ei}(x)=\int^{x}_{-\infty}e^{t}/t\
{\rm d}t$, and $C$ is a constant determined by proper smooth conditions. The
angular momentum for this Gaussian state is

$\displaystyle L_{\zeta}=\int(n_{+}-n_{-})\cos\theta{\rm
d}\Omega=\frac{2}{\alpha}(1-\alpha\coth{\alpha}).$ (91)

The fluid velocity field for this state is

$\displaystyle u_{\theta}$
$\displaystyle=\frac{1}{\sin\theta}\frac{\partial\psi}{\partial\phi}=0,$ (92)
$\displaystyle u_{\phi}$
$\displaystyle=-\frac{\partial\psi}{\partial\theta}=\frac{\alpha}{4\pi\sinh\alpha\sin\theta}\left[\frac{2\cosh({\alpha\cos\theta})}{\alpha}+C\right].$
(93)

Here we impose the zero velocity condition at poles, namely
$u_{\phi}(\theta=0)=u_{\phi}(\theta=\pi)=0$, giving rise that
$C=-2\cosh(\alpha)/\alpha$. Figure 6 shows the vorticity distribution, the
energy and the angular momentum as functions of $\alpha$. Note that the
condition $u_{\phi}(\theta=\pi/2)=0$ leads to unphysical results
$u_{\phi}(\theta=0)=u_{\phi}(\theta=\pi)=\infty$ for $\alpha\neq 0$.

Figure 6: (a) Vorticity $\sigma=n_{+}-n_{-}$ for the Gaussian state at
$\alpha=-8$. (b) and (c) show the energy and the angular momentum of as
functions of $\alpha$, respectively.

Similar to the fluid velocity field for the dipole state, for the Gaussian
state the peak fluid velocity appears on the equator for small values of
$|\alpha|$ and moves towards to poles as $|\alpha|$ exceeds a certain value of
$|\alpha_{c}|$ (Fig. 7), where $|\alpha_{c}|$ satisfies
${\alpha_{c}}^{2}-\cosh(\alpha_{c})+1=0$. The amplitude of the fluid velocity
on the equator is $|u_{\phi}(\theta=\pi/2)|=(1-1/\cosh\alpha)/(2\pi)$ and
reaches the maximum value $1/2\pi$ when $|\alpha|\to\infty$.

Figure 7: The amplitude of the fluid velocity field $u_{\phi}$ for different
values of $|\alpha|$.

##  V Microcanonical Monte Carlo Simulations

In this section, we show numerical results of Monte Carlo (MC) simulations of
a large number of point vortices on a unit sphere. The simulations are
performed for a range of increasing energies and at fixed angular momentum
$L_{\zeta}=0$.

###  V.1 Sandwich state and the quadrupole moment

In our microcanonical MC simulations, we adopt the general scheme developed in
Refs. Smith (1989); Smith and O’Neil (1990); Yu  _et al._ (2016) which are
based on the demon algorithm Creutz (1983). In particular, we closely follow
the algorithm developed in Ref. Yu  _et al._ (2016) for a neutral vortex point
vortex system. We find that the sandwich state emerges as increasing energy. A
comparison between the analytical predictions of the mean field theory and the
microcanonical MC sampling for point vortices is shown in Fig. 8. The
numerical results for the values of $\nu$, $Q_{0}$ and the upper bound of the
quadrupole moment are in good agreement with the mean field predictions.

Figure 8: A comparison between the mean field theory and MC simulations for
$N=1000$ point vortices. (a) and (b) show results from the mean field theory
and Monte Carlo sampling for the scaled vorticity $\tilde{\sigma}$,
respectively. Here $\widetilde{\sigma}=\sigma$ for $E=0.005$, $E=0.01$, and
$\widetilde{\sigma}=\sigma/20$ for $E=0.1$. The purpose for introducing the
scaled vorticity $\tilde{\sigma}$ is to present the large energy range on a
single color map. (c) The quadrupole moment $Q$ for increasing energy $E$
(circle red) and the mean field prediction Eq. (56) (solid blue). The
quadrupole moment is well bounded by the maximum possible value $Q^{\rm
max}_{s}$ predicted by the mean field theory. The inset in (c) shows the
$N$-dependence behavior of $Q$ in the vicinity of the clustering transition.
For finite $N$, the transition energy $E_{c}$ is finite and approaches to zero
as $N\rightarrow\infty$. The finite values of $Q\sim 1/\sqrt{N}$ below the
transition energy is nearly energy independent and is due to uncorrelated
fluctuations. The data presented here are taken in runs of $40000$ MC steps at
each value of energy.

###  V.2 Sandwich state vs. quadrupole state

Our theory predicts that the sandwich state has higher entropy than the
quadrupole state (see Sec. IV.2), for the fixed values of energy, number of
vortices, and zero angular momentum. In order to verify this analytical result
numerically, we perform the MC sampling starting from a quadrupole
configuration at a moderate energy. We find that the vortex distribution
saturates at the sandwich state [Fig.9(a)(b)], indicating that indeed the
sandwich state has greater statistical weight than the quadrupole state. This
can be further verified by measuring directly the entropy of the initial and
final distributions for a certain range of energies [Fig.9(c)]. Here the
initial quadrupole distribution is obtained through sampling of the quadrupole
mode $\sigma_{21}$ (or $\sigma_{22}$).

Figure 9: (a) A point vortex system containing $N=1000$ vortices which is
initially prepared at a quadrupole state with $E=0.06$ and $L_{\zeta}=0$. (b)
The finial state after $160000$ MC steps and the final state is the sandwich
state. (c) A comparison between the entropy of the initial quadrupole state
and the final sandwich state. The sphere is divided into $40\times 40$ cells
by longitude and latitude (these cells have different area). The entropy is
calculated through Eq. (94). Here the entropy is negative because of the
artificial choice of the minimum area in the phase space, and only the entropy
difference between the two states has physical meaning.

The entropy of a point vortex system confined on a domain can be measured
numerically. We divide the domain into a large number of cells with size
$a_{i}$, where $i$ denotes the cell index. Each cell contains a sufficient
number of vortices. We denote the number of point vortices in a cell of each
species as $N_{i,\hat{\sigma}}$ with $\hat{\sigma}=\pm$. The number of states
is then given by

$\displaystyle\mathcal{W}\left(\\{N_{i,\hat{\sigma}}\\}\right)=\prod_{\hat{\sigma}}\left\\{\prod_{i}\frac{N_{\hat{\sigma}}!}{N_{i,\hat{\sigma}}!}\left(\frac{a_{i}}{\Omega}\right)^{N_{i,\hat{\sigma}}}\right\\},$
(94)

where is $\Omega$ is the area of domain Joyce and Montgomery (1973); Maestrini
and Salman (2019). As it follows the entropy per vortex is
$S=(1/N)\log\mathcal{W}$. We apply this method to vortices on a unit sphere
and compare the entropy between the sandwich state and the quadrupole state
[Fig.9(c)].

##  VI Conclusion and discussion

We investigate Onsager vortex clustered states of point vortices confined on a
sphere. We find that for vortex distributions with zero angular momentum, the
sandwich state is the most probable state and is characterized by the
quadrupole moment tensor. The mean field analytical predictions agree well
with the Monte Carlo simulations. It is worthwhile mentioning that Ref. Kanai
and Guo (2021) shows that the Gross-Pitaevskii dynamics drives an initial
uniform vortex state to the end quadruple state through the evaporative
heating process. This result is not necessary inconsistent with the our
findings. Since the quadruple distribution also solves the self-consistent
equation and hence it is a local maximum entropy state. In the situation
described in Ref. Kanai and Guo (2021), vortices may be dynamically trapped in
a local equilibrium state. To explore the entropy barrier between the sandwich
state and the quadruple state and its effect on the dynamical transitions
between the two states deserves future investigations.

###  Acknowledgement

We thank Y. Xiong and C. Ma 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. U2330401).

##  Appendix A Appendix

##  Appendix B Normolization dependence of the onset of the supercondensation

The value of $\beta$ at which the supercondensation occurs depends on the
normalization condition. Here we show the method we used to determine the
value of $\beta_{s}$ for the sandwich state [see Fig. 10(a)].

Figure 10: (a) The $\beta-e^{\gamma_{+}}$ curve is determined by normalization
Eq. (73) and touches the $\beta=\beta_{s}=-16\pi$ curve when the
supercondensation occurs. (b) The $\beta-e^{\gamma_{-}}$ curve is determined
by the normalization Eq. (73) ($n_{+}\rightarrow n_{-}$). The condition $\beta
e^{\gamma_{-}}<-5/2$ ensures $n^{\prime}_{-}(\rho)|_{\rho=0}=0$.

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# On a new generalization of metric spaces

Mohamed Jleli, Bessem Samet

###### Abstract.

In this paper, we introduce the $\mathcal{F}$-metric space concept, which
generalizes the metric space notion. We define a natural topology
$\tau_{\mathcal{F}}$ in such spaces and we study their topological properties.
Moreover, we establish a new version of the Banach contraction principle in
the setting of $\mathcal{F}$-metric spaces. Several examples are presented to
illustrate our study.

###### Key words and phrases:

$\mathcal{F}$-metric space; topological properties; Banach contraction
principle

###### 2010 Mathematics Subject Classification:

54E50; 54A20; 47H10

##  1\. Introduction

A metric on a nonempty set $X$ is a mapping $d:X\times X\to[0,+\infty)$
satisfying the following properties:

  * (i)

$d(x,y)=0$ if and only if $x=y$.

  * (ii)

$d(x,y)=d(y,x)$.

  * (iii)

$d(x,y)\leq d(x,z)+d(z,y)$.

If $d$ is a metric on $X$, then the pair $(X,d)$ is said to be a metric space.
The theory of metric spaces is the general theory which underlies several
branches of mathematical analysis, as real analysis, complex analysis,
multidimensional calculus, etc.

In recent years, many interesting generalizations (or extensions) of the
metric space concept appeared. Czerwik [3] introduced the notion of a
$b$-metric. Khamsi and Hussain [8] reintroduced this notion under the name
metric-type. In [4], Fagin et al. introduced the notion of $s$-relaxedp
metric. Note that any $s$-relaxedp metric is a $b$-metric, but the converse is
not true in general (see [9]). Gähler [5] introduced the notion of a 2-metric,
which is a mapping defined on the product set $X\times X\times X$, and
satisfying certain conditions. Gähler claimed that a 2-metric is a
generalization of the usual notion of a metric. However, different authors
showed that no relations between these two concepts exist (see, for example
[6]). A more appropriate notion of generalized metric space was introduced by
Mustafa and Sims [11] under the name G-metric space. In [2], Branciari
suggested a new generalization of the metric notion by replacing the triangle
inequality (iii) by a more general one involving four points. Matthews [10]
introduced the notion of a partial metric as a part of the study of
denotational semantics of dataflow networks. Recently, we introduced [7] the
concept of JS-metric, where the triangle inequality is replaced by a
$\limsup$-condition. For more details about the above cited concepts and other
generalizations of the metric notion, we refer the reader to the nice book [9]
by Kirk and Shahzad.

In this paper, we introduce a new generalization of the metric space notion,
which we call an $\mathcal{F}$-metric space. We compare our concept with
existing generalizations from the literature. Next, we define a natural
topology $\tau_{\mathcal{F}}$ on these spaces, and we study their topological
properties. Moreover, a new version of the Banach contraction principle is
established in the setting of $\mathcal{F}$-metric spaces.

The paper is organized as follows. In Section 2, the concept of
$\mathcal{F}$-metric spaces is introduced. We show that any metric space is an
$\mathcal{F}$-metric space but the converse is not true in general, which
confirms that our concept is more general than the standard metric concept.
Moreover, we compare our proposed notion with previous generalizations of
metric spaces. More precisely, we show that any $s$-relaxedp-metric space is
an $\mathcal{F}$-metric space (see Example 2.2). Further, we provide an
example of an $\mathcal{F}$-metric space that cannot be an $s$-relaxedp-metric
space (see Example 2.4), which confirms that the class of $\mathcal{F}$-metric
spaces is more large than the class of $s$-relaxedp-metric spaces. A
comparison with b-metric spaces is also considered. We show that there exist
$\mathcal{F}$-metric spaces that are not b-metric spaces (see Example 2.2) and
there exist b-metric spaces that are not $\mathcal{F}$-metric spaces (see
Proposition 2.1). In Section 3, we introduce the notion of
$\mathcal{F}$-metric boundedness, which is used to provide a characterization
of $\mathcal{F}$-metrics (see Theorem 3.1). In Section 4, a topology
$\tau_{\mathcal{F}}$ is introduced on $\mathcal{F}$-metric spaces using the
concept of balls. It is well-known that in standard metric spaces, the closed
ball is closed with respect to the topology defined via balls (or equivalently
the sequential topology). In our situation, we do not know whether closed
balls are closed with respect to $\tau_{\mathcal{F}}$. However, we provide a
sufficient condition so that any closed ball is closed with respect to
$\tau_{\mathcal{F}}$ (see Proposition 4.3). Additional topological properties
are also discussed, as compactness, completeness, etc. In Section 5, we
establish the Banach contraction principle in the setting of
$\mathcal{F}$-metric spaces (see Theorem 5.1).

##  2\. A generalized metric space

Let $\mathcal{F}$ be the set of functions $f:(0,+\infty)\to\mathbb{R}$
satisfying the following conditions:

  * ($\mathcal{F}_{1}$)

$f$ is non-decreasing, i.e., $0<s<t\implies f(s)\leq f(t)$.

  * ($\mathcal{F}_{2}$)

For every sequence $\\{t_{n}\\}\subset(0,+\infty)$, we have

$\lim_{n\to+\infty}t_{n}=0\Longleftrightarrow\lim_{n\to+\infty}f(t_{n})=-\infty.$

We generalize the concept of metric spaces as follows.

######  Definition 2.1.

Let $X$ be a nonempty set, and let $D:X\times X\to[0,+\infty)$ be a given
mapping. Suppose that there exists $(f,\alpha)\in\mathcal{F}\times[0,+\infty)$
such that

  * (D1)

$(x,y)\in X\times X,\,D(x,y)=0\Longleftrightarrow x=y$.

  * (D2)

$D(x,y)=D(y,x)$, for all $(x,y)\in X\times X$.

  * (D3)

For every $(x,y)\in X\times X$, for every $N\in\mathbb{N}$, $N\geq 2$, and for
every $\displaystyle(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$, we
have

$D(x,y)>0\implies f(D(x,y))\leq
f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+\alpha.$

Then $D$ is said to be an $\mathcal{F}$-metric on $X$, and the pair $(X,D)$ is
said to be an $\mathcal{F}$-metric space.

Observe that any metric on $X$ is an $\mathcal{F}$-metric on $X$. Indeed, if
$d$ is a metric on $X$, then it satisfies (D1) and (D2). On the other hand, by
the triangle inequality, for every $(x,y)\in X\times X$, for every
$N\in\mathbb{N}$, $N\geq 2$, and for every
$\displaystyle(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$, we have

$d(x,y)\leq\sum_{i=1}^{N-1}d(u_{i},u_{i+1}),$

which yields,

$d(x,y)>0\implies\ln\left(d(x,y)\right)\leq\ln\left(\sum_{i=1}^{N-1}d(u_{i},u_{i+1})\right).$

Then $d$ satisfies (D3) with $f(t)=\ln t$, $t>0$, and $\alpha=0$.

In the following, some examples of $\mathcal{F}$-metric spaces which are not
metric spaces are presented.

######  Example 2.1.

Let $X=\mathbb{N}$, and let $D:X\times X\to[0,+\infty)$ be the mapping defined
by

(2.4) $\displaystyle D(x,y)=\left\\{\begin{array}[]{lll}(x-y)^{2},&\mbox{ if
}&(x,y)\in[0,3]\times[0,3],\\\ \\\ |x-y|,&\mbox{ if
}&(x,y)\not\in[0,3]\times[0,3],\end{array}\right.$

for all $(x,y)\in X\times X$. It can be easily seen that $D$ satisfies (D1)
and (D2). However, $D$ doesn’t satisfy the triangle inequality, since

$d(1,3)=4>1+1=d(1,2)+d(2,3).$

Hence, $D$ is not a metric on $X$. Further, let us fix a certain $(x,y)\in
X\times X$ such that $D(x,y)>0$. Let $\displaystyle(u_{i})_{i=1}^{N}\subset
X$, where $N\in\mathbb{N}$, $N\geq 2$, and $(u_{1},u_{N})=(x,y)$. Let

$I=\\{i=1,2,\cdots,N-1:\,(u_{i},u_{i+1})\in[0,3]\times[0,3]\\}$

and

$J=\\{1,2,\cdots,N-1\\}\backslash I.$

Therefore, we have

$\displaystyle\sum_{i=1}^{N-1}D(u_{i},u_{i+1})$ $\displaystyle=$
$\displaystyle\sum_{i\in I}D(u_{i},u_{i+1})+\sum_{j\in J}D(u_{j},u_{j+1})$
$\displaystyle=$ $\displaystyle\sum_{i\in I}(u_{i+1}-u_{i})^{2}+\sum_{j\in
J}|u_{j+1}-u_{j}|.$

Next, we discuss two possible cases.
Case 1: If $(x,y)\not\in[0,3]\times[0,3]$. In this case, we have

$\displaystyle D(x,y)$ $\displaystyle=$ $\displaystyle|x-y|$
$\displaystyle\leq$ $\displaystyle\sum_{i=1}^{N-1}|u_{i+1}-u_{i}|$
$\displaystyle=$ $\displaystyle\sum_{i\in I}|u_{i+1}-u_{i}|+\sum_{j\in
I}|u_{j+1}-u_{j}|.$

On the other hand, observe that

$|u_{i+1}-u_{i}|\leq(u_{i+1}-u_{i})^{2},\quad i\in I.$

Therefore, we deduce that

$\displaystyle D(x,y)$ $\displaystyle\leq$ $\displaystyle\sum_{i\in
I}(u_{i+1}-u_{i})^{2}+\sum_{j\in J}|u_{j+1}-u_{j}|$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{N-1}D(u_{i},u_{i+1}).$

Case 2: If $(x,y)\in[0,3]\times[0,3]$. In this case, we have

$\displaystyle D(x,y)$ $\displaystyle=$ $\displaystyle|x-y|^{2}$
$\displaystyle\leq$ $\displaystyle 3|x-y|$ $\displaystyle\leq$ $\displaystyle
3\left(\sum_{i\in I}|u_{i+1}-u_{i}|+\sum_{j\in J}|u_{j+1}-u_{j}|\right)$
$\displaystyle\leq$ $\displaystyle 3\left(\sum_{i\in
I}|u_{i+1}-u_{i}|^{2}+\sum_{i\in J}|u_{j+1}-u_{j}|\right)$ $\displaystyle=$
$\displaystyle 3\sum_{i=1}^{N-1}D(u_{i},u_{i+1}).$

Next, combining the above cases, we deduce that for every $(x,y)\in X\times
X$, for every $N\in\mathbb{N}$, $N\geq 2$, and for every
$\displaystyle(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$, we have

(2.5) $D(x,y)>0\implies D(x,y)\leq 3\sum_{i=1}^{N-1}D(u_{i},u_{i+1}),$

which yields

$D(x,y)>0\implies\ln(D(x,y))\leq\ln\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+\ln
3.$

This proves that $D$ satisfies (D3) with $f(t)=\ln t$, $t>0$, and $\alpha=\ln
3$. Then $D$ is an $\mathcal{F}$-metric on $X$.

######  Example 2.2 (The class of $s$-relaxedp metrics).

Let $d:X\times X\to[0,+\infty)$ be an $s$-relaxedp metric on $X$ (see [4]),
i.e., $d$ satisfies (D1), (D2), and

  * (S)

There exists $K\geq 1$ such that for every $(x,y)\in X\times X$, for every
$N\in\mathbb{N}$, $N\geq 2$, and for every
$\displaystyle(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$, we have

$d(x,y)\leq K\sum_{i=1}^{N-1}d(u_{i},u_{i+1}).$

Then $d$ satisfies (D3) with $f(t)=\ln t$, $t>0$, and $\alpha=\ln K$. As
consequence, any $s$-relaxedp metric on $X$ is an $\mathcal{F}$-metric on $X$.

######  Remark 2.1.

Note that from (2.5), the mapping $D$ defined by (2.4) is an $s$-relaxedp
metric on $X$ with $K=3$.

######  Example 2.3 (The class of bounded 2-metric spaces).

Let $\sigma:X\times X\times X\to[0,+\infty)$ be a mapping satisfying the
following conditions:

  * ($\sigma_{1}$)

$(a,b)\in X\times X,\,a\neq b\implies\exists\,c\in X:\,\sigma(a,b,c)\neq 0$.

  * ($\sigma_{2}$)

For all $(a,b,c)\in X\times X\times X$, $\sigma(a,b,c)=0$ if and only if at
least two elements from $\\{a,b,c\\}$ are equal.

  * ($\sigma_{3}$)

$(a,b,c)\in X\times X\times X\implies\sigma(a,b,c)=\sigma(u,v,w)$, where
$\\{u,v,w\\}$ is any permutation of $\\{a,b,c\\}$.

  * ($\sigma_{4}$)

For all $(a,b,c)\in X\times X\times X$, we have

$\sigma(a,b,c)\leq\sigma(a,b,d)+\sigma(b,c,d)+\sigma(c,a,d).$

Then $\sigma$ is called a 2-metric on $X$, and $(X,\sigma)$ is called a
2-metric space (see [5]). Moreover, suppose that $\displaystyle\sup_{x,y,z\in
X}\sigma(x,y,z)<+\infty$. In this case, $(X,\sigma)$ is said to be a bounded
2-metric space. Define the mapping $D_{\sigma}:X\times X\to[0,+\infty)$ by

$D_{\sigma}(x,y)=\sup_{a\in X}\sigma(a,x,y),\quad(x,y)\in X\times X.$

It was proved in [1] that $D_{\sigma}$ is an s-relaxedp metric on $X$ with
$K=2$. Therefore, $D_{\sigma}$ is an $\mathcal{F}$-metric on $X$.

The next example shows that the class of $\mathcal{F}$-metrics is more large
than the class of $s$-relaxedp metrics.

######  Example 2.4.

Let $X=\mathbb{N}$, and let $D:X\times X\to[0,+\infty)$ be the mapping defined
by

(2.9) $\displaystyle
D(x,y)=\left\\{\begin{array}[]{lll}\exp\left(|x-y|\right),&\mbox{ if }&x\neq
y,\\\ \\\ 0,&\mbox{ if }&x=y,\end{array}\right.$

for all $(x,y)\in X\times X$. It can be easily seen that $D$ satisfies (D1)
and (D2).

First, let us prove that $D$ cannot be an $s$-relaxedp metric. We argue by
contradiction, by supposing that $D$ satisfies the condition (S) of Example
2.2 with a certain $K\geq 1$. Therefore, we have

$D(2n,0)\leq K\left(D(2n,n)+D(n,0)\right),\quad n\in\mathbb{N}^{*},$

that is,

$\exp(n)\leq 2K,\quad n\in\mathbb{N}^{*}.$

Passing to the limit as $n\to+\infty$, we obtain a contradiction. Therefore,
$D$ is not an $s$-relaxedp metric.

Next, we shall prove that $D$ belongs to the class of $\mathcal{F}$-metrics.
Let

$f(t)=\frac{-1}{t},\quad t>0.$

It can be easily seen that $f\in\mathcal{F}$. In order to check (D3), let us
fix $(x,y)\in X\times X$ with $D(x,y)>0$. For every $N\in\mathbb{N}$, $N\geq
2$, and for every $\displaystyle(u_{i})_{i=1}^{N}\subset X$ with
$(u_{1},u_{N})=(x,y)$, we have

$\displaystyle 1+f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)-f(D(x,y))$
$\displaystyle=1-\frac{1}{\displaystyle\sum_{i=1:N-1,\,u_{i+1}\neq
u_{i}}\exp\left(|u_{i+1}-u_{i}|\right)}+\frac{1}{\exp\left(|x-y|\right)}$
$\displaystyle\geq 1-1+\frac{1}{\exp\left(|x-y|\right)}$ $\displaystyle\geq
0.$

Therefore, we have

$f(D(x,y))\leq f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+1.$

This proves that $D$ satisfies (D3) with $f(t)=\frac{-1}{t}$, $t>0$, and
$\alpha=1$. Then $D$ is an $\mathcal{F}$-metric.

It was proved in [4] (see also [9]) that there is a b-metric space that is not
an s-relaxedp metric space for any $K\geq 1$. We shall prove an analogous
result for the case of $\mathcal{F}$-metric spaces. First, recall that a
mapping $d:X\times X\to[0,+\infty)$ is said to be a b-metric on $X$ if it
satisfies (D1), (D2), and

  * (S)’

There exists $K\geq 1$ such that

$d(x,y)\leq K\left(d(x,z)+d(z,y)\right),\quad(x,y,z)\in X\times X\times X.$

Observe that (S)$\implies$ (S)’. Therefore, any s-relaxedp metric is a
b-metric. However, as we mentioned before, the converse is not true in
general.

######  Proposition 2.1.

There is a b-metric space that is not an $\mathcal{F}$-metric space.

###### Proof.

Let $X=[0,1]$, and let $d:X\times X\to[0,+\infty)$ be the mapping defined by

$d(x,y)=(x-y)^{2},\quad(x,y)\in X\times X.$

It can be easily seen (see, for example [9]) that $d$ is a b-metric on $X$
with constant $K=2$. Suppose that there exists
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ such that $d$ satisfies (D3). Let
$n\in\mathbb{N}^{*}$, and let

$u_{i}=\frac{i}{n},\quad i=0,2,\cdots,n.$

By (D3), we have

$f(d(0,1))\leq
f\left(d(0,u_{1})+d(u_{1},u_{2})+\cdots+d(u_{n-1},1)\right)+\alpha,\quad
n\in\mathbb{N}^{*},$

i.e.,

$f(1)\leq f\left(\frac{1}{n}\right)+\alpha,\quad n\in\mathbb{N}^{*}.$

On the other hand, by ($\mathcal{F}_{2}$), we have

$\lim_{n\to+\infty}f\left(\frac{1}{n}\right)+\alpha=-\infty,$

which is a contradiction. ∎

######  Remark 2.2.

We proved in Example 2.4 that the mapping $D$ defined by (2.9) is an
$\mathcal{F}$-metric on $X$ but it is not an $s$-relaxedp metric. It can be
easily seen that $D$ is not also a b-metric on $X$.

##  3\. Characterization of $\mathcal{F}$-metrics

In this section, we introduce the concept of $\mathcal{F}$-metric boundedness,
which will be used later to give a characterization of $\mathcal{F}$-metrics.

######  Definition 3.1.

Let $X$ be a nonempty set, and let $D:X\times X\to[0,+\infty)$ be a given
mapping satisfying (D1) and (D2). We say that the pair $(X,D)$ is
$\mathcal{F}$-metric bounded with respect to
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$, if there exists a metric $d$ on
$X$ such that

(3.1) $(x,y)\in X\times X,\,D(x,y)>0\implies f(d(x,y))\leq f(D(x,y))\leq
f(d(x,y))+\alpha.$

We have the following result.

######  Theorem 3.1.

Let $X$ be a nonempty set, and let $D:X\times X\to[0,+\infty)$ be a given
mapping satisfying (D1) and (D2). Let
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$, and suppose that $f$ is
continuous from the right. Then the following statements are equivalent:

  * (i)

$(X,D)$ is an $\mathcal{F}$-metric on $X$ with $(f,\alpha)$ defined above.

  * (ii)

$(X,D)$ is $\mathcal{F}$-metric bounded with respect to $(f,\alpha)$.

###### Proof.

(i)$\implies$ (ii): Assume that $(X,D)$ is an $\mathcal{F}$-metric on $X$ with
respect to $(f,\alpha)$. Let us define the mapping $d:X\times X\to[0,+\infty)$
by

$d(x,y)=\inf\left\\{\sum_{i=1}^{N-1}D(u_{i},u_{i+1}):\,N\in\mathbb{N},\,N\geq
2,\,(u_{i})_{i=1}^{N}\subset X,\,(u_{1},u_{N})=(x,y)\right\\},$

for all $(x,y)\in X\times X$. We shall prove that $d$ is a metric on $X$.
Since $D(x,x)=0$, for all $x\in X$, it follows from the definition of $d$ that

$d(x,x)=0,\quad x\in X.$

Now, let $(x,y)\in X\times X$ be such that $x\neq y$. Suppose that $d(x,y)=0$.
Let $\varepsilon>0$, by the definition of $d$, there exist $N\in\mathbb{N}$,
$N\geq 2$, and $(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$ such
that

$\sum_{i=1}^{N-1}D(u_{i},u_{i+1})<\varepsilon.$

By ($\mathcal{F}_{1}$), we obtain

(3.2) $f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)\leq f(\varepsilon).$

On the other hand, by (D3), we have

(3.3) $f(D(x,y))\leq f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+\alpha.$

Using (3.2) and (3.3), we obtain

$f(D(x,y))\leq f(\varepsilon)+\alpha,\quad\varepsilon>0.$

But, using ($\mathcal{F}_{2}$), we have

$\lim_{\varepsilon\to 0^{+}}\left(f(\varepsilon)+\alpha\right)=-\infty,$

which is a contradiction. Therefore, we have $d(x,y)>0$. From the definition
of $d$ and (D2), it can be easily seen that $d(x,y)=d(y,x)$, for all $(x,y)\in
X\times X$. In order to check the triangle inequality, let $x,y$ and $z$ be
three given points in $X$, and let $\rho>0$. By the definition of $d$, there
exist two chains of points $x=u_{1},u_{2},\cdots,u_{n}=y$ and
$y=u_{n},u_{n+1},\cdots,u_{m}=z$ such that

$\sum_{i=1}^{n-1}D(u_{i},u_{i+1})<d(x,y)+\rho$

and

$\sum_{i=n}^{m-1}D(u_{i},u_{i+1})<d(y,z)+\rho.$

Adding the above inequalities, we obtain

$d(x,z)\leq\sum_{i=1}^{m-1}D(u_{i},u_{i+1})<d(x,y)+d(y,z)+2\rho,\quad\rho>0.$

Passing to the limit as $\rho\to 0^{+}$, we get

$d(x,z)\leq d(x,y)+d(y,z).$

As consequence, we deduce that $d$ is a metric on $X$. Next, we shall prove
that $d$ satisfies (3.1). Let $(x,y)\in X\times X$ be such that $D(x,y)>0$.
From the definition of $d$, it is clear that

$d(x,y)\leq D(x,y),$

which implies from ($\mathcal{F}_{1}$) that

(3.4) $f(d(x,y))\leq f(D(x,y)).$

Let $\varepsilon>0$. By the definition of $d$, there exist $N\in\mathbb{N}$,
$N\geq 2$, and $(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$ such
that

$\sum_{i=1}^{N-1}D(u_{i},u_{i+1})<d(x,y)+\varepsilon.$

By ($\mathcal{F}_{1}$), we obtain

$f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)\leq f(d(x,y)+\varepsilon).$

Using (D3) and the above inequality, we get

$f(D(x,y))\leq f(d(x,y)+\varepsilon)+\alpha,\quad\varepsilon>0.$

Passing to the limit as $\varepsilon\to 0^{+}$, and using the right continuity
of $f$, we obtain

(3.5) $f(D(x,y))\leq f(d(x,y))+\alpha.$

By (3.4) and (3.5), we have

$f(d(x,y))\leq f(D(x,y))\leq f(d(x,y))+\alpha.$

Then (3.1) is satisfied and $(X,D)$ is $\mathcal{F}$-metric bounded with
respect to $(f,\alpha)$.
(ii)$\implies$(i): Suppose that $(X,D)$ is $\mathcal{F}$-metric bounded with
respect to $(f,\alpha)$, that is, there exists a certain metric $d$ on $X$
such that (3.1) is satisfied. We have just to prove that $D$ satisfies (D3).
Let $(x,y)\in X\times X$ be such that $D(x,y)>0$. Let $N\in\mathbb{N}$, $N\geq
2$, and $(u_{i})_{i=1}^{N}\subset X$ with $(u_{1},u_{N})=(x,y)$. Since $d$ is
a metric on $X$, the triangle inequality yields

(3.6) $d(x,y)\leq\sum_{i=1}^{N-1}d(u_{i},u_{i+1}).$

On the other hand, using ($\mathcal{F}_{1}$) and the fact that

$(u,v)\in X\times X,\,D(u,v)>0\implies f(d(u,v))\leq f(D(u,v)),$

we deduce that

(3.7) $d(u,v)\leq D(u,v),\quad(u,v)\in X\times X.$

By (3.6) and (3.7), we obtain

$d(x,y)\leq\sum_{i=1}^{N-1}D(u_{i},u_{i+1}),$

which implies by ($\mathcal{F}_{1}$) that

$f(d(x,y))+\alpha\leq f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+\alpha.$

Using the above inequality and the fact that

$f(D(x,y))\leq f(d(x,y))+\alpha,$

we deduce that

$f(D(x,y))\leq f\left(\sum_{i=1}^{N-1}D(u_{i},u_{i+1})\right)+\alpha.$

Therefore, (D3) is satisfied and $(X,D)$ is an $\mathcal{F}$-metric on $X$. ∎

######  Remark 3.1.

Observe that from the proof of Theorem 3.1, the right continuity assumption
imposed on $f$ is used only to prove that (i)$\implies$ (ii). However, for any
$f\in\mathcal{F}$, we have (ii)$\implies$ (i).

##  4\. Topological $\mathcal{F}$-metric spaces

In this section, we discuss a natural topology defined on $\mathcal{F}$-metric
spaces.

######  Definition 4.1.

Let $(X,D)$ be an $\mathcal{F}$-metric space. A subset $\mathcal{O}$ of $X$ is
said to be $\mathcal{F}$-open if for every $x\in\mathcal{O}$, there is some
$r>0$ such that $B(x,r)\subset\mathcal{O}$, where

$B(x,r)=\\{y\in X:\,D(x,y)<r\\}.$

We say that a subset $\mathcal{C}$ of $X$ is $\mathcal{F}$-closed if
$X\backslash\mathcal{C}$ is $\mathcal{F}$-open. We denote by
$\tau_{\mathcal{F}}$ the family of all $\mathcal{F}$-open subsets of $X$.

The following result can be proved easily.

######  Proposition 4.1.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Then $\tau_{\mathcal{F}}$ is a
topology on $X$.

######  Proposition 4.2.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Then, for any nonempty subset
$A$ of $X$, the following statements are equivalent:

  * (i)

$A$ is $\mathcal{F}$-closed.

  * (ii)

For any sequence $\\{x_{n}\\}\subset A$, we have

$\lim_{n\to+\infty}D(x_{n},x)=0,\,x\in X\implies x\in A.$

###### Proof.

Assume that $A$ is $\mathcal{F}$-closed, and let $\\{x_{n}\\}$ be a sequence
in $A$ such that

(4.1) $\lim_{n\to+\infty}D(x_{n},x)=0,$

where $x\in X$. Suppose that $x\in X\backslash A$. Since $A$ is
$\mathcal{F}$-closed, $X\backslash A$ is $\mathcal{F}$-open. Therefore, there
exists some $r>0$ such that $B(x,r)\subset X\backslash A$, i.e. $B(x,r)\cap
A=\emptyset$. On the other hand, by (4.1), there exists some $N\in\mathbb{N}$
such that

$D(x_{n},x)<r,\quad n\geq N,$

i.e.

$x_{n}\in B(x,r),\quad n\geq N.$

Hence, $x_{N}\in B(x,r)\cap A$, which leads to a contradiction. Therefore, we
deduce that $x\in A$, and (i)$\implies$(ii) is proved. Conversely, assume that
(ii) is satisfied. Let $x\in X\backslash A$. We have to prove that there is
some $r>0$ such that $B(x,r)\subset X\backslash A$. We argue by contradiction
by supposing that for every $r>0$, there exists $x_{r}\in B(x,r)\cap A$. This
implies that for any $n\in\mathbb{N}^{*}$, there exists $x_{n}\in
B(x,\frac{1}{n})\cap A$. Then

$\\{x_{n}\\}\subset A,\,\lim_{n\to+\infty}D(x_{n},x)=0.$

By (ii), this implies that $x\in A$, which is a contradiction with $x\in
X\backslash A$. Hence, $A$ is $\mathcal{F}$-closed, and (ii) $\implies$(i). ∎

######  Proposition 4.3.

Let $(X,D)$ be an $\mathcal{F}$-metric space, $a\in X$, and $r>0$. We denote
by $\mathbf{B}(a,r)$ the subset of $X$ defined by

$\mathbf{B}(a,r)=\\{x\in X:\,D(a,x)\leq r\\}.$

Suppose that for every sequence $\\{x_{n}\\}\subset X$, we have

(4.2) $\lim_{n\to+\infty}D(x_{n},x)=0,\,x\in X\implies
D(x,y)\leq\limsup_{n\to+\infty}D(x_{n},y),\,y\in X.$

Then $\mathbf{B}(a,r)$ is $\mathcal{F}$-closed.

###### Proof.

Let $\\{x_{n}\\}\subset\mathbf{B}(a,r)$ be a sequence such that

$\lim_{n\to+\infty}D(x_{n},x)=0,$

for a certain $x\in X$. From proposition 4.2, we have to prove that
$x\in\mathbf{B}(a,r)$. By the definition of $\mathbf{B}(a,r)$, we have

$D(x_{n},a)\leq r,\quad n\in\mathbb{N}.$

Passing to the supremum limit as $n\to+\infty$ and using (4.2), we obtain

$D(x,a)\leq\limsup_{n\to+\infty}D(x_{n},a)\leq r,$

which yields $x\in\mathbf{B}(a,r)$. Therefore, $\mathbf{B}(a,r)$ is
$\mathcal{F}$-closed. ∎

######  Remark 4.1.

Proposition 4.3 provides only a sufficient condition ensuring that
$\mathbf{B}(a,r)$ is $\mathcal{F}$-closed. An interesting problem consists to
find a necessary and sufficient condition under which $\mathbf{B}(a,r)$ is
$\mathcal{F}$-closed.

######  Definition 4.2.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. We denote by $\overline{A}$ the closure of $A$ with respect to the
topology $\tau_{\mathcal{F}}$, i.e. $\overline{A}$ is the intersection of all
$\mathcal{F}$-closed subsets of $X$ containing $A$. Clearly, $\overline{A}$ is
the smallest $\mathcal{F}$-closed subset which contains $A$.

######  Proposition 4.4.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Then, for any nonempty subset
$A$ of $X$, we have

$x\in\overline{A},\,r>0\implies B(x,r)\cap A\neq\emptyset.$

###### Proof.

Let $(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ be such that (D3) is
satisfied. Let us define the set

$A^{\prime}=\\{x\in X:\,\mbox{ for any }r>0,\mbox{ there exists }a\in
A:\,D(x,a)<r\\}.$

By (D1), it can be easily seen that $A\subset A^{\prime}$. Next, we shall
prove that $A^{\prime}$ is $\mathcal{F}$-closed. Let $\\{x_{n}\\}$ be a
sequence in $A^{\prime}$ such that

(4.3) $\lim_{n\to+\infty}D(x_{n},x)=0,\quad x\in X.$

Let $r>0$. By ($\mathcal{F}_{2}$), there exists some $\delta_{r}>0$ such that

(4.4) $0<t<\delta_{r}\implies\mu(t)<\mu(r)-\alpha.$

On the other hand, by (4.3), there exists some $N\in\mathbb{N}$ such that

$D(x_{n},x)<\frac{\delta_{r}}{3},\quad n\geq N.$

Since $x_{N}\in A^{\prime}$, there exists $a\in A$ such that

$D(x_{N},a)<\frac{\delta_{r}}{3}.$

If $D(x,a)>0$, by (D3), we have

$f(D(x,a))\leq f(D(x,x_{N})+D(x_{N},a))+\alpha\leq
f\left(\frac{2\delta_{r}}{3}\right)+\alpha.$

But by (4.4), since $\frac{2\delta_{r}}{3}<\delta_{r}$, we obtain

$f\left(\frac{2\delta_{r}}{3}\right)<f(r)-\alpha.$

Hence,

$f(D(x,a))<f(r),$

which implies from ($\mathcal{F}_{1}$) that $D(x,a)<r$. Therefore, in all
cases, we have

$D(x,a)<r,$

which yields $x\in A^{\prime}$. Then by Proposition 4.2, $A^{\prime}$ is
$\mathcal{F}$-closed, which contains $A$. Then $\overline{A}\subset
A^{\prime}$, which yields the desired result. ∎

######  Definition 4.3.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $\\{x_{n}\\}$ be a sequence
in $X$. We say that $\\{x_{n}\\}$ is $\mathcal{F}$-convergent to $x\in X$ if
$\\{x_{n}\\}$ is convergent to $x$ with respect to the topology
$\tau_{\mathcal{F}}$, i.e. for every $\mathcal{F}$-open subset
$\mathcal{O}_{x}$ of $X$ containing $x$, there exists some $N\in\mathbb{N}$
such that $x_{n}\in\mathcal{O}_{x}$, for all $n\geq N$. In this case, we say
that $x$ is the limit of $\\{x_{n}\\}$.

The following result follows immediately from the above definition and the
definition of $\tau_{\mathcal{F}}$.

######  Proposition 4.5.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $\\{x_{n}\\}$ be a sequence
in $X$, and $x\in X$. The following statements are equivalent:

  * (i)

$\\{x_{n}\\}$ is $\mathcal{F}$-convergent to $x$.

  * (ii)

$\displaystyle\lim_{n\to+\infty}D(x_{n},x)=0$.

The next result shows that the limit of an $\mathcal{F}$-convergent sequence
is unique.

######  Proposition 4.6.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $\\{x_{n}\\}$ be a sequence
in $X$. Then

$(x,y)\in X\times
X,\,\lim_{n\to+\infty}D(x_{n},x)=\lim_{n\to+\infty}D(x_{n},y)=0\implies x=y.$

###### Proof.

Let $(x,y)\in X\times X$ be such that

$\lim_{n\to+\infty}D(x_{n},x)=\lim_{n\to+\infty}D(x_{n},y)=0.$

Suppose that $x\neq y$, i.e. (from (D1)) $D(x,y)>0$. By (D3), there exists
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ such that

$f(D(x,y))\leq f(D(x,x_{n})+D(x_{n},y))+\alpha,\quad\mbox{ for all }n.$

On the other hand, using (D2) and ($\mathcal{F}_{2}$), we have

$\lim_{n\to+\infty}f(D(x,x_{n})+D(x_{n},y))+\alpha=\lim_{n\to+\infty}f(D(x_{n},x)+D(x_{n},y))+\alpha=-\infty,$

which is a contradiction. Therefore, we have $x=y$. ∎

######  Definition 4.4.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $\\{x_{n}\\}$ be a sequence
in $X$.

  * (i)

We say that $\\{x_{n}\\}$ is $\mathcal{F}$-Cauchy, if

$\lim_{n,m\to+\infty}D(x_{n},x_{m})=0.$

  * (ii)

We say that $(X,D)$ is $\mathcal{F}$-complete, if every $\mathcal{F}$-Cauchy
sequence in $X$ is $\mathcal{F}$-convergent to a certain element in $X$.

######  Example 4.1.

Let $X=\mathbb{N}$, and let $D:X\times X\to[0,+\infty)$ be the mapping defined
by (2.9). It was shown in Example 2.4 that $(X,D)$ is an $\mathcal{F}$-metric
space with $f(t)=\frac{-1}{t}$, $t>0$, and $\alpha=1$. We shall prove that
$(X,D)$ is a $\mathcal{F}$-complete. Let $\\{x_{n}\\}\subset X$ be an
$\mathcal{F}$-Cauchy sequence. This means that

$\lim_{n,m\to+\infty}D(x_{n},x_{m})=0.$

Therefore, there exists $N\in\mathbb{N}$ such that

$D(x_{n},x_{m})<\frac{1}{2},\quad n,m\geq N.$

Suppose that for some $n,m\geq N$, we have $x_{n}\neq x_{m}$. By the
definition of $D$, and using the above inequality, we obtain

$1\leq D(x_{n},x_{m})=\exp(|x_{n}-x_{m}|)<\frac{1}{2},$

which is a contradiction. Then, we deduce that

$x_{n}=x_{N},\quad n\geq N,$

which implies that

$\lim_{n\to+\infty}D(x_{n},x_{N})=0,$

i.e., $\\{x_{n}\\}$ is $\mathcal{F}$-convergent to $x_{N}$. As consequence,
$(X,D)$ is $\mathcal{F}$-complete.

######  Proposition 4.7.

Let $(X,D)$ be an $\mathcal{F}$-metric space. If $\\{x_{n}\\}\subset X$ is
$\mathcal{F}$-convergent, then it is $\mathcal{F}$-Cauchy.

###### Proof.

Let $(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ be such that (D3) is
satisfied. Let $x\in X$ be such that

(4.5) $\lim_{n\to+\infty}D(x_{n},x)=0.$

Let $\varepsilon>0$ be fixed. By ($\mathcal{F}_{2}$), we know that there
exists some $\delta>0$ such that

(4.6) $0<t<\delta\implies f(t)<f(\varepsilon)-\alpha.$

On the other hand, by (4.5), there exists some $N\in\mathbb{N}$ such that

(4.7) $D(x_{n},x)+D(x_{m},x)<\delta,\quad n,m\geq N.$

Let $n,m\geq N$. We discuss two cases.
Case 1: If $x_{m}=x_{n}$. In this case, by (D1), we have

$D(x_{n},x_{m})=0<\varepsilon.$

Case 2: If $x_{m}\neq x_{n}$. In this case, from (4.7), we have

$0<D(x_{n},x)+D(x_{m},x)<\delta.$

Therefore, by (4.6), we have

$f(D(x_{n},x)+D(x_{m},x))<f(\varepsilon)-\alpha.$

Now, using (D3), we obtain

$f(D(x_{n},x_{m}))\leq f(D(x_{n},x)+D(x_{m},x))+\alpha<f(\varepsilon),$

which implies from ($\mathcal{F}_{1}$) that

$D(x_{n},x_{m})<\varepsilon.$

As consequence, we have

$D(x_{n},x_{m})<\varepsilon,\quad n,m\geq N,$

which yields

$\lim_{n,m\to+\infty}D(x_{n},x_{m})=0,$

i.e. $\\{x_{n}\\}$ is $\mathcal{F}$-Cauchy. ∎

Next, we discuss the compactness on $\mathcal{F}$-metric spaces.

######  Definition 4.5.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. We say that $A$ is $\mathcal{F}$-compact if $A$ is compact with respect
to the topology $\tau_{\mathcal{F}}$ on $X$.

######  Proposition 4.8.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. Then, the following statements are equivalent:

  * (i)

$A$ is $\mathcal{F}$-compact.

  * (ii)

For any sequence $\\{x_{n}\\}\subset A$, there exist a subsequence
$\\{x_{n_{k}}\\}$ of $\\{x_{n}\\}$ and $x\in A$ such that

$\lim_{k\to+\infty}D(x_{n_{k}},x)=0.$

###### Proof.

Suppose that $A$ is $\mathcal{F}$-compact. It can be easy seen that any
decreasing sequence of nonempty $\mathcal{F}$-closed subsets of $A$ have a
nonempty intersection. Let $\\{x_{n}\\}$ be a sequence in $A$. For every
$n\in\mathbb{N}$, let

$C_{n}=\\{x_{m}:\,m\geq n\\}.$

Observe that $C_{n+1}\subset C_{n}$, for every $n\in\mathbb{N}$, which yields
$\displaystyle\\{\overline{C_{n}}\\}_{n\in\mathbb{N}}$ is a decreasing
sequence of nonempty $\mathcal{F}$-closed subsets of $A$. Therefore, there is
some $x$ that belongs to
$\displaystyle\bigcap_{n\in\mathbb{N}}\overline{C_{n}}$. Next, let
$\varepsilon>0$ be fixed. Since $x\in\overline{C_{0}}$, by Proposition 4.4,
there exist $n_{0}\geq 0$ and $x_{n_{0}}\in A$ such that
$D(x_{n_{0}},x)<\varepsilon$. continuing this process, for any
$k\in\mathbb{N}$, there exist $n_{k}\geq k$ and $x_{n_{k}}\in A$ such that
$D(x_{n_{k}},x)<\varepsilon$. Therefore, we have

$\lim_{k\to+\infty}D(x_{n_{k}},x)=0.$

On the other hand, since $A$ is $\mathcal{F}$-compact, then it is
$\mathcal{F}$-closed, and $x\in A$. Then we proved that (i)$\implies$(ii).
Conversely, suppose that (ii) is satisfied. Let
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ be such that (D3) is satisfied.
First, we claim that

(4.8) $\forall\,r>0,\,\exists\,(x_{i})_{i=1,\cdots,n}\subset
A:\,A\subset\bigcup_{i=1,\cdots,n}B(x_{i},r).$

We argue by contradiction, by supposing that there exists $r>0$ such that for
any finite number of elements $(x_{i})_{i=1,\cdots,n}\subset A$, we have

$A\not\subset\bigcup_{i=1,\cdots,n}B(x_{i},r).$

Let $x_{1}\in A$ be an arbitrary element. Then

$A\not\subset B(x_{1},r),$

i.e. there exists $x_{2}\in A$ such that

$D(x_{1},x_{2})\geq r.$

Again, we have

$A\not\subset B(x_{1},r)\cup B(x_{2},r),$

i.e. there exists $x_{3}\in A$ such that

$D(x_{i},x_{3})\geq r,\quad i=1,2.$

Continuing this process, by induction, we can construct a sequence
$\\{x_{n}\\}\subset A$ such that

$D(x_{n},x_{m})\geq r,\quad n,m\in\mathbb{N}^{*}.$

Observe that in this case, it is not possible to extract from $\\{x_{n}\\}$
any $\mathcal{F}$-Cauchy subsequence, so (from Proposition 4.7), any
$\mathcal{F}$-convergent subsequence. Then we obtain a contradiction with
(ii), which proves (4.8). Next, let $\\{\mathcal{O}_{i}\\}_{i\in I}$ be an
arbitrary family of $\mathcal{F}$-open subsets of $X$ such that

(4.9) $A\subset\bigcup_{i\in I}\mathcal{O}_{i}.$

We claim that

(4.10) $\exists\,r_{0}>0:\,\forall\,x\in A,\,\exists\,i\in
I:\,B(x,r_{0})\subset\mathcal{O}_{i}.$

We argue by contradiction by supposing that for any $r>0$, there exists
$x_{r}\in A$ such that $B(x_{r},r)\not\subset\mathcal{O}_{i}$, for all $i\in
I$. In particular, for all $n\in\mathbb{N}^{*}$, there exists $x_{n}\in A$
such that $B\left(x_{n},\frac{1}{n}\right)\not\subset\mathcal{O}_{i}$, for all
$i\in I$. By (ii), we can extract a subsequence $\\{x_{n_{k}}\\}$ from
$\\{x_{n}\\}$ such that

(4.11) $\lim_{k\to+\infty}D(x_{n_{k}},x)=0,$

for a certain $x\in A$. On the other hand, by (4.9), there exists some $j\in
I$ such that $x\in\mathcal{O}_{j}$. Since $\mathcal{O}_{j}$ is an
$\mathcal{F}$-open subset of $X$, there exists some $r_{0}>0$ such that
$B(x,r_{0})\subset\mathcal{O}_{j}$. Next, for any $n_{k}\in\mathbb{N}^{*}$,
and for any $z\in B\left(x_{n_{k}},\frac{1}{n_{k}}\right)$, we have

$D(x,z)>0\implies f(D(x,z))\leq
f(D(x,x_{n_{k}})+D(x_{n_{k}},z))+\alpha<f\left(D(x,x_{n_{k}})+\frac{1}{n_{k}}\right)+\alpha.$

By (4.11) and ($\mathcal{F}_{2}$), there exists $K\in\mathbb{N}^{*}$ such that

$f\left(D(x,x_{n_{k}})+\frac{1}{n_{k}}\right)<f(r_{0})-\alpha,\quad k\geq K,$

which yields

$D(x,z)>0\implies f(D(x,z))<f(r_{0}).$

Therefore, by ($\mathcal{F}_{1}$), we obtain

$D(x,z)<r_{0}.$

Thus we have

$B\left(x_{n_{k}},\frac{1}{n_{k}}\right)\subset B(x,r_{0}),\quad
n_{k}\in\mathbb{N}^{*},$

which implies

$B\left(x_{n_{k}},\frac{1}{n_{k}}\right)\subset\mathcal{O}_{j},\quad
n_{k}\in\mathbb{N}^{*}.$

Observe that we obtain a contradiction with the fact that
$B\left(x_{n},\frac{1}{n}\right)\not\subset\mathcal{O}_{i}$, for all $i\in I$.
Then (4.10) holds. Further, by (4.8), there exists
$\displaystyle(x_{p})_{p=1,\cdots,n}\subset A$ such that

$A\subset\bigcup_{p=1,\cdots,n}B(x_{p},r_{0}).$

But by (4.10), for any $p=1,\cdots,n$, there exists $i(p)\in I$ such that
$B(x_{p},r_{0})\subset\mathcal{O}_{i(p)}$, which yields

$A\subset\bigcup_{p=1,\cdots,n}\mathcal{O}_{i(p)}.$

Therefore, $A$ is $\mathcal{F}$-compact, and (ii)$\implies$(i). ∎

######  Definition 4.6.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. The subset $A$ is called sequentially $\mathcal{F}$-compact, if for any
sequence $\\{x_{n}\\}\subset A$, there exist a subsequence $\\{x_{n_{k}}\\}$
of $\\{x_{n}\\}$ and $x\in A$ such that

$\lim_{k\to+\infty}D(x_{n_{k}},x)=0.$

######  Definition 4.7.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. The subset $A$ is called $\mathcal{F}$-totally bounded, if

$\forall\,r>0,\,\exists\,(x_{i})_{i=1,\cdots,n}\subset
A:\,A\subset\bigcup_{i=1,\cdots,n}B(x_{i},r).$

From the proof of Proposition 4.8, we deduce the following result.

######  Proposition 4.9.

Let $(X,D)$ be an $\mathcal{F}$-metric space. Let $A$ be a nonempty subset of
$X$. Then

  * (i)

$A$ is $\mathcal{F}$-compact if and only if $A$ is sequentially
$\mathcal{F}$-compact.

  * (ii)

If $A$ if $\mathcal{F}$-compact, then $A$ is $\mathcal{F}$-totally bounded.

##  5\. Banach contraction principle on $\mathcal{F}$-metric spaces

In this section, we establish a new version of Banach contraction principle on
the setting of $\mathcal{F}$-metric spaces.

######  Theorem 5.1.

Let $(X,D)$ be an $\mathcal{F}$-metric space, and let $g:X\to X$ be a given
mapping. Suppose that the following conditions are satisfied:

  * (i)

$(X,D)$ is $\mathcal{F}$-complete.

  * (ii)

There exists $k\in(0,1)$ such that

$D(g(x),g(y))\leq kD(x,y),\quad(x,y)\in X\times X.$

Then $g$ has a unique fixed point $x^{*}\in X$. Moreover, for any $x_{0}\in
X$, the sequence $\\{x_{n}\\}\subset X$ defined by

(5.1) $x_{n+1}=g(x_{n}),\quad n\in\mathbb{N},$

is $\mathcal{F}$-convergent to $x^{*}$.

###### Proof.

First, observe that $g$ has at most one fixed point. Indeed, if $(u,v)\in
X\times X$ are two fixed points of $g$ with $u\neq v$, i.e.

$D(u,v)>0,\,\,g(u)=u,\,\,g(v)=v,$

then from (ii), we have

$D(u,v)=D(g(u),g(v))\leq kD(u,v)<D(u,v),$

which is a contradiction.

Next, let $(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ be such that (D3) is
satisfied. Let $\varepsilon>0$ be fixed. By ($\mathcal{F}_{2}$), there exists
$\delta>0$ such that

(5.2) $0<t<\delta\implies f(t)<f(\varepsilon)-\alpha.$

Let $x_{0}\in X$ be an arbitrary element. Let $\\{x_{n}\\}\subset X$ be the
sequence defined by (5.1). Without restriction of the generality, we may
suppose that $D(x_{0},x_{1})>0$. Otherwise, $x_{0}$ will be a fixed point of
$g$. It can be easily seen that from (ii), we have

$D(x_{n},x_{n+1})\leq k^{n}D(x_{0},x_{1}),\quad n\in\mathbb{N},$

which yields

$\sum_{i=n}^{m-1}D(x_{i},x_{i+1})\leq\frac{k^{n}}{1-k}D(x_{0},x_{1}),\quad
m>n.$

Since

$\lim_{n\to+\infty}\frac{k^{n}}{1-k}D(x_{0},x_{1})=0,$

there exists some $N\in\mathbb{N}$ such that

(5.3) $0<\frac{k^{n}}{1-k}D(x_{0},x_{1})<\delta,\quad n\geq N.$

Hence, by(5.2) and $(\mathcal{F}_{1}$), we have

(5.4) $f\left(\sum_{i=n}^{m-1}D(x_{i},x_{i+1})\right)\leq
f\left(\frac{k^{n}}{1-k}D(x_{0},x_{1})\right)<f(\varepsilon)-\alpha,\quad
m>n\geq N.$

Using (D3) and (5.4), we obtain

$D(x_{n},x_{m})>0,\,m>n\geq N\implies f(D(x_{n},x_{m}))\leq
f\left(\sum_{i=n}^{m-1}D(x_{i},x_{i+1})\right)+\alpha<f(\varepsilon),$

which implies by ($\mathcal{F}_{1}$) that

$D(x_{n},x_{m})<\varepsilon,\quad m>n\geq N.$

This proves that $\\{x_{n}\\}$ is $\mathcal{F}$-Cauchy. Since $(X,D)$ is
$\mathcal{F}$-complete, there exists $x^{*}\in X$ such that $\\{x_{n}\\}$ is
$\mathcal{F}$-convergent to $x^{*}$, i.e.

(5.5) $\lim_{n\to+\infty}D(x_{n},x^{*})=0.$

We shall prove that $x^{*}$ is a fixed point of $g$. We argue by contradiction
by supposing that $D(g(x^{*}),x^{*})>0$. By (D3), we have

$f(D(g(x^{*}),x^{*}))\leq
f(D(g(x^{*}),g(x_{n}))+D(g(x_{n}),x^{*}))+\alpha,\quad n\in\mathbb{N}.$

Using (ii) and ($\mathcal{F}_{1}$), we obtain

$f(D(g(x^{*}),x^{*}))\leq f(kD(x^{*},x_{n})+D(x_{n+1},x^{*}))+\alpha,\quad
n\in\mathbb{N}.$

On the other hand, using ($\mathcal{F}_{2}$) and (5.5), we have

$\lim_{n\to+\infty}f(kD(x^{*},x_{n})+D(x_{n+1},x^{*}))+\alpha=-\infty,$

which is a contradiction. Therefore, we have $D(g(x^{*}),x^{*})=0$, i.e.
$g(x^{*})=x^{*}$. As consequence, $x^{*}\in X$ is the unique fixed point of
$g$. ∎

######  Corollary 5.1.

Let $(X,D)$ be an $\mathcal{F}$-metric space, and
$(f,\alpha)\in\mathcal{F}\times[0,+\infty)$ be such that (D3) is satisfied.
Let $g:B(x_{0},r)\to X$ be a given mapping, where $x_{0}\in X$ and $r>0$.
Suppose that the following conditions are satisfied:

  * (i)

(4.2) is satisfied.

  * (ii)

$(X,D)$ is $\mathcal{F}$-complete.

  * (iii)

There exists $k\in(0,1)$ such that

$D(g(x),g(y))\leq kD(x,y),\quad(x,y)\in B(x_{0},r)\times B(x_{0},r).$

  * (iv)

There exists $0<\varepsilon<r$ such that

$f\left(k\varepsilon+D(x_{0},g(x_{0}))\right)\leq f(\varepsilon)-\alpha.$

Then $g$ has a fixed point.

###### Proof.

Let $0<\varepsilon<r$ be such that (iv) is satisfied. First, we shall prove
that

(5.6) $g(\mathbf{B}(x_{0},\varepsilon))\subset\mathbf{B}(x_{0},\varepsilon).$

Let $x\in\mathbf{B}(x_{0},\varepsilon)$, i.e.

$D(x_{0},x)\leq\varepsilon.$

Suppose that $D(g(x),x_{0})>0$. By (D3), we have

$f(D(g(x),x_{0}))\leq f(D(g(x),g(x_{0}))+D(g(x_{0}),x_{0}))+\alpha.$

Using ($\mathcal{F}_{1}$), (iii), and (iv), we obtain

$\displaystyle f(D(g(x),x_{0}))$ $\displaystyle\leq$ $\displaystyle
f(kD(x,x_{0})+D(g(x_{0}),x_{0}))+\alpha$ $\displaystyle\leq$ $\displaystyle
f(k\varepsilon+D(g(x_{0}),x_{0}))+\alpha$ $\displaystyle\leq$ $\displaystyle
f(\varepsilon).$

Hence, by ($\mathcal{F}_{1}$), we have $D(g(x),x_{0})\leq\varepsilon$, which
yields $g(x)\in\mathbf{B}(x_{0},\varepsilon)$. Therefore, we proved (5.6).
Further, the mapping
$g:\mathbf{B}(x_{0},\varepsilon)\to\mathbf{B}(x_{0},\varepsilon)$ is well-
defined, and satisfies the Banach contraction. On the other hand, since (4.2)
is satisfied, by Proposition 4.3, we know that $\mathbf{B}(x_{0},\varepsilon)$
is $\mathcal{F}$-closed, so from (i), it is $\mathcal{F}$-complete. Finally
the result follows from Theorem 5.1. ∎

Acknowledgements. The second author extends his appreciation to Distinguished
Scientist Fellowship Program (DSFP) at King Saud University (Saudi Arabia).

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Mohamed Jleli
Department of Mathematics, College of Science, King Saud University, Riyadh
11451, Saudi Arabia.
E-mail<EMAIL_ADDRESS>

Bessem Samet
Department of Mathematics, College of Science, King Saud University, Riyadh
11451, Saudi Arabia.
E-mail<EMAIL_ADDRESS>