# An entropic approach to analyze phase transitions in the $q=3$ Potts model

L. S. Ferreira<EMAIL_ADDRESS>Departamento de Física, Universidade
Federal de Minas Gerais, Av. Antônio Carlos 6627, 31270-901, Belo Horizonte,
Minas Gerais, Brazil  L. N. Jorge  Instituto Federal do Mato Grosso - Campus
Cáceres, Av. dos Ramires s/n, 78200-000, Cáceres, MT, Brazil  C. J. DaSilva
Instituto Federal de Goiás, Rua 76, Centro, Goiânia - GO, Brazil  A. A.
Caparica  Instituto de Física, Universidade Federal de Goiás, Av. Esperança
s/n, 74.690-900, Goiânia, GO, Brazil

(August 27, 2024)

###### Abstract

Boltzmann’s microcanonical entropy is the link between statistical physics and
thermodynamics, forasmuch as the behavior of any thermodynamic quantity is
directly related to the number of microscopic configurations. Accordingly, in
this work, we investigate the behavior of the logarithm of the density of
states of the three-state Potts model with an external field applied to one of
the states using joint entropic simulations based on the Wang-Landau
algorithm. Our analysis reveals that the microcanonical entropy curve is
detachable, and each resulting path is related to the formation of clusters.
Such a description is consistent with the energy-entropy argument related to
the inception of a phase transition. When the external field is reversed and
strong, the observed phase transition is from an ordered configuration to
cluster formations. The behavior of the microcanonical inverse temperature
indicates both first and second-order phase transitions occurring at different
temperatures for high values of the external field.

##  I Introduction

Phase transitions have been attracting attention for a long time. Its aspects
are outward from microscopic to cosmic dimensions [1, 2, 3]. However, its
description is not as prompt as its observation. For instance, the theoretical
description of what happens at the triple point of water or its critical
behavior is not straightforward[4]. The fluctuations are immense, and the
visualization of a phase transition is subtle.

To tackle phase transitions theoretically, one can define an order parameter
to characterize a particular phase[5]. This quantity has a non-zero value in
an ordered phase and is null in the disordered one. However, for a second-
order phase transition, the order parameter only signals the loss of the
system order, unable to determine which configuration the system attains.

It is well-known that at the critical point the system comprises two
indistinguishable phases, as a result of the formation of clusters of
different sizes. In a real system, this discretion relies on the visualization
of critical opalescence due to the scattering of light at all frequencies, due
to the different sizes of the phases[6, 7]. Configurations at the critical
point certainly do not satisfy the order parameter, so they can easily be
confused with disordered configurations. A common feature at the critical
point is the dependence of thermodynamic properties on the size of the system,
called the finite-size effect[8, 9].

While theoretical descriptions do not provide a complete description, entropic
simulations have proven to be a powerful tool to study phase transitions in
several models [10, 11, 12, 13, 14]. Their findings have disclosed new results
that need a scrutinized analysis to provide a coherent interpretation. An
example is the ambiguous behavior of the phase transition for the spin-1
Baxter-Wu model[15]. A previous study showed that the critical behavior of
quantities related to first and second-order phase transitions are equally
satisfied. Furthermore, using a technique of tearing the densities of states
curve, it was possible to observe that the ferromagnetic ground state
undergoes a first-order phase transition and the ferrimagnetic ground state a
second-order one.

Another feature evidenced by entropic simulations is the behavior of
probability distributions of the microcanonical energy. For the Baxter-Wu
model with a crystalline field [16], this quantity presents an unusual noise
classified as a finite-size effect. However, this statement has no proof up to
now. Other models also showed similar behavior, such as the ZGB-like
model[17], polymer chain [18], q Potts model [19], $q=10$ and $20$ Potts[20],
and the three-dimensional Ising model [21]. To this extent, such behavior
lacks a competent investigation.

The $q$-states Potts model has been extensively studied not only due to its
theoretical interest but also for its practical applications. For $q\leq 4$,
with no external magnetic field, the model can be exactly solved and is known
to have a second-order phase transition with well-defined critical exponents
that depends on $q$ [22, 23]. For $q>4$, the phase transition is first order.
Within the entropic simulations scenario, the Potts model is used to test
improvements in the algorithm[24, 25, 13, 26, 27, 28, 29]. Surungan [27]
investigated the $q=4$ Potts model with the presence of invisible states using
entropic simulations. They used the inverse of the microcanonical temperature,
$\frac{\Delta S}{\Delta E}$, to determine the order of the phase transition.
For the case where there is no invisible state in the lattice, the initial
behavior of the curve does not match the expected result. The same is true for
the spin 1 Baxter-Wu model [15].

Furthermore, when an intense and reverse external field acts to one of the $q$
states, the phase transition is analogous to the model with $q-1$ states with
no external field applied [30, 31], i.e., the critical exponents and critical
temperature are the same.

In this paper, we address the phenomena of phase transition of the $q=3$ Potts
model using entropic simulations. More precisely, we will probe the logarithm
of the density of states in the region close to the ground state energies by
checking the configurations and analyzing the behavior of probability
distribution of the microcanonical energy and the inverse microcanonical
temperature. These quantities reveal the microcanonical behavior of the system
and can clarify questions about the phase transition.

##  II Model

The $q$-states Potts model has been at the forefront of scientific research in
Statistical Mechanics since its inception.[32]. It represents an extension of
the ferromagnetic Ising model for the case $q>2$. In this model, the sites of
a two-dimensional lattice can contain one of the $q$ possible states,
represented by the spin variable $\sigma$, and interact ferromagnetically if
two neighboring sites are equal. We can write the Hamiltonian as,

$\mathcal{H}=-J\sum_{\langle
i,j\rangle}\delta_{\sigma_{i}\sigma_{j}}-H\sum_{i}\delta_{1,\sigma_{i}},$ (1)

where the first sum runs over the nearest neighbors and the second across the
entire lattice. $\delta$ is the Kronecker delta, being 1 when the states of
neighboring sites are equal and zero otherwise. $H$ represents the magnitude
of an external field. It acts only on state 1, causing this state to be
energetically favorable when $H>0$. Otherwise, it causes an energy increase
due to the presence of this state in the lattice.

The ground state is formed by configurations where the lattice is completely
filled with only one of the states. Its energy is independent of $q$ and
corresponds to $E_{g}=-2JL^{2}$ for $H<0$. When $H>0$ the minimum energy is
$E_{g}=-(2J+H)L^{2}$ and the lattice is filled by state 1.

##  III Computational details

To study the behavior of the density of states we use a modified version of
the Wang-Landau algorithm [33, 34], called entropic simulation, due to its
ability to estimate the logarithm of the density of states ($\ln g(E))$)
directly, which, except for a constant, is the Boltzmann entropy,
$S(E)\equiv\ln g(E)$. This new version aims to improve the estimation of
microcanonical averages and save computational time[35, 36, 37]. The
computational details are presented below.

A new configuration is defined as changing the state of a site, chosen at
random, to any of the possible states with equal probability. A Monte Carlo
(MC) step is counted after $L^{2}$ new configuration attempts. The initial
configuration is arbitrary, so we start from a ground state configuration, due
to the ease of assigning the energy and order parameter to this configuration.
Initially the density of states is not known, so we assign a unit to all
possible energy levels of the system, $g(E)=1$, making $S(E)=0$. The
functioning of the algorithm is based on the creation and monitoring of an
energy histogram $H(E)$. The random walk through the energy space is from
$E_{min}$ to $E_{max}$ with probability

$p(E\rightarrow E^{\prime})=\min\\{\exp[(S(E)-S(E^{\prime}))],1\\},$ (2)

where $E^{\prime}$ and $E$ are the energies of the new and current
configurations, respectively. After a MC step $S(E)$ and $H(E)$ are updated,
$S(E)\rightarrow S(E)+\ln f_{i}$ and $H(E)\rightarrow H(E)+1$, where $f_{i}$
is the so-called modification factor, which initial value is $f_{0}\equiv
e=2.71828\cdots$. After a certain amount of MC steps the energy histogram
flatness is checked. It is considered flat if $H(E)>0.8\langle H\rangle$, for
all energy levels, where $\langle H\rangle$ is the mean value of the
histogram. If it is not flat the random walk continues. Otherwise, we have an
estimate for $S(E)$, completing a Wang-Landau level. To obtain $S(E)$ with
greater precision the modification factor is refined $f_{i+1}=\sqrt{f_{i}}$
and the simulation continues with the random walk. From $f_{i}=f_{7}$ the
microcanonical averages start to be accumulated.

To finish a simulation run, we follow the evolution of the parameter

$\varepsilon=|T_{c}(t)-T_{c}(0)|,$ (3)

where $T_{c}(0)$ and $T_{c}(t)$ are temperatures of the maximum of the
specific heat obtained with $S(E)$ at the beginning of the WL level and when
the flatness criterion is checked. If during an entire WL level this parameter
remains less than $10^{-4}$ the simulation is halted. To save computational
time, we start the verification at the eighth level, $f_{7}$. Identifying the
Hamiltonian sums, Eq. 1, as

$\displaystyle E_{1}=$
$\displaystyle-\sum_{<i,j>}\delta_{\sigma_{i}\sigma_{j}}$ (4) $\displaystyle
E_{2}=$ $\displaystyle-\sum_{i}\delta_{1,\sigma_{i}},$ (5)

we can obtain the logarithm of the joint density of states $\ln
g(E_{1},E_{2})$ [38, 39, 40, 16, 10], using the same simulation scheme, just
exchanging $\ln g(E)$ by $\ln g(E_{1},E_{2})$, where $E=JE_{1}+HE_{2}$ . We
can obtain $\ln g(E)_{J,H}$ from $\ln g(E_{1},E_{2})$ using the equation

$\ln g(E)_{J,H}=\ln\left(\sum_{E_{1},E_{2}}\delta(E-JE_{1}-HE_{2})e^{\ln
g(E_{1},E_{2})}\right),$ (6)

where $\delta(E-JE_{1}-HE_{2})$ is the Dirac delta that filters out the
energies $E_{1}$ and $E_{2}$ that satisfy $E=JE_{1}+HE_{2}$. Thus, the
estimation of the logarithm of the joint density of states allows us to
investigate the microcanonical entropy for any external field.

##  IV Results

Figure 1: Inspection of the logarithm of the density of states for $L=12$. (a)
Full density of states. (b) Detailed view of the logarithm of the density of
states close to the ground state.  Figure 2: Logarithm of the density of
states separated into four densities. The labels $GS$, $GS+4$, $GS+6$, $GS+7$
and $GS+9$, $GS$ represents the ground state energy and the numbers are the
changes in energy from the ground state.  Figure 3: Configurations of the
excited levels.

###  IV.1 Case $H=0$

Special attention must be paid to the logarithm of the density of states in
the region close to the ground state. There one can notice a very peculiar
behavior, which can be seen in Fig. 1 for $L=12$. Looking at the logarithm of
the density of states without much care one would say that there is nothing
special, Fig. 1(a), but when we look closer, Fig. 1(b), we notice that there
is an irregularity in the logarithm of the density of states.

This same behavior is found in the Ising model[41] and spin 1/2 Baxter-Wu
model. In order to understand what is going on, we investigated the
configurations of the first four excited levels. They are labeled as $GS+4$,
$GS+6$, $GS+7$ and $GS+9$. $GS$ represents the ground state energy and the
numbers are the changes in energy from the ground state. The results for size
$L=12$ can be seen in Fig. 1(b).

The first excited state has the configuration whose energy is equal to the
ground state plus four ($GS+4$), as in Fig. 3(a). It is obtained by taking a
ground state configuration and changing the state of one of the lattice sites.
The same to the configuration whose energy is $GS+6$ (Fig. 3(b)): starting
from the ground state configuration, two neighboring sites in the lattice are
changed to the same state. If the neighboring sites are in different states,
the energy becomes $GS+7$, as in 3(c). The energy configuration $GS+9$ is
obtained by altering the states of three neighboring sites of the ground state
configuration, with two equal states (Fig. 3(d)).

Due to the constraints on the positions of the chosen sites and states in the
configurations $GS+6$, $GS+7$ and $GS+9$, the number of configurations with
these energies will be lower than a configuration where they are not required.
An example is the configuration where two non-neighboring sites are changed in
the ground state configuration, generating a configuration whose energy is
$GS+8$. This fact can be seen in the curve of the density of states in Fig. 2.

When analyzing the following energies we observe the same behavior for these
four configurations. For energy $GS+8$, the configuration keeps the exchanges
distant, while for $GS+10$, $GS+11$ and $GS+13$ the chosen sites remain
neighbors, changing only the states. We observe that configurations of similar
characteristics are obtained by varying the energy of $\Delta E=4$ units
starting from the first four excited states.

From these observations we can conclude that the cluster formations minimize
the energy, i.e., when grouping equal states, the energy is less than keeping
them apart. While the formation of clusters is avoided, the entropy is
maximized, i.e., it increases the number of possible configurations. We see
that different configurations are obtained by minimizing energy and maximizing
entropy. By the energy-entropy argument, some phase transition must exist at
some intermediate temperature[42]. Such analysis suggests that other lattice
models may also contain analogous behaviour of the density of states, such as
the Ising model.

We also realized that some of these energies can only be obtained if there is
a specific dynamic to reach certain configurations. Apparently, each of these
configurations develop independent densities of states that come together at a
certain energy. In order to visualize this, we show the logarithm of the
density of states starting from these four excited states and restricting the
energy variation to 4. The result is shown in Fig 2, confirming our
expectation. We will label $\ln g_{1}$, $\ln g_{2}$, $\ln g_{3}$ and $\ln
g_{4}$ the logarithms of densities starting from the states $GS$, $GS+6$,
$GS+7$ and $GS+9$, respectively.

We expect this behavior to be reflected in the probability distribution of
energy

$P(E,T)=\frac{g(E)e^{-E/k_{B}T}}{Z},$ (7)

where $g(E)$ is the density of states, $k_{B}$ is the Boltzmann’s constant and
$Z$ is the partition function. Rewriting in terms of the logarithm of the
density of states we have

$P(E,T)=\frac{e^{-F/k_{B}T}}{Z},$ (8)

where $F=E-k_{B}T\ln g(E)$ is the Helmholtz free energy.

Figure 4: Distribution of energy probabilities without separating the
logarithm of the density of states.

To calculate the probability distribution of energy we use $k_{B}=1$ and
neglect the partition function. Furthermore, we use a normalization factor
$e^{-\lambda(T)}$, where $\lambda$ returns the maximum of $F$ in the fixed
temperature energy interval, so that the probability distribution is written
as

$P(E,T)=e^{-F/k_{B}T-\lambda}.$ (9)

Due to the normalization factor, the maximum of the probability distribution
is equal to unit. The value of the factor $\lambda$ is obtained using the
logarithm of the density of states for all energy levels. The typical behavior
of this quantity is shown in Fig. 4(a) for $T=0.91$. As can be seen, their
behavior is peculiar when we use all energy levels.

Restricting the energy variation to $\Delta E=4$ and starting $P(E,T)$ from
the energies of the first four excited states we obtain the curves shown in
Fig. 4(b). In Fig. 5 we show a sequence of temperatures.

Figure 5: Probability distribution of energy separating the logarithm of the
density of states into four curves shown for four different temperatures. In
(a) $T=0.91$, (b) $T=0.95$, (c) $T=0.99$ and (d) $T=1.02$.

One can see that each curve of the logarithm of the density of states
generates a well-behaved probability distribution, with a Gaussian like shape.
The heights of the distributions show that, at low temperatures, the system is
more likely to change its state to a configuration where the energy is not as
low as possible. Regarding the configurations, the system does not present
cluster formation. As the temperature increases, the presence of clusters is
favored. Then the presence of clusters from several different states becomes
more probable. As the temperature approaches the phase transition, the four
curves tend to merge and their width tends to increase. The above description
can be seen through the GIF that shows the most probable configurations as we
increase the temperature (t-16.0.0.gif file, sent in attachment). To a better
visualization of this procedure, check the supplemental material.

To identify finite-size effects, we performed entropic simulations estimating
the density of states for the sizes $L=\leavevmode\nobreak\
32,\leavevmode\nobreak\ 64$ and $80$. In Fig. 6 (a) we show the behavior of
the energy probability distribution at temperatures
$T=0.80,\leavevmode\nobreak\ 0.70$ and $0.60$. In 6(b), showing the energy
probability distributions for $T=1.15$ and $L=\leavevmode\nobreak\
32,\leavevmode\nobreak\ 64$ and $80$, we realized that the larger the lattice
size, the smaller the temperature at which the four energy probability curves
collapse and that the collapsed curves, of all lattice sizes, are together
after to the critical temperature of the system. An explanation for this
behavior follows from the fact that an energy variation $\Delta E=4$ becomes
very small compared to the total energy range $2L^{2}$, yielding $\frac{\Delta
E}{2L^{2}}=2L^{-2}$. At the thermodynamic limit we have
$\lim_{L\rightarrow\infty}\frac{\Delta E}{2L^{2}}=0$, so we expect the curves
to collapse at very low energy, as the entire analysis is carried out in the
region close to the ground state. However, as we are dealing with finite
systems, all the effects observed here must be taken into account when
tackling such quantities.

Figure 6: Dependency of the energy probability distribution with the lattice
size $L=32,\leavevmode\nobreak\ 64$ and $80$. In (a), the behavior for lattice
sizes at different temperatures and in (b) at the same temperature.  Figure 7:
Inverse of the microcanonical temperature for $L=32$. (a) No separation of the
logarithms of the densities of states and (b) separated.

Another quantity that also displays a significant difference under the above
procedure is the microcanonical temperature $\frac{1}{T}=\frac{\Delta
S}{\Delta E}$. In Fig 7(a) we show the behavior of this quantity without
taking into account the different density of states curves. We noticed that at
low energies the behavior of the curve does not agree with the expected one,
$T\rightarrow 0\Rightarrow\frac{1}{T}\rightarrow\infty$. The same behavior was
obtained by Surungan et. al.[27] for the $q=4$ Potts model and Fiore and
DaSilva [20] for the $q=10$ and $q=20$ Potts model. In fig 7(b), the curves
represent the individual derivative of each of the four curves. In this case,
the energy variation is $\Delta E=4$, regardless of the chosen density of
states. Initially the curves are separated and, as in the case of the energy
probability distribution, they merge at the transition temperature. The
divergence $\frac{1}{T}\rightarrow\infty$ is only obtained in the
thermodynamic limit[29], whose dependence is logarithmic.

###  IV.2 Case $H<0$

Figure 8: Logarithm of the density of states for q=3 and H=-1, -5 and -10 .

Now we turn to what happens when we add a negative external field, i.e., keep
out particles that interact with the field. Starting with $H=-1$, Fig. 8(a)
shows the detail of the density of states with the four openings of the ground
state. Here we have the levels $GS$, $GS+4$, $GS+5$, $GS+6$ and $GS+11$. The
ground state is formed by one of the states that are not affected by the
external field. The first excited state $GS+4$ is reached by changing the
state from one site to one that does not interact with the field. If the new
state interacts with the field it will correspond to the second excited state
$GS+5$.

The third excited state corresponds to the formation of clusters that do not
interact with the external field. The beginning of the fourth density of
states occurs when we have the conversion of three sites in the lattice. Two
of them with states other than the ground state and not interacting with the
field and one of them interacting with the field. Another possibility is the
presence of three states interacting with the field and forming clusters.

In Fig. 8(a), one can see that the line related to the ground state has a
greater number of states. All the others, as they contains specifications
regarding the shape and number of particles in each state, have a lower number
of states. This behavior is valid up to the transition region between the
phases of the system. It is noteworthy that the logarithm of the density of
states in this interval is highly dependent on the number of sites, since the
number of ways of rearranging the system with a fixed number of particles is
proportional to the number of particles. The most likely configurations at
different temperatures are shows in the supplementary material.

Investigating the logarithm of the density of states for $H=-5$ and $-10$ (see
Fig. 8 (b) and (c), respectively) we note that there is a well-defined
separation between the lines. $GS+4$ and $GS+6$ are present in all fields,
indicating that the energies do not depend on the field value, i.e., the
states present in these configurations do not interact with the external
field. On the other hand, $GS+9$ and $GS+15$ for $H=-5$ and $GS+21$ and
$GS+23$ for $H=-10$ represent the entry of states interacting with the field,
without and with formation of clusters, respectively. We note that the larger
the field the further apart these lines become.

In Fig. 9 (a) and (b) we show the inverse of the microcanonical temperature
for $H=-5$ and $-10$, respectively. One can see two distinct aspects. The
first is the joining of the curves with states interacting with the field
($\ln g_{3}$ and $\ln g_{4}$), the joining of the non-interacting curves ($\ln
g_{1}$ and $\ln g_{2}$), forming two distinct curves. When this happens, we
have the formation of a plateau. The two curves seem to come together at the
critical temperature, but come apart again. The second is the junction of the
four curves. For $H=-10$, this effect is better visualized. As the curves
meet, a second plateau appears. At this point, a lattice site is equally
likely to contain any of the $q$ states, that is, the configuration is
disordered.

Figure 9: Inverse of microcanonical temperature for $q=3$, in (a) $H=-5$ and
(b) $H=-10$.

###  IV.3 Case $H>0$

When $H>0$, no finite-size effects is observed in the thermodynamic
quantities. However, the logarithm of the density of states has a large number
of independent lines, as can be seen in Fig. 10. When $H=1$ (see Fig 10 (a)),
we have a behavior similar to the previous cases: five independent curves can
be obtained from the logarithm of the density of states. Now the variation
between the points is 5 and the beginning of each curve are in $GS$, $GS+8$,
$GS+9$, $GS+11$ and $GS+12$. Solely the lines corresponding to $GS$, $GS+9$
and $GS+12$ were presented for a better visualization, since the curves are
very close to each other. A very useful comment about the configurations to
this case are made in the supplementary material.

Figure 10: Logarithm of the density of states for q=3 and H=1, 5 and 10.

For $H=5$, Fig 10 (b), the number of curves increases to 9. The start of each
curve happens at $GS$, $GS+16$, $GS+17$, $GS+23$, $GS+24$, $GS+28$, $GS+30$,
$GS+31$ and $GS+38$ and the variation between the points is 9. For $H=10$, Fig
10 (c), we have 13 curves starting at $GS$, $GS+26$, $GS+27$, $GS+38$,
$GS+48$, $GS+50$ , $GS+51$, $GS+60$, $GS+61$, $GS+62$, $GS+72$, $GS+73$ and
$GS+85$ with variation of 14 between the points.

In all cases analyzed for $H>0$ we observe that the curves tend to form a
single curve, so that at the maximum temperature of the specific heat the most
likely energy is the one where the curves of the logarithm of the density of
states become equal. Observing the probability distribution of energy at
different temperatures, shown in Fig. 11, we see that the different curves at
low temperatures become equal at the maximum specific heat temperature.

Figure 11: Probability distribution of energy separating the logarithm of the
density of states into different curves shown for different temperatures. In
(a) $H=1$, (b) $H=5$, and (c) $H=10$.

The high number of curves interferes in the analysis of the behavior of the
energy probability distribution, not allowing the visualization of double
peaks in the curves with lower probabilities. To verify the occurrence of
double peaks, we check the inverse of the microcanonical temperature for the
first two curves of the logarithm of the density of states, shown in Fig. 12.
In Fig. 12 (a) for $H=1$, the temperature at which the curves meet is coherent
with that of the maximum specific heat. When $H=5$, Fig. 12 (b), the curve
referring to $\ln g_{1}$ presents a first-order transition behavior while the
second a second-order transition behavior, with the junction of the two curves
taking place after the evidence of the two transitions. At $H=10$, the two
curves show a first-order transition behavior. The omitted curves show a
behavior similar to the ones presented in each case. The omission allows a
better visualization of the first and second order behavior.

Figure 12: Configurations of the excited levels.

##  V Conclusions

In this work we have analyzed the logarithm of the density of states of the
$q=3$ Potts model with an external field applied to one of the states. Using
entropic simulations with two parameters, it was possible to observe the
behavior of the microcanonical entropy, the probability distribution of the
energy and the inverse of the microcanonical temperature for different
external field values. The results show that the logarithm of the density of
states near the ground state presents a behavior that indicates the presence
of several curves that can be detached, which are related to the formation of
clusters or not.

It was verified that the probability distribution and the inverse of the
microcanonical temperature show the expected behavior when taking into account
the different curves of the logarithm of the density of states. The
microcanonical entropy is formed by different curves that come together after
a certain temperature. We verified that, for $H=0$, this effect occurs for
lattices of size $L=16,\leavevmode\nobreak\ 32,\leavevmode\nobreak\ 64$ and
$80$, suggesting that it can be observed in all lattice sizes considered here.

For negative external fields, we observed the separation of two transitions:
one related to the formation of clusters and other to the entry of states
interacting with the external field. The first case is related to a critical
transition and the second to a order-disorder one.

For positive external fields, where there no indication of a phase transition
in the thermodynamic quantities, the microcanonical entropy possesses several
independent curves, with the number of lines depending on the value of the
external field. Thus, the probability distribution shows several curves that
compose a gaussian-shaped envelope, and phase transition effects are not
observed. The inverse of the microcanonical temperature showed that the curves
independently present first-order phase transition signals at different
temperatures.

Moreover, the visualization of this effect will help in the study of models
with a more complex energy landscape, where the estimation of the
microcanonical entropy is obtained in different energy ranges through
simulation in parallel and then joined by observing the inverse of the
microcanonical temperature. This procedure is called Replical Exchange Wang-
Landau[43, 44, 45]. As shown in this work, the energy region where the phase
transition occurs is the same where the curves meet.

In summary, the logarithm of the density of states estimated by the entropic
simulations is formed by different curves related to the formation of clusters
and the phase transition characteristics may be present in situations where
thermodynamic quantities such as specific heat and susceptibility do not
present finite-size effects.

###### Acknowledgements.

We thanks the J. A. Plascak for comments. We thank the computer support from
LaMCAD/UFG. L. S. Ferreira acknowledge the support by CNPq.

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