theory, we found that the leading order dynamics in this regime are actually
governed by the formal second order term at $O(\varepsilon^{1/2})$ (recall
equation (38)), since the formal leading order term is exponentially small due
to a delay effect. Lemma 3.16 describes a delayed loss of stability in the
symmetric approximation with $\mu(x)\equiv 0$. The requirement (40) and the
bound in (41) together imply that initial conditions with $v(0)=-\rho_{in}$
remain exponentially small up to $v(T)\approx\rho_{in}$. In order to prove
that a hard onset of patterned states occurs near $v=\rho_{in}$ we would also
need to provide a lower bound on the norm of $\Psi(\cdot,T)$. Such an approach
has been successfully applied in the context of scalar reaction-diffusion
problems in [44, 67], but the analysis for equation (4) is left for future
work.

As in classical modulation theory, rigorous bounds for the error associated to
the approximation are necessary in order to relate the dynamics of the
approximation $\Psi$ and solutions to the SH problem (4). We provided detailed
error estimates in $H_{ul}^{\theta}$ spaces in Theorem 3.20. Different bounds
were obtained in the general case $\mu(x)\neq 0$ and the symmetric case
$\mu(x)\equiv 0$. In the first case we were able to control the error up to a
distance of $O(\varepsilon^{1/2})$ from the static bifurcation point at $v=0$,
i.e. up to the exit section $\Delta^{mid}$ in chart $\mathcal{K}_{2}$. In the
symmetric case $\mu(x)\equiv 0$ we were able to show that the error remains
exponentially small up to an $O(1)$ distance from $v=0$, but estimate obtained
in Assertion (ii) is likely to be sub-optimal due to the unwanted growth of
the residual. It is conjectured that a better bound can be obtained if the
algebraic control over the residual is improved to exponential control; see
again Remark 3.21. Finally, we note once more that the error estimates of
Theorem 3.20 apply for a rather large set of initial conditions
$u(x,0)=u^{\ast}(x)$. We require only that $\|u^{\ast}\|_{H_{ul}^{\theta}}\leq
K$ for some constant $K>0$ which is small but $O(1)$ as $\varepsilon\to 0$.
This is an improvement on known results for the static SH equation in e.g.
[29, 75], which only apply for initial conditions that are $O(\delta)$ as
$\delta\to 0$; see again the classical result in Theorem 2.1 and the
discussion which follows it. This is because of additional contraction
undergone by solutions in the fast-slow setting during the time spend in the
asymptotically stable regime with $v(t)<0$.

Finally, we presented rigorous results on the dynamics of SH solutions in
Theorem 3.22. Assertion (i) shows that a large space of SH solutions (all
those with initial conditions $(u(x,0),v(0))=(u^{\ast}(x),-\rho_{in})$ such
that $\|u^{\ast}\|_{H_{ul}^{\theta}}\leq K$) have the same asymptotic form at
$\Delta^{mid}$, i.e. at a distance of $O(\varepsilon^{1/2})$ past the static
bifurcation point. Assertion (ii) characterises the delayed stability loss in
the symmetric case $\mu(x)\equiv 0$. A lower bound on the delay time is given,
but limited by the (likely sub-optimal) bound on the error estimates in
Theorem 3.20. Nevertheless, this result constitutes, to the best of our
knowledge, the first rigorous and global result on the existence of delayed
stability loss in systems featuring a slow passage through a Turing
bifurcation.

## Acknowledgements

SJ and CK acknowledge funding from the SFB/TRR 109 Discretization and Geometry
in Dynamics grant. FH and CK acknowledge support of the EU within the TiPES
project funded the European Unions Horizon 2020 research and innovation
programme under grant agreement No. 820970. CK has also been supported by a
Lichtenberg Professorship of the VolkswagenStiftung.

## Data availability

Data sharing not applicable to this article as no datasets were generated or
analysed during the current study.

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##  Appendix A Technical estimates

In the following we provide a number of known results (or slight adaptations
thereof) that are useful for the proofs in Sections 5 and 6.

#### Estimate methods in $H_{ul}^{\theta}$

Functions approximated by the blow-up approximations $\Psi_{GL}$ and
$\Psi_{n}$ defined in Section 3.1 depend on two distinct spatial scales $x$
and $\bar{x}$, where the latter is defined via (15). In order to to estimate
the norm of functions like
$\Psi_{GL}(x,t)=r(t)(A(\bar{x},\bar{t})\mathrm{e}^{ix}+c.c.)$ in
$H_{ul}^{\theta}$ spaces, we use the following scaling property to relate the
norms of $\|x\mapsto u(x)\|_{H_{ul}^{\theta}}$ and $\|x\mapsto
u(rx)\|_{H_{ul}^{\theta}}$ for a non-negative scaling factor $r\geq 0$.

######  Lemma A.1.

c.f. [82, Ch. 10] Let $u\in H_{ul}^{\theta+1}$ and $r\geq 0$. Then

$\|x\mapsto u(rx)\|_{H_{ul}^{\theta}}\leq C\|u\|_{H_{ul}^{\theta+1}}.$

###### Proof.

It suffices to show that $\|x\mapsto u(rx)\|_{L_{ul}^{2}}\leq
C\|u\|_{H_{ul}^{1}}$, which can be done by a direct calculation. Applying the
definition of $\|\cdot\|_{L_{ul}^{2}}$ in (11) we obtain

$\begin{split}\|u(r\cdot)\|_{H_{ul}^{\theta}}&=\sup_{y\in\mathbb{R}}\left(\int_{y}^{y+1}|u(rx)|^{2}dx\right)^{1/2}\\\
&\leq\sup_{y\in\mathbb{R}}\left\\{\left(\int_{y-1/2}^{y+1/2}dx\right)^{1/2}\sup_{x\in[y-1/2,y+1/2]}|u(rx)|\right\\}\leq\|u\|_{C_{b}^{0}}\leq
C\|u\|_{H_{ul}^{1}},\end{split}$

where we used a Sobolev embedding in the final inequality, c.f. [82, Lemma
8.3.11] and Lemma A.3 below. ∎

The next result comes from multiplier theory, and is useful in order to
estimate the norm of semigroup and evolution family operators in uniformly
local Sobolev spaces $H_{ul}^{\theta}$. Multiplier theory takes advantage of
the Fourier transform properties of Sobolev spaces in order to provide methods
for estimating norms of translation invariant operators in $H_{ul}^{\theta}$,
which is typically a very difficult task in physical space. An operator
$M:H_{ul}^{q}\to H_{ul}^{\theta}$ is a called a multiplier if the
corresponding Fourier operator $\widehat{M}:\mathbb{R}\to\mathbb{C}$ is a
multiplication operator, i.e. if $M=\mathscr{F}^{-1}(\widehat{M}\mathscr{F})$.
It is known that the the norms for multipliers on Sobolev spaces $H^{\theta}$
can be bounded in terms of the $C_{b}^{0}(\mathbb{R},\mathbb{C})$ norm (see
e.g. [82, Lemma 8.3.6]). The following result extends this to multipliers on
uniformly local Sobolev spaces $H_{ul}^{\theta}$. Although it was originally
shown in [73], we present here it in the formulation of [65, 74, 82], along
with a specialisation to the particular case of interest for the proofs in
Sections 5 and 6.

######  Lemma A.2.

[82, Lemma 8.3.7] For $\theta,q\geq 0$ and
$w_{\theta-q}(\xi)=(1+\xi^{2})^{(\theta-q)/2}\widehat{M}(\xi)\in
C^{2}_{b}(\mathbb{R},\mathbb{C})$, the map $M_{ul}:H_{ul}^{q}\to
H_{ul}^{\theta}$, $u\mapsto\mathscr{F}^{-1}(\widehat{M}\mathscr{F}u)$ is well-
defined and satisfies

$\|Mu\|_{H_{ul}^{\theta}}\leq
C\|w_{\theta-q}\|_{C_{b}^{2}(\mathbb{R},\mathbb{C})}\|u\|_{H_{ul}^{q}},$

where the constant $C$ depends on $q$ and $\theta$ but not on $\widehat{M}$.
For the case $q=\theta$ occurring frequently in this work,
$w_{\theta-q}(\xi)=w_{0}(\xi)=\widehat{M}(\xi)$ such that

$\|M_{ul}u\|_{H_{ul}^{\theta}}\leq
C\|w_{0}\|_{C_{b}^{2}(\mathbb{R},\mathbb{C})}\|u\|_{H_{ul}^{\theta}},$

i.e. $\|M_{ul}u\|_{H_{ul}^{\theta}\to H_{ul}^{\theta}}\leq
C\|w_{0}\|_{C_{b}^{2}(\mathbb{R},\mathbb{C})}$.

Finally, we have the following result, which extends a well-known result on
the multiplicative closure of Sobolev spaces $H^{\theta}$ to the uniformly
local Sobolev spaces $H_{ul}^{\theta}$.

######  Lemma A.3.

[82, Lemma 8.3.11] Fix $\theta>1/2$. Then $H_{ul}^{\theta}$ is an algebra with

$\|uv\|_{H_{ul}^{\theta}}\leq 9\|u\|_{H_{ul}^{\theta}}\|v\|_{H_{ul}^{\theta}}$

for all $u,v\in H_{ul}^{\theta}$, and can be continuously embedded in
$C_{b,unif}^{0}$. In particular, Sobolev’s embedding theorem gives

$\|u\|_{C_{b}^{\theta}}\leq\|u\|_{H_{ul}^{\theta+1}}.$

#### Integral identities and inequalities

The following Grönwall-type inequality from [42] is frequently used in
Sections 5 and 6 in order to control the norms of integral equations obtained
after applying the variation of constants formula.

######  Lemma A.4.

[42, Lemma 2.8] Let $\alpha\in\mathbb{R}$, $\beta,N,T>0$ and $\gamma\in(0,1]$.
Assume that $a,b:[0,T]\to[0,\infty)$ are continuous, the derivative
$b^{\prime}$ is locally integrable in $L^{2}$ and that the function
$[t\mapsto\mathrm{e}^{-\alpha t/\beta}b(t)]$ is non-decreasing. If

$a(t)\leq
b(t)+N\int_{0}^{t}\frac{\mathrm{e}^{\alpha(t-s)/\beta}}{\beta^{\gamma}(t-s)^{1-\gamma}}a(s)ds,$

for all $t\in[0,T]$, then

$a(t)\leq
2b(0)\mathrm{e}^{\tilde{\alpha}t/\beta}+2\int_{0}^{t}\left(b^{\prime}(s)-\frac{\alpha}{\beta}b(s)\right)\mathrm{e}^{\tilde{\alpha}(t-s)/\beta}ds,$

where $\tilde{\alpha}=\alpha+2N^{1/\gamma}(2/\gamma)^{(1-\gamma)/\gamma}$, for
all $t\in[0,T]$. In the special case that $\beta=\gamma=1$ and
$b(t)=C\mathrm{e}^{\alpha t}$ for some constant $C>0$, which arises frequently
in this work, we have

$a(t)\leq 2C\mathrm{e}^{(\alpha+2N)t},$

for all $t\in[0,T]$.

Finally, we provide an integral identity and a number of basic properties of
the upper incomplete gamma function

$\Gamma(a,z):=\int_{z}^{\infty}s^{a-1}\mathrm{e}^{-s}ds,$ (165)

which arises frequently in the proofs in Sections 5 and 6. We refer to [90,
Ch. 11.2] for more background and properties on incomplete gamma functions.
The asymptotics

$\Gamma(a,z)\sim\mathrm{e}^{-z}z^{a-1}\left(1+O(z^{-1})\right),\qquad|z|\to\infty,$
(166)

are particularly useful for our purposes. The integral identity presented in
the following result is also helpful for many calculations.

######  Lemma A.5.

Let $\alpha,\beta,\gamma\in\mathbb{R}$ and $t\in[0,|\beta^{-1}|)$. Then

$Q(t)=\int_{0}^{t}\frac{\mathrm{e}^{-\alpha s}}{(1+\beta
s)^{\gamma}}ds=\frac{\mathrm{e}^{\alpha/\beta}}{\beta}\left(\frac{\beta}{\alpha}\right)^{1-\gamma}\left[\Gamma\left(1-\gamma,\frac{\alpha}{\beta}\right)-\Gamma\left(1-\gamma,\frac{\alpha}{\beta}(1+\beta
t)\right)\right],$ (167)

which is increasing on $t\in[0,|\beta^{-1}|)$.

###### Proof.

This can be shown by a direct calculation using the substitution
$\eta=\alpha(1+\beta s)/\beta$ and the definition of $\Gamma(\cdot,\cdot)$ in
(165), or verified by directly differentiating the right-hand side in (167)
and appealing to the fundamental theorem of calculus. The increasing property
follows from the fact that

$Q^{\prime}(t)=\frac{\mathrm{e}^{-\alpha t}}{(1+\beta t)^{\gamma}}>0$

for all $t\in[0,|\beta^{-1}|)$. ∎


# Dynamic phase transition of charged dilaton black holes

Jie-Xiong Mo, Shan-Quan Lan111shanquanlan@126.com(corresponding author)
Institute of Theoretical Physics, Lingnan Normal University, Zhanjiang,
524048, Guangdong, China

###### Abstract

Dynamic phase transition of charged dilaton black holes is investigated in
this paper. We introduce the Gibbs free energy landscape and calculate the
corresponding $G_{L}$ for the dilaton black hole. On the one hand, we
numerically solve the Fokker-Planck equation constrained by only the
reflecting boundary condition. The effects of dilaton gravity on the
probabilistic evolution of dilaton black holes are quite obvious. Firstly, the
horizon radius difference between the large dilaton black hole and small
dilaton black hole increases with the parameter $\alpha$. Secondly, with the
increasing of $\alpha$, the system needs much more time to achieve a
stationary distribution. Thirdly, the value which $\rho(r_{l},t)$ and
$\rho(r_{s},t)$ finally attain varies with the parameter $\alpha$. On the
other hand, resolving Fokker-Planck equation constrained by both the
reflecting boundary condition and absorbing boundary condition, we investigate
the first passage process of dilaton black holes. The initial peak decays more
slowly with the increase of $\alpha$, which can also be witnessed from the
slow down of the decay of $\Sigma(t)$ (the sum of the probability that the
black hole system having not finished a first passage by time $t$). Moreover,
the time corresponding to the single peak of the first passage time
distribution is also found to increase with the parameter $\alpha$.
Considering all the observations mentioned above, the dilaton field does slow
down the dynamic phase transition process between the large black hole and
small black hole.

dynamic phase transition charged dilaton AdS black holes Fokker-Planck
equation

###### pacs:

04.70.Dy, 04.70.-s

##  I Introduction

Dilaton gravity Gibbons1 ; Garfinkle ; Koikawa  has received considerable
attention within the community for its great physical significance. Firstly,
it serves as the low energy limit of string theory. One can recover Einstein
gravity with a non-minimally coupled dilaton field and other fields in the low
energy limit. Secondly, there exist one or more Liouville-type potentials in
its action. And these potentials can be caused by spacetime supersymmetry
breaking in ten dimensions. Thirdly, it has been disclosed that the dilaton
field may affect both the casual structure and thermodynamic properties of
black holes. Due to the above physical importance, various black hole
solutions were presented with their thermodynamic properties discussed
Gibbons1 \- zhangchengyong . Interestingly, it was argued in Refs.
gaochangjun1 ; gaochangjun2 ; gaochangjun3  that the cosmological constant is
found to be coupled to Liouville-type potential in the dilaton theory.

Among these solutions, a class of $(n+1)$-dimensional topological black hole
solutions were obtained Sheykhi2  in Einstein-Maxwell-dilaton theory, which
are neither asymptotically flat nor anti-de Sitter. Thermodynamic properties
of these black holes were carefully analyzed. Moreover, the effect of dilaton
field on thermal stability was disclosed Sheykhi2 . $P-V$ criticality was
investigated in Refs. zhaoren ; Sheykhi1 . Provided the coupling constant of
dilaton gravity is less than one, the critical values of relevant physical
quantities are physical Sheykhi1 . Ref. Sheykhi13  probed the phase transition
and thermodynamic geometry while Ref. Sheykhi16  further disclosed the
existence of zeroth-order phase transition. Both the two point correlation
function and the entanglement entropy of dilaton black holes were studied by
one of us xiong10 , providing an alternative perspective to observe the
critical phenomena in dilaton black holes. With the tool of Ruppeiner
geometry, Ref. zhangshaojun  tried to provide the microscopic explanation for
phase transitions of charged dilatonic black holes.

Although many attempts have been made as stated above, the dynamical phase
transition of this typical class of dilaton black hole solution proposed in
Ref. Sheykhi2  and the possible kinetics behind it remains to be probed. So it
is the main motivation of this paper to complete this “missing puzzle piece”
concerning the phase transition of charged dilaton black holes. Dynamic phase
transition of black holes has been a newly emerged hot topic ever since the
creative works reporting the kinetics of the Hawing-Page phase transition
liran1  and the van der Waals like phase transition liran2  recently. For the
latter, the transition between the large black hole and the small black hole
due to thermal fluctuation can be viewed as stochastic process, which can be
probed via the tool of Fokker-Planck equation liran2 . Soon after, dynamic
phase transition of Gauss-Bonnet black holes liran3 ; weishaowen2 ;
weishaowen3  and charged AdS black holes surrounded by quinteness dark energy
xiong11  were investigated. More interestingly, it was even suggested in a
very recent work liran4  that the turnover of the kinetics can be used to
probe the microstructure of black holes. It is certainly of interest to
generalize these pioneer works to dilaton black holes considering the great
physical importance of dilaton gravity stated in the first paragraph of this
paper. It is expected that novel features will be disclosed about the dynamic
phase transition of black holes.

The organization of this paper is as follows. Sec.II is devoted to a short
review of thermodynamic properties of charged dilaton black holes. In Sec.III,
we will investigate the dynamic phase transition of charged dilaton black
holes. We will further probe the first passage process for the phase
transition of charged dilaton black holes in Sec.IV. Conclusions will be
presented in Sec.V.

##  II A short review on thermodynamic properties of charged dilaton black
holes

The Einstein-Maxwell-Dilaton action in $(n+1)$-dimensional spacetime reads
Chan

$S=\frac{1}{16\pi}\int
d^{n+1}x\sqrt{-g}\left(\mathcal{R}-\frac{4}{n-1}(\nabla\Phi)^{2}-V(\Phi)-(e^{-4\alpha\Phi/(n-1)}F_{\mu\nu}F^{\mu\nu}\right),\\\
$ (1)

where $\mathcal{R}$, $\Phi$ and $F_{\mu\nu}$ denote respectively the Ricci
scalar curvature, the dilaton field and the electromagnetic field tensor. The
parameter $\alpha$ describes the strength of coupling between the
electromagnetic field and the dilaton field.

To obtain a solution, one needs to take a specific form for the potential of
the dilaton field $V(\Phi)$. Here, the potential of the dilaton field is
chosen as $V(\Phi)=2\Lambda_{0}e^{2\zeta_{0}\Phi}+2\Lambda e^{2\zeta\Phi}$
Chan ; Sheykhi4 ; Sheykhi2 ; Sheykhi3 . Note that if one consider the power-
law Maxwell field Sheykhi8 , an additional term should be added to the
potential mentioned above.

One can take the general form of metric as Sheykhi2

$ds^{2}=-f(r)dt^{2}+\frac{1}{f(r)}dr^{2}+r^{2}R^{2}(r)h_{ij}dx^{i}dx^{j},\\\ $
(2)

where $h_{ij}dx^{i}dx^{j}$ corresponds to $(n-1)$-dimensional hypersurface
with constant scalar curvature $(n-1)(n-2)k$. $k$ can be taken as
$-1,\,0,\,1$, corresponding to hyperbolic, flat and spherical constant
curvature hypersurface respectively.

Making the ansatz $R=e^{2\alpha\Phi/(n-1)}$, the corresponding solution was
derived Sheykhi2  as

$\displaystyle f(r)$ $\displaystyle=$
$\displaystyle-\frac{k(n-2)(1+\alpha^{2})^{2}b^{-2\gamma}r^{2\gamma}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}-\frac{m}{r^{(n-1)(1-\gamma)-1}}+\frac{2\Lambda
b^{2\gamma}(1+\alpha^{2})^{2}r^{2(1-\gamma)}}{(n-1)(\alpha^{2}-n)}$ (3)
$\displaystyle+\frac{2q^{2}(1+\alpha^{2})^{2}b^{-2(n-2)\gamma}r^{2(n-2)(\gamma-1)}}{(n-1)(n+\alpha^{2}-2)},$
$\displaystyle\Phi(r)$ $\displaystyle=$
$\displaystyle\frac{(n-1)\alpha}{2(\alpha^{2}+1)}\ln\left(\frac{b}{r}\right),$
(4)

where $b$ is an arbitrary constant and $\gamma$ is related to $\alpha$ by
$\gamma=\alpha^{2}/(\alpha^{2}+1)$. And the coefficients in the expression of
$V(\Phi)$ read Sheykhi2

$\Lambda_{0}=\frac{k(n-1)(n-2)\alpha^{2}}{2b^{2}(\alpha^{2}-1)},\;\zeta_{0}=\frac{2}{\alpha(n-1)},\;\zeta=\frac{2\alpha}{n-1}.\\\
$ (5)

The mass, the charge, Hawking temperature and the entropy was obtained
respectively as Sheykhi2 ; zhaoren

$\displaystyle M$ $\displaystyle=$
$\displaystyle\frac{b^{(n-1)\gamma}(n-1)\omega_{n-1}m}{16\pi(\alpha^{2}+1)},$
(6) $\displaystyle Q$ $\displaystyle=$
$\displaystyle\frac{\omega_{n-1}q}{4\pi},$ (7) $\displaystyle T$
$\displaystyle=$
$\displaystyle\frac{-(1+\alpha^{2})}{2\pi(n-1)}\Big{[}\frac{k(n-2)(n-1)}{2b^{2\gamma}(\alpha^{2}-1)}r_{+}^{2\gamma-1}+\Lambda
b^{2\gamma}r_{+}^{1-2\gamma}+q^{2}b^{-2(n-2)\gamma}r_{+}^{(2n-3)(\gamma-1)-\gamma}\Big{]},$
(8) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{\omega_{n-1}b^{(n-1)\gamma}r_{+}^{(n-1)(1-\gamma)}}{4}.$
(9)

where the parameter $m$ can be derived via $f(r_{+})=0$.

Considering the demand that the electric potential $A_{t}$ should be finite at
infinity and the parameter $m$ should vanish at spacial infinity, restrictions
on $\alpha$ for dilaton black holes coupled with power-law Maxwell field was
obtained Sheykhi8  as $0\leq\alpha^{2}<n-2$ for the case
$\frac{1}{2}<p<\frac{n}{2}$ while $2p-n<\alpha^{2}<n-2$ for the case
$\frac{n}{2}<p<n-1$, where $p$ describes the nonlinearity of the
electromagnetic field. For the Maxwell field here, $p=1$. And the
corresponding restriction reduces to be $0\leq\alpha^{2}<n-2$.

##  III Dynamic phase transition of charged dilaton black holes

To probe the dynamic phase transition of black holes, one should introduce the
“Gibbs free energy landscape” and define the corresponding Gibbs free energy
$G_{L}$ as $G_{L}=H-T_{E}S$. Here, $T_{E}$ denotes the temperature of the
ensemble. As suggested in the pioneer work liran1 ; liran2 ; weishaowen2 ,
$G_{L}$ possesses the following properties. Firstly, only when $T_{E}=T$,
$G_{L}$ describes a real black hole. Secondly, only the extremal points of
$G_{L}$ make sense. The local maximum and minimum denote respectively the
unstable and stable black hole phases. Thirdly, the small-large black hole
phase transition corresponds to the double wells of $G_{L}$ having the same
depth.

Via Eqs. (6) and (9), $G_{L}$ of the dilaton black hole can be calculated as

$\displaystyle G_{L}$ $\displaystyle=$
$\displaystyle\frac{\omega_{n-1}b^{(n-1)\gamma}r_{+}^{n-2+(1-n)\gamma}}{16\pi}\Big{\\{}\Big{[}\frac{2b^{-2(n-2)\gamma}q^{2}r_{+}^{2(n-2)(\gamma-1)}}{(n-1)(\alpha^{2}+n-2)}-\frac{k(n-2)b^{-2\gamma}r^{2\gamma}}{(\alpha^{2}-1)(\alpha^{2}+n-2)}+\frac{16\pi
Pb^{2\gamma}r^{2(1-\gamma)}}{(n-1)(n-\alpha^{2})}\Big{]}$ (10)
$\displaystyle\times(n-1)(1+\alpha^{2})-4\pi r_{+}T_{E}\Big{\\}},$

where we have introduced the extended phase space and identify the
thermodynamic pressure as $P=-\Lambda/8\pi$.

Based on $G_{L}$, we can utilize the following Fokker-Planck equation Zwanzig
; Lee1 ; Lee2 ; wang1 ; Bryngelson  to investigate the probabilistic evolution
of the black hole after the thermal fluctuation

$\frac{\partial\rho(r,t)}{\partial t}=D\frac{\partial}{\partial
r}\left\\{e^{-\beta G_{L}(T,P,r)}\frac{\partial}{\partial r}\left[e^{\beta
G_{L}(T,P,r)}\rho(r,t)\right]\right\\},$ (11)

where $\rho(r,t)$ denotes the probability distribution. The parameter
$\beta=\frac{1}{k_{B}T}$ with $k_{B}$ as the Boltzman constant. The diffusion
coefficient $D=\frac{k_{B}T}{\zeta}$ with $\zeta$ denoting the dissipation
coefficient. Without loss of generality, one can set $\zeta$ and $k_{B}$ as
one. For the sake of conciseness, we use the notation $r$ to replace the
notation $r_{+}$ for the black hole horizon radius here and hereafter.

Before solving the above equation, one should impose both the boundary
condition and the initial condition first. The boundary conditions can be
divided into two categories, reflecting boundary condition and absorbing
boundary condition. As suggested in Ref. liran2 , the reflecting boundary
condition can be considered as $\beta
G^{\prime}(r)\rho(r,t)+\rho^{\prime}(r,t)\big{|}_{r=r_{0}}=0$ while absorbing
boundary condition can be considered as $\rho(r_{m},t)=0$ (The meaning of
$r_{m}$ would be discussed in the next paragraph). The initial condition can
be chosen as Gaussian wave pocket located at the small/large dilaton black
hole state respectively. Namely,
$\rho(r,0)=\frac{1}{\sqrt{\pi}a}e^{-(r-r_{l(s)})^{2}/a^{2}}$, with $r_{l}$ and
$r_{s}$ denote the horizon radius of large dilaton black hole and small
dilaton black hole respectively liran2 .

We numerically solve Eq.(11) constrained by both the initial condition and the
reflecting boundary condition mentioned above. Fig.1 shows the time evolution
of the probability distribution $\rho(r,t)$ of charged dilaton AdS black
holes, where the thermodynamic pressure is chosen as $P=0.003$. In the top,
middle and bottom rows of Fig.1, the parameter characterizing the strength of
coupling between the electromagnetic field and the dilaton field $\alpha$ is
chosen to be $0.4,0.2,0$ respectively to compare the effect of dilaon gravity
on the probability evolution.

For the left three graphs of Fig.1, the initial condition is chosen as
Gaussian wave pocket located at the large dilaton black hole state. This can
be reflected in the graphs that the initial peak of $\rho(r,t)$ corresponds to
the radius of the large dilaton black hole. With the time evolution, the
initial peak is decreasing while another peak corresponds to the radius of the
small dilaton black hole is booming. Finally, these two peaks reach a
stationary distribution that they equal to each other at the first sight. This
clearly exhibits the dynamic process of how the initial large dilaton black
hole evolves into the small dilaton black hole. So the left three graphs
correspond to the case the large dilaton black hole evolves dynamically into
small dilaton black hole.

By contrast, the initial condition is chosen as Gaussian wave pocket located
at the small dilaton black hole state for the right three graphs of Fig.1. The
initial peak locating at the small dilaton black hole is decaying while
another peak corresponding to the radius of the large dilaton black hole is
increasing with the time evolution. At the long time limit, these two peaks
corresponding to the large (small) black holes respectively approach to a
stationary distribution. So the right three graphs of Fig.1 correspond to the
case the small dilaton black hole evolves dynamically into large dilaton black
hole.

Figure 1: Time evolution of $\rho(r,t)$ for charged dilaton black hole with
$b=1,q=1,P=0.003$. (a) $\alpha=0.4$ (b) $\alpha=0.4$ (c) $\alpha=0.2$ (d)
$\alpha=0.2$ (e) $\alpha=0$ (f) $\alpha=0$. For the left three graphs, the
initial condition is chosen as Gaussian wave pocket located at the large
dilaton black hole state while for the right three graphs, the initial
condition is chosen as Gaussian wave pocket located at the small dilaton black
hole state.

Figure 2: Time evolution of $\rho(r_{l},t)$ and $\rho(r_{s},t)$ for charged
dilaton black hole with $b=1,q=1,P=0.003$. (a) $\alpha=0.4$ (b) $\alpha=0.4$
(c) $\alpha=0.2$ (d) $\alpha=0.2$ (e) $\alpha=0$ (f) $\alpha=0$. For the left
three graphs, the initial condition is chosen as Gaussian wave pocket located
at the large dilaton black hole state while for the right three graphs, the
initial condition is chosen as Gaussian wave pocket located at the small
dilaton black hole state.

Comparing the top two graphs ($\alpha=0.4$), the middle two graphs
($\alpha=0.2$) with the bottom two graphs ($\alpha=0$) in Fig.1, the effect of
dilaton gravity is quite obvious. Firstly, the horizon radius difference
between the large dilaton black hole and small dilaton black hole increases
with the parameter $\alpha$. Secondly, with the increasing of $\alpha$, the
system needs much more time to achieve a stationary distribution. Note that
these two observations do not depend on the initial condition.

Fig.2 provides a complementary but much more quantitative picture of the time
evolution of $\rho(r_{l},t)$ and $\rho(r_{s},t)$. As can be seen from the left
three graphs of Fig.2, $\rho(r_{l},t)$ is finite while $\rho(r_{s},t)$ equals
to zero at first. Then $\rho(r_{l},t)$ starts to decrease while
$\rho(r_{s},t)$ begins to increase. Finally, they attain the same value. The
value can be read from the graph, $0.229$ for the case of $\alpha=0.4$,
$0.250$ for the case of $\alpha=0.2$ and $0.297$ for the case of $\alpha=0$.
This difference can be viewed as the third kind of effect (besides the two
observations stated in the former paragraph) of the dilaon gravity on the
dynamic phase transition of black holes. The situation for the right three
graphs is quite the reverse. At the beginning, $\rho(r_{s},t)$ is finite while
$\rho(r_{l},t)$ equals to zero. $\rho(r_{s},t)$ decays with time while
$\rho(r_{l},t)$ increases with time. In the end, they also reach the same
value, as in the left three graphs.

##  IV First passage process for the phase transition of charged dilaton black
holes

To further probe the dynamic phase transition of dilaton black holes, one can
also consider first passage process. This process describes how the initial
black hole state approach the intermediate transition state for the first
time. To numerically investigate such a process, both the reflecting boundary
condition and absorbing boundary condition should be imposed. Note that
absorbing boundary condition $\rho(r_{m},t)=0$ liran2  is considered at the
the intermediate transition state whose horizon radius denoted as $r_{m}$.

Resolving the Fokker-Planck equation constrained by these two types of
boundary conditions, we show the first passage process of charged dilaon black
holes intuitively in Fig.3. Similarly, the initial condition is chosen as
Gaussian wave pocket located at the large dilaton black hole state for the
left three graphs while the initial condition is chosen as Gaussian wave
pocket located at the small dilaton black hole state for the right two graphs.
No matter what the initial black hole state is, the initial peaks decreases
very quickly. We also compare the evolution for different choices of parameter
$\alpha$. In the top, middle and bottom rows of Fig.3, $\alpha$ is chosen
respectively as $0.4,0.2,0$. As can be witnessed from Fig.3, the initial peak
decays more slowly with the increase of $\alpha$, consistent with the findings
in Fig.1 and Fig.2.

The time needed for the first passage process is denoted as the first passage
time. Its distribution $F_{p}(t)$ can be obtained via liran1

$F_{p}(t)=-\frac{d\Sigma(t)}{dt}.$ (12)

where $\Sigma(t)$ is the sum of the probability that the black hole system
having not finished a first passage by time $t$.

Fig.4 shows the time evolution of $\Sigma(t)$. Irrespective of the initial
state, $\Sigma(t)$ decays very fast. The effect of dilaton gravity can also be
obviously witnessed. $\alpha$ is chosen respectively as $0.4,0.2,0$ for the
top, middle and bottom rows of Fig.4. With the increasing of $\alpha$, the
decay of $\Sigma(t)$ slows down.

$F_{p}(t)$ can also be derived via Eqs. (11) and (12) as liran2

$F_{p}(t)=-D\frac{\partial\rho(r,t)}{\partial r}\Big{|}_{r_{m}},$ (13)

The distribution of first passage time is depicted in Fig.5. It is shown that
there exists one single peak in all the graphs of $F_{p}(t)$. This observation
suggests that the first passage process takes place very fast. And the time
corresponding to the peak increases with the parameter $\alpha$.

Figure 3: Time evolution of $\rho(r,t)$ in the first passage process of
charged dilaton black hole with $b=1,q=1,P=0.003$. (a) $\alpha=0.4$ (b)
$\alpha=0.4$ (c) $\alpha=0.2$ (d) $\alpha=0.2$ (e) $\alpha=0$ (f) $\alpha=0$.
For the left three graphs, the initial condition is chosen as Gaussian wave
pocket located at the large dilaton black hole state while for the right three
graphs, the initial condition is chosen as Gaussian wave pocket located at the
small dilaton black hole state.

Figure 4: Time evolution of $\Sigma(t)$ for charged dilaton black hole with
$b=1,q=1,P=0.003$. (a) $\alpha=0.4$ (b) $\alpha=0.4$ (c) $\alpha=0.2$ (d)
$\alpha=0.2$ (e) $\alpha=0$ (f) $\alpha=0$. For the left three graphs, the
initial condition is chosen as Gaussian wave pocket located at the large
dilaton black hole state while for the right three graphs, the initial
condition is chosen as Gaussian wave pocket located at the small dilaton black
hole state.

Figure 5: $F_{p}(t)$ for charged dilaton black hole with $b=1,q=1,P=0.003$.
(a) $\alpha=0.4$ (b) $\alpha=0.4$ (c) $\alpha=0.2$ (d) $\alpha=0.2$ (e)
$\alpha=0$ (f) $\alpha=0$. For the left three graphs, the initial condition is
chosen as Gaussian wave pocket located at the large dilaton black hole state
while for the right three graphs, the initial condition is chosen as Gaussian
wave pocket located at the small dilaton black hole state.

##  V Conclusions

We probe the effect of dilaon gravity on the dynamic phase transition of black
holes by investigating the case of charged dilaton black holes. Specifically,
we consider the the variation of the parameter $\alpha$ describing the
strength of coupling between the electromagnetic field and the dilaton field.

We introduce the Gibbs free energy landscape and calculate the corresponding
$G_{L}$ of the dilaton black hole. Based on $G_{L}$, we can numerically solve
the Fokker-Planck equation. To probe the probabilistic evolution of dilaton
black holes, we first consider the reflecting boundary condition. When the
initial condition is chosen as the large dilaton black hole, the initial peak
of $\rho(r,t)$ corresponding to the radius of the large dilaton black hole is
decreasing with time while another peak corresponding to the radius of the
small dilaton black hole is booming with time. Finally, these two peaks reach
a stationary distribution where $\rho(r_{l},t)$ and $\rho(r_{s},t)$ attain the
same value. This process clearly illustrates how the initial large dilaton
black hole evolves dynamically into the small dilaton black hole. Similar
analysis hold for the case that the initial condition is chosen as the small
dilaton black hole. The effect of dilaton gravity on the probabilistic
evolution of dilaton black holes is quite obvious. Firstly, the horizon radius
difference between the large dilaton black hole and small dilaton black hole
increases with the parameter $\alpha$. Secondly, with the increasing of
$\alpha$, the system needs much more time to achieve a stationary
distribution. Note that these two observations do not depend on the initial
condition. Thirdly, the value which $\rho(r_{l},t)$ and $\rho(r_{s},t)$ attain
varies with the parameter $\alpha$ ($0.229$ for the case of $\alpha=0.4$,
$0.250$ for the case of $\alpha=0.2$ and $0.297$ for the case of $\alpha=0$).

To further probe the dynamic phase transition of dilaton black holes, we
consider first passage process which describes how the initial black hole
state approach the intermediate transition state for the first time. Besides
the reflecting boundary condition, absorbing boundary condition should also be
imposed here. Resolving the Fokker-Planck equation constrained by these two
types of boundary conditions, we show the first passage process of charged
dilaon black holes intuitively. No matter what the initial black hole state
is, the initial peak decreases very quickly. We compare the evolution for
different choices of $\alpha$. The initial peak decays more slowly with the
increase of $\alpha$. We also study the distribution of the first passage time
$F_{p}(t)$ and the sum of the probability $\Sigma(t)$ that the black hole
system having not finished a first passage by time $t$. Irrespective of the
initial state, $\Sigma(t)$ decays very fast. The effect of dilaton gravity can
also be obviously witnessed. With the increasing of $\alpha$, the decay of
$\Sigma(t)$ slows down. There exists one single peak in all the graphs of
$F_{p}(t)$, suggesting that the first passage process takes place very fast.
And the time corresponding to the peak is found to increase with the parameter
$\alpha$.

To conclude, the dilaton field slows down the dynamic phase transition process
between the large black hole and the small black hole. It would also be
interesting to probe the effect of dilaton field on the dynamic process of
novel phase transition disclosed in Ref. Sheykhi16 .

## Acknowledgements

Shan-Quan Lan is supported by National Natural Science Foundation of China
(Grant No.12005088) and Lingnan Normal University Project with Grant
Nos.YL20200203, ZL1930. The authors are also in part supported by Guangdong
Basic and Applied Basic Research Foundation, China (Grant No.2021A1515010246).

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