Kramer’s Escape Rate and Phase Transition Dynamics
in AdS Black Holes with Dark Structures
Jafar Sadeghi ⋆111Email<EMAIL_ADDRESS>Mohammad Ali S. Afshar ⋆222Email:
<EMAIL_ADDRESS>Saeed Noori Gashti†,⋆333Email:
<EMAIL_ADDRESS>
Mohammad Reza Alipour ⋆444Email<EMAIL_ADDRESS>
⋆Department of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
†School of Physics, Damghan University, P. O. Box 3671641167, Damghan, Iran

###### Abstract

Traditional static methods in phase transition studies offer insights into
black hole thermodynamics but miss dynamic aspects. Kramers’ escape rate,
central to our research, has a somewhat dynamic approach to phase transition.
We explore free energy landscapes for black holes influenced by ’dark’ and
’dark +strings’ entities, assessing how additional parameters affect escape
and transition rates during phase transitions from the small black holes to
the large black holes. In our dual-perspective analysis, we first consider the
escape rate as a function of the black hole radius, demonstrating a natural
increase in the escape rate up to a Maximum point, followed by a decrease due
to a reaction mechanism. From this viewpoint, the escape rate from smaller to
larger black holes will consistently be greater than the reverse. From the
second perspective, we consider the escape rate as a function of temperature
(pressure). In both models under study, initially, the behavior of the
probability diagrams is akin to the radius-based behavior. However, with an
increase in temperature (and corresponding pressure), we reach a specific and
distinct point, where the systems will move towards either smaller or larger
black holes with the same probability. Beyond this temperature (pressure), the
behavior of the models diverges. In the nonlinear model, an inversion occurs
where the escape rate from larger black holes to smaller ones becomes
predominant. This inversion area seems to be significantly reduced compared to
the charged black hole model, although it is still discernible in the
temperature-based escape rate diagram. In other words, within the temperature
range studied, where the likelihood of radiative conditions and movement from
a large black hole to a small black hole is less, It seems that the addition
of dark parameters minimizes the occurrence of this transition, indicating the
subtle influence of dark structures on the phase transition dynamics between
black holes. In the string model, although the trend of increasing pressure in
the transition from a large black hole to a smaller one is greater than the
reverse, it seems that the transition process from a small black hole to a
larger one, with decreasing pressure, completely prevents the ’low probable’
transition from a large black hole to a smaller one. This is an interesting
and distinctive observation.
Keywords: Phase Transition, Free Energy Landscape, Kramer’s Escape Rate, Dark
Structures.

###### Contents

  1. 1 Introduction
  2. 2  Theoretical preliminaris
    1. 2.1 Kramer’s escape rate
    2. 2.2 Different shapes for free energy landscape
      1. 2.2.1 Gibbs Free Energy Landscape
      2. 2.2.2 Landau free energy
      3. 2.2.3  Thermal potential
  3. 3 Non-linear magnetic-charged AdS black hole surrounded by quintessence in the perfect fluid dark matter background
    1. 3.0.1 Behavioral comparison of landscape free energy
    2. 3.0.2 Kramer’s escape rate
  4. 4 4D AdS Einstein-gauss-bonnet-Yang-Mills black hole with a cloud of strings
    1. 4.0.1  The landscape free energy
    2. 4.0.2  Kramer’s escape rate
  5. 5 Discussion and comparison
  6. 6 Conclusion
  7. 7 Appendix A: Behavioral sequence of the free energy landscape for (EGB-YM-CS) black hole

##  1 Introduction

From ancient times to the present, the existence of similar behavioral
patterns and recurring elements in various physical systems, ranging from
subatomic to cosmic scales, has been abundantly observed. This could be
interpreted as a mysterious sign of a fractal structure inherent in the
essence and fabric of nature. However, it can also be considered an
intentionally hidden shortcut, ingeniously and knowingly embedded in the
labyrinth of nature to guide us step by step in deciphering complexities that
we cannot grasp all at once. Among the structures that have benefited most
from the analogy of behavioral patterns for theoretical modeling, due to their
inaccessibility for practical study, are black holes. The comparison of the
gravitational field treatment of a black hole and its surrounding world, as a
system and environment, with the behavior of a thermodynamic ensemble is
indeed one of the most valuable achievements of this modeling. A resemblance
that has led to black hole thermodynamics becoming a pioneering and
fundamental branch in the evolution of our understanding not only of the
structural nature of black holes, but also of the cosmos itself. If we were to
highlight the three principal keys that paved the way for the application of
thermodynamics and statistical mechanics to the study of black holes, we would
undoubtedly mention Hawking temperature, Bekenstein-Hawking entropy, and the
conversion of the cosmological constant to pressure. These concepts led to the
formulation of the four fundamental laws of black hole thermodynamics [1, 2,
3, 4]. With a slight indulgence, one might also add the exchange and
conversion of enthalpy to mass to these golden keys [5].

The increasing study of black hole models based on thermodynamics and
statistical mechanics has gradually revealed that, apparently, the best model
that aligns most closely with current data and computational capabilities, and
environmental conditions for studying black hole behavior, is the Van der
Waals model (VdW). For instance, in canonical ensembles, a black hole phase
transition was found in the background of charged rotating AdS black holes,
reminiscent of the liquid-to-gas phase transition of the (VdW) fluid [6, 7,
8]. It was also observed that there is a significant resemblance between the
(VdW) liquid-gas system and the charged black hole in the extended phase space
[9, 10], or another study showed the critical behavior of charged black hole
in fixed charge ensemble coincides exactly with that of (VdW) liquid-gas
system [11]. Undoubtedly, phase transition is one of the most fundamental
phenomena in classical thermodynamics, which, through extensive study of
various structures of matter, has led to a profound understanding of the true
behavior of systems in the past. The question that quickly arose was whether,
given the (VdW)-like thermodynamic behavior of black holes, these structures,
considering their quantum-relativistic structural pattern, could exhibit
quasi-material behavior and phase transitions. And if so, what could be the
cause of the initiation of this process? The answer to the first part of the
question was subsequently obtained, as it was determined that there exists a
Hawking-Page phase transition between a large stable black hole and pure
thermal radiation [12]. Of course, numerous other studies were conducted that
even confirmed the interesting phase transition behaviors for black holes [13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28]. However, the
second part of the question remains unclear to us until now.

Although the temperature variation graph of a system can itself be a reliable
criterion for phase transition detection, one of the tools that can be used to
study the behavioral range of thermodynamic phase transition with greater
precision is the free energy landscape. Free energy landscape provides a way
to visualize and understand how systems, transition between different states
over time. The dependence of free energy on temperature causes temperature
changes to significantly alter the free energy landscape. Consequently,
studying these apparent changes can provide a better understanding of the
system’s behavior and the formation of transitions between different phase
states. In thermodynamics, free energy is rewritten in various forms depending
on the system’s conditions. However, since the interplay between temperature
and pressure is often of primary concern in black hole structures, and since
enthalpy offers a better and simpler description of energy transfer at a
specific pressure, the Gibbs free energy function is a better candidate among
the various definitions of free energy for studying black holes. But the point
that was hidden in the study of the phase transition behavior of black holes
using Gibbs free energy was that, apparently, in this form of study, no
attention was paid to the rate and mode of transition from one state to
another. In other words, in these studies, we did not address the following
points in the phase transition process from small to large black holes:
• At what rate do these processes occur?
• Is the rate of transition constant across different temperature ranges?
• Does the initial state (small black hole) completely vanish at the end of
the transition?
Also, could there be hidden points in the leakage from the small to the large
black hole that we are unaware of?
These and similar questions that address the dynamics of phase transitions
provide important information about the phase transitions of black holes,
which apparently cannot be obtained in the initial analyses of Gibbs free
energy. It seems that we were considering the phase transition and its results
in a static manner, only at the end points and equilibrium points of Gibbs.
Recent studies have observed that phase transition under thermal fluctuations
can be examined through the Fokker-Planck equation in non-equilibrium physics,
based on the free energy landscape, particularly the Gibbs form. Accordingly,
it has been observed that in black holes, a transition occurs from small to
large black holes and vice versa, influenced by temperature and the height of
the potential barrier in the free energy landscape [29, 30, 31, 32]. According
to the studies done, it seemed that the Gibbs free energy is in the form of an
effective potential it plays the role of a driving force that leads to the
phase transition of the black hole [33]. Based on this, a special focus was
formed on the Gibbs energy landscape, which, according to the structure of the
Fokker-Planck equation, tried to investigate some kinetics of the phase
transition by calculating the mean first passage time in several black
holes[34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56]. Of course, other efforts were also made in this direction
based on the rewriting of free energy in terms of Landau free energy[57, 58]
or thermal potential [59, 60, 61, 62].

One might pose the question: How has the connection between stochastic-form
equations and phase transitions, as well as thermodynamic equations, been
established? In response, it should be noted that since some thermodynamic
processes in the system are driven by stochastic fluctuations, it would be
logical to employ the method of analyzing non-equilibrium stochastic processes
to study the occurrence of inherently non-equilibrium thermodynamic processes.
Building on this foundation, in this paper, we consider two black holes
influenced by dark energy, namely:
$\bullet$ Non-Linear Magnetic-Charged AdS black hole surrounded by
Quintessence, in the background of Perfect Fluid Dark matter (NLM-C-Q-PFD)
[63].
$\bullet$ 4D AdS Einstein-Gauss-Bonnet-Yang-Mills Black hole with a Cloud of
Strings (EGB-YM-CS) [64].

Initially, we calculate the various forms of free energy mentioned and
demonstrate that all three forms will exhibit nearly identical behavior in the
canonical ensemble. Subsequently, using Kramer’s escape rate method, we will
examine the characteristics of their first-order phase transition rate in
terms of variations in radius, pressure, or temperature. We will then compare
our results with those of black holes studied using the same method [59, 60,
61, 62] to determine the influence of the added parameters.
The present paper is organized as follows: In Section 2, we explore the
theoretical preliminaries to thoroughly study and introduce various concepts,
such as Kramer’s escape rate and different forms of the free energy landscape.
This includes the Gibbs free energy landscape, Landau free energy, and thermal
potential. In Section 3, we conduct a behavioral comparison of the free energy
landscape and Kramer’s escape rate for (NLM-C-Q-PFD) AdS black hole. Section 4
presents an accurate calculation for the (NLM-C-Q-PFD) AdS black hole. In
Section 5, we fully analyze the concepts discussed in the previous two
sections. Finally, in Section 6, we comprehensively present the results of our
work.

##  2  Theoretical preliminaris

###  2.1 Kramer’s escape rate

The study of Brownian particle leakage or escape under thermal fluctuations
from a stable or metastable potential, known as the Kramers problem, is a
fascinating topic that has garnered attention across most branches of physics
[62]. This subject, in addition to the diverse results it has yielded
throughout its study, has led to the calculation of a relationship known as
the Kramers escape rate. We will briefly address this for a better
understanding here.

Figure 1: The possibility of leakage due to Brownian motion or thermal
fluctuation

As we can see in Fig (1), the point $x_{min}$ is a local minimum such that
particles are in equilibrium at this point and maintain their original
position under small perturbations. Without loss of generality and for
simplicity, we can point out a few things here. For instance, we assume only
one direction of escape, which may occur through the nearest local maximum
located at $x_{max}$. On the left side of $x_{min}$, the potential is such
that it does not have any additional extremes. After the local maximum at
$x_{max}$, on the right side, there may be another local or global minimum, or
even the potential may be unbounded below, which is irrelevant. This is
because the escape rate should be independent of the shape of the potential on
the right side of $x_{max}$. When we add the effect of thermal baths to the
system, the particles are affected by the two competing effects of fluctuation
and dissipation. Random forces arising from thermal fluctuations push the
particle away from its initial position and allow it to eventually climb the
potential barrier. On the other hand, damping tends to slow the particle down
and makes it more difficult to return to the equilibrium point. There are
three possible situations here.
• Very Low Damping: In this scenario, particles experience minimal frictional
forces. The escape rate is determined by the energy diffusion regime, where
the particle’s inertia plays a significant role. The particle can oscillate
many times near the potential well’s boundary before escaping, and the escape
rate is relatively low due to the lack of significant damping forces.
• Intermediate Damping: The escape rate in this regime is more complex to
describe, as it involves a balance between inertial effects and damping
forces. The system oscillates less than before, but it can still exist, and
the escape rate is still not significant.
• Over-damped: In the over-damped case, the damping forces are strong, and the
system experiences high frictional forces, which dominate over inertial
effects. The particle moves slowly away from the potential well, and the
escape rate is governed by the rate at which the system can overcome the
barrier height compared to the thermal energy. The escape process is akin to a
slow, diffusive motion over the barrier, and the escape rate is higher than in
the very low or Intermediate damping case and is significant.
Since we intend to simulate this model according to the phase transition
behavior of the black hole, then among the above options, the over-damping
mode seems to be the appropriate option, since due to the strengthening of the
combined effect in this mode, the system is no longer stable and The
probability of the particle exiting the well will start to grow. That is,
after a sufficient period of time, we can expect the particle to have almost
passed through the barrier. The escape rate is closely related to the inverse
of the average time required for the first passage of the local maximum
potential, which is called the ”mean first passage time”, which was noted in
articles [35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51,
52, 53, 54, 55, 56]. It should be noted that the escape rate is different from
the diffusion rate to the next minimum [65, 66, 67, 68, 69, 70].
Let us consider the the Fokker-Planck equation [67, 70, 71],

$\frac{\partial}{\partial t}P\\!\left(x,t\right)=-\frac{\partial}{\partial
x}J\\!\left(x,t\right),$

where the $P(x,t)$ is the probability distribution of particles and current J
is defined as,

$J=-\frac{\left(\frac{\partial}{\partial
x}U\\!\left(x,t\right)\right)P\\!\left(x,t\right)}{\gamma}-\mathrm{D}\left(\frac{\partial}{\partial
x}P\\!\left(x,t\right)\right),$

here $U(x)$ is a potential field, $\gamma$ is the friction coefficient and D
is the diffusion coefficient. The current can be rewritten as follows,

$J=-\mathrm{D}{\mathrm{e}}^{-\frac{U}{k_{B}T}}\frac{\partial}{\partial
x}({\mathrm{e}}^{-\frac{U}{k_{B}T}}P),$

$k_{B}$ is the Boltzmann constant and T is the temperature. To integrate, we
rewrite the above relation as follows,

$\frac{\partial}{\partial
x}({\mathrm{e}}^{-\frac{U}{k_{B}T}}P)=-\frac{J{\mathrm{e}}^{\frac{U}{k_{B}T}}}{\mathrm{D}}.$

By integrating from the above equation and taking into account the fact that
on the left side of the equation, the contribution of the distribution term P
in $x_{min}$ dominates over $x_{max}$, we can conclude,

$J=\frac{\mathrm{D}{\mathrm{e}}^{\frac{U\left(x_{\min}\right)}{k_{B}T}}P\\!\left(x_{\min}\right)}{\int_{x_{\min}}^{x}{\mathrm{e}}^{-\frac{U\left(x\right)}{k_{B}T}}dx}.$

Now, if we consider the upper limit of the integral of the denominator in the
above equation at a point on the right side of $x_{max}$ and using the
expansion around $x_{max}$ and after some calculations, we will have,

$\int_{x_{\min}}^{x}{\mathrm{e}}^{-\frac{U\left(x\right)}{k_{B}T}}dx=\
\sqrt{\frac{2\pi
k_{B}T}{{|\frac{d^{2}}{dx^{2}}U(x_{max})|}}}\,{\mathrm{e}}^{\frac{U\left(x_{\max}\right)}{k_{B}T}}.$

If we define p as the probability of the particle being inside the well, then
with some calculations we can find that,

$p=P\\!\left(x_{\min}\right)\ \sqrt{\frac{2\pi
k_{B}T}{{|\frac{d^{2}}{dx^{2}}U(x_{min})|\\!\left(x\right)|}}}.$

Finally, using the definition of Kramers escape rate($r_{k}=J/P$), we will
have,

$r_{k}=\frac{\mathrm{D}\sqrt{{|\frac{d^{2}}{dx^{2}}U(x_{max})\\!}{\frac{d^{2}}{dx^{2}}U(x_{min})\\!|}}\,{\mathrm{e}}^{-\frac{U\left(x_{\max}\right)-U\left(x_{\min}\right)}{k_{B}T}}}{2\pi
k_{B}T}.$

When the system reaches thermal equilibrium, D can be considered constant[55],
and with a little simplification, the above relationship can be rewritten as
follows[56],

$r_{k}=\frac{\sqrt{{|\frac{d^{2}}{dx^{2}}U(x_{max})\\!}{\frac{d^{2}}{dx^{2}}U(x_{min})\\!|}}\,{\mathrm{e}}^{-\frac{U\left(x_{\max}\right)-U\left(x_{\min}\right)}{D}}}{2\pi}.$
(1)

Note that the formula can only be applied for $\Delta U>>k_{B}T$.

###  2.2 Different shapes for free energy landscape

Free energy landscape provides a way to visualize and understand how systems,
transition between different states over time. In simple terms, the free
energy landscape is a graphical representation where each point on the
landscape corresponds to a possible state of the system. From the vantage
point of free energy, the exploration of phase transition conditions becomes
accessible through the scrutiny of the emergence and stability of distinct
phases. Furthermore, the characterization and analysis of the free energy
landscape are profoundly influenced by its relationship with the order
parameters. Taking the radius of a black hole’s event horizon as an exemplary
order parameter, one can articulate the free energy landscape. Consequently,
the phase transition of a black hole can be interpreted as a stochastic
thermal fluctuation of said order parameter, offering a novel perspective on
the underlying physics [33]. In the following, since we want to study the
phase transition from the perspective of the free energy landscape and in a
different structure under specific thermal fluctuations, we will introduce
different forms of this energy that can help us in this way.

####  2.2.1 Gibbs Free Energy Landscape

The Gibbs free energy function, renowned for its intrinsic reliance on
enthalpy and temperature, emerges as the special tool for probing the phase
transitions within a black hole’s architecture, particularly under constant
pressure conditions. The swallowtail behavior, manifested graphically as a
function of temperature, stands as a prevalent methodology for the exploration
of phase transitions. Within the formalism of the free energy perspective, the
Gibbs free energy is delineated as a continuous function that is intimately
connected to the system’s order parameter, thereby providing a robust
framework for the analysis of phase behavior,

$G=H-T_{H}S,$ (2)

which $T_{H}$ is Hawking temperature, H is enthalpy and S is entropy. Also
considering the correspondence between mass and enthalpy in the extended phase
of a black hole ($H\equiv M$), this equation can be rewritten in terms of
mass.
During the intricate process of phase transition, particularly of the first
order, an assortment of black holes of varying sizes-small, medium, and large-
along with transitional states, frequently emerges. Consequently, it is
feasible to conceptualize a collection of these black hole space-times, each
with an arbitrary horizon radius, at a specified temperature ”T”. For a better
understanding, it must be stated that one of the most important
characteristics of a black hole is its temperature, or more specifically, the
Hawking temperature. The mere existence of temperature, regardless of how it
is defined, implies the possibility of thermal fluctuations within the system.
Consider this as a background. Now, the black hole under study has been
excited for some environmental reason and begins a phase transition process.
At both the initial and ends of this phase transition, there are two locally
stable black holes, one of which is globally more stable than the other,
setting the stage for this transition. However, an important point regarding
these two states is that both must necessarily be solutions to Einstein’s
field equations. The golden key to defining a set of black holes lies here.
Although the system must end up in a state that is necessarily a solution to
Einstein’s equations at both ends, there is no need to strictly follow the
static field equations during the transition from the initial phase to the
final phase, since the system is in an unstable transitional phase. Therefore,
considering a predefined energy range and with the condition that the chosen
paths do not destroy the overall stability of the system, the system can be
allowed to traverse any desired path from the origin to the end. This freedom
of choice and multiple paths could serve as a gateway for the introduction of
probability and statistical methods. Hence, by selecting an arbitrary
indistinctive set within this framework, which acts as a canonical ensemble
for us, a statistical framework for analysis can be presented. It is
noteworthy that the temperature ”T” of the ensemble may differ from the
Hawking temperature associated with black holes Notably, the temperature ”T”
of the ensemble may diverge from the Hawking temperature associated with black
holes. Each of the space-time configurations within this ensemble is
characterized by a distinct Gibbs free energy. Hence, we refer to this
construct as the generalized Gibbs result or, more descriptively, the Gibbs
free energy landscape.

$G_{L}=M-TS.$ (3)

The terminology ’on-shell’ and ’off-shell’ delineates the aforementioned
temperature discrepancy. ’Off-shell’ encompasses all transitions occurring at
temperature ”T”, which, despite conforming to the laws of thermodynamics, do
not necessarily constitute solutions to Einstein’s field equations [33]. It is
imperative to recognize that solely the extremal points on the $G_{L}$ signify
the authentic black hole phases that comply with the Einstein field equations.
Furthermore, within this context, the local maxima and minima are indicative
of the unstable and locally stable black hole phases, respectively [72].

####  2.2.2 Landau free energy

The Landau free energy constitutes another showy path for scrutinizing the
phase transition dynamics within the system’s free energy landscape. While
using this method assumes heightened significance in the context of second-
order phase transitions- wherein the bifurcation of the Landau functional’s
local minimum epitomizes the thermodynamic system’s second-order phase
transition- it is also pertinent to first-order phase transitions.
Specifically, the transmutation of the global minimum is emblematic of a
first-order phase transition, underscoring the multifaceted nature of the
Landau-free energy in characterizing phase behavior. This function can be
written as follows, [57]

$L=\int F\\!\left(X,P,T\right)dX,$ (4)

that in the delineated relationship, P denotes pressure, T represents
temperature, and X serves as an auxiliary variable. However, in the realm of
black hole thermodynamics, pressure holds a distinctive advantage. Contrary to
other thermodynamic variables such as temperature, mass, and volume -which are
solely functions of the event horizon’s radius- pressure not only contributes
to the equation of state, inextricably linked to the radius, but also assumes
an independent role based on the definition of the cosmic constant in the
studied model. This endows it with a unique identity within the thermodynamic
landscape. Consequently, preserving the essence of generality, the ”F”
function can be reformulated as follows,

$F\\!\left(X,P,T\right)=P-f\\!\left(X,T\right).$ (5)

Given that ”L” is an energy function, it inherently embodies the fundamental
characteristic of energy minimization. This implies that the system invariably
seeks a trajectory that reduces the energy expenditure required for
transformation. Therefore, among the triad of system parameters $(X,P,T)$ that
influence the ”F” function, only the configurations that minimize ”L” will
accurately reflect the system’s state.

$\frac{d}{dX}L\ =F\\!\left(X,P,T\right)=0.$ (6)

The important point before the end of this section is that the most familiar
form of the characteristic ”X” in Landau’s definition is the volume, to be
able to use the probabilistic method for this function, we must consider the
volume off-shell

####  2.2.3  Thermal potential

Another form of landscape energy that can be used to investigate phase
transition based on thermal fluctuation is the model that was introduced in
[59] under the title of thermal potential.

$U=\int\left(T_{H}\ -T\ \right)dS,$ (7)

which $T_{H}$ is Hawking temperature,$T$ is ’Off-shell’ temperature, and $S$
is entropy that has appeared here in the role of a variable. Since In
thermodynamics, the vigor of thermal motion is quantified by the product of
temperature and entropy. Consequently, the aforementioned thermal potential
serves as an approximate gauge for the extent to which all conceivable states
within the canonical ensemble deviate from the genuine black hole state.
When a black hole system is situated within a potential field $U$, it is
subject to thermodynamic fluctuations. The resultant stochastic behavior of
black holes within such a potential field can offer insights into certain
thermodynamic attributes of black holes. Furthermore, the thermal potential’s
utility extends to its ability to indicate system stability through the
analysis of concavity and convexity at critical points.

##  3 Non-linear magnetic-charged AdS black hole surrounded
by quintessence in the perfect fluid dark matter background

In this article, we went to black holes that somehow interact with the
structure of matter and dark energy, so that we can study the effect of the
presence of parameters caused by the dark structure on the first-order phase
transition. So the metric of (NLM-C-Q-PFD) black holeis as follows [63],

$f\\!\left(r\right)=1-\frac{2M\,r^{2}}{Q^{3}+r^{3}}+\frac{\alpha\ln\\!\left(\frac{r}{{|\alpha|}}\right)}{r}-\frac{c_{q}}{r^{3\varepsilon+1}}+\frac{8\pi
P\,r^{2}}{3}.$

For the main quantities of this model i.e. mass $M$, Hawking temperature
$T_{H}$, entropy $S$, pressure $P$ and volume $V$ we will have (note that as
long as there is no specific definition of the radius, it will be $r=r_{H}$ in
calculations) [63],

$M=\frac{\left(Q^{3}+r^{3}\right)\left(1+\frac{\alpha\ln\left(\frac{r}{|\alpha|}\right)}{r}-\frac{c_{q}}{r^{3\varepsilon+1}}+\frac{8\pi
P\,r^{2}}{3}\right)}{2r^{2}},$ (8)
$T_{H}=\frac{\frac{-2Q^{3}+r^{3}}{r}+\frac{3c_{q}\varepsilon\left(r^{3}+\frac{Q^{3}\left(\varepsilon+1\right)}{\varepsilon}\right)}{r^{3\varepsilon+2}}+\frac{\alpha
Q^{3}\left(1-3\ln\left(\frac{r}{|\alpha|}\right)\right)}{r^{2}}+\alpha r+8\pi
P\,r^{4}}{4\pi\left(Q^{3}+r^{3}\right)},$ (9) $S=\pi
r^{2}\left(1-\frac{2Q^{3}}{r^{3}}\right),$ (10) $P=\frac{3\alpha
Q^{3}\ln\\!\left(\frac{r}{|\alpha|}\right)-3r^{3-3\varepsilon}c_{q}\varepsilon-3Q^{3}c_{q}\left(\varepsilon+1\right)r^{-3\varepsilon}+\left(4\pi
T\,r^{2}-\alpha+2r\right)Q^{3}+4T\pi r^{5}-r^{4}-\alpha r^{3}}{8\pi r^{6}},$
(11) $V=\frac{4\pi\left(Q^{3}+r^{3}\right)}{3},$ (12)

where $\alpha$ denotes the intensity of the PFDM, $c_{q}$ and $\varepsilon$
are the quintessence parameters and Q is the magnetic charge.

####  3.0.1 Behavioral comparison of landscape free energy

Now according to equations (2), (3), (4) and (7) for Gibbs and Landau free
energy function and also the thermal potential of this model, we will have,

$\begin{split}&\textbf{G}=\mathcal{A}+\mathcal{B},\\\
&\mathcal{A}=\frac{\left(Q^{3}+r^{3}\right)\left(3r+3\alpha\ln\\!\left(\frac{r}{|\alpha|}\right)-3c_{q}r^{-1-3\varepsilon}r+8\pi
P\,r^{3}\right)}{6r^{3}}\\\
&\mathcal{B}=\frac{\left(-\frac{2Q^{3}-r^{3}}{r}+3c_{q}\left(Q^{3}\varepsilon+r^{3}\varepsilon+Q^{3}\right)r^{-2-3\varepsilon}+\frac{\alpha
Q^{3}\left(1-3\ln\left(\frac{r}{|\alpha|}\right)\right)}{r^{2}}+\alpha r+8\pi
P\,r^{4}\right)\left(2Q^{3}-r^{3}\right)}{4r\left(Q^{3}+r^{3}\right)}\\\
\end{split}$ (13)
$G_{L}=\frac{\left(Q^{3}+r^{3}\right)\left(1+\frac{\alpha\ln\left(\frac{r}{{|\alpha|}}\right)}{r}-\frac{c_{q}}{r^{1+3\varepsilon}}+\frac{8\pi
P\,r^{2}}{3}\right)}{2r^{2}}-T\pi r^{2}\left(1-\frac{2Q^{3}}{r^{3}}\right)$
(14)
$\begin{split}&\textbf{L}=PX-\frac{\mathcal{C}+\mathrm{D}}{48Q^{3}\pi^{2}-36\pi
X},\\\
&\mathcal{C}=36\left(Q^{3}\pi^{\frac{7}{3}}-\frac{\pi^{\frac{4}{3}}X}{4}\right)2^{\frac{2}{3}}T\left(-4\pi
Q^{3}+3X\right)^{\frac{2}{3}}+9\,2^{\frac{1}{3}}X\pi^{\frac{2}{3}}\left(-4\pi
Q^{3}+3X\right)^{\frac{1}{3}},\\\ &\mathrm{D}=6\ln\\!\left(-4\pi
Q^{3}+3X\right)\pi X\alpha-18X\pi^{\varepsilon+1}c_{q}\left(-\pi
Q^{3}+\frac{3X}{4}\right)^{-\varepsilon}-8Q^{3}\pi^{2}\alpha\left(\ln\\!\left(\pi\right)+3\ln\\!\left({|\alpha|}\right)+2\ln\\!\left(2\right)\right),\\\
\end{split}$ (15) $\begin{split}&\textbf{U}=\mathcal{E}-T\pi
r^{2}\left(1-\frac{2Q^{3}}{r^{3}}\right),\\\
&\mathcal{E}=\frac{3c_{q}\left(-Q^{3}-r^{3}\right)r^{-3\varepsilon}+3\alpha\left(Q^{3}+r^{3}\right)\ln\\!\left(r\right)+8\pi
P\,r^{6}-3\alpha Q^{3}\ln\\!\left(|\alpha|\right)+3Q^{3}r+3r^{4}}{6r^{3}}\\\
.\end{split}$ (16)

Now we will go to a special state that is formed based on the arbitrary
parameter setting. For this purpose, by considering the values
$\alpha=0.6,c_{q}=0.2,\varepsilon=-2/3,Q=1$ for the parameters and with a
little calculation for the critical quantities, we will have,

$\begin{split}&T_{c}=0.02426755451,\hskip 19.91684ptP_{c}=0.00456781599,\hskip
19.91684ptr_{c}=2.594697206,\\\ &V_{c}=77.36141969,\hskip
19.91684ptU_{c}=1.00399626317,\hskip 19.91684ptG_{c}=1.02312988617.\\\
\end{split}$ (17)

According to the usual procedure, we convert the parameters into dimensionless
form so that the results can be compared with other studies in addition to
convenience,

$t=\frac{T}{T_{c}},\hskip 19.91684ptp=\frac{P}{P_{c}},\hskip
19.91684ptx=\frac{r}{r_{c}},\hskip
19.91684pt\widetilde{G}=\frac{G}{G_{c}},\hskip
19.91684pt{\widetilde{G}_{L}}=\frac{G_{L}}{G_{c}},\hskip
19.91684ptl=\frac{L}{G_{c}},\hskip 19.91684ptu=\frac{U}{U_{c}}.$ (18)

Before any calculations, it is better first compare the various forms of free
energy definitions, namely $G_{L}$, U, and L. Theoretically, from the
canonical perspective the number of particles and mass should remain constant,
consequently only the entropy-temperature and pressure-volume terms will play
a role in the first law of thermodynamics. Additionally, given that in
Landau’s free energy, $X$ can be considered equivalent to volume, it appears
that ultimately, all three introduced forms for the free energy landscape must
coincide. In Fig (2a) and (2b), for the model under study, we have plotted
these three definitions once according to their general structure and once
based on their dimensionless state. Practically, in terms of the shape
structure, all three forms have the same trend, but there are also some
visible differences, some of which may be attributed to computational
precision. Moreover, since Landau’s free energy has dimensions of energy, we
used the $G_{c}$ to render it dimensionless, which seems to have increased the
degree of displacement, although the shape has been preserved. An intriguing
point that comes to mind before concluding this discussion is that in
statistical ensembles, such as the grand canonical ensemble where terms
related to the chemical potential play a more prominent role, the complexities
and structural differences of these three definitions may become more apparent
compared to this scenario. Concerning the above discussion and as seen in Fig
(2), all the shapes of the free energy landscape (with a little tolerance for
displacement) will practically have the same shape structure in the canonical
view. Accordingly, the choice and use of each do not seem to have an advantage
over the others. For this reason, we will henceforth utilize them under the
designation of the free energy landscape, without loss of generality.

Figure 2:  Comparison of the Gibbs landscape free energy ($G_{L}$), thermal
potential ($U$), and Landau free energy ($L$) in the diagram in two states,
i.e., with dimension in fig (2a) and dimensionless in fig (2b)

Now, to be able to interpret the behavior of the energy function, we need to
have the graph of temperature changes Fig (3a) and Gibbs free energy in terms
of temperature Fig (3b).

Figure 3: The graph of T against x for free parameters in fig (3a) and Gibbs
Free energy versus temperature in fig (3b)

To better interpret the behavior of free energy, we initially turn to the
swallowtail fig (3b). In this figure, the line (A-N) represents the small
black hole, and( C-J ) represents the large black hole, both of which are
locally stable with positive heat capacity, and the line (N-C) with negative
heat capacity represents the intermediate black hole, which is unstable. As a
classical principle, we know that systems always tend to choose states with
the least amount of energy. This principle also holds for the free energy
landscape, meaning the system will move towards its minimum. In other words,
the minima of the energy function correspond to stable states, and its maxima
represent unstable states. Considering the overall shape in Fig (4), it is
also important to note that in a system with multiple local minima, each of
these minima, although locally stable relative to their immediate vicinity,
may act as unstable states globally relative to the primary minimum. Even if
the system is initially in these states, over time it will progress towards
the ultimate global minimum.
In Fig (4a), as is evident, the potential at point A has a global minimum,
which belongs to the branch of the small black hole. Over time, with the
increase in temperature, in Fig (4b), the local extrema B (minimum) and C
(maximum) begin to form. However, they are still in the form of globally
unstable states, and the system tends to maintain its position at the primary
minimum on the branch of the small black hole. The situation in Fig (4c) is
similar to (4b), with the difference that the local extrema have fully formed.
With a gradual increase in temperature, the system progresses towards the
formation of a large and stable black hole, as shown in Fig (4d). Yet, the
small black hole also exists in a stable state, indicating a situation where
two equivalent global minima are created. This state corresponds to the
isotherm line (H-I) in Fig (3b), which seemingly has a maximum on the branch
of the intermediate black hole (I) and a global minimum (H). However, in
reality, (H) is located on both the small and large black holes, meaning that
a state is created where both the small and large black holes appear
simultaneously and stably. In other words, a coexistence phase between the two
stable black holes is formed. Further, the process of temperature increase
will strengthen the large black hole and weaken the small black hole, Fig(4e)
and (4f). The point that can be mentioned at the end of this discussion is
related to the second-order phase transition. When we study the system at a
temperature higher than the critical temperature, Fig(4g), the system will
follow a path similar to (I-H-J) in Fig (3b), and practically it is directly
and continuously transferred from the small black hole to the large one.

Figure 4: Behavioral sequence of energy in terms of x for different
temperatures

####  3.0.2 Kramer’s escape rate

An important point to focus on initially is the difference in concept between
this section and the previous one. Contrary to the previous section, here we
do not intend to study the influence of temperature on free energy and the
process of transition from a small to a large black hole during a first-order
phase transition. Instead, we aim to examine, in light of the concepts
previously discussed for Kramer’s escape rate -which represents the rate of
particle escape under thermal fluctuations from a known local minimum in the
potential- whether conditions might arise that could reverse the transition
process from a small to a large black hole? Typically, we expect a transition
from an unstable local minimum to a stable global minimum. However, the
question arises whether there is a range in which, due to environmental
conditions such as changes in temperature or pressure, the probability of a
reverse process, namely the transition from a large to a small black hole,
could precede the primary and natural process, causing systems to move in
reverse from the large to the small black hole. To better understand, Fig (5a)
illustrates a specific instance of a free energy landscape at a defined
pressure of $p=0.875692$. For this particular scenario, we examine two states:
1\. The investigation of leakage or the probability of transition from the
unstable local minimum alpha to the stable global minimum gamma, which is
conventionally expected to occur with an increase in temperature.
2\. The examination of the likelihood that, under certain environmental
conditions such as increased temperature or pressure, the reverse process,
namely the transition from gamma to alpha, could become the dominant process,
even if only in a small region.

Figure 5: The graph of $G_{L}$ against x for free parameters in fig (5a) and T
versus x in fig (5b) which are determined as the locations of ($\alpha$)
intersection (red and orange), ($\beta$) intersection (blue and orange) and
($\gamma$) intersection (green and orange)

Now, using equation (1), we calculate the escape rate of each of the possible
transitions, i.e. ($\alpha\rightarrow\beta$) and ($\gamma\rightarrow\beta$),
the result of which can be seen in figure (6).

Figure 6: (6a): $r_{k}$ with respect to $x$, (6b): $r_{k}$ with respect to $t$
and (6c): Enlarged detail, bottom part of diagram (6b)

In Figure 6a, we have plotted the escape rate as a function of distance. For a
better interpretation of this graph, it is advisable to read it concurrently
with the energy versus distance graph fig (5a). Initially, as energy increases
from $\alpha$ towards $\beta$, we should witness an increase in the escape
growth, which, as confirmed by both figures, should peak at a distance of
1.014. Then, with a gradual decrease in energy from $\beta$ towards $\gamma$,
we should observe a decreasing trend in the probability. Considering the
relatively significant height difference between $\gamma\rightarrow\beta$ and
$\alpha\rightarrow\beta$, we expect that throughout the entire range studied,
the dominant process should always be the transition from a small to a large
black hole, which the graph precisely confirms our classical understanding.
In Figure 6b, we have depicted the escape rate as a function of temperature t.
At first glance, the process appears similar to the previous case; that is,
initially, with an increase in temperature and considering the small radius,
we should observe an increase in the escape rate until it reaches a peak, and
then, despite the temperature increase, due to the radius enlargement and
energy decrease, we should witness a decreasing trend in probability, which
the shape of the graph confirms. However, an analysis of the end section of
this graph, which we have illustrated in Figure 6c, reveals that contrary to
the previous case, it represents a completely different situation within a
small range.
At ( t = 0.8976 ), the two escape rate curves intersect, which, makes their
occurrence probabilities equivalent. This means that at this temperature,
black holes have the discretion to choose whether to follow the classic
process from a small to a large black hole or to undertake a reverse process.
Subsequently, it is the reverse process that becomes dominant, meaning that,
although this range may seem exceedingly small, the system nonetheless
exhibits a propensity within this limited interval to transition from a large
black hole to a small black hole, a tendency that may seem contrary to
classical expectations.

##  4 4D AdS Einstein-gauss-bonnet-Yang-Mills black hole with a cloud of
strings

Since we tried to explain the contents as much as possible in the previous
sections, considering the computational similarities, in this section, we will
only mention the basic relationships, and we will bring the similar and
unnecessary diagrams in Appendix A. The second selected model is the (EGB-YM-
CS) black hole. So the metric of 4D AdS Einstein-gauss-bonnet-yang-mills black
hole with a cloud of strings is as follows[64],

$f\\!\left(r\right)=1+\frac{r^{2}\left(1-\sqrt{1-4g\left(\frac{8\pi
P}{3}+\frac{q^{2}}{r^{4}}-\frac{2M}{r^{3}}+\frac{a}{r^{2}}\right)}\right)}{2g}.$

For the main quantities of this model i.e. mass $M$, Hawking temperature
$T_{H}$, entropy $S$, pressure $P$ and volume $V$ we will have[64],

$M=\frac{8r^{4}P\pi+\left(3a+3\right)r^{2}+3q^{2}+3g}{6r},$ (19)
$T_{H}=\frac{8r^{4}P\pi+r^{2}\left(a+1\right)-q^{2}-g}{4\left(r^{2}+2g\right)r\pi},$
(20) $S=r^{2}\pi+4\ln\\!\left(r\right)g\pi,$ (21) $P=\frac{4\pi T\,r^{3}+8\pi
Tgr-r^{2}a+q^{2}-r^{2}+g}{8r^{4}\pi},$ (22) $V=\frac{4\pi r^{3}}{3},$ (23)

where q is the charge of the black hole, ”a” is related to the density of
clouds of strings and comes from the energy-momentum tensor of the clouds of
strings and g, which is displayed with g instead of $\alpha$ to avoid mistakes
with clouds of strings parameter, is a positive Gauss-Bonnet coupling
constant.

####  4.0.1  The landscape free energy

According to equations (2), (3), (4) and (7) for Gibbs and Landau free energy
function and thermal potential of this model, we will have,

$\textbf{G}=\frac{4P\pi
r^{3}}{3}+\frac{ar}{2}+\frac{g}{2r}+\frac{r}{2}+\frac{q^{2}}{2r}+\frac{\left(-8r^{4}P\pi-r^{2}\left(a+1\right)+q^{2}+g\right)\left(4\left(\frac{2\left(r-1\right)}{r+1}+\frac{2\left(r-1\right)^{3}}{3\left(r+1\right)^{3}}\right)g+r^{2}\right)}{4\left(r^{2}+2g\right)r},$
(24) $G_{L}=\frac{4P\pi
r^{3}}{3}+\frac{ar}{2}+\frac{g}{2r}+\frac{r}{2}+\frac{q^{2}}{2r}-\left(4g\ln\\!\left(r\right)+r^{2}\right)T\pi,$
(25)
$\textbf{U}=\frac{8r^{4}P\pi+\left(3a+3\right)r^{2}+3q^{2}+3g}{6r}-\left(r^{2}\pi+4\ln\\!\left(r\right)g\pi\right)T,$
(26)
$\textbf{L}=PX-\frac{2\,6^{\frac{2}{3}}\left(-\frac{3\pi^{\frac{1}{3}}6^{\frac{2}{3}}\left(a+1\right)X^{\frac{2}{3}}}{16}+\pi^{\frac{5}{3}}Tg6^{\frac{1}{3}}\ln\\!\left(X\right)X^{\frac{1}{3}}+\frac{9\pi\left(TX-\frac{2q^{2}}{3}-\frac{2g}{3}\right)}{8}\right)}{9\pi^{\frac{2}{3}}X^{\frac{1}{3}}},$
(27)

Now we will go to a special state that is formed based on the arbitrary
parameter setting. For this purpose, by considering the values
$a=1.5,q=0.5,g=0.5$ for the parameters and with a little calculation for the
critical quantities, we will have,

$\begin{split}&T_{c}=0.0978,\hskip 19.91684ptP_{c}=0.0075,\hskip
19.91684ptr_{c}=2.269260985,\\\ &V_{c}=48.94878867,\hskip
19.91684ptU_{c}=1.28320692868,\hskip 19.91684ptG_{c}=1.28662690146,\\\
\end{split}$ (28)

In this model, we still use relations (17) to make the equations
dimensionless.

####  4.0.2  Kramer’s escape rate

In this black hole, unlike the previous case where we examined a specific
state of the potential of the system at a certain pressure in terms of
temperature changes, we are looking to investigate the changes in the rate of
escape, at a constant temperature for a certain state of the system, in terms
of pressure. For this purpose, we consider the temperature t = 0.937, which is
equivalent to the energy state of the system in Fig (7a).

Figure 7: The graph of $G_{L}$ against x for free parameters in fig (7a) and p
versus x in fig (7b) which are determined as the locations of ($\alpha$)
intersection (red and orange), ($\beta$) intersection (blue and orange) and
($\gamma$) intersection (green and orange)

Now, using equation (1), we calculate the escape rate of each of the possible
transitions, i.e. ($\alpha\rightarrow\beta$) and ($\gamma\rightarrow\beta$),
the result of which can be seen in fig (8).

Figure 8: (8a): $r_{k}$ with respect to $x$, (8b): $r_{k}$ with respect to $p$
and (8c): Enlarged detail, bottom part of diagram (8b)

As it is evident from Fig (8c), at (p = 0.8384), again the two escape rate
curves intersect, which makes their occurrence probabilities equivalent. It
appears that after this convergence point, a process entirely different from
the nonlinear or charged model takes shape. This means that quantitatively,
the pressure associated with the transition from a large black hole to a small
one increases more than in the reverse case. However, this increase in
pressure for the transition from a large to a small black hole does not
actually cause the corresponding probability to increase as well, leading not
to the dominance of the large to small black hole trend. As can be seen from
the figure, if we draw an isobar, we find that $r_{k1}$ remains larger than
$r_{k2}$. This may indicate a greater harmony and alignment of this model with
the known natural laws of physics

##  5 Discussion and comparison

In this section, for a better understanding, comparison, and interpretation of
the results, it is first beneficial to delve a little into the concept of
transition from small to large black holes (or large to small). The transition
from a small to a large black hole can generally occur through the accretion
of matter and energy from the surrounding environment of the black hole, or it
can also occur through the merging of black holes, which may be accompanied by
intensified emission of gravitational waves. However, from a thermodynamic
perspective, this transition seems to encompass any physical process that
facilitates changes in temperature, pressure, and other thermodynamic
parameters in such a way that it leads to an increase in the radius of the
event horizon and consequently, the enlargement of the black hole, which,
given the super-gravitational structure of black holes, is a highly probable
process. The reverse process, the transition from a large to a small black
hole, can occur through processes such as Hawking radiation, where black holes
emit particles due to quantum effects near the event horizon. Over time, this
radiation can lead to a reduction in the mass and size of the black hole and
its eventual evaporation. From a thermodynamic perspective, this phenomenon is
equivalent to any process that creates a change in thermodynamic parameters
leading to a reduction in the radius of the horizon and the shrinking of the
black hole.
As observed in Figs (5a) and (7a), our black hole initially resides in a
stable local minimum, alpha, which globally is considered an unstable minimum.
Consequently, as one would naturally expect, with the slightest thermal
perturbation, and corresponding pressure, the system increasingly tends to
transition towards the global minimum,$\gamma$, signifying an increase in
radius. As the radius increases and the system begins to move towards a larger
black hole, according to the reaction principle, it is natural that processes
that prevent excessive radial growth and uncontrolled instability also form.
This leads to the fact that the escape rate after reaching a peak, a
decreasing trend will ensue, and the system will potentially stabilize at the
global minimum (Fig (6c)).
However, in (NLM-C-Q-PFD) modle, when examining the escape rate diagram as a
function of temperature (or pressure), we find that when the temperature (and
corresponding pressure) exceeds a certain limit, a small range emerges where a
reverse process forms and prevails Fig (6c). In other words, With $r_{k2}$
prevailing over $r_{k1}$, a situation is created that allows black holes to
move from $\gamma$ to $\beta$. It seems that the transition from $\gamma$ to
$\beta$ is less consistent with physical conditions. This is because, within
the same energy structure and without topological changes in the energy
function (that is, maintaining the gamma state as the global minimum and alpha
as the local minimum), the system prefers to reach an $\alpha$ state with a
smaller radius and consequently reduced entropy, which is at a higher
energetic level. Therefore, from our perspective, this situation can be termed
’low probable.’ In our opinion, based on the results, it appears that the
larger this region is, meaning the reverse process begins at lower
temperatures or pressures, it may indicate an incomplete structural evolution
of the black hole model to fully conform with the laws of nature. For example,
in the Reissner-Nordstrm black hole [48], as a basic model that includes only
electric charge, this region is significantly larger and practically quite
evident. As we know, due to the choice of mass and charge size, this model has
repeatedly shown non-conformity with standard physical conditions and
cosmological conjectures [61,62]. In other words, from comparing the results
of the Reissner-Nordstrm model with the (NLM-C-Q-PFD) black hole, it seems
that the addition of parameters influenced by dark structure (energy and
matter) has led to the black hole behavior approaching a more natural and
orderly structure. The situation regarding the (EGB-YM-CS) structure is
completely different from the previous state. It appears that after the
convergence point of the two probability diagrams, which gives black holes an
equal choice to move between two minima, the transition from $\gamma$ to
$\beta$, in terms of increasing pressure changes, overtakes the transition
from $\alpha$ to $\beta$. However, this increase in pressure, according to
diagram (8c), is not accompanied by an increased probability of transition;
that is, if we draw an isobar in any region after the convergence, we will see
that the transition probability $r_{k1}$ remains greater than $r_{k2}$ . It
seems that the transition from $\alpha$ to $\beta$ after the convergence
point, with decreasing pressure, prevents the ’low probable’ process and
displays a complete harmony with nature.

##  6 Conclusion

The study of black hole phase transitions based on the Gibbs free energy
function is an important and influential part of advancing our understanding
of black hole thermodynamics. However, a point that existed in this
conventional method was that it seemed only static phase transitions were
considered, leaving questions about the dynamics of phase transitions
unanswered.
The proposition that the dynamics of phase transitions in black holes are
controlled by thermodynamic and statistical forces, followed by the study of
phase transitions according to the stochastic Langevin equation or the Fokker-
Planck equation [33], and considering the ’mean first passage time’ in some
studies [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 56], and then the introduction of Kramer’s escape rate in
the study of phase transitions [60], motivated us to examine and compare
different forms of the free energy landscape and then study Kramer’s escape
rate in the first-order phase transitions of more complex black holes. These
black holes are specifically formed based on cosmic and stringy properties,
meaning they interact more with the features of dark structures. Our goal is
to see what similarities or differences the addition of dark structure and
stringy parameters create in the behavior of these black holes compared to the
simpler model of the charged black hole.
According to this, in this article, we examined the escape rate from two
perspectives. Initially, we considered it in terms of distance or the radius
of the black hole and observed that black holes exhibit natural behavior
consistent with environmental evidence. That is, the escape rate initially
increases with the radius until it reaches a certain maximum, but this
increase is not uncontrolled or permanent. A physical feedback mechanism
causes this probability to gradually decrease, ending at a specific radius
tolerable for the black hole’s survival. What is evident in comparison is that
the escape rate and leakage from the small to the large black hole always
dominate over the reverse. However,in the (NLM-C-Q-PFD) case, the escape rate
diagram as a function of temperature (pressure) indicates something else.
Meaning that the diagram initially, at its peak, and even near the end, shows
behavior similar to the escape rate diagram as a function of distance. But
suddenly, near the end, at a relatively high temperature (pressure), it takes
on a different behavior. Specifically, at a certain temperature (pressure),
the probability of transitioning from small to large and vice versa becomes
equal. This implies that at this temperature, black holes have the ’choice’ to
select their desired path towards either the small or large black hole. After
this temperature, in a region, albeit small, the process completely reverses,
meaning the probability of transitioning from the large to the small black
hole becomes dominant, and black holes prefer to take the path from the large
to the small black hole! But in the case of (EGB-YM-CS), after the
intersection point of the graphs, the high probable transition still maintains
its preference over the ’low probable’ transition and, in our opinion, shows a
more perfect harmony with nature.We have addressed this in the discussion and
comparison section. What we found in comparison with the charged black hole
model is that in the charged model, this region is quite evident and
noticeable. However, with the addition of parameters related to dark
structure, this region-which we consider to be less physical-will either
disappear (EGB-YM-CS) or become so small that radial changes due to their
insignificance do not appear in the radius diagram, but still exist in the
escape rate diagram (NLM-C-Q-PFD) as a function of temperature (pressure). The
question of whether the existence of this region, albeit small, for a black
hole has a physical interpretation and is considered natural behavior, and how
it should be justified, is something that requires further study of various
models. However, in conclusion, it seems that as the theoretical models of
black holes evolve and more empirical and observable factors such as dark
structure are added to the model, there will exist greater conformity with the
’pattern of repetition’ mentioned at the beginning of the introduction.

##  7 Appendix A: Behavioral sequence of the free energy landscape for (EGB-
YM-CS) black hole

The swallow tail representation of the Gibbs free energy landscape in terms of
t and the behavioral sequence of the potential function in terms of distance
and temperature for the (EGB-YM-CS) black hole is shown in Fig (10), (11).

Figure 9: The Gibbs free energy landscape in terms of t

Figure 10: Behavioral sequence of energy in terms of x for different
temperatures

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