#  Gravothermal catastrophe and critical dimension in a $D$-dimensional
asymptotically AdS spacetime

Hiroki
Asami111Email:mailto:asami.hiroki.b3<EMAIL_ADDRESS>Division of Particle and Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan  Chul-Moon
Yoo222Email:mailto:yoo<EMAIL_ADDRESS>Division of Particle and Astrophysical Science, Graduate School of Science,
Nagoya University, Nagoya 464-8602, Japan

###### Abstract

We investigate the structure and stability of the thermal equilibrium states
of a spherically symmetric self-gravitating system in a $D$–dimensional
asymptotically Anti-de Sitter(AdS) spacetime. The system satisfies the
Einstein-Vlasov equations with a negative cosmological constant. Due to the
confined structure of the AdS potential, we can construct thermal equilibrium
states without any artificial wall in the asymptotically AdS spacetime.
Accordingly, the AdS radius can be regarded as the typical size of the system.
Then the system can be characterized by the gravothermal energy and AdS radius
normalized by the total particle number. We investigate the catastrophic
instability of the system in a $D$–dimensional spacetime by using the turning
point method. As a result, we find that the curve has a double spiral
structure for $4\leq D\leq 10$ while it does not have any spiral structures
for $D\geq 11$ as in the asymptotically flat case confined by an adiabatic
wall. Irrespective of the existence of the spiral structure, there exist upper
and lower bounds for the value of the gravothermal energy. This fact indicates
that there is no thermal equilibrium solution outside the allowed region of
the gravothermal energy. This property is also similar to the asymptotically
flat case.

##  1 Introduction

Since the seminal work of Antonov [1], the thermodynamical and statistical
properties of the self-gravitating many-particle systems have attracted much
attention. Antonov investigated thermal equilibrium states of the self-
gravitating system surrounded by a spherical adiabatic wall of radius $R$ in
the microcanonical ensemble 333In the microcanonical ensemble, the thermal
relaxation leads the extremum Boltzmann-Gibbs entropy state with fixed energy
and particle number in the wall. The adiabatic wall plays a role in preventing
the particles from diffusion.. In Ref [1], it has been shown that the
Boltzmann-Gibbs entropy has no global maximum but may have a local maximum.
Furthermore, it has been also shown that the second variation of the entropy
with respect to the distribution function can be positive if the density
contrast, which is given by the ratio between the mass density at the center
and the edge, is greater than 709. This result indicates that the density
contrast must not exceed 709 for the thermal equilibrium state to be stable.
Lynden-Bell and Wood [2] confirmed and extended the results of [1] in
accordance with Poincare’s stability criterion [3]. Ref. [3] argued that a
series of equilibria can increase the number of unstable modes only if it
passes a turning point [4, 5, 6, 7]. They found the existence of the minimum
value of the gravothermal energy associated with the turning point which is
identical to the critical point found by Antonov in the microcanonical
ensemble.

This catastrophic instability of the self-gravitating systems is called the
gravothermal catastrophe, which is a consequence of the negative specific heat
of gravity. The negative specific heat makes the core of the system be heated
when it transfers energy to the surrounding halo. Once it begins, the energy
current carried by particles continues to be transferred from the core to the
halo, and fragmentation will occur. In the Newtonian case, this catastrophic
instability does not appear if the thermal energy dominates over the
gravitational potential energy and the confinement is sufficiently effective.
This is because the adiabatic wall puts external pressure on the system and
makes the specific heat positive. This situation corresponds to the case where
the density contrast is smaller than 709, and it can be realized by increasing
the thermal energy under fixed values of the particle number and the size of
the confining wall.

To perform the relativistic extension, we need to take into account the
contribution of the thermal energy to the total mass. Thus the increasing
thermal energy raises the compactness of the system defined by the total mass
divided by the radius. In general relativity, the system cannot avoid
gravitational collapse if it is sufficiently compact [8, 9, 10, 11, 12, 13,
14, 15]. As a result, the series of equilibria has a double spiral structure,
and the two catastrophic instabilities appear in association with two turning
points. For a relativistic perfect fluid with a linear equation of state, the
dependence of the stability on the number of spacetime dimensions has been
studied in Ref. [15]. In an asymptotically flat spacetime, Chavanis concluded
that the system has the critical dimension $D_{\ast}$ such that the system has
a double spiral structure for $D<D_{\ast}$ and not for $D\geq D_{\ast}$. In an
asymptotically AdS spacetime, motivated by the Hawking-Page phase transition,
Refs. [16, 17] investigated the stability of self-gravitating radiation and
obtained the same result, i.e., the double spiral structure vanishes above the
critical dimension $D_{\ast}$.

Asymptotically AdS spacetimes have also been attractive in the context of the
AdS/CFT correspondence [18, 19, 20] and the gravitational turbulent
instability [21]. Since an asymptotically AdS spacetime has a confined
structure, perturbations of matter fields in the spacetime may not dissipate
to infinity. Due to the nonlinearity of the field equations, they interact
with each other and may form black holes even if the perturbations are
arbitrarily small. Despite many works on the turbulent instability [22, 23,
24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35], the final fate has not been
clarified yet because the time evolution and final states are dependent on
symmetries, dimensions and boundary conditions of the spacetime [36, 37]. In
addition to the case of matter fields, the stability of the asymptotically AdS
Einstein-Vlasov system is also investigated in Ref. [38, 39] (see also [40,
41]) 444 Besides, the stability of the massless Einstein–Vlasov system with an
inner mirror has been investigated in Ref. [41] and Ref. [40] analyzed the
stability of steady states of the spherically symmetric Einstein-Vlasov system
in a flat spacetime.. However, the conditions of black formation and the final
fate are still not completely clear [36, 37]. Since the self-gravitating many-
particle system might be regarded as a macroscopic model for the excitations
in an asymptotically AdS spacetime, the analysis in our paper might be helpful
to guess the final states of asymptotically AdS spacetimes in general
dimensions.

This paper is organized as follows. In section 2, we derive the basic
equations for the thermal equilibrium states. The details of the derivation
are shown in Appendix A and calculations for physical quantities are shown in
Appendix B. We present the results in section 3. In section 3.1, we show the
parameter dependence of the radial profile of the system. After the brief
review on the turning point method in section 3.2, we show the main results of
the stability analysis in section 3.3-3.5. Section 4 is devoted to a summary
and conclusion.

Throughout this paper, we use the geometrized units in which both the speed of
light and gravitational constant in $D$–dimension are unity, $c=G_{D}=1$.

##  2 Field equations and thermodynamical quantities

###  2.1 Physical quantities of the self-gravitating system

We consider a self-gravitating system of massive point particles in a
$D$–dimensional spacetime. For simplicity, setting the rest mass of each
particle to unity, we can identify the number of particles with the total rest
mass. Focusing on a static and spherically symmetric spacetime, the metric is
given by

$\displaystyle
g_{\mu\nu}\differential{x^{\mu}}\differential{x^{\nu}}=-\mathrm{e}^{2\mu(r)}\differential{t^{2}}+\mathrm{e}^{2\nu(r)}\differential{r^{2}}+r^{2}\differential\Omega_{D-2}^{2},$
(2.1)

where

$\displaystyle\differential\Omega_{D-2}^{2}=\differential\theta_{1}^{2}+\sin^{2}\theta_{1}\differential\theta_{2}^{2}+\sin^{2}\theta_{1}\sin^{2}\theta_{2}\differential\theta_{3}^{2}+\cdots+\sin^{2}\theta_{1}\sin^{2}\theta_{2}\cdots\sin^{2}\theta_{D-3}\differential\theta_{D-2}^{2}.$
(2.2)

Introducing the one-particle distribution function $f(x^{\mu},p^{i})$ in the
mean field approximation, the energy-momentum tensor of this system is written
as

$\displaystyle T_{\mu\nu}:=\int dV_{p}p_{\mu}p_{\nu}f(x^{\mu},p^{i}),$ (2.3)

where $\differential{V_{p}}$ is the invariant volume element in the momentum
space. The on-shell condition $p^{2}+1=0$ leads to

$\displaystyle dV_{p}$
$\displaystyle=2\sqrt{-g}\delta(p^{2}+1)\theta(\varepsilon)\differential[D]{p}$
$\displaystyle=\frac{\sqrt{-g}}{\varepsilon}\differential{p}^{r}\wedge\differential{p}^{\theta_{1}}\wedge\cdots\wedge\differential{p}^{\theta_{D-2}},$
(2.4)

where $\delta$ and $\theta$ are the delta function and the Heaviside’s step
function, respectively, and $\varepsilon:=-p_{t}$ is the energy of the
particle. The energy density and the pressure are defined by
$\rho(r):=-T_{t}^{\ t}$ and $p(r):=T_{r}^{\ r}$, respectively. We introduce
the quasi–local mass $M(r)$ as

$\displaystyle M(r):=S_{D-1}\int_{0}^{r}\differential{u}u^{D-2}\rho(u),$ (2.5)

where $S_{D-1}$ is the surface area of the $D$-sphere:

$\displaystyle S_{D-1}:=\frac{2\pi^{\frac{D-1}{2}}}{\Gamma(\frac{D-1}{2})},$
(2.6)

with $\Gamma$ being the Gamma function. For a given function of the
distribution function as $\mathcal{F}={{\mathcal{F}}\pqty{f}}$, the current

$\displaystyle{{F^{\mu}}\bqty{{{\mathcal{F}}\pqty{f}}}}=\int\differential{V_{p}}p^{\mu}{{\mathcal{F}}\pqty{f}}$
(2.7)

carries the density ${{\mathcal{F}}\pqty{f}}$ and we can define the charge $F$
as

$\displaystyle{{F}\bqty{{{\mathcal{F}}\pqty{f}}}}:=\int_{\Sigma}\differential{\Sigma_{\mu}}F^{\mu},$
(2.8)

where $\Sigma$ is a constant time hyper-surface. By substituting
${{\mathcal{F}}\pqty{f}}=f$, we can define the number of particles within the
radius $r$ as

$\displaystyle
N(r):=\evaluated{{{F}\bqty{\mathcal{F}}}}_{{{\mathcal{F}}\pqty{f}}=f}=\int_{0}^{r}\differential{u}u^{D-2}n(u),$
(2.9)

where $n(r):=\mathrm{e}^{\mu+\nu}N^{t}$ with
$N^{\mu}:=\evaluated{{{F^{\mu}}\bqty{\mathcal{F}}}}_{{{\mathcal{F}}\pqty{f}}=f}$.
We also define the Boltzmann-Gibbs (BG) entropy as

$\displaystyle
S(r):=\evaluated{{{F}\bqty{\mathcal{F}}}}_{{{\mathcal{F}}\pqty{f}}=-f\pqty{\log
f-1}}=\int_{0}^{r}\differential{u}u^{D-2}s(u),$ (2.10)

where $s(r):=\mathrm{e}^{\mu+\nu}S^{t}$ with
$S^{\mu}:=\evaluated{{{F^{\mu}}\bqty{\mathcal{F}}}}_{{{\mathcal{F}}\pqty{f}}=-f\pqty{\log
f-1}}$.

###  2.2 Einstein’s equations

The Einstein’s equations with a negative cosmological constant
$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi T_{\mu\nu}$ reduce to the following
equations:

$\displaystyle-16\pi r^{2}\rho(r)\mathrm{e}^{2\nu}$
$\displaystyle=(D-3)(D-2)+\mathrm{e}^{2\nu}\pqty{-(D-3)(D-2)+2\Lambda
r^{2}}-2(D-2)r\nu^{\prime},$ (2.11a) $\displaystyle 16\pi
r^{2}p(r)\mathrm{e}^{2\nu}$
$\displaystyle=(D-3)(D-2)+\mathrm{e}^{2\nu}\pqty{-(D-3)(D-2)+2\Lambda
r^{2}}+2(D-2)r\mu^{\prime}.$ (2.11b)

Integrating Eq. (2.11a) with the central regularity: $\nu(0)=0$, we obtain

$\displaystyle\mathrm{e}^{-2\nu}=1-\frac{kM(r)}{r^{D-3}}+\frac{r^{2}}{L^{2}},\quad
k:=\frac{16\pi}{(D-2)S_{D-1}},$ (2.12)

where $L$ is the AdS radius defined as

$\displaystyle L:=\sqrt{-\frac{(D-1)(D-2)}{2\Lambda}}.$ (2.13)

By using Eq. (2.12), Eq. (2.11b) becomes

$\displaystyle\mu^{\prime}=\frac{\frac{k(D-3)M(r)}{2r^{D-3}}+\frac{r^{2}}{L^{2}}+\frac{8\pi
r^{2}p(r)}{D-2}}{r\pqty{1-\frac{kM(r)}{r^{D-3}}+\frac{r^{2}}{L^{2}}}}.$ (2.14)

If the central values of the energy density $\rho_{c}$ and the pressure
$p_{c}$ do not vanish, we can normalize the physical quantities by using them
together with $\mu_{c}:=\mu(0)$ . Introducing the typical length scale
$\ell:=\pqty{S_{D-1}\rho_{c}}^{-1/2}$ and defining $x:=r/\ell$,
$y:=\mu-\mu_{c}$, $\tilde{\rho}:=\rho/\rho_{c}$, $\tilde{n}:=n/n_{c}$ and
$\tilde{p}:=p/p_{c}$, we obtain the mass and the particle number as

$\displaystyle
M(r)=\ell^{D-3}\int_{0}^{x}\differential{u}u^{D-2}\tilde{\rho}(u),\quad
N(r)=\ell^{D-3}\frac{n_{c}}{\rho_{c}}\int_{0}^{x}\differential{u}u^{D-2}\tilde{n}(u).$
(2.15)

Then introducing $\tilde{M}(x):=M(r)/\ell^{D-3}$ and
$\tilde{N}(x):=N(r)/\ell^{D-3}$, we can rewrite Eq. (2.14) as

$\displaystyle
y^{\prime}=\frac{\frac{k(D-3)\tilde{M}(x)}{2x^{D-3}}+\frac{x^{2}}{\lambda^{2}}+\frac{8\pi
w_{c}x^{2}\tilde{p}(x)}{(D-2)S_{D-1}}}{x\pqty{1-\frac{k\tilde{M}(x)}{x^{D-3}}+\frac{x^{2}}{\lambda^{2}}}},$
(2.16)

where $\lambda:=L/\ell$ and $w_{c}:=p_{c}/\rho_{c}$. By numerically solving
Eq. (2.16) together with

$\displaystyle\derivative{\tilde{M}}{x}=x^{D-2}\tilde{\rho},$ (2.17)

we can get a solution. The boundary conditions are given by
${{y}\pqty{0}}={{y^{\prime}}\pqty{0}}={{\tilde{M}}\pqty{0}}=0$ due to the
definition of ${{y}\pqty{x}}$ and ${{\tilde{M}}\pqty{x}}$. We note that Eq.
(2.16) is valid for $D\geq 3$, and the $D=3$ case must be independently
treated. We focus on the $D\geq 4$ cases in this paper.

###  2.3 Thermal equilibrium states

In the isolated system, the thermal equilibrium states are given as extremal
entropy states with a fixed total mass and total particle number. Introducing
the positive numbers $\alpha$ and $\beta$ as the Lagrange multipliers, the
condition, which is called the Gibbs relation, is written as

$\displaystyle\variation{S}+\alpha\variation{N}-\beta\variation{M}=0,$ (2.18)

where $S:=\lim_{r\to\infty}{{S}\pqty{r}},\ N:=\lim_{r\to\infty}{{N}\pqty{r}}$
and $M:=\lim_{r\to\infty}{{M}\pqty{r}}$ are the total entropy, the total
particle number and the total mass, respectively. As is shown in Appendix A,
this condition leads to the Maxwell-Jüttner (MJ) distribution function
$f=\exp\pqty{\alpha-\beta\varepsilon}$ in the $D$–dimensional static and
spherically symmetric spacetime. As is shown in Appendix B, for MJ
distribution, performing the integral over the momentum space, we obtain the
energy density, the pressure and the number density as follows:

$\displaystyle\rho(r)$
$\displaystyle=2\mathrm{e}^{\alpha}\pqty{\frac{2\pi}{z}}^{\frac{D}{2}-1}\pqty{\frac{D-1}{z}K_{\frac{D}{2}}(z)+K_{\frac{D}{2}-1}(z)},$
(2.19a) $\displaystyle p(r)$
$\displaystyle=\frac{\mathrm{e}^{\alpha}}{\pi}\pqty{\frac{2\pi}{z}}^{\frac{D}{2}}K_{\frac{D}{2}}(z),$
(2.19b) $\displaystyle n(r)$
$\displaystyle=2\mathrm{e}^{\alpha+\nu}\pqty{\frac{2\pi}{z}}^{\frac{D}{2}-1}K_{\frac{D}{2}}(z),$
(2.19c)

where $z=\beta\mathrm{e}^{\mu}\eqqcolon\beta\varepsilon_{0}$. By using the
normalized variables, they are rewritten as

$\displaystyle\tilde{\rho}(x)$
$\displaystyle=\frac{(D-1)K_{\frac{D}{2}}(z)\mathrm{e}^{-y}+z_{c}K_{\frac{D}{2}-1}(z)}{(D-1)K_{\frac{D}{2}}(z_{c})+z_{c}K_{\frac{D}{2}-1}(z_{c})}\mathrm{e}^{-\pqty{\frac{D}{2}-1}y},$
(2.20a) $\displaystyle\tilde{p}(x)$
$\displaystyle=\frac{K_{\frac{D}{2}}(z)}{K_{\frac{D}{2}}(z_{c})}\mathrm{e}^{-\frac{D}{2}y},$
(2.20b) $\displaystyle\tilde{n}(x)$
$\displaystyle=\frac{K_{\frac{D}{2}}(z)}{K_{\frac{D}{2}}(z_{c})}\pqty{1-\frac{k\tilde{M}(r)}{x^{D-3}}+\frac{x^{2}}{\lambda^{2}}}^{-\frac{1}{2}}\mathrm{e}^{-\pqty{\frac{D}{2}-1}y}$
(2.20c)

and

$\displaystyle w_{c}$
$\displaystyle=\frac{K_{\frac{D}{2}}(z_{c})}{(D-1)K_{\frac{D}{2}}(z_{c})+z_{c}K_{\frac{D}{2}-1}(z_{c})},$
(2.21)

where $z_{c}:=\beta\mathrm{e}^{\mu_{c}}$, which is the inverse temperature
including the effect of the lapse function at the center. Therefore the
parameter $z_{c}$ corresponds to the red-shft factor at the center and it
characterizes the relativistic effect of the system. We can also rewrite
$\tilde{M}(x)$ and $\tilde{N}(x)$ as

$\displaystyle\tilde{M}(x)=\int_{0}^{x}\differential{z}z^{D-2}\tilde{\rho}(z),\quad\tilde{N}(x)=z_{c}w_{c}\int_{0}^{x}\differential{z}z^{D-2}\tilde{n}(z)$
(2.22)

because $n_{c}/\rho_{c}=n_{c}/p_{c}\cdot p_{c}/\rho_{c}=z_{c}w_{c}$ is
satisfied. Normalizing the physical quantities by the particle number, we
obtain

$\displaystyle\tilde{E}$ $\displaystyle\coloneqq\frac{E}{N}=\frac{\int
dxx^{D-2}\tilde{\rho}}{z_{c}w_{c}\int dxx^{D-2}\tilde{n}}-1,$ (2.23a)
$\displaystyle\tilde{L}^{D-3}$
$\displaystyle\coloneqq\frac{L^{D-3}}{N}=\frac{\lambda^{D-3}}{z_{c}w_{c}\int
dxx^{D-2}\tilde{n}},$ (2.23b) $\displaystyle\tilde{r}^{D-3}$
$\displaystyle\coloneqq\frac{r^{D-3}}{N}=\frac{x^{D-3}}{z_{c}w_{c}\int
dxx^{D-2}\tilde{n}},$ (2.23c)

where $E:=M-N$ is the energy except for the rest mass of the particles, which
is called the gravothermal energy.

##  3 Results

###  3.1 Parameter dependence of the profile

Fig. 1 shows the energy density profile as a function of $x$ for each
parameter set $\pqty{z_{c},\lambda}$ in the $D=5$ case. Fig. 1(a) indicates
that the system becomes more compact as $\lambda$ decreases. This is because
the AdS potential works as a wall for the particles and the AdS radius
$\lambda$ characterizes the size of the system. Fig. 1(b) indicates that the
system becomes more compact as $z_{c}$ increases. As mentioned above, since
the parameter $z_{c}$ characterizes the central redshift factor, it can be
regarded as an indicator of the significance of the relativistic gravitational
effect in the system. That is, in the low temperature limit $z_{c}\to\infty$,
the Newtonian limit can be realized. Actually, the compactness parameter,
which is also an indicator of the relativistic effect, gets smaller as $z_{c}$
increases as is shown in Fig. 2(b), and the relativistic effect of the system
gets weaker.

(a) $\lambda$ dependence for $z_{c}=1$.

(b) $z_{c}$ dependence for $\lambda=1$.

Fig. 1:  The parameter dependence of the energy density for $D=5$.

Fig. 2 also shows that the total mass is constant for $x\gg\lambda$ because
${{\tilde{M}}\pqty{x}}/x^{2}\sim x^{-2}$. In a $D$-dimensional spacetime, the
compactness parameter behaves as ${{\tilde{M}}\pqty{x}}/x^{D-3}\sim x^{D-3}$
and the total mass becomes constant (see Fig. 3 for the $D=10$ case). These
results reflect the confined structure of the self-gravitating system in the
asymptotically AdS spacetime due to the AdS potential. Actually, both the
total mass and the total particle number diverge for $\Lambda=0$ without an
artificial wall, which is shown numerically in Ref. [39] for the $D=4$ case.
For $x\ll 1$, $\derivative*{\tilde{\rho}}{x}\sim\derivative*{y}{x}=0$ and
${{\tilde{M}}\pqty{x}}\sim x^{D-1}$ for $x\ll 1$, so that the compactness
parameter behaves as ${{\tilde{M}}\pqty{x}}/x^{2}\sim x^{2}$ independently of
the number of dimensions.

(a) $\lambda$ dependence for $z_{c}=1$.

(b) $z_{c}$ dependence for $\lambda=1$.

Fig. 2:  The parameter dependence of the compactness parameter for $D=5$. It
behaves as $\tilde{M}(x)/x^{2}\sim x^{2}$ for $x\ll 1$ and
$\tilde{M}(x)/x^{2}\sim x^{-2}$ for $x\gg\lambda$.

(a) $\lambda$ dependence for $z_{c}=1$.

(b) $z_{c}$ dependence for $\lambda=1$.

Fig. 3:  The parameter dependence of the compactness parameter for $D=10$. It
behaves as ${{\tilde{M}}\pqty{x}}/x^{7}\sim x^{2}$ for $x\ll 1$ and
${{\tilde{M}}\pqty{x}}/x^{7}\sim x^{-7}$ for $x\gg\lambda$.

###  3.2 Turning point methods and parameter sets

Since the Einstein’s equations (2.16) and (2.17) are uniquely determined by
the parameters $\pqty{z_{c},\lambda}$, they give a sequence of the solutions
as a two parameter family of $\pqty{z_{c},\lambda}$. In order to investigate
the thermodynamical stability of each solution, we regard
$(\tilde{E},\tilde{L})$ as a parameter set specifying an equilibrium solution.
Defining $\gamma:=\log\tilde{L}$ and considering solutions with a fixed value
of $\gamma$, these solutions reduce to a one-parameter family. Introducing the
sharpness parameter $\sigma:=-\log\tilde{\rho}(\lambda)$ as the remaining one
parameter, the physical quantities listed in Eq. (2.23) can be expressed by
$\tilde{E}=\tilde{E}(\sigma)$, $\tilde{L}=\tilde{L}(\sigma)$ and
$\tilde{r}=\tilde{r}(\sigma)$. By using these values, we can define

$\displaystyle(\bar{E},\bar{\beta})=\evaluated{\pqty{\frac{\lambda^{D-3}{{\tilde{E}}\pqty{x}}}{{{\tilde{N}}\pqty{x}}^{2}},\frac{\beta{{\tilde{N}}\pqty{x}}}{\lambda^{D-3}}}}_{x\to\infty}$
(3.1)

for each value of $\gamma$, and of course, $\bar{E}$ and $\bar{\beta}$ are
parametrized by $\sigma$ i.e.,
$(\bar{E},\bar{\beta})=(\bar{E}(\sigma),\bar{\beta}(\sigma))$. For a fixed
value of $\gamma$, the series of equilibria draws a one-parameter curve in the
$(\bar{E},\bar{\beta})$ space.

The series of equilibria (3.1) specifies the thermodynamical stability of the
self-gravitating system, and the stability can be checked through the turning
point method[3, 4, 5]. According to the turning point method, the stability of
the system with a fixed total particle number can change at points where the
curve is vertical in the $(\bar{E},\bar{\beta})$ space. These points are
called turning points. The turning point method also says that the series of
equilibria gets more unstable if it passes the turning point in the counter-
clockwise direction.

###  3.3 Double spiral structure and the catastrophic instabilities

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 4:  Fig. 4(a) shows the series of equilibria and Fig. 4(b) shows the
sharpness parameter $\sigma(\bar{E})$ with $\gamma=2.7$ in the $D=4$ case. We
can see that the curve has a double spiral structure and $\sigma$ increases
from the middle to the both sides. The parameter $\bar{E}$ oscillates along
the curve towards the both ends, and the curve forms spirals.

Fig. 4 shows the inverse temperature $\bar{\beta}$ as a function of the
gravothermal energy $\bar{E}$ of the series of equilibria with $\gamma=2.7$
for the 4-dimensional case, and the sharpness parameter $\sigma(\bar{E})$ is
also plotted. The curve has a double spiral structure and we call the spiral
in the high energy region “hot spiral” and one in the low energy region “cold
spiral”, respectively. It would be noted that a similar structure also appears
in the case of self-gravitating particles inside an artificial wall with a
vanishing cosmological constant [12, 13]. The structure implies that the
system has two catastrophic instabilities due to the negative specific heat of
gravity. Since each spiral is spiraling in the counter-clockwise direction
towards the center, the series of equilibria gets more unstable every time it
passes a turning point. As is shown in Fig. 4(b), the sharpness parameter
$\sigma$ increases towards the center of each spiral and the particle
distribution gets sharper. The two spirals are characterized by the total mass
compactness $M/L$. Since we can rewrite
$M/L=N/L(1+N\bar{E}/L)=\mathrm{e}^{-\gamma}(1+\mathrm{e}^{-\gamma}\bar{E})$,
given a fixed value of $\gamma$, we can say the system is more compact for a
larger value of $\bar{E}$. Based on this fact, we can understand the physical
meaning of the instability on each spiral. On the cold spiral, the compactness
of the system is relatively small. That implies particles can take energy away
from the central region and the system causes a gravothermal catastrophe
corresponding to the Newtonian one. By contrast, the compactness of the system
is relatively large on the hot spiral. This behavior is peculiar to the
relativistic system and it indicates the hot spiral reflects the strong non-
linear gravity. If the gravity is sufficiently strong, the system cannot
support the configuration of the particles and experiences a catastrophic
gravitational collapse.

The double spiral structure ensures the existence of another instability. As
is clear from the figures, the existence of the two spirals delimits the
possible parameter region of $\bar{E}$ in a finite region. The fact tells us
that no equilibrium solutions exist outside the region, which is called the
“strong instability” in contrast with the weak instability associated with a
turning point. Therefore, both cold and hot spirals ensure two kinds of
catastrophic instability.

###  3.4 Configuration of curves and parameter dependence for $4\leq D\leq 10$

In the case of $4\leq D\leq 10$, the curve for thermal equilibrium states also
has a double spiral structure. Fig. 5 shows the inverse temperature
$\bar{\beta}(\bar{E})$ and the sharpness parameter $\sigma(\bar{E})$ for
$\gamma=2.65$ in the 5-dimensional case. The embedded figures in Fig. 5(a) are
enlarged figures of the cold and hots spirals. We can see that both of spirals
are counter-clockwise towards the center. Therefore, as in the 4-dimensional
case, the system gets more unstable as the state approaches the center of each
spiral.

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 5:  The inverse temperature and the sharpness parameter for a series of
equilibrium states in the 5-dimensional case. We set $\gamma=2.65$ and the
curve has a double spiral structure as in the 4-dimensional case.

The parameter $\gamma=\log\tilde{L}^{D-3}$ denotes the rest mass compactness
of the system. In the dilute limit, i.e. $\gamma\to\infty$, the hot spiral
moves to the right infinitely, and the series of equilibria has only the cold
spiral. In the dilute limit, the gravitational interaction becomes relatively
less effective, and the Newtonian approximation would be valid. Then only the
Newtonian catastrophic instability is realized on the cold spiral.

Let us consider how the configuration of the curve changes as $\gamma$
decreases. Two spirals approach each other as $\gamma$ decreases (schematic
figures are shown in Fig. 6), and the cold spiral intersects with the main
part of the curve between the two spirals.

(a) Large $\gamma$.

(b) Small $\gamma$.

Fig. 6:  Schematic figures for the transition of configuration of the double
spiral structure. The series of equilibria has no intersection in the
$(\bar{E},\bar{\beta})$ plane for relatively large $\gamma$. As $\gamma$
decreases, the cold spiral moves to the right in the $(\bar{E},\bar{\beta})$
plane, and it intersects with the main part of the curve between the cold and
hot spirals.

In this process, a set of stationary points for $\bar{E}$ and $\bar{\beta}$ on
each spiral approaches a common single point, and they merge at the “merge
point” $\gamma_{\mathrm{m}}\simeq 2.2525$.

Fig. 7:  Schematic figures for the series of equilibria at the merge point. At
the merge point, a set of stationary points for $\bar{E}$ and $\bar{\beta}$ on
each spiral contract to a common single point. The numerical result
corresponding to the merge point is shown in Fig. 8.

Fig. 8 shows the configuration of the curves in the $(\bar{E},\bar{\beta})$
and $(\bar{E},\sigma)$ planes for $\gamma=2.2525$.

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 8:  The curves of equilibrium states in the $(\bar{E},\bar{\beta})$ and
$(\bar{E},\sigma)$ planes for $\gamma\simeq 2.2525$. The cold and hot spirals
merge at this point.

For a value of $\gamma$ smaller than $\gamma_{\rm m}$, the curve of equilibria
reconnects and separate into two sequences as shown in Fig. 9. We can see that
one of sequences forms a loop and the other does not.

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 9:  The curves of equilibrium states in the $(\bar{E},\bar{\beta})$ and
$(\bar{E},\sigma)$ planes for $\gamma\simeq 2.25$. In this parameter region,
the series of equilibria separate into two pieces and two curves exist in
$(\bar{E},\sigma)$ and $(\sigma,\bar{\beta})$ planes.

As $\gamma$ gets even smaller, the sequence without a loop vanishes and the
series of equilibria consists of only one loop (see Fig. 10). We call this
point the “loop point” and the value of $\gamma$ is
$\gamma_{\mathrm{\ell}}\simeq 2.2343$.

(a) Separated sequences.

(b) A loop structure.

Fig. 10:  Schematic figures around the loop point $\gamma_{\ell}$. The
sequence without loop structure shrinks and vanishes at this point. After the
loop point, the other sequence shrinks until it reaches to the vanishing point
$\gamma_{\mathrm{v}}$.

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 11:  The curves of equilibrium states in the $(\bar{E},\bar{\beta})$ and
$(\bar{E},\sigma)$ planes for $\gamma\simeq 2.23$. The series has only the
sequence with a loop.

If we continue to make $\gamma$ smaller, the sequence with a loop shrinks and
vanishes at the “vanishing point” $\gamma_{\mathrm{v}}\simeq 2.1369$. At the
vanishing point, the series of equilibria becomes just a point. Therefore the
thermal equilibrium state exists only at a unique gravothermal energy
$\bar{E}_{\ast}\simeq 18.17$. For $\gamma>\gamma_{\mathrm{v}}$, the system has
no thermal equilibrium states, and strongly unstable.

In the $4\leq D\leq 10$ case, the behaviors of the series of equilibria are
qualitatively similar to the 5-dimensional case. The series of equilibria has
a double spiral structure and three critical points of $\gamma$. Therefore,
the system has weak and strong instabilities associated with the turning
points on both of cold and hot spirals. In any case, the catastrophic
instabilities are caused by the negative specific heat of gravity.

###  3.5 Configuration of curves and parameter dependence in $D\geq 11$ case

For $D\geq 11$, the behavior of the series of equilibria is dramatically
different from the case for $4\leq D\leq 10$. Fig. 12 shows the inverse
temperature $\bar{\beta}(\bar{E})$ and the sharpness parameter
$\sigma(\bar{E})$ for $\gamma=2.6$ in the 11-dimensional case. We cannot see
any spiral structures and stationary points disappear. This result implies the
system does not have any weak instabilities due to the negative specific heat.
In this case, the system has neither merge point nor loop point while it has a
vanishing point around $\gamma_{\mathrm{v}}\simeq 2.2271$.

Similar critical behaviors have been reported for the systems confined by an
artificial adiabatic wall [42, 16, 17]. In our case, particles are confined by
not an adiabatic wall but an AdS potential. We found that the critical
dimension is exactly the same as the system confined by an adiabatic wall. The
system still has the strong instabilities associated with the endpoints of the
curve. This behavior is also qualitatively similar to the system with an
adiabatic wall.

(a) Inverse temperature: $\bar{\beta}=\bar{\beta}(\bar{E})$.

(b) Sharpness parameter: $\sigma=\sigma(\bar{E})$.

Fig. 12:  The inverse temperature and the sharpness parameter for a series of
equilibrium states in the 11-dimensional case for $\gamma=2.6$. The series
does not have any spiral structures.

##  4 Conclusion

We have analyzed the structure and stability of the thermal equilibrium states
of a self-gravitating system in a $D$–dimensional asymptotically AdS
spacetime. The thermal equilibrium can be realized because of the confinement
of the AdS potential and we use the turning point method to investigate the
stability. We concluded that the properties of the stability are qualitatively
similar to the relativistic system confined by an artificial wall [12, 13, 16,
17].

In a $D$–dimensional spacetime, the equilibria can be parameterized by two
parameters contained in the solution for the Einstein’s field equations. Thus
the series of equilibria draws a two-dimensional surface in the space of the
rest mass compactness parameter $\gamma$, the gravothermal energy $\bar{E}$
and the inverse temperature $\bar{\beta}$. Given a fixed value of $\gamma$,
the series reduces to a one-dimensional curve in the $(\bar{E},\bar{\beta})$
plane. As a result, the curve has a double spiral structure for $4\leq D\leq
10$ while it does not have any spirals for $D\geq 11$.

For $4\leq D\leq 10$, the system has two weak catastrophic instabilities
corresponding to two spirals. The cold spiral, which is located in the lower
energy region, implies the gravothermal catastrophe associated with the
fragmentation of the system. The other spiral called a hot spiral, implies the
catastrophic gravitational collapse. These properties indicate the existence
of the strong instability since the allowed region of gravothermal energy is
bounded by the turning points. For $4\leq D\leq 10$, the configuration of the
curve in the $(\bar{E},\bar{\beta})$ plane is dependent on the value of
$\gamma$, and there are three critical points $\gamma_{\mathrm{m}}$,
$\gamma_{\mathrm{\ell}}$ and $\gamma_{\mathrm{v}}$. For
$\gamma>\gamma_{\mathrm{m}}$, the series of equilibria has a double spiral
structure. Two spirals approach each other as $\gamma$ gets smaller. When the
value of $\gamma$ gets smaller than $\gamma_{\mathrm{m}}$, the curve separates
into two sequences of equilibria. One of sequences vanishes at
$\gamma=\gamma_{\mathrm{\ell}}$. Then, the series consists of only one loop
for $\gamma_{\mathrm{\ell}}<\gamma<\gamma_{\mathrm{v}}$ and there is no
equilibrium states for $\gamma>\gamma_{\mathrm{v}}$.

For $D\geq 11$, the system has no weak instabilities but strong instabilities
corresponding to the endpoints of the curve in the $(\bar{E},\bar{\beta})$
plane. Since the curve does not have any spiral structures, the system has
neither the merge point nor the loop point. The series of equilibria shrinks
as $\gamma$ decreases and vanishes similarly to the cases for $4\leq D\leq
10$, and therefore it has a vanishing point $\gamma_{\mathrm{v}}$ as a
critical point.

At the end, we conclude that the many-particle system has parameter regions
with stable configuration. These configurations might well describe final
fates of the turbulent instability in asymptotically AdS spacetimes. It should
be noted that, however, the solutions of the many-particle system are not
necessarily corresponding to stability islands discussed in Refs. [36, 37]
because the system should be regarded as a macroscopic model after the non-
linear turbulent phenomena. Apparently, we need further investigations to
clarify implication of our results in the turbulent instability in
asymptotically AdS spacetimes.

## Acknowledgments

This work was supported in part by JSPS KAKENHI Grant Nos. JP19H01895 (CY),
JP20H05850 (CY) and JP20H05853 (CY). This work was also supported by JST
SPRING, Grant Number JPMJSP2125 (HA). The author (HA) would like to take this
opportunity to thank the “Interdisciplinary Frontier Next-Generation
Researcher Program of the Tokai Higher Education and Research System.”

##  Appendix Appendix A Derivation of the thermal equilibrium

In this appendix, from the extremal entropy condition (2.18), we first derive
the thermal equilibrium condition for the distribution function $f$ with the
following general expression of the entropy:

$\displaystyle{{S}\bqty{{{\mathcal{S}}\pqty{f}}}}:=\int_{\Sigma}\differential{\Sigma_{\mu}}S^{\mu},\quad{{S^{\mu}}\bqty{{{\mathcal{S}}\pqty{f}}}}=\int\differential{V_{p}}p^{\mu}{{\mathcal{S}}\pqty{f}}.$
(A.1)

Then we show that the condition for the BG entropy leads to the MJ
distribution.

###  Appendix A.1 Coordinate systems in the momentum space

First, let us introduce the local inertial frame whose metric components are
related to $g_{\mu\nu}$ given in Eq. (2.1) by

$\displaystyle\eta_{\hat{\mu}\hat{\nu}}=\mathcal{L}^{\mu}_{\
\hat{\mu}}\mathcal{L}^{\nu}_{\ \hat{\nu}}g_{\mu\nu},$ (A.2)

where $\eta_{\hat{\mu}\hat{\nu}}=\mathrm{diag}\pqty{-1,1,\cdots,1}$ and

$\displaystyle\pqty{\mathcal{L}^{\mu}_{\
\hat{\mu}}}=\mathrm{diag}\pqty{\mathrm{e}^{-\mu},\mathrm{e}^{-\nu},\frac{1}{r},\cdots,\frac{1}{r\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{D-3}}}.$
(A.3)

In this frame, the invariant volume element in the momentum space is written
as

$\displaystyle dV_{p}$
$\displaystyle=2\delta(p^{2}+1)\theta(\hat{\varepsilon})\differential[D]{p}$
$\displaystyle=\frac{1}{\hat{\varepsilon}}\differential{p}^{\hat{r}}\wedge\differential{p}^{\hat{\theta}_{1}}\wedge\cdots\wedge\differential{p}^{\hat{\theta}_{D-2}},$
(A.4)

where $\hat{\varepsilon}:=-p_{\hat{t}}$. The variation of the distribution
function $f$ is taken as a function of $p_{\hat{\mu}}$, that is,
$p_{\hat{\mu}}$ is fixed in the variation.

For convenience, we also introduce the coordinates
$(\varepsilon,J,\psi_{1},\psi_{2},\cdots,\psi_{D-3})$ in the momentum space as
follows:

$\displaystyle p^{\hat{\theta}_{1}}$
$\displaystyle=\frac{\sqrt{J}}{r}\cos\psi_{1},$ $\displaystyle
p^{\hat{\theta}_{2}}$
$\displaystyle=\frac{\sqrt{J}}{r}\sin\psi_{1}\cos\psi_{2},$
$\displaystyle\hskip 5.0pt\vdots$ $\displaystyle p^{\hat{\theta}_{D-3}}$
$\displaystyle=\frac{\sqrt{J}}{r}\sin\psi_{1}\sin\psi_{2}\cdots\sin\psi_{D-4}\cos\psi_{D-3},$
$\displaystyle p^{\hat{\theta}_{D-2}}$
$\displaystyle=\frac{\sqrt{J}}{r}\sin\psi_{1}\sin\psi_{2}\cdots\sin\psi_{D-4}\sin\psi_{D-3},$
(A.5)

where $J$ is the square of the total angular momentum. The volume element is
written as

$\displaystyle\differential{V_{p}}=\frac{J^{\frac{D-4}{2}}}{2r^{D-2}\hat{\varepsilon}}\differential{p^{\hat{r}}}\wedge\differential{J}\wedge\differential{\Omega_{D-3}}$
(A.6)

and the relevant domain for $p^{\hat{r}}$ and $J$ is given by
$\Bqty{(p^{\hat{r}},J)|-\infty<p^{\hat{r}}<\infty,0<J<\infty}$. Regarding
$\hat{\varepsilon}$ as an independent variable, the invariant volume element
is given by

$\displaystyle\differential{V_{p}}=\frac{J^{\frac{D-4}{2}}}{r^{D-2}}\pqty{{\hat{\varepsilon}}^{2}-\pqty{1+\frac{J}{r^{2}}}}^{-\frac{1}{2}}\differential{\hat{\varepsilon}}\wedge\differential{J}\wedge\differential{\Omega_{D-3}},$
(A.7)

and the relevant region is given by
$\Bqty{(\hat{\varepsilon},J)|1<\hat{\varepsilon}<\infty,0<J<r^{2}({\hat{\varepsilon}}^{2}-1)}$
due to the on-shell condition, where we multiplied the factor two for the
degeneracy of the sign of $p^{\hat{r}}$.

###  Appendix A.2 Variation of physical quantities

Since the system is isolated, we fix the total mass and the total particle
number. For this purpose, we calculate the variation of such quantities and
the entropy in $f\to f+\delta f$. Hereafter, the subscript $E$ denotes the
equilibrium state, such as $\mu_{E},\ \nu_{E}$ and $\rho_{E}$, and we consider
the perturbation around it, that is, $\mu=\mu_{E}+\delta\mu\ etc.$ with
respect to the perturbation $\delta f$.

The perturbation of $\nu$, say $\delta\nu$, is determined by Eq. (2.11a)
through the energy density as follows:

$\displaystyle\derivative{r}(r^{D-3}\mathrm{e}^{-2\nu_{E}}\variation{\nu})=\frac{8\pi}{D-2}r^{D-2}\variation{\rho},$
(A.8)

where the perturbation of the energy density is given by

$\displaystyle\delta\rho:={{\rho}\bqty{f+\delta
f}}-{{\rho}\bqty{f}}=\int\differential{V_{p}}\hat{\varepsilon}^{2}\delta f.$
(A.9)

Therefore, the perturbation $\delta\nu$ is not independent of $\delta f$
through the Einstein’s equations. However, it is convenient to formally treat
them as independent perturbations in the following calculations.

The perturbation of ${{F}\bqty{f}}$ defined by Eq. (2.8) is

$\displaystyle\delta{F}:$ $\displaystyle={{F}\bqty{f+\delta f}}-{{F}\bqty{f}}$
$\displaystyle=\int\differential{\Gamma}\hat{\varepsilon}\delta\pqty{\mathrm{e}^{\nu}\mathcal{F}}$
$\displaystyle=\int\differential{\Gamma}\mathrm{e}^{\nu}\hat{\varepsilon}\bqty{\pqty{\mathcal{F}\delta\nu+\partialderivative{\mathcal{F}}{f}\delta
f}+\order{\delta{f}^{2}}},$ (A.10)

where
$\differential{\Gamma}:=\differential[D-1]{x}\differential{V_{p}}=\differential{r}\wedge\pqty{r\differential{\theta_{1}}}\wedge\cdots\wedge\pqty{r\sin\theta_{1}\sin\theta_{2}\cdots\sin\theta_{D-3}\differential\theta_{D-2}}\wedge\differential{V_{p}}$
is the measure in the phase space.

###  Appendix A.3 Derivation

Up to the first order, the equilibrium condition (2.18) becomes

$\displaystyle\int\differential{r}r^{D-2}\int\differential{V_{p}}\Bqty{\mathrm{e}^{\nu}\hat{\varepsilon}\pqty{\mathcal{S}\variation{\nu}+\partialderivative{\mathcal{S}}{f}\variation{f}}+\alpha\mathrm{e}^{\nu}\hat{\varepsilon}\pqty{f\variation{\nu}+\variation{f}}-\beta\hat{\varepsilon}^{2}\variation{f}}=0.$
(A.11)

Introducing

$\displaystyle
s^{\ast}:=\partialderivative{\mathcal{S}}{f}+\alpha-\beta\varepsilon=\partialderivative{\mathcal{S}}{f}+\alpha-\hat{\beta}\hat{\varepsilon}$
(A.12)

with $\hat{\beta}:=\mathrm{e}^{\mu}\beta$, thermal equilibrium condition
(A.11) is rewritten as

$\displaystyle\int\differential{r}r^{D-2}\int\differential{V_{p}}\pqty{\mathcal{S}\variation{\nu}+s^{\ast}\variation{f}+\alpha
f\variation{\nu}}\mathrm{e}^{\nu}\hat{\varepsilon}+\int\differential{r}r^{D-2}\beta\pqty{\mathrm{e}^{\mu+\nu}-1}\int\differential{V_{p}}\hat{\varepsilon}^{2}\variation{f}=0.$
(A.13)

The second term becomes

$\displaystyle\int\differential{r}r^{D-2}\beta\pqty{\mathrm{e}^{\mu+\nu}-1}\int\differential{V_{p}}\hat{\varepsilon}^{2}\variation{f}$
$\displaystyle=$
$\displaystyle\int\differential{r}\beta\pqty{\mathrm{e}^{\mu+\nu}-1}\frac{\pqty{D-2}}{8\pi}\derivative{r}(r^{D-3}\mathrm{e}^{-2\nu}\variation{\nu})$
$\displaystyle=$
$\displaystyle-\frac{\pqty{D-2}\beta}{8\pi}\int\differential{r}\pqty{\mu^{\prime}+\nu^{\prime}}\mathrm{e}^{\mu-\nu}r^{D-3}\variation{\nu}$
$\displaystyle=$
$\displaystyle-\beta\int\differential{r}r^{D-2}\mathrm{e}^{\mu+\nu}\pqty{\rho+p}\variation{\nu},$
(A.14)

where we used the conservation of the energy-momentum:

$\displaystyle\mu^{\prime}+\nu^{\prime}=\frac{8\pi}{D-2}r\mathrm{e}^{2\nu}\pqty{\rho+p}.$
(A.15)

Then the condition (A.13) becomes

$\displaystyle\int\differential{r}r^{D-2}\int\differential{V_{p}}\pqty{H\variation{\nu}+s^{\ast}\variation{f}}\mathrm{e}^{\nu}\hat{\varepsilon}-\beta\int\differential{r}r^{D-2}\mathrm{e}^{\mu+\nu}\pqty{\rho+p}\variation{\nu}=0,$
(A.16)

where $H=\mathcal{S}+\alpha f$.

We use the identity for $G={{G}\pqty{\varepsilon,J}}$:

$\displaystyle\int\differential{V_{p}}\evaluated{\partialderivative{G}{\varepsilon}}_{J}\pqty{p_{t}}^{i}\pqty{p_{r}}^{j}=-i\int\differential{V_{p}}G\pqty{p_{t}}^{i-1}\pqty{p_{r}}^{j}-\pqty{j-1}\mathrm{e}^{-2\mu+2\nu}\int\differential{V_{p}}G\pqty{p_{t}}^{i+1}\pqty{p_{r}}^{j-2}.$
(A.17)

The proof of this identity is given in Appendix A.4. Substituting
$(i,j)=(1,2)$ into Eq. (A.17), we obtain

$\displaystyle\rho+p=$
$\displaystyle-\int\differential{V_{p}}p_{t}p^{t}f+\int\differential{V_{p}}p_{r}p^{r}f$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-2\nu}\Bqty{\int\differential{V_{p}}\pqty{p_{r}}^{2}f+\mathrm{e}^{-2\mu+2\nu}\int\differential{V_{p}}\pqty{p_{t}}^{2}f}$
$\displaystyle=$
$\displaystyle-\mathrm{e}^{-2\nu}\int\differential{V_{p}}\pqty{p_{t}}\pqty{p_{r}}^{2}\partialderivative{f}{\varepsilon}\evaluated{}_{J}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{\mu}\int\differential{V_{p}}\hat{\varepsilon}\
p_{r}p^{r}\partialderivative{f}{\varepsilon}\evaluated{}_{J}.$ (A.18)

Therefore,

$\displaystyle\beta\pqty{\rho+p}=$
$\displaystyle\beta\mathrm{e}^{\mu}\int\differential{V_{p}}\hat{\varepsilon}\
p_{r}p^{r}\partialderivative{f}{\varepsilon}\evaluated{}_{J}$ $\displaystyle=$
$\displaystyle\int\differential{V_{p}}p_{r}p^{r}\beta\varepsilon\partialderivative{f}{\varepsilon}\evaluated{}_{J}.$
(A.19)

Since

$\displaystyle\partialderivative{\mathcal{S}}{\varepsilon}\evaluated{}_{m,J}=\partialderivative{\mathcal{S}}{f}\partialderivative{f}{\varepsilon}\evaluated{}_{J}=\pqty{s^{\ast}-\alpha+\beta\varepsilon}\partialderivative{f}{\varepsilon}\evaluated{}_{J},$
(A.20) $\displaystyle\beta\pqty{\rho+p}=$
$\displaystyle\int\differential{V_{p}}p_{r}p^{r}\pqty{\partialderivative{\mathcal{S}}{\varepsilon}\evaluated{}_{J}+\alpha\partialderivative{f}{\varepsilon}\evaluated{}_{m,J}-s^{\ast}\partialderivative{f}{\varepsilon}\evaluated{}_{J}}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{-2\nu}\int\differential{V_{p}}\pqty{p_{r}}^{2}\partialderivative{H}{\varepsilon}\evaluated{}_{J}-\int\differential{V_{p}}p_{r}p^{r}s^{\ast}\partialderivative{f}{\varepsilon}\evaluated{}_{J}.$
(A.21)

Substituting $(i,j)=(0,2)$ into Eq. (A.17), we obtain

$\displaystyle\beta\pqty{\rho+p}=$
$\displaystyle\mathrm{e}^{-\mu}\int\differential{V_{p}}\hat{\varepsilon}H-\int\differential{V_{p}}p_{r}p^{r}s^{\ast}\partialderivative{f}{\varepsilon}\evaluated{}_{J}.$
(A.22)

Therefore, Eq. (A.16) becomes

$\displaystyle 0$
$\displaystyle=\int\differential{r}r^{D-2}\bqty{\int\differential{V_{p}}\pqty{H\variation{\nu}+s^{\ast}\variation{f}}\mathrm{e}^{\nu}\hat{\varepsilon}-\beta\mathrm{e}^{\mu+\nu}\pqty{\rho+p}\variation{\nu}}$
$\displaystyle=\int\differential{r}r^{D-2}\int\differential{V_{p}}\pqty{\hat{\varepsilon}\variation{f}+p_{r}p^{r}\mathrm{e}^{\mu}\partialderivative{f}{\varepsilon}\evaluated{}_{J}\variation{\nu}}\mathrm{e}^{\nu}s^{\ast}.$
(A.23)

Since $\variation{f}$ and $\variation{\nu}$ are dependent variations, above
condition for an arbitrary variation leads to the condition $s^{\ast}=0$,
i.e.,

$\displaystyle\partialderivative{\mathcal{S}}{f}+\alpha-\beta\varepsilon=0.$
(A.25)

For the BG entropy, the condition becomes $-\log f+\alpha-\beta\varepsilon=0$,
which leads to the MJ distribution function.

###  Appendix A.4 Proof of Eq. (A.17)

Using the coordinates with Eq. (A.7), the left-hand side of Eq. (A.17) becomes

$\displaystyle\int\differential{V_{p}}\evaluated{\partialderivative{G}{\varepsilon}}_{J}\pqty{p_{t}}^{i}\pqty{p_{r}}^{j}$
$\displaystyle=\int\differential{\varepsilon}\wedge\differential{J}\wedge\differential{\Omega_{D-3}}\frac{J^{\frac{D-4}{2}}}{r^{2(D-2)}}\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{-\frac{1}{2}}\evaluated{\partialderivative{G}{\varepsilon}}_{J}\pqty{p_{t}}^{i}\pqty{p_{r}}^{j}.$
(A.26)

Since

$\displaystyle\evaluated{\partialderivative{\varepsilon}}_{J}\pqty{\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{-\frac{1}{2}}\pqty{p_{t}}^{i}\pqty{p_{r}}^{j}}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{j(-\mu+\nu)}\evaluated{\partialderivative{\varepsilon}}_{J}\pqty{\pqty{-\varepsilon}^{i}\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{\frac{j-1}{2}}}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{j(-\mu+\nu)}\pqty{-i\pqty{-\varepsilon}^{i-1}\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{\frac{j-1}{2}}-\pqty{j-1}\pqty{-\varepsilon}^{i+1}\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{\frac{j-3}{2}}}$
$\displaystyle=$
$\displaystyle\pqty{\varepsilon^{2}-\varepsilon_{0}^{2}\pqty{1+\frac{J}{r^{2}}}}^{-\frac{1}{2}}\bqty{-i\pqty{p_{t}}^{i-1}\pqty{p_{r}}^{j}-\pqty{j-1}\mathrm{e}^{-2\mu+2\nu}\pqty{p_{t}}^{i+1}\pqty{p_{r}}^{j-2}},$
(A.27)

integrating by parts, we get Eq. (A.17). $\square$

##  Appendix Appendix B The integration over the momentum space

We derive the explicit expressions for the physical quantities by performing
the integration over the momentum space. In order to simplify the
integrations, we transform the integral variables as $J\to
s:=J/J_{\mathrm{max}}$, where
$J_{\mathrm{max}}:=r^{2}({\hat{\varepsilon}}^{2}-1)$ is the upper bound for
the integration of $J$. In the coordinate system
$\pqty{\hat{\varepsilon},s,\psi_{1},\psi_{2},\cdots,\psi_{D-3}}$, the
invariant volume element is rewritten as

$\displaystyle
dV_{p}=\frac{s^{\frac{D-4}{2}}}{\sqrt{1-s}}({\hat{\varepsilon}}^{2}-1)^{\frac{D-3}{2}}\differential{\hat{\varepsilon}}\wedge\differential{s}\wedge\differential{\Omega_{D-3}},$
(B.1)

where $1\leq\hat{\varepsilon}<\infty$ and $0\leq s\leq 1$. If the distribution
function takes the form $f={{f}\pqty{\hat{\varepsilon},J}}$, the energy
density is given by

$\displaystyle\rho(r)$ $\displaystyle=\int dV_{p}\
\hat{\varepsilon}^{2}f(\hat{\varepsilon},J)$
$\displaystyle=S_{D-2}\int_{1}^{\infty}\differential{\hat{\varepsilon}}\int_{0}^{1}\differential{s}\frac{s^{\frac{D-4}{2}}(\hat{\varepsilon}^{2}-1)^{\frac{D-3}{2}}}{\sqrt{1-s}}\hat{\varepsilon}^{2}f,$
(B.2)

and the pressure becomes

$\displaystyle p(r)$ $\displaystyle=\int dV_{p}\
(p^{\hat{r}})^{2}f(\hat{\varepsilon},J)$
$\displaystyle=S_{D-2}\int_{1}^{\infty}\differential{\hat{\varepsilon}}\int_{0}^{1}\differential{s}s^{\frac{D-4}{2}}\sqrt{1-s}(\hat{\varepsilon}^{2}-1)^{\frac{D-1}{2}}f.$
(B.3)

For the MJ distribution function
$f(\hat{\varepsilon})=\exp(\alpha-z\hat{\varepsilon})$, by using

$\displaystyle\int_{0}^{1}\differential{s}\frac{s^{\frac{D-4}{2}}}{\sqrt{1-s}}$
$\displaystyle=\frac{\sqrt{\pi}\Gamma(\frac{D-2}{2})}{\Gamma(\frac{D-1}{2})},$
$\displaystyle\int_{0}^{1}\differential{s}s^{\frac{D-4}{2}}\sqrt{1-s}$
$\displaystyle=\frac{\sqrt{\pi}\Gamma(\frac{D-2}{2})}{2\Gamma(\frac{D+1}{2})},$
(B.4)

we obtain

$\displaystyle\rho(r)$
$\displaystyle=S_{D-1}\mathrm{e}^{\alpha}\int_{1}^{\infty}\differential{\hat{\varepsilon}}(\hat{\varepsilon}^{2}-1)^{\frac{D-3}{2}}\hat{\varepsilon}^{2}\mathrm{e}^{-z\hat{\varepsilon}},$
(B.5a) $\displaystyle p(r)$
$\displaystyle=\frac{S_{D-1}}{D-1}\mathrm{e}^{\alpha}\int_{1}^{\infty}\differential{\hat{\varepsilon}}(\hat{\varepsilon}^{2}-1)^{\frac{D-1}{2}}\mathrm{e}^{-z\hat{\varepsilon}}.$
(B.5b)

Introducing the modified Bessel function of the second kind
${{K_{n}}\pqty{z}}$:

$\displaystyle{{K_{n}}\pqty{z}}=\frac{\sqrt{\pi}\pqty{\frac{z}{2}}^{n}}{\Gamma\pqty{n+\frac{1}{2}}}\int_{1}^{\infty}\differential{t}(t^{2}-1)^{n-\frac{1}{2}}\mathrm{e}^{-zt},$
(B.6)

the energy density and the pressure become

$\displaystyle\rho(r)$
$\displaystyle=2\mathrm{e}^{\alpha}\pqty{\frac{2\pi}{z}}^{\frac{D-2}{2}}\pqty{\frac{D-1}{z}K_{\frac{D}{2}}(z)+K_{\frac{D-2}{2}}(z)},$
(B.7a) $\displaystyle p(r)$
$\displaystyle=\frac{\mathrm{e}^{\alpha}}{\pi}\pqty{\frac{2\pi}{z}}^{\frac{D}{2}}K_{\frac{D}{2}}(z).$
(B.7b)

For the particle number, we can rewrite its density as

$\displaystyle n(r):=$
$\displaystyle\mathrm{e}^{-\mu+\nu}\int\differential{V_{p}}\varepsilon\mathrm{e}^{\alpha-\beta\varepsilon}$
$\displaystyle=$
$\displaystyle\mathrm{e}^{\alpha+\nu}\frac{2\pi^{\frac{D-1}{2}}}{\Gamma(\frac{D-1}{2})}\int_{1}^{\infty}\differential{\hat{\varepsilon}}({\hat{\varepsilon}}^{2}-1)^{\frac{D-3}{2}}{\hat{\varepsilon}}\mathrm{e}^{-z{\hat{\varepsilon}}}$
$\displaystyle=$ $\displaystyle
2\mathrm{e}^{\alpha+\nu}\pqty{\frac{2\pi}{z}}^{\frac{D-2}{2}}K_{\frac{D}{2}}(z),$
(B.8)

where we used

$\displaystyle\int_{1}^{\infty}\differential{{\hat{\varepsilon}}}\pqty{{\hat{\varepsilon}}^{2}-1}^{n-\frac{1}{2}}{\hat{\varepsilon}}\mathrm{e}^{-z{\hat{\varepsilon}}}=$
$\displaystyle\frac{1}{2n+1}\int_{1}^{\infty}\differential{{\hat{\varepsilon}}}\derivative{{\hat{\varepsilon}}}\pqty{\pqty{{\hat{\varepsilon}}^{2}-1}^{n+\frac{1}{2}}}\mathrm{e}^{-z{\hat{\varepsilon}}}$
$\displaystyle=$
$\displaystyle\frac{1}{2n+1}\bqty{\pqty{{\hat{\varepsilon}}^{2}-1}^{n+\frac{1}{2}}\mathrm{e}^{-z{\hat{\varepsilon}}}}_{1}^{\infty}+\frac{z}{2n+1}\int_{1}^{\infty}\differential{{\hat{\varepsilon}}}\pqty{{\hat{\varepsilon}}^{2}-1}^{n+\frac{1}{2}}\mathrm{e}^{-z{\hat{\varepsilon}}}$
$\displaystyle=$
$\displaystyle\frac{\Gamma(n+\frac{1}{2})}{\sqrt{\pi}\pqty{\frac{z}{2}}^{n}}K_{n+1}(z).$
(B.9)

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