#  RIKEN-iTHEMS-Report-23
Deconfinement transition in the revolving bag model

Kazuya Mameda  Keiya Takizawa  Department of Physics, Tokyo University of
Science, Tokyo 162-8601, Japan  RIKEN iTHEMS, RIKEN, Wako 351-0198, Japan

###### Abstract

Based on the bag model, we revisit the deconfinement phase transition under
rotation. On top of the usual rotational energy for noninteracting particles,
we perturbatively analyze the revolution effect of the hadron bag, i.e., of
the potential confining quarks. The revolution effect can be
phenomenologically translated into the rotational correction to the QCD vacuum
energy or the gluon condensate. We demonstrate that if the revolution effect
is (is not) taken into account, the transition temperature increases
(decreases) as the angular velocity increased. The ‘revolving bag model’
provides a feasible explanation of the recent lattice simulations showing,
contrary to effective models, that rotation favors the confined phase.

###### keywords:

hadronic matter , quarks and gluons , deconfinement phase transition ,
rotation

##  1 Introduction

Motivated by the measurements of strong vorticities in heavy-ion collisions
[1], the thermodynamic properties of quantum chromodynamics (QCD) under
rotation have been one of the fascinating topics in high-energy physics.
Various effective models and perturbative approaches predict that rotation
promotes chiral symmetry restoration [2, 3, 4, 5, 6] and deconfinement [7, 8,
9, 10, 11, 12]. On the other hand, lattice QCD simulations cannot be directly
available for rotating systems, because of the sign problem [13].
Nevertheless, the simulation with an imaginary angular velocity could be
extrapolated to that with a real angular velocity, as long as the finite
system-size is properly taken into account [11]. Such analyses indicate, in
conflict with the model prediction, that rotation favors confinement [14, 15,
16, 17]. This result is also supported by a lattice simulation with the Taylor
expansion in terms of real angular velocity [18].

This discrepancy suggests missing pieces in effective models and perturbative
approaches under rotation. In usual arguments, the rotational energy of
noninteracting particles plays the most essential role to determine the
thermodynamics under rotation [19]. In other words, it is assumed that there
is no rotational modification in the interactions mediated by gluons, and
model parameters mimicking them. This could however be an oversimplification,
as is for the magnetic response of QCD [20, 21, 22, 23, 24, 25]. Rather,
rotation, unlike electromagnetic field, directly couples with arbitrary fields
including gluon and even ghost [26, 27, 28, 29]. It is hence conceivable that
under rotation, such overlooked ingredients revise the usual model prediction.

The purpose of this work is to phenomenologically verify the above scenario,
based on the bag model [30, 31], which incorporates the two fundamental
features of QCD, i.e., confinement and asymptotic freedom. In the bag model,
the deconfinement transition is determined through the pressure competition
involving the bag constant, i.e., the contribution from the QCD vacuum energy.
The value of the bag constant is characterized predominantly by the kinetic
energy of quarks confined inside a hadron bag [32, 33]. The most significant
perspective is that when rotation is applied to the bag model, it affects not
only the quarks but also the bag itself. Then, the rotation (more precisely,
revolution) of the confining potential modifies the energy spectra of the
quarks, and consequently the bag constant. This is a great advantage of the
‘revolving bag model’, compared with other models where details of
interactions are already integrated out and subsumed into model parameters
(see Ref. [34] for an exception). In this work, we show that the revolution
effect largely changes the critical temperature.

##  2 Pressures of noninteracting gases

The first building block of our model is the pressures of noninteracting
gases. To evaluate them, we consider the rotating cylindrical system with the
radius $R$ and rigid angular velocity $\boldsymbol{\omega}=\omega\hat{z}$.
Because of the relativistic causality constraint $\omega R\leq 1$, we need to
keep $R$ finite. Under the Dirichlet-type boundary conditions, the energy
dispersions for massless quarks ($i=q$), gluons ($i=g$) and pions ($i=\pi$)
are [35, 36, 37]

$\epsilon_{i}=\sqrt{\bigl{(}\xi_{\nu_{i},k}/R\bigr{)}^{2}+p_{z}^{2}}-\omega\mu_{i},$
(1)

where $p_{z}$ is the continuous momentum along the $z$-direction, and
$\mu_{i}$ is the total angular momentum;
$\mu_{q}=\pm\frac{1}{2},\pm\frac{3}{2},\cdots$ and $\mu_{g},\mu_{\pi}=0,\pm
1,\pm 2,\cdots$. We defined $\nu_{q}=|\mu_{q}|-\frac{1}{2}$, $\nu_{g}=\mu_{g}$
and $\nu_{\pi}=\mu_{\pi}$, and $\xi_{\nu,k}=\xi_{-\nu,k}$ is the $k$-th
smallest zero of the Bessel function of the first kind, $J_{\nu}(\xi)$. As
long as $\omega R\leq 1$ is imposed, the above dispersion is lower-bounded by
$\mu_{q}=\frac{1}{2}$, $\mu_{g}=0$ and $\mu_{\pi}=0$. Besides, while $\mu_{i}$
and $-\mu_{i}$ are degenerate for $\omega=0$, they are not for $\omega>0$.

In the standard manner, one straightforwardly derives the pressures. The only
difference is the transverse momentum phase space modified as [37]

$\int\frac{{\mathrm{d}}p_{x}{\mathrm{d}}p_{y}}{(2\pi)^{2}}\to\frac{1}{\pi
R^{2}}\sum_{\mu_{i},k}.$ (2)

Thus the pressures become

$P_{i}=\eta_{i}\,d_{i}\,T\int\frac{{\mathrm{d}}p_{z}}{2\pi}\frac{1}{\pi
R^{2}}\sum_{\mu_{i},k}\ln\Bigl{[}1+\eta_{i}\,{\mathrm{e}}^{-\epsilon_{i}/T}\Bigr{]}$
(3)

with $\eta_{q}=1$, $\eta_{g}=\eta_{\pi}=-1$,
$d_{q}=2d_{\mathrm{spin}}N_{\mathrm{f}}N_{\mathrm{c}}$,
$d_{g}=d_{\mathrm{spin}}(N_{\mathrm{c}}^{2}-1)$,
$d_{\pi}=N_{\mathrm{f}}^{2}-1$ and $d_{\mathrm{spin}}=2$. In this work, we
take $N_{\mathrm{f}}=2$ and $N_{\mathrm{c}}=3$, namely, $d_{q}=24$, $d_{g}=16$
and $d_{\pi}=3$. The pressures (3) are also reproduced from the radial
integration of the local expression in Ref. [7] (with discretized transverse
momenta). Figure 1 shows the $R$-dependence of $P_{i}$’s at $\omega=0$ and
$T=150\,\mathrm{MeV}$. The pressures are normalized by the corresponding
Stefan–Boltzmann (SB) limits:
$(P_{q}^{\mathrm{SB}},P_{g}^{\mathrm{SB}},P_{\pi}^{\mathrm{SB}})=(7d_{q}/8,d_{g},d_{\pi})\times\pi^{2}T^{4}/90$.
At large $R$, we confirm the convergent behavior to the SB limit. The
deviations from 1 are caused by the finite-size effect.

Figure 1: Pressures of noninteracting gases for $\omega=0$ in the finite-size
cylinder.

##  3 Quarks confined in a revolving bag

Let us now study the rotational effect on the hadron bag. For this purpose, we
consider a spherical bag embedded into the cylindrical system, as depicted in
Fig. 2. Their radii are $R_{\mathrm{b}}$ and $R$, respectively. The bag
revolves with angular velocity $\boldsymbol{\omega}=\omega\hat{z}$, so that it
corotates with the above noninteracting gases. We define $\rho\leq R$ as the
difference between the bag center and the $z$-axis. It is convenient to employ
$x^{\mu}$ being the coordinate comoving with the bag, instead of the inertial
coordinate $x^{\prime\mu}$. We set $(x,y)$ and $(x^{\prime},y^{\prime})$ to be
spanned on the same plane $z=0$, and $(x,y)=(0,0)$ is defined as the bag
center. Then, the coordinate transformation from $x^{\prime\mu}$ to $x^{\mu}$
is similar to that to the usual rigidly rotating coordinate. Hence, one can
adopt the following metric tensor $g_{\mu\nu}$ and vierbein ${e^{a}_{\mu}}$
[13]:

$\begin{split}&g_{tt}=1-\omega^{2}(X^{2}+y^{2}),\quad g_{tx}=\omega y,\quad
g_{ty}=-\omega X,\\\ &e^{0}_{t}=e^{1}_{x}=e^{2}_{y}=e^{3}_{z}=1,\quad
e^{0}_{x}=\omega y,\quad e^{0}_{y}=-\omega X\end{split}$ (4)

with $X:=x+\rho$. Other components of $g_{\mu\nu}$ and ${e^{a}_{\mu}}$ are
equivalent to the Minkowski metric $\eta_{\mu\nu}$ and zero, respectively.

We analyze the energy spectra of massless quarks confined inside the revolving
bag. In the bag coordinate $x^{\mu}$, the Dirac equation becomes
${\mathrm{i}}\partial_{t}\psi=(H_{0}+H_{1}+H_{2})\psi$ with

$\begin{split}&H_{0}=\gamma^{0}\bigl{[}-{\mathrm{i}}\,\boldsymbol{\gamma}\cdot\boldsymbol{\nabla}+U(r)\bigr{]},\quad
H_{1}=-\omega J_{z}\end{split}$ (5)

and

$\begin{split}H_{2}&={\mathrm{i}}\omega\rho\partial_{y}\\\
&\;=\sqrt{\frac{2\pi}{3}}\omega\rho\biggl{[}-(Y^{-1}_{1}+Y^{1}_{1})\,\partial_{r}\\\
&\qquad-\frac{1}{r}(Y^{-1}_{1}-Y^{1}_{1})L_{z}+\frac{1}{r}\frac{1}{\sqrt{2}}Y^{0}_{1}(L_{+}+L_{-})\biggr{]}.\end{split}$
(6)

Here $U(r)$ is the infinite spherical potential well at $r=R_{\mathrm{b}}$
with $r=|\boldsymbol{r}|$ and $\boldsymbol{r}=(x,y,z)$. This imitates the
confining potential. The angular momentum operators are
$\boldsymbol{J}=\boldsymbol{L}+\boldsymbol{S}$,
$\boldsymbol{L}=-{\mathrm{i}}\,\boldsymbol{r}\times\boldsymbol{\nabla}$,
$\boldsymbol{S}=\frac{{\mathrm{i}}}{2}\boldsymbol{\gamma}\times\boldsymbol{\gamma}$,
and $L_{\pm}=L_{x}\pm{\mathrm{i}}L_{y}$. Both $H_{1}$ and $H_{2}$ are regarded
as the rotational energy. While the former is from the angular momentum
$J_{z}$, the latter is due to $-{\mathrm{i}}\rho\partial_{y}$. We emphasize
that without $U(r)$, the latter could be eliminated by the translation along
the $x$-axis. In this model, hence, $H_{2}$ represents the bag revolution
effect, or the rotational correction to the confining potential.

Figure 2: Coordinate on the revolving spherical bag embedded in the cylinder.

For $\omega=0$, the Dirac equation is analytically solved [31] (see also Refs.
[32, 33]). In the Dirac representation, the positive-energy solution is
$\psi^{(0)}={\mathrm{e}}^{-{\mathrm{i}}E^{(0)}t}\varphi^{(0)}(\boldsymbol{r})$
with $E^{(0)}=p$ and

$\begin{split}&\varphi^{(0)}(\boldsymbol{r})=\frac{1}{\mathcal{N}}\begin{pmatrix}j_{\ell}(pr)\chi_{\kappa,\ell,\mu}(\hat{\boldsymbol{r}})\\\
{\mathrm{i}}\,\mathrm{sgn}(\kappa)j_{\ell^{\prime}}(pr)\chi_{-\kappa,\ell^{\prime},\mu}(\hat{\boldsymbol{r}})\end{pmatrix},\end{split}$
(7)

where $j_{\ell}(pr)$ is the spherical Bessel function. The factor
$\mathcal{N}$ is the normalization constant to fulfill
$\int{\mathrm{d}}^{3}\boldsymbol{r}\,\varphi^{{(0)}{\dagger}}\varphi^{(0)}=1$.
The quantum numbers are defined as $\boldsymbol{J}^{2}\psi=j(j+1)\psi$,
$J_{z}\psi=\mu\psi$, and $K\psi=-\kappa\psi$ with
$K=\gamma^{0}(\boldsymbol{L}\cdot\boldsymbol{S}+1)$,
$j=\frac{1}{2},\frac{3}{2},\cdots$,
$\mu=\pm\frac{1}{2},\pm\frac{3}{2},\cdots,\pm j$ and
$\kappa=\pm(j+\frac{1}{2})$. We introduced
$\ell=\ell^{\prime}+1=j+\frac{1}{2}=\kappa$ for $\kappa>0$ and
$\ell+1=\ell^{\prime}=j+\frac{1}{2}=-\kappa$ for $\kappa<0$. Also
$\chi(\hat{\boldsymbol{r}})$ with $\hat{\boldsymbol{r}}=\boldsymbol{r}/r$ is
the spinor spherical harmonics:

$\begin{split}&\chi_{\kappa,\ell,\mu}(\hat{\boldsymbol{r}})=\frac{1}{\sqrt{2\ell+1}}\begin{pmatrix}\mathcal{A}_{\kappa,\ell,\mu}\,Y_{\ell}^{\mu-\frac{1}{2}}(\hat{\boldsymbol{r}})\\\
\mathcal{B}_{\kappa,\ell,\mu}\,Y_{\ell}^{\mu+\frac{1}{2}}(\hat{\boldsymbol{r}})\end{pmatrix},\end{split}$
(8)

where the coefficients are
$-\mathcal{A}_{\kappa>0,\ell,\mu}=\mathcal{B}_{\kappa<0,\ell,\mu}=\mathcal{C}^{-}_{\ell,\mu}$
and
$\mathcal{A}_{\kappa<0,\ell,\mu}=\mathcal{B}_{\kappa>0,\ell,\mu}=\mathcal{C}^{+}_{\ell,\mu}$
with $\mathcal{C}_{\ell,\mu}^{\pm}=\sqrt{\ell\pm\mu+1/2}$. Under the MIT
boundary condition, the radial momentum is discretized as
$p=\zeta_{k,\kappa,j}/R_{\mathrm{b}}$ with $\zeta=\zeta_{k,\kappa,j}$ being
the $k$-th smallest solution of

$j_{\ell}(\zeta)=-\mathrm{sgn}(\kappa)j_{\ell+1}(\zeta).$ (9)

The energy dispersion reads

$E_{N}^{(0)}=\frac{\zeta_{n}}{R_{\mathrm{b}}},\quad
N=\\{n,\mu\\}=\\{k,\kappa,j,\mu\\},$ (10)

which is degenerate in terms of $\mu$. There are two ground states as

$E_{\mathrm{g}}^{(0)}:=E_{N^{\pm}_{\mathrm{g}}}^{(0)}=\frac{\zeta_{n_{\mathrm{g}}}}{R_{\mathrm{b}}},\
\
N_{\mathrm{g}}^{\pm}=\\{n_{\mathrm{g}},\pm\tfrac{1}{2}\\}=\\{1,-1,\tfrac{1}{2},\pm\tfrac{1}{2}\\}.$
(11)

For $\omega\neq 0$, we need to implement a perturbation theory. We can treat
$H^{\prime}=H_{1}+H_{2}$ as the time-independent perturbation, under the
conditions $\omega R_{\mathrm{b}}\ll 1$ and $\omega\rho\leq\omega R\ll 1$. The
matrix element of $H^{\prime}$ is

$H^{\prime}_{MN}=\int_{r\,\leq
R_{\mathrm{b}}}{\mathrm{d}}^{3}\boldsymbol{r}\,\varphi^{{(0)}{\dagger}}_{M}H^{\prime}\varphi^{(0)}_{N},$
(12)

where $\varphi^{(0)}_{N}$ is given by Eq. (7). This is evaluated with the help
of the following angular integral formula [38]:

$\begin{split}&\int{\mathrm{d}}\Omega\,Y_{\ell_{1}}^{m_{1}}Y_{\ell_{2}}^{m_{2}}(Y_{\ell_{3}}^{m_{3}})^{*}\\\
&=\biggl{[}\frac{(2\ell_{1}+1)(2\ell_{2}+1)}{4\pi(2\ell_{3}+1)}\biggr{]}^{1/2}\,C^{\ell_{1}\ell_{2}\ell_{3}}_{000}C^{\ell_{1}\ell_{2}\ell_{3}}_{m_{1}m_{2}m_{3}}\end{split}$
(13)

with the Clebsch–Gordon coefficient

$C^{\ell_{1}\ell_{2}\ell_{3}}_{m_{1}m_{2}m_{3}}=\bigl{\langle}{\ell_{1}}\,{m_{1}};{\ell_{2}}\,{m_{2}}\big{|}{\ell_{1}}\,{\ell_{2}};{\ell_{3}}\,{m_{3}}\bigr{\rangle}.$
(14)

Because of the degeneracy in Eq. (10), the energy corrections are calculated
from the degenerate perturbation theory. Let $N_{1}$ and $N_{2}$ be labels
belonging to the same degenerate subspace, namely, $N_{1}=\\{n,\mu_{1}\\}$ and
$N_{2}=\\{n,\mu_{2}\\}$ for a given $n$. The first-order energy correction is
the eigenvalue of $H^{\prime}_{N_{1}N_{2}}$. From the selection rules imposed
by Eq. (LABEL:eq:three-Y), one finds that the matrix element is already
diagonalized as
$H^{\prime}_{N_{1}N_{2}}=(H_{1})_{N_{1}N_{2}}=-\omega\mu_{1}\delta_{\mu_{1}\mu_{2}}$.
Thus, we get

$E^{(1)}_{N}=-\omega\mu.$ (15)

This is the usual rotational energy, and $E^{(0)}_{N}+E^{(1)}_{N}$ becomes the
exact energy dispersion when $\rho=0$. One can prove that
$E^{(0)}_{N}+E^{(1)}_{N}>0$ holds for arbitrary $N$, as long as $\omega
R_{\mathrm{b}}\leq 1$ is imposed [39].

The degeneracy is completely resolved up to the first-order perturbation.
Hence, the second-order energy correction reads

$E^{(2)}_{N}=\sum_{\bar{n}\neq
n}\sum_{\bar{\mu}}\frac{H^{\prime}_{N\bar{N}}H^{\prime}_{\bar{N}N}}{E_{N}^{(0)}-E_{\bar{N}}^{(0)}}$
(16)

with $\bar{N}=\\{\bar{n},\bar{\mu}\\}$. The above summation over all states
for $\bar{n}\neq n$ excludes the diagonal element
$(H_{1})_{\bar{N}N}=-\omega\mu\,\delta_{\bar{n}n}\delta_{\bar{\mu}\mu}$. The
energy correction for the two unperturbed ground states $N_{\mathrm{g}}^{\pm}$
is numerically computed as

$E_{\mathrm{g}}^{(2)}:=E_{N^{\pm}_{\mathrm{g}}}^{(2)}=-\frac{\alpha}{R_{\mathrm{b}}}(\omega\rho)^{2},\quad\alpha\simeq
1.07.$ (17)

The two states $N_{\mathrm{g}}^{\pm}$ receive the same second-order
correction. This is because $\omega^{2}$ does not distinguish the spin-up and
spin-down, while the first-order correction (15) does.

##  4 Bag constant

In the bag model, the proton ground-state energy $E_{p}$ is described as

$E_{p}=E_{q}+E_{\mathrm{other}}+\frac{4\pi}{3}R_{\mathrm{b}}^{3}b.$ (18)

Here $E_{q}$ is the kinetic energy of three ground-state quarks and
$E_{\mathrm{other}}$ contains the interaction among quarks, the Casimir
energy, the effect of meson clouds etc. The last term is the bulk energy with
the bag constant $b$. For $\omega=0$, we identify
$E_{q}=3E^{(0)}_{\mathrm{g}}$ and $E_{p}=M_{p}$ with the proton mass $M_{p}$,
and parametrize $E_{\mathrm{other}}=Z/R_{\mathrm{b}}$ for the dimensional
reason.

To generalize Eq. (18) to the finite $\omega$ case, we implement two
assumptions. First, since $E_{q}$ is more dominant than $E_{\mathrm{other}}$,
it is likely that the same is true for their rotational corrections. As the
first attempt to pursue the revolution effect of the bag model, we here assume
$E_{\mathrm{other}}$ to be unchanged under rotation. Second, the spin-rotation
coupling contributes to the ground-state energy of protons, unlike the orbit-
rotation one111 Strictly speaking, because of $\omega R\leq 1$, the spectrum
should become gapped as in Eq. (1). This is however irrelevant to the
following analysis. Indeed, the ground state is still the mode with zero
orbital angular momentum, and the finite-size effect is of $O((M_{p}R)^{-2})$,
which is negligible for our parameters leading to $M_{p}R=50$. . It is then
seems that we face the difficult question of how to construct the
spin-$\frac{1}{2}$ proton with three spin-$\frac{1}{2}$ quarks and spin-$1$
gluons [40]. However, a term linear in $\omega$ is prohibited from emerging in
Eq. (18), because $b$ is a pressure, a time-reversal even quantity. Therefore,
it is necessary that the spin-rotation coupling of the proton are totally
canceled by that of quarks, i.e., Eq. (15) (and by that from
$E_{\mathrm{other}}$, if any). Although the underlying physics of the
cancelation is elusive in the present phenomenology, we just suppose it.

Eventually, the only relevant modification in Eq. (18) is the second-order
energy correction (17). Replacing $E^{(0)}_{\mathrm{g}}$ with
$E^{(0)}_{\mathrm{g}}+E^{(2)}_{\mathrm{g}}$, we arrive at

$M_{p}=\frac{C-3\alpha(\omega\rho)^{2}}{R_{\mathrm{b}}}+\frac{4\pi}{3}R_{\mathrm{b}}^{3}b.$
(19)

We introduced the parameter $C=3\zeta_{n_{\mathrm{g}}}+Z>0$, which we will
choose so that an appropriate critical temperature is reproduced for
$\omega=0$. Minimizing $M_{p}$ with respect to $R_{\mathrm{b}}$, we identify
$b=b(\rho)$, which is the bag pressure balanced by the internal energy. Up to
$O((\omega\rho)^{2})$, this is

$b(\rho)=B_{0}\biggl{[}1+\frac{9\alpha}{C}(\omega\rho)^{2}\biggr{]},$ (20)

where $B_{0}:=C/(4\pi R_{\mathrm{b}0}^{4})$ with
$R_{\mathrm{b}0}:=4C/(3M_{p})$ is the bag constant for $\omega=0$.

For later convenience, we also derive the global bag constant $B$ from
$b(\rho)$. Then, we need the radial distribution of the bag inside the
cylinder with radius $R$. In the simplest case with the homogeneously
distributed bag, the bag constant is

$B=\frac{1}{\pi
R^{2}}\int_{0}^{R}2\pi\rho\,{\mathrm{d}}\rho\,b(\rho)=B_{0}\biggl{[}1+\frac{9\alpha}{2C}(\omega
R)^{2}\biggr{]}.$ (21)

This is a presumable expression because $\omega R$ is the quantity to
characterize the strength of rotation. If the bag is distributed
inhomogeneously, $B$ is of course modified. Nevertheless, owing to the
centrifugal force, such a distribution should be a monotonically increasing
function of $\rho$. As a result, the correction in $B$ should become larger.
In other words, Eq. (21) is the form involving the minimal revolution effect
of the bag, and we utilize it in the following analysis. Also we demonstrate
that even such a minimal effect drastically affects the phase transition.

By the definition of $B$, Eq. (21) hints that the QCD vacuum is affected by
rotation, in sharp contrast with the vacuum of noninteracting systems [37]. It
is difficult to validate this fact within the bag model. Focusing on energy
spectra, however, we can phenomenologically examine the rotational response of
the QCD vacuum. In general, the rotational effect becomes visible even at zero
temperature and zero density, if the rotational energy overcomes the infrared
energy gap [37]. Unlike in noninteracting systems, such a situation is
realized in magnetized systems [2, 41], where the Landau quantization pushes
down the gap [42]. A similar argument could be applicable to the QCD vacuum,
which is effectively viewed as a color-magnetic medium; this interpretation
explains asymptotic freedom and qualitatively illustrates the vacuum energy
[43, 44, 45, 32, 46]. Therefore, the strong interaction enables the QCD vacuum
to receive the visible rotational effect. The above picture matches the
intuition that while the perturbative vacuum has nothing to rotate, the QCD
vacuum is a nonempty state.

##  5 Critical temperature

In the bag model, the critical temperature $T_{\mathrm{c}}(\omega)$ is the
solution of

$\begin{split}&P_{\pi}(T,\omega)=P_{q}(T,\omega)+P_{g}(T,\omega)-B(\omega),\end{split}$
(22)

where $P_{i}$ and $B$ are given by Eqs. (3) and (21), respectively. For
simplicity, we ignored other subdominant excitations in the hadronic phase.
Instead of Eq. (LABEL:eq:P_balance), one can determine $T_{\mathrm{c}}$ from
the local pressure competition, as in Ref. [7]. However, we have numerically
checked that the qualitative behavior of $T_{\mathrm{c}}$ is irrelevant to the
radial position, except for the vicinities of the center and the boundary
edge. For this reason, we focus on the global pressure competition
(LABEL:eq:P_balance).

Figure 3:  Critical temperature $T_{\mathrm{c}}$ as a function of angular
velocity $\omega$ for the homogeneously distributed bag.

Figure 3 shows the numerical solutions. The solid line is
$T_{\mathrm{c}}(\omega)$ from the revolving bag model with $B(\omega)$. The
dash line is the result from the usual static bag, namely, the solution of Eq.
(LABEL:eq:P_balance) under the replacement $B(\omega)\to B_{0}$. We set
$R=10\,\mathrm{fm}$, $M_{p}=1\,\mathrm{GeV}$ and $C=2.51$, which lead to the
standard value $B_{0}^{1/4}\simeq 200\,\mathrm{MeV}$. For $\omega=0$, the
solution is $T_{\mathrm{c}}(\omega=0)\simeq 146\,\mathrm{MeV}$. This is, due
to the finite-size effect (see Fig. 1), slightly larger than that in the SB
limit:
$T_{\mathrm{c}}^{\mathrm{SB}}=[(\frac{7}{8}d_{q}+d_{g}-d_{\pi})\pi^{2}/90]^{-1/4}B_{0}^{1/4}\simeq
144\,\mathrm{MeV}$.

For $\omega\neq 0$, the behavior of $T_{\mathrm{c}}$ strongly depends on
whether the bag revolution is taken into account or not. For the static bag
with $B_{0}$, rotation decreases $T_{\mathrm{c}}$. The underlying reason is
clarified from the $\omega$-dependence of $P_{i}$. Up to $O(\omega^{2})$, the
pressure correction $\Delta P_{i}:=P_{i}(T,\omega)-P_{i}(T,0)$ is written as
the familiar form of the rotational energy density of rigid body:

$\Delta P_{i}=\frac{1}{2V}\omega^{2}I_{i},\quad
I_{i}:=V\frac{{\mathrm{d}}^{2}P_{i}(T,0)}{{\mathrm{d}}\omega^{2}}>0,$ (23)

where $I_{i}$ is the moment of inertia with the system volume $V=\pi
R^{2}\int{\mathrm{d}}z$. Also $I_{\pi}<I_{q}+I_{g}$ holds because of the large
difference of $d_{i}$’s. In Eq. (LABEL:eq:P_balance) with $\omega\neq 0$,
hence, the net pressure correction $\Delta P_{\mathrm{net}}:=\Delta
P_{q}+\Delta P_{g}-\Delta P_{\pi}>0$ should be compensated by the suppression
of the temperature, i.e., $T_{\mathrm{c}}(\omega\neq
0)<T_{\mathrm{c}}(\omega=0)$.

The revolving bag with $B(\omega)$ yields the opposite behavior, namely, the
enhancement of $T_{\mathrm{c}}$ by arbitrary $\omega\neq 0$. In the revolving
bag case, Eq. (LABEL:eq:P_balance) receives the modification not only from
$\Delta P_{\mathrm{net}}$ but also from $\Delta
B:=B(\omega)-B_{0}=9B_{0}\alpha(\omega R)^{2}/(2C)$. From the same argument as
before, one finds that $T_{\mathrm{c}}(\omega\neq 0)>T_{\mathrm{c}}(\omega=0)$
is a consequence of the competition $\Delta P_{\mathrm{net}}<\Delta B$. The
origin of $\Delta B$ is the rotational effect on the confining potential,
which emulates an interaction mediated by gluons. This is consistent with the
following observation in Fig 3; the revolving (static) bag supports the
lattice simulations (the effective approaches), where the rotational effect on
the nonperturbative gluonic contribution is (is not) taken into account.

Another interpretation of our result is found from the relation between the
bag constant and the QCD vacuum energy. Let us rewrite $\Delta
P_{\mathrm{net}}<\Delta B$ into

$I_{q}+I_{g}-I_{\pi}<I_{\mathrm{b}}:=\frac{18\alpha}{C}\int{\mathrm{d}}VB_{0}\rho^{2}.$
(24)

The above $I_{\mathrm{b}}$ is the moment of inertia of the QCD vacuum energy,
which is dominated by the gluon condensate [47, 48] through the trace anomaly.
Therefore, for the rotational enhancement of $T_{\mathrm{c}}$, it is essential
that the gluon condensate bears a large positive moment of inertia (see Ref.
[49, 50] for a similar but different argument). Such a situation would also be
realized in lattice QCD simulations under rotation.

##  6 Summary

We analyzed the deconfinement transition of the bag model under rotation. The
finding is that the critical temperature is enhanced with angular velocity
increased, as a result of the revolution of the confining potential, or more
fundamentally, the moment of inertia of the gluon condensate. This plausibly
expounds the lattice QCD simulations indicating that rotation favors
confinement. In our model, we adopt a homogeneous bag distribution as the
simplest case, which involves the minimal revolution effect. In realistic
cases with an inhomogeneous bag distribution, hence, the confined phase should
be more favored by rotation. Our result also exhibits the importance of more
rudimentary studies, such as the field-theoretical perturbation theory under
rotation [51]. Furthermore, nonperturbative analyses are ineludible in order
to rigorously uncover the QCD vacuum property under rotation.

## Acknowledgements

The authors thank Yuki Fujimoto, Xu-Guang Huang, Takeshi Morita, Artem Roenko
and Katsuhiko Suzuki for valuable comments.

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# The Beta Flexible Weibull Distribution

Beih S. El-Desouky, Abdelfattah Mustafa111Corresponding author:
<EMAIL_ADDRESS>and Shamsan AL-Garash

Department of Mathematics, Faculty of Science, Mansoura University, Mansoura
35516, Egypt.

###### Abstract

We introduce in this paper a new generalization of the flexible Weibull
distribution with four parameters. This model based on the Beta generalized
(BG) distribution, Eugene et al. [7], they first using the BG distribution for
generating new generalizations. This new model is called the beta flexible
Weibull BFW distribution. Some statistical properties such as the mode, the
$r$th moment, skewness and kurtosis are derived. The moment generating
function and the order statistics are obtained. Moreover, the estimations of
the parameters are given by maximum likelihood method and the Fisher’s
information matrix is derived. Finally, we study the advantage of the BFW
distribution by an application using real data set.

Keywords: Beta Flexible Weibull; Beta Generalized distribution; Modified
Flexible Weibull Distribution; Beta Weibull Distribution; Maximum Likelihood
Method.

2010 MSC: 60E05, 62N05, 62H10

##  1 Introduction

In recent years appeared many generalizations which based on the Weibull (WD)
distribution, Weibull[20], such as Modified Weibull (MW) distribution
submitted by Lai et al. [10], and Sarhan et al. [16], generalized modified
Weibull (GMW) distribution, Carrasco et al. [4]. Kumaraswamy Weibull (KW)
distribution, Cordeiro et al. [5], exponentiated modified Weibull extension
(EMWE) distribution, Sarhan et al. [17]. and flexible Weibull extension (FWE)
distribution, introduced by Bebbington et al. [2].

We said a random variable $X$ has a flexible Weibull (FW) distribution with
two parameter, if it’s cumulative distribution function (CDF) is given as
follows

$F_{FW}(x;\alpha,\beta)=1-\exp\left\\{-e^{\alpha
x-\frac{\beta}{x}}\right\\},\quad\alpha,\beta\;\text{and}\;\;x>0.$ (1.1)

Moreover the probability density function (PDF) corresponding Eq. (7) is given
by

$f_{FW}(x;\alpha,\beta)=(\alpha+\frac{\beta}{x^{2}})e^{\alpha
x-\frac{\beta}{x}}\exp\left\\{-e^{\alpha x-\frac{\beta}{x}}\right\\},\quad
x>0.$ (1.2)

Some generalizations of the flexible Weibull distribution were discussed
recently, such as exponentiated flexible Weibull extension (EFWE)
distribution, Weibull generalized flexible Weibull extension (WG-FWE)
distribution, Mustafa et al. [11], kumaraswamy flexible Weibull extension
(KFWE) distribution, El Damcese et al. [6], inverse flexible Weibull extension
(IFWE) distribution and exponentiated generalized flexible Weibull extension
(EG-FWE) distribution, Mustafa et al. [12].

In this paper, we give a new generalization of the flexible Weibull
distribution. This new generalization is called beta flexible Weibull (BFW)
distribution, using class of beta generalized (BG) distribution, Eugene et al.
[7]. By substituting $F(x,\varphi)$ given in Eq. (1.3) to be
$F_{FW}(x,\alpha,\beta)$ given in Eq. (7).
If $F(x,\varphi)$ the baseline CDF of a random variable, then the beta
generalized distribution is given by

$F_{BG}(x;p,q,\varphi)=I_{F(x;\varphi)}(p,q)=\frac{1}{B(p,q)}\int\limits_{0}^{F(x,\varphi)}u^{p-1}(1-u)^{q-1}du,\quad
p,q>0,$ (1.3)

where $\varphi$ is the parameter vector,
$I_{y}(p,q)=\frac{B_{y}(a,q)}{B(a,q)}$ is the incomplete beta function ratio,
and

$B_{y}(p,q)=\int\limits_{0}^{y}u^{p-1}(1-u)^{q-1}du.$

The PDF of the BG distribution corresponding Eq. (1.3) is

$f_{BG}(x;p,q,\varphi)=\frac{f(x;\varphi)}{B(p,q)}F(x;\varphi)^{p-1}\left[1-F(x;\varphi)\right]^{q-1}.$
(1.4)

The reliability function and failure rate function of the BG distribution
corresponding Eq.(1.3) are given, respectively, by the relations

$S_{BG}(x;p,q,\varphi)=1-F_{BG}(x;p,q,\varphi)=1-I_{F(x;\varphi)}(p,q),$ (1.5)

and

$h_{BG}(x;p,q,\varphi)=\frac{f(x;\varphi)F(x;\varphi)^{p-1}\left[1-F(x;\varphi)\right]^{q-1}}{B(p,q)\left[1-I_{F(x;\varphi)}(p,q)\right]}.$
(1.6)

Using expression $B(p,q)=\frac{\Gamma(p).\Gamma(q)}{\Gamma(p+q)}$, and the
series representation

$(1-u)^{q-1}=\sum_{i=0}^{q-1}\frac{(-1)^{i}\Gamma(q)}{i!\Gamma(q-i)}u^{i},\quad\text{for
real non-integer}\quad|u|<1,\quad q>0,$

the CDF of the BG distribution in Eq. (1.3) can be rewritten as follows

$F_{BG}(x;p,q,\varphi)=\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[F(x;\varphi)\right]^{p+i}}{i!(p+i)\Gamma(q-i)},\quad
p,q>0.$ (1.7)

Recently the class of BG distribution Eq. (1.3) has been receiving
considerable attention. This distribution was firstly studied by Eugene et al.
[7]. Many researchers considered different forms of distribution function
$F(x,\varphi)$ given in Eq. (1.3) and studied their properties. Beta normal
(BN) distribution, Eugene et al. [7], beta gumbel (BG) distribution has been
introduced by Nadarajah and Kotz [13], beta Weibull (BW) distribution, Famoye
et al. [8], beta modified Weibull (BMW) distribution, Silva et al. [19] and
beta exponential (BE) distribution, Nadarajah et al. [14].

This paper is arranged as follows, the distribution function, density function
and failure rate function of the BFW distribution are defined in Section 2.
Some statistical properties including, the mode, $rth$ moment, skewness and
kurtosis are presented in Sections 3. The moment generating function is
derived in Section 4. The order statistics is discussed in Section 5. The
maximum likelihood estimation MLEs of the parameters is obtained in Section 6.
Real data set are analyzed in Section 7. Moreover, we discuss the results and
compare it with existing distributions. Finally, we give a conclusion in
Section 8.

##  2 The Beta Flexible Weibull Distribution

We define the BFW($\alpha,\beta,p,q$) distribution, by substituting
$F(x,\varphi)$ in Eq. (1.7) to be the distribution function
$F_{FW}(x;\alpha,\beta)$ of the FWD in Eq. (7). So the CDF of the Beta
Flexible Weibull distribution is

$F_{BFW}(x;\eta)=\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p+i}}{i!\Gamma(q-i)(p+i)},\;\forall x>0,$ (2.1)

where $\eta=(\alpha,\beta,p,q)$, $\alpha,\beta,p>0$ and $q>0$.
Replacing $F_{FW}(x;\alpha,\beta)$ in Eq. (7) and $f_{FW}(x;\alpha,\beta)$ in
Eq. (1.2) by the $F(x,\varphi)$ and $f(x;\varphi)$ in Eq. (1.4), respectively,
the PDF corresponding Eq.(2.1) can be obtained as follows

$f_{BFW}(x;\eta)=\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-qe^{\alpha x-\frac{\beta}{x}}}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}.$ (2.2)

The reliability function $S_{BFW}(x)$ and failure rate $h_{BFW}(x)$ function
of $X$ having the BFWD$(\eta)$, respectively, are given by

$S_{BFW}(x;\eta)=1-\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p+i}}{i!\Gamma(q-i)(p+i)},\;\forall x>0,$ (2.3)

and

$h_{BFW}(x;\eta)=\frac{\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-qe^{\alpha x-\frac{\beta}{x}}}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}}{1-\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p+i}}{i!\Gamma(q-i)(p+i)}}.$ (2.4)

Also, the reversed failure rate $r_{BFW}(x)$ and cumulative-failure rate
$H_{BFW}(x)$ functions of $X$ having the BFWD$(\eta)$, respectively, are given
by

$r_{BFW}(x;\eta)=\frac{\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-qe^{\alpha x-\frac{\beta}{x}}}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}}{\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p+i}}{i!\Gamma(q-i)(p+i)}},$ (2.5)

and

$H_{BFW}(x;\eta)=\int_{0}^{x}h_{BFW}(t)dt=\int_{0}^{x}\frac{\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{t^{2}}\right)e^{\alpha
t-\frac{\beta}{t}}e^{-qe^{\alpha t-\frac{\beta}{t}}}\left[1-e^{-e^{\alpha
t-\frac{\beta}{t}}}\right]^{p-1}}{1-\frac{\Gamma(p+q)}{\Gamma(p)}\sum_{i=0}^{q-1}\frac{(-1)^{i}\left[1-e^{-e^{\alpha
t-\frac{\beta}{t}}}\right]^{p+i}}{i!\Gamma(q-i)(p+i)}}dt.$ (2.6)

Figures 1, 2 and 3 display functions the CDF, PDF, $S_{BFW}(x)$, $h_{BFW}(x)$,
$r_{BFW}(x)$ and $H_{BFW}(x)$ of the BFWD$(\eta)$ for different values of
parameters.

Figure 1: Shows the CDF and PDF of the BFWD for different values of parameters

Figure 2: Shows the $S_{BFW}(x)$ and $h_{BFW}(x)$ of the BFWD for different
values of parameters

Figure 3: Shows the $r_{BFW}(x)$ and $H_{BFW}(x)$ of the BFWD for different
values of parameters

##  3 Statistical Properties

Some statistical properties for the BFWD$(\eta)$, such as the mode, the $r$th
moment, skewness and kurtosis are given as follows.

###  3.1 The Mode of the BFWD

The mode of the BFWD$(\eta)$ can be obtained by differentiating its
probability density function with respect to $x$ in Eq. (2.2) and equating it
to zero.

$f^{{}^{\prime}}(x;\eta)=0.$

So the mode of the BFWD$(\eta)$ is solution of the following relation

$\displaystyle\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}e^{\alpha
x-\frac{\beta}{x}}e^{-qe^{\alpha x-\frac{\beta}{x}}}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}\times$
$\displaystyle\left\\{\frac{-2\beta}{x^{3}}+(\alpha+\frac{\beta}{x^{2}})^{2}\left[1-e^{\alpha
x-\frac{\beta}{x}}\left(q+\frac{(p-1)}{e^{e^{\alpha
x-\frac{\beta}{x}}}-1}\right)\right]\right\\}=0.$ (3.1)

The BFWD$(\eta)$ has exactly only one peak, so this generalization is a
unimodal. Figure (2) shows that Eq. (3.1) has only one solution. We can solve
it numerically.

###  3.2 The Moments

The $rth$ moment for BFWD$(\eta)$ is given by Theorem (3.1). The gamma
function with negative integer is defined by

$\Gamma(-r)=\frac{(-1)^{r}}{r!}\left[\phi(r)-\gamma\right],$

where $\gamma$ denote Euler’s constant which is defined as

$\gamma=\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(\frac{1}{k}-\ln(n)\right),\qquad\phi(r)=\sum_{i=1}^{r}\frac{1}{i},$

see Fisher and Kilicman [9].

######  Theorem 3.1.

If $X$ random variable having the BFWD$(\eta)$, so the $r$th moment of $X$, is
given by

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(p+q)}{\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{(-1)^{n+m}(q+n)^{m}(m+1)^{r+2\ell+1}\alpha^{\ell}\beta^{r+\ell+1}}{n!m!\ell!\Gamma(p-n)}\times$
(3.2) $\displaystyle\hskip
85.35826pt\left[\alpha\Gamma(-r-\ell-1)+\frac{\Gamma(-r-\ell+1)}{(m+1)^{2}\beta}\right].$

###### Proof.

The $r$th moment of the random variable $X$ with the PDF is given by

$\mu_{r}^{{}^{\prime}}=\int\limits_{0}^{\infty}x^{r}f(x)dx.$ (3.3)

Using PDF for the BFWD$(\eta)$ from Eq. (2.2) into Eq. (3.3) we get

$\mu_{r}^{{}^{\prime}}=\int_{0}^{\infty}x^{r}\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{x^{2}}\right)e^{\alpha
x-\frac{\beta}{x}}e^{-qe^{\alpha x-\frac{\beta}{x}}}\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}dx,$

where the $\left[1-e^{-e^{\alpha x-\frac{\beta}{x}}}\right]^{p-1}$ can be
written as

$\left[1-e^{-e^{\alpha
x-\frac{\beta}{x}}}\right]^{p-1}=\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(p)}{n!\Gamma(p-n)}e^{-ne^{\alpha
x-\frac{\beta}{x}}},$

then we get

$\mu_{r}^{{}^{\prime}}=\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\sum_{n=0}^{\infty}\frac{(-1)^{n}\Gamma(p)}{n!\Gamma(p-n)}\int_{0}^{\infty}x^{r}\left[\alpha+\frac{\beta}{x^{2}}\right]e^{\alpha
x-\frac{\beta}{x}}e^{-(q+n)e^{\alpha x-\frac{\beta}{x}}}dx,$

the function $e^{-(q+n)e^{\alpha x-\frac{\beta}{x}}}$, can be written as

$e^{-(q+n)e^{\alpha
x-\frac{\beta}{x}}}=\sum_{m=0}^{\infty}\frac{(-1)^{m}(q+n)^{m}}{m!}e^{m(\alpha
x-\frac{\beta}{x})},$

we obtain

$\mu_{r}^{{}^{\prime}}=\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{(-1)^{n+m}\Gamma(p)(q+n)^{m}}{n!m!\Gamma(p-n)}\int_{0}^{\infty}x^{r}\left[\alpha+\beta
x^{-2}\right]e^{(m+1)\alpha x}e^{-(m+1)\frac{\beta}{x}}dx,$

using series expansion of $e^{(m+1)\alpha x}$,

$e^{(m+1)\alpha
x}=\sum_{\ell=0}^{\infty}\frac{(m+1)^{\ell}\alpha^{\ell}x^{\ell}}{\ell!},$

we have

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{(-1)^{n+m}\Gamma(p)(q+n)^{m}(m+1)^{\ell}\alpha^{\ell}}{n!m!\ell!\Gamma(p-n)}\times$
$\displaystyle\hskip 85.35826pt\left[\int_{0}^{\infty}\alpha
x^{r+\ell}e^{-(m+1)\frac{\beta}{x}}dx+\int_{0}^{\infty}\beta
x^{r+\ell-2}e^{-(m+1)\frac{\beta}{x}}dx\right],$

where gamma function is defined by ( see Zwillinger [21]),

$\Gamma(\nu)=x^{\nu}\int_{0}^{\infty}e^{-tx}t^{\nu-1}dt,\quad x>0.$

Finally the $r$th moment of the BFWD$(\eta)$ can be obtained as follows

$\displaystyle\mu_{r}^{{}^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(p+q)}{\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}\frac{(-1)^{n+m}(q+n)^{m}(m+1)^{r+2\ell+1}\alpha^{\ell}\beta^{r+\ell+1}}{n!m!\ell!\Gamma(p-n)}\times$
$\displaystyle\hskip
85.35826pt\left[\alpha\Gamma(-r-\ell-1)+\frac{\Gamma(-r-\ell+1)}{(m+1)^{2}\beta}\right].$

This completes the proof.

∎

Furthermore, using the first four moments in Eq. (3.2), the measures of
skewness and kurtosis of the BFWD$(\eta)$ can be expressed in the following
relations, (see Bowley [3]).
The skewness of the BFWD$(\eta)$

$Sk(X)=\frac{E(X^{3})-3\mu E(X^{2})+2\mu^{3}}{\sigma^{3}}.$ (3.4)

The kurtosis of the BFWD$(\eta)$

$Ku(X)=\frac{E(X^{4})-4\mu E(X^{2})+6\mu^{2}E(X^{2})-3\mu^{4}}{\sigma^{4}}.$
(3.5)

##  4 The Moment Generating Function

We derive the moment generating function MGF of the BFWD$(\eta)$ by Theorem
(4.1) .

######  Theorem 4.1.

If $X$ is a random variable having the BFWD$(\eta)$, so the moment generating
function $MGF$ of $X$ is given by

$\displaystyle M_{X}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(p+q)}{\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{r=0}^{\infty}\frac{(-1)^{n+m}(q+n)^{m}(m+1)^{r+2\ell+1}\alpha^{\ell}\beta^{r+\ell+1}t^{r}}{n!m!\ell!r!\Gamma(p-n)}\times$
(4.1) $\displaystyle\hskip
85.35826pt\left[\alpha\Gamma(-r-\ell-1)+\frac{\Gamma(-r-\ell+1)}{(m+1)^{2}\beta}\right].$

###### Proof.

The moment generating function $MGF$ of the random variable $X$ with the PDF
is

$M_{X}(t)=\int\limits_{0}^{\infty}e^{tx}f(x)dx.$ (4.2)

Since

$e^{tx}=\sum_{r=0}^{\infty}\frac{t^{r}x^{r}}{r!},$

then we get

$M_{X}(t)=\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\int_{0}^{\infty}x^{r}f(x)dx.=\sum_{r=0}^{\infty}\frac{t^{r}}{r!}\mu_{r}^{{}^{\prime}}.$
(4.3)

Using $\mu_{r}^{{}^{\prime}}$ from Eq. (3.2) into Eq. (4.3) the $MGF$ of the
BFWD$(\eta)$ can be obtained as follows

$\displaystyle M_{X}(t)$ $\displaystyle=$
$\displaystyle\frac{\Gamma(p+q)}{\Gamma(q)}\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{r=0}^{\infty}\frac{(-1)^{n+m}(q+n)^{m}(m+1)^{r+2\ell+1}\alpha^{\ell}\beta^{r+\ell+1}t^{r}}{n!m!\ell!r!\Gamma(p-n)}\times$
$\displaystyle\hskip
85.35826pt\left[\alpha\Gamma(-r-\ell-1)+\frac{\Gamma(-r-\ell+1)}{(m+1)^{2}\beta}\right].$

This completes the proof. ∎

##  5 The Order Statistics

Suppose $X_{1:n},X_{2:n},\cdots,X_{n:n}$ denote the order statistics obtained
from a random sample $X_{1}$, $X_{2}$, $\cdots$, $X_{n}$ taken from a
continuous population with distribution function $F_{BFW}(x;\eta)$ and density
function $f_{BFW}(x;\eta)$, so the PDF of $X_{r:n}$ can be written as

$f_{r:n}(x;\eta)=\frac{1}{B(r,n-r+1)}\left[F_{BFW}(x;\eta)\right]^{r-1}\left[1-F_{BFW}(x;\eta)\right]^{n-r}f_{BFW}(x;\eta),$
(5.1)

where $F_{BFW}(x;\eta)$ and $f_{BFW}(x;\eta)$ are the CDF and PDF of the
BFWD$(\eta)$ given by Eq. (2.1) and Eq. (2.2), respectively. Can be defined
first order statistics as $X_{1:n}=\min(X_{1},X_{2},\cdots,X_{n})$, and the
last order statistics as $X_{n:n}=\max(X_{1},X_{2},\cdots,X_{n})$, where
$0<F_{BFW}(x;\eta)<1,$ $\forall x>0.$
The binomial expansion of $[1-F_{BFW}(x;\eta)]^{n-r}$ can be expressed as

$\left[1-F_{BFW}(x;\eta)\right]^{n-r}=\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}[F_{BFW}(x;\eta)]^{i}.$ (5.2)

Using Eq. (5.2) in Eq. (5.1), we get

$f_{r:n}(x;\eta)=\frac{1}{B(r,n-r+1)}f_{BFW}(x;\eta)\sum_{i=0}^{n-r}\begin{pmatrix}n-r\\\
i\end{pmatrix}(-1)^{i}\left[F_{BFW}(x;\eta)\right]^{i+r-1}.$ (5.3)

Using Eq. (2.1) and Eq. (2.2) in Eq. (5.3), we obtain

$f_{r:n}(x;\eta)=\sum_{i=0}^{n-r}\frac{(-1)^{i}n!}{i!(r-1)!(n-r-i)!}\left[F_{BFW}(x,\eta)\right]^{i+r-1}f_{BFW}(x;\eta).$
(5.4)

Equation (5.4) shows that the $f_{r:n}(x;\eta)$ is the weighted average of the
BFWD$(\eta)$ with various shape parameters.

##  6 Parameters Estimation

Parameters estimation of the BFWD$(\alpha,\beta,p,q)$, point and interval
estimation are derived by using maximum likelihood estimation MLE method with
a complete sample.

###  6.1 Maximum likelihood estimation

Let $x_{1},x_{2},\cdots,x_{n}$ denote a random sample of complete data from
the BFWD$(\eta)$. The Likelihood function $L$ is defined as follows

$L=\prod_{i=1}^{n}f(x_{i};\vartheta).$ (6.1)

Using Eq.(2.2) in Eq.(6.1), we have

$L=\prod_{i=1}^{n}\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\left(\alpha+\frac{\beta}{x_{i}^{2}}\right)e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}e^{-qe^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left[1-e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right]^{p-1}.$

The log-likelihood function is

$\mathcal{L}=n\left[\ln\left(\frac{\Gamma(p+q)}{\Gamma(p).\Gamma(q)}\right)\right]+\sum_{i=1}^{n}\ln\left(\alpha+\frac{\beta}{x_{i}^{2}}\right)+\sum_{i=1}^{n}\left(\alpha
x_{i}-\frac{\beta}{x_{i}}\right)-q\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+(p-1)\sum_{i=1}^{n}\ln\left(1-e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right).$ (6.2)

The MLE of the parameters $\alpha,\beta,p$ and $q$ are obtained by
differentiating the function $\mathcal{L}$ in Equation (6.2) with respect to
$\alpha,\beta,p$ and $q$, then equating it to zero, as follows

$\displaystyle\frac{\partial\mathcal{L}}{\partial\alpha}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\frac{x_{i}^{2}}{\beta+\alpha
x_{i}^{2}}+\sum_{i=1}^{n}x_{i}-q\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+(p-1)\sum_{i=1}^{n}\frac{x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{e^{e^{\alpha x_{i}-\frac{\beta}{x_{i}}}}-1}=0,$
(6.3) $\displaystyle\frac{\partial\mathcal{L}}{\partial\beta}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{1}{\beta+\alpha
x_{i}^{2}}-\sum_{i=1}^{n}\frac{1}{x_{i}}+q\sum_{i=1}^{n}\frac{1}{x_{i}}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-(p-1)\sum_{i=1}^{n}\frac{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{x_{i}\left(e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}-1\right)}=0,$ (6.4)
$\displaystyle\frac{\partial\mathcal{L}}{\partial p}$ $\displaystyle=$
$\displaystyle n\psi_{p}(p+q)-n\psi(p)+\sum_{i=1}^{n}\ln\left(1-e^{-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\right)=0,$ (6.5)
$\displaystyle\frac{\partial\mathcal{L}}{\partial q}$ $\displaystyle=$
$\displaystyle n\psi_{q}(p+q)-n\psi(q)-\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}=0.$ (6.6)

The MLEs can be obtained by solving the equations, (6.3)-(6.6), numerically
for $\alpha,\beta,p$ and $q$ by using Mathcad or Maple software.

###  6.2 Asymptotic confidence bounds

The asymptotic confidence intervals can be obtained by using variance
covariance matrix $I^{-1}$, since the parameters ($\alpha,\beta,p,q$) are
positive and the $I^{-1}$ is inverse of the observed information matrix given
by

$\displaystyle\mathbf{I^{-1}}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}-\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial
p}&-\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial q}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial
p}&-\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial q}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial
p\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial
p\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial
p^{2}}&-\frac{\partial^{2}\mathcal{L}}{\partial p\partial q}\\\
-\frac{\partial^{2}\mathcal{L}}{\partial
q\partial\alpha}&-\frac{\partial^{2}\mathcal{L}}{\partial
q\partial\beta}&-\frac{\partial^{2}\mathcal{L}}{\partial q\partial
p}&-\frac{\partial^{2}\mathcal{L}}{\partial
q^{2}}\end{array}\right)^{-1}=\left(\begin{array}[]{cccc}var(\hat{\alpha})&cov(\hat{\alpha},\hat{\beta})&cov(\hat{\alpha},\hat{p})&cov(\hat{\alpha},\hat{q})\\\
cov(\hat{\beta},\hat{\alpha})&var(\hat{\beta})&cov(\hat{\beta},\hat{p})&cov(\hat{\beta},\hat{q})\\\
cov(\hat{p},\hat{\alpha})&cov(\hat{p},\hat{\beta})&var(\hat{p})&cov(\hat{p},\hat{q})\\\
cov(\hat{q},\hat{\alpha})&cov(\hat{q},\hat{\beta})&cov(\hat{q},\hat{p})&var(\hat{q})\end{array}\right),$
(6.15)

where

$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha^{2}}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{4}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}-q\sum_{i=1}^{n}x_{i}^{2}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}+(p-1)\sum_{i=1}^{n}x_{i}H_{i},$ (6.17)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial\beta}$
$\displaystyle=$
$\displaystyle-\sum_{i=1}^{n}\frac{x_{i}^{2}}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}+q\sum_{i=1}^{n}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}-(p-1)\sum_{i=1}^{n}H_{i},$ (6.18)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial p}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{\left[e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}-1\right]},$ (6.19)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\alpha\partial q}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}x_{i}e^{\alpha
x_{i}-\frac{\beta}{x_{i}}},$ (6.20)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta^{2}}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}\frac{1}{\left(\beta+\alpha
x_{i}^{2}\right)^{2}}-q\sum_{i=1}^{n}\frac{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{x_{i}^{2}}+(p-1)\sum_{i=1}^{n}\frac{H_{i}}{x_{i}^{2}},$
(6.21) $\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial p}$
$\displaystyle=$ $\displaystyle-\sum_{i=1}^{n}\frac{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{x_{i}\left[e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}-1\right]},$ (6.22)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial\beta\partial q}$
$\displaystyle=$ $\displaystyle\sum_{i=1}^{n}\frac{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}{x_{i}},$ (6.23)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial p^{2}}$ $\displaystyle=$
$\displaystyle n\frac{\partial^{2}}{\partial
p^{2}}\ln\left[\Gamma(p+q)\right]-n\frac{\partial^{2}}{\partial
p^{2}}\ln\left[\Gamma(p)\right],$ (6.24)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial p\partial q}$
$\displaystyle=$ $\displaystyle n\frac{\partial^{2}}{\partial p\partial
q}\ln\left[\Gamma(p+q)\right],$ (6.25)
$\displaystyle\frac{\partial^{2}\mathcal{L}}{\partial q^{2}}$ $\displaystyle=$
$\displaystyle n\frac{\partial^{2}}{\partial
q^{2}}\ln\left[\Gamma(p+q)\right]-n\frac{\partial^{2}}{\partial
q^{2}}\ln\left[\Gamma(q)\right],$ (6.26)

where

$H_{i}=\frac{x_{i}e^{\alpha x_{i}-\frac{\beta}{x_{i}}}\left[e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}\left(1-e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}\right)-1\right]}{\left[e^{e^{\alpha
x_{i}-\frac{\beta}{x_{i}}}}-1\right]^{2}}.$

Moreover, the $(1-\lambda)100\%$ confidence intervals of the four parameters
can be obtained by using variance matrix as follows

$\hat{\alpha}\pm
Z_{\frac{\lambda}{2}}\sqrt{var(\hat{\alpha})},\quad\hat{\beta}\pm
Z_{\frac{\lambda}{2}}\sqrt{var(\hat{\beta})},\quad\hat{p}\pm
Z_{\frac{\lambda}{2}}\sqrt{var(\hat{p})},\quad\hat{q}\pm
Z_{\frac{\lambda}{2}}\sqrt{var(\hat{q})},$

where $\lambda>0$, and $Z_{\frac{\lambda}{2}}$ denote the upper
$(\frac{\lambda}{2})$-th percentile of standard normal distribution.

##  7 Application

In this Section, we will analysis a real data set using the BFWD
$(\alpha,\beta,p,q)$ and compare it with the other fitted distributions such
as flexible Weibull extension distribution (FWED), Weibull distribution (WD),
Modified Weibull distribution (MWD), Reduced Additive Weibull distribution
(RAWD) and Extended Weibull distribution (EWD) by using some criteria
statistical such as K–S statistics ( Kolmogorov Smirnov), Akaike information
criterion $AIC$ and Akaike Information criterion with correction $AICC$, see
[1]. Moreover, we use Bayesian information criterion $BIC$, see [18].

We will use the data given by Salman et al. [15], in Table 1.

Table 1: Time between failures of secondary reactor pumps, (thousands of
hours) [15]

2.160 | 0.746 | 0.402 | 0.954 | 0.491 | 6.560 | 4.992 | 0.347
---|---|---|---|---|---|---|---
0.150 | 0.358 | 0.101 | 1.359 | 3.465 | 1.060 | 0.614 | 1.921
4.082 | 0.199 | 0.605 | 0.273 | 0.070 | 0.062 | 5.320 |

Table 2 gives MLEs of parameters of the BFWD$(\eta)$ and the Statistics K–S.
The values of $AIC$, $AICC$, $BIC$, $HQIC$ and the log-likelihood functions
$\mathcal{L}$ are in Table 3.

Table 2: MLEs and K–S of parameters for secondary reactor pumps.

Model | $\hat{\alpha}$ | $\hat{\beta}$ | $\hat{p}$ | $\hat{q}$ | K–S
---|---|---|---|---|---
FWD | 0.0207 | 2.5875 | – | – | 0.1342
WD | 0.8077 | 13.9148 | – | – | 0.1173
MWD | 0.1213 | 0.7924 | 0.0009 | – | 0.1188
RAWD | 0.0070 | 1.7292 | 0.0452 | – | 0.1619
EWD | 0.4189 | 1.0212 | 10.2778 | – | 0.1057
BFWD | 0.052 | 0.024 | 35.077 | 20.328 | 0.1151

Table 3: Log-likelihood $\mathcal{L}$, $AIC$, $AICC$,$BIC$ and $HQIC$ values
of models fitted.

Model | $\mathcal{L}$ |  -2 $\mathcal{L}$ | AIC | AICC | BIC | HQIC
---|---|---|---|---|---|---
FWD | -83.3424 | 166.6848 | 170.6848 | 171.2848 | 172.95579 | 171.2559
WD | -85.4734 | 170.9468 | 174.9468 | 175.5468 | 177.21779 | 175.5179
MWD | -85.4677 | 170.9354 | 176.9354 | 178.1986 | 180.34188 | 177.7921
RAWD | -86.0728 | 172.1456 | 178.1456 | 179.4088 | 181.55208 | 179.0023
EWD | -86.6343 | 173.2686 | 179.2686 | 180.5318 | 182.67508 | 180.1253
BFWD | -30.768 | 61.5360 | 69.5360 | 71.7582 | 74.0780 | 70.6783

We find that the BFWD$(\eta)$ with the parameters $(\alpha,\beta,p,q)$ gives a
better fit to data set than the previous generalizations of Weibull
distributions such as a flexible Weibull distribution FWD, Weibull
distribution WD, Modified Weibull distribution MWD, Reduced Additive Weibull
distribution RAWD and Extended Weibull distribution EWD. It has the largest
likelihood function $L$, and the smallest criteria values $AIC$, $AICC$,
$BIC$, $HQIC$ and K–S as shown in Tables 2 and 3.

Replace the MLEs of the unknown parameters for BFWD$(\alpha,\beta,p,q)$ into
Eq. (6.15), we can obtained estimation of the variance covariance matrix as
follows

$I_{0}^{-1}=\left(\begin{array}[]{cccc}2.123\times 10^{-3}&9.575\times
10^{-4}&-2.748&-1.6\\\ 9.575\times 10^{-4}&5.558\times 10^{-4}&-1.415&-0.81\\\
-2.748&-1.415&3.912\times 10^{3}&2.256\times 10^{3}\\\ -1.6&-0.81&2.256\times
10^{3}&1.304\times 10^{3}\\\ \end{array}\right).$

The confidence intervals for approximately 95% two sided of the unknown
parameters $\alpha,\beta,p$ and $q$ are $\left[0,0.142\right]$,
$\left[0,0.07\right]$, $\left[0,157.671\right]$ and $\left[0,91.105\right]$,
respectively.

From, Figures 4 and 5 can be seen that the log-likelihood function
$\mathcal{L}$ have unique solution.

Figure 4: The profile of the log-likelihood function of $\alpha,\beta$.

Figure 5: The profile of the log-likelihood function of $p$ and $q$.

Figure 6, represents the estimation for the reliability function $S(x)$, by
using the Kaplan-Meier method and its fitted parametric estimations when the
distribution is assumed to be FWD, WD, MWD, RAWD, EWD and BFWD are computed
and plotted in the following shape.

Figure 6: Display the Kaplan-Meier estimate of the reliability function for
the data.

Figure 7 gives the shape of the distribution function for the FWD, WD, MWD,
RAWD, EWD and BFWD after that the unknown parameters included in each
distribution are replaced by their maximum likelihood estimation MLE

Figure 7: The Fitted cumulative distribution function for the data.

##  8 Conclusion

We studied a new generalized distribution, based on the beta generated method.
This new generalization is called the beta flexible Weibull BFWD distribution
. Its definition and some of statistical properties are studied. The maximum
likelihood method is used for estimating parameters. The advantage of the BFWD
is interpreted by an application using real data. Moreover, it is shown that
the beta flexible Weibull BFWD distribution fits better than existing
generalizations of the Weibull distribution.

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