# Thermodynamics of the Schwarzschild-AdS Black Hole with a Minimal Length

Authors: Yan-Gang Miao; Yu-Mei Wu
Journal: Advances in High Energy Physics
 (2017)

---

## Abstract

Using the mass-smeared scheme of black holes, we study the thermodynamics of black holes. Two interesting models are considered. One is the self-regular Schwarzschild-AdS black hole whose mass density is given by the analogue to probability densities of quantum hydrogen atoms. The other model is the same black hole but whose mass density is chosen to be a rational fractional function of radial coordinates. Both mass densities are in fact analytic expressions of theδ-function. We analyze the phase structures of the two models by investigating the heat capacity at constant pressure and the Gibbs free energy in an isothermal-isobaric ensemble. Both models fail to decay into the pure thermal radiation even with the positive Gibbs free energy due to the existence of a minimal length. Furthermore, we extend our analysis to a general mass-smeared form that is also associated with the δ-function and indicate the similar thermodynamic properties for various possible mass-smeared forms based on the δ-function.


---

## Body

## 1. Introduction

Nonrenormalizability is a well-known puzzle when gravity is combined with quantum theory. Among many attempts to solve the problem, the idea of a nonperturbative quantum gravity theory is attractive. Recently, Dvali and collaborators propelled [1, 2] this idea by putting forward the “UV self-complete quantum gravity” in which the production of micro black holes is assumed to play a leading role in scattering at the Planck energy scale. According to the Heisenberg uncertainty relation, an incident photon needs to have higher energy in order to locate a target particle more accurately. When the photon energy is comparable to that of the particle, it is likely to create a new particle. If the energy of the incident photon gets further higher, say the Planck energy, so huge energy confined in a small scale may create a black hole, the so-called the hoop conjecture [3]. After the formation of micro black holes, the higher the incident photon energy is, the bigger the black hole is, which makes the probe meaningless. As a result, the existence of a minimum length may be deduced by the horizon radius of an extreme black hole.Regular black holes, that is, the black holes without singularity at the origin, whose research could be traced back to Bardeen’s brilliant work [4], appeared in [5, 6] on the discussions of mass definition, casual structure, and other related topics. In 2005, for the purpose of constructing the noncommutative geometry inspired Schwarzschild black hole, Nicolini, Smailagic, and Spallucci suggested [7] that the energy-momentum tensor should be modified in the right-hand side of Einstein’s equations, such that it describes a kind of anisotropic fluid rather than the perfect fluid, where the original point-like source depicted by the δ-function should be replaced by a mass-smeared distribution, while no changes should be made in the left-hand side. By replacing the δ-function form of the point-like source by the Gaussian form1 and then solving the modified Einstein equations, they obtained a self-regular Schwarzschild metric and established a new relation between the mass and the horizon radius (i.e., the existence of a minimum radius and its corresponding minimum mass). They also discussed the relevant thermodynamic properties and deduced the vanishing temperature in the extreme configuration and thus eliminated the unfavorable divergency of the Hawking radiation. Besides, there is a correction to the entropy in the near-extreme configuration. Since then, a lot of related researches have been carried out, such as those extending to high dimensions [8, 9], introducing the AdS background [10], quantizing the mass of black holes [11], and generalizing to other types of black holes [12, 13].In this paper, we at first introduce the AdS spacetime background to the self-regular Schwarzschild black hole and then investigate equations of state and phase transitions of the self-regular Schwarzschild-AdS black hole with two specific mass densities, where one mass density is assumed [14] by an analogue to a quantum hydrogen atom and the other to a collapsed shell. Although the mass densities of the two models are different, both of them can be regarded as a smeared point-like particle and then be depicted by an analytic expression of the δ-function. Next, we extend our discussions to a general mass-smeared form which can be written as an unfixed analytic expression of the δ-function and explain the similarity in thermodynamic properties for different mass distributions previously studied in [10, 15, 16].The paper is organized as follows. In Section2, we derive the relations between masses and horizons of the two models and give the limits of vacuum pressure under the consideration of the hoop conjecture. Then, we analyze the thermodynamic properties of the two models in detail in Section 3. We turn to our analysis in Section 4 for the self-regular Schwarzschild-AdS black hole with a general mass distribution. Finally, we give a brief summary in Section 5.We adopt the Planck units in this article:ħ=c=G=kB=1.


## 2. Two Specific Models

### 2.1. Mass Density Based on Analogy between Black Holes and Quantum Hydrogen Atoms

Based on our recent work in which we made an analogy between black holes and quantum hydrogen atoms [14], we choose the probability density of the ground state of hydrogen atoms as the mass density of black holes:(1)ρr=Me-r/a8πa3,where M is the total mass of black holes and a with the length dimension will be seen to be related to a minimal length through the extreme configuration discussed below. We plot the mass density in Figure 1 and obtain ρ(0)=M/(8πa3), which means that the origin is no longer singular. Now the mass distribution of black holes takes the form(2)Mr=∫0rρr~4πr~2dr~=M1-e-r/a1+ra+r22a2,which, divided by M, can be understood as a kind of step functions with continuity (see Figure 1(b)). It is easy to see that such a function M(r)/M goes to the step function under the limit a→0. Correspondingly, as done in [7, 10], we work out the self-regular Schwarzschild-AdS metric through solving the modified Einstein equations associated with the above mass distribution:(3)ds2=-1-2Mrr+r2b2dt2+1-2Mrr+r2b2-1dr2+r2dΩ2,where b represents the curvature radius of the AdS spacetime and has the relation with the vacuum pressure as follows:(4)P≡-Λ8π=38πb2.One can easily find that the self-regular metric goes back to the ordinary Schwarzschild-AdS one if r≫a and that on the other hand it goes to the de Sitter form if r approaches zero:(5)ds2=-1-M3a3r2+r2b2dt2+1-M3a3r2+r2b2-1dr2+r2dΩ2,where the equivalent cosmological constant can be understood as2  Λ~≡M/a3-3/b2. Thus the singularity is avoided since the positive cosmological constant represents the negative (outward) vacuum pressure that prevents the black hole being less than a certain scale from collapsing. From g00=0, one can have the relation between the total mass M and the horizon radius rH:(6)M=rH21+rH2b21-1+rHa+rH22a2e-rH/a-1,which is shown in Figure 2 under different vacuum pressure.Figure 1
Plots of the mass density equation (1) (a) and of the mass distribution equation (2) (b). Note that when we take the limit a→0, except the origin with M(0)=0, the mass ratio M(r)/M approaches 1 at other radii due to r/a→∞, which indicates that the mass distribution goes to the step function.
(a) (b)Figure 2
Plot of the relation ofM with respect to rH. The solid curves correspond to the relation equation (6), the dashed curves correspond to the relation of the ordinary Schwarzschild-AdS black hole with M=(rH/2)(1+rH2/b2), and each vertical line lies at the extremal radius of the curve with the same color. The blue, purple, cyan, and green colors correspond to the cases of b/a=10, 19.46 (the critical vacuum pressure), 30, and ∞, respectively.From Figure2, we observe that there is a lower bound M0 for the black hole mass, which is regarded as the mass of the extreme black hole. For the case of M>M0, the black hole possesses two horizons; for the case of M=M0, the two horizons coalesce into one, rH=r0, as the extremal radius, that is, the minimal length since no black holes are smaller than the extreme black hole. The extremal radius depends on the vacuum pressure, which can be determined by ∂M/∂rHrH=r0=0, where(7)∂M∂rH=aerH/a2a3erH/a-1b2+3rH2-2a2rHb2+3rH2-arH2b2+3rH2-rH3b2+rH2b2-2a2erH/a-1+2arH+rH22.It is difficult to express r0 as the function of b since the equation is transcendental, so we solve b with respect to r0 instead:(8)b=r0-3+2r03/a32+2r0/a+r02/a2+r03/a3-2er0/a.Considering the hoop conjecture [3] where the mean radius r¯ is supposed to be not greater than the extremal radius r0, we have estimated the ranges of both b and r0 in [14] through the numerical fitting. It is easy to calculate(9)r¯=1M∫0∞rρr4πr2dr=3a.The hoop conjecture implies r¯≤r0, which ensures the formation of a black hole. Using (8) and the hoop conjecture, one can obtain the exact ranges of the ratios, b/a: [9.996,∞), r0/a: [3,3.384), and M/a: (2.575,2.835], respectively. We note that a smaller b corresponds to an extreme black hole with a smaller radius but a larger mass; that is, when b/a=9.996, we have r0/a=3 and M/a=2.835. The reason is that a smaller b represents the larger vacuum pressure which makes the matter denser [14].


### 2.2. Mass Density Based on a Collapsed Shell

Besides collapsed cores, like the first model, a collapsed shell is also an important model of black hole formation, which is usually dealt with as a massive membrane without thickness. However, if the minimal length is considered, the shell is supposed to be described with a smeared distribution [15]. One can take the following rational fractional function of radial coordinators as the mass density of black holes:(10)ρr=75Ml03r32πl03+5r33,where l0 is a quantity with the length dimension which can be regarded as the minimal length and M the total mass of black holes. The relation is plotted in Figure 3.Figure 3
Plots of the mass density equation (10) (a) and of the mass distribution equation (11) (b) of the collapsed shell.
(a) (b)The mass distribution of this model,M(r), thus has the form(11)Mr=∫0rρr~4πr~2dr~=25Mr6l03+5r32,which, like (2), can also be understood as a kind of step functions with continuity and goes to the step function under the limit l0→0. By solving g00=0, we get the total mass M as the function of the horizon radius rH:(12)M=b2+rH2l03+5rH3250b2rH5,which is plotted in Figure 4. As done for the first model, one can also determine the allowed ranges of the ratios for the extreme configuration in this model: b/l0: [3.24, ∞), r0/l0: [0.943, 1), and M/l0: (0.720, 0.784], respectively, when the hoop conjecture is considered. Moreover, the data lead to the same result as in the first model; that is, a smaller extreme black hole is heavier than a greater one.Figure 4
Plot of the relation ofM with respect to rH. Each vertical line lies at the extremal radius of the curve with the same color.


## 2.1. Mass Density Based on Analogy between Black Holes and Quantum Hydrogen Atoms

Based on our recent work in which we made an analogy between black holes and quantum hydrogen atoms [14], we choose the probability density of the ground state of hydrogen atoms as the mass density of black holes:(1)ρr=Me-r/a8πa3,where M is the total mass of black holes and a with the length dimension will be seen to be related to a minimal length through the extreme configuration discussed below. We plot the mass density in Figure 1 and obtain ρ(0)=M/(8πa3), which means that the origin is no longer singular. Now the mass distribution of black holes takes the form(2)Mr=∫0rρr~4πr~2dr~=M1-e-r/a1+ra+r22a2,which, divided by M, can be understood as a kind of step functions with continuity (see Figure 1(b)). It is easy to see that such a function M(r)/M goes to the step function under the limit a→0. Correspondingly, as done in [7, 10], we work out the self-regular Schwarzschild-AdS metric through solving the modified Einstein equations associated with the above mass distribution:(3)ds2=-1-2Mrr+r2b2dt2+1-2Mrr+r2b2-1dr2+r2dΩ2,where b represents the curvature radius of the AdS spacetime and has the relation with the vacuum pressure as follows:(4)P≡-Λ8π=38πb2.One can easily find that the self-regular metric goes back to the ordinary Schwarzschild-AdS one if r≫a and that on the other hand it goes to the de Sitter form if r approaches zero:(5)ds2=-1-M3a3r2+r2b2dt2+1-M3a3r2+r2b2-1dr2+r2dΩ2,where the equivalent cosmological constant can be understood as2  Λ~≡M/a3-3/b2. Thus the singularity is avoided since the positive cosmological constant represents the negative (outward) vacuum pressure that prevents the black hole being less than a certain scale from collapsing. From g00=0, one can have the relation between the total mass M and the horizon radius rH:(6)M=rH21+rH2b21-1+rHa+rH22a2e-rH/a-1,which is shown in Figure 2 under different vacuum pressure.Figure 1
Plots of the mass density equation (1) (a) and of the mass distribution equation (2) (b). Note that when we take the limit a→0, except the origin with M(0)=0, the mass ratio M(r)/M approaches 1 at other radii due to r/a→∞, which indicates that the mass distribution goes to the step function.
(a) (b)Figure 2
Plot of the relation ofM with respect to rH. The solid curves correspond to the relation equation (6), the dashed curves correspond to the relation of the ordinary Schwarzschild-AdS black hole with M=(rH/2)(1+rH2/b2), and each vertical line lies at the extremal radius of the curve with the same color. The blue, purple, cyan, and green colors correspond to the cases of b/a=10, 19.46 (the critical vacuum pressure), 30, and ∞, respectively.From Figure2, we observe that there is a lower bound M0 for the black hole mass, which is regarded as the mass of the extreme black hole. For the case of M>M0, the black hole possesses two horizons; for the case of M=M0, the two horizons coalesce into one, rH=r0, as the extremal radius, that is, the minimal length since no black holes are smaller than the extreme black hole. The extremal radius depends on the vacuum pressure, which can be determined by ∂M/∂rHrH=r0=0, where(7)∂M∂rH=aerH/a2a3erH/a-1b2+3rH2-2a2rHb2+3rH2-arH2b2+3rH2-rH3b2+rH2b2-2a2erH/a-1+2arH+rH22.It is difficult to express r0 as the function of b since the equation is transcendental, so we solve b with respect to r0 instead:(8)b=r0-3+2r03/a32+2r0/a+r02/a2+r03/a3-2er0/a.Considering the hoop conjecture [3] where the mean radius r¯ is supposed to be not greater than the extremal radius r0, we have estimated the ranges of both b and r0 in [14] through the numerical fitting. It is easy to calculate(9)r¯=1M∫0∞rρr4πr2dr=3a.The hoop conjecture implies r¯≤r0, which ensures the formation of a black hole. Using (8) and the hoop conjecture, one can obtain the exact ranges of the ratios, b/a: [9.996,∞), r0/a: [3,3.384), and M/a: (2.575,2.835], respectively. We note that a smaller b corresponds to an extreme black hole with a smaller radius but a larger mass; that is, when b/a=9.996, we have r0/a=3 and M/a=2.835. The reason is that a smaller b represents the larger vacuum pressure which makes the matter denser [14].


## 2.2. Mass Density Based on a Collapsed Shell

Besides collapsed cores, like the first model, a collapsed shell is also an important model of black hole formation, which is usually dealt with as a massive membrane without thickness. However, if the minimal length is considered, the shell is supposed to be described with a smeared distribution [15]. One can take the following rational fractional function of radial coordinators as the mass density of black holes:(10)ρr=75Ml03r32πl03+5r33,where l0 is a quantity with the length dimension which can be regarded as the minimal length and M the total mass of black holes. The relation is plotted in Figure 3.Figure 3
Plots of the mass density equation (10) (a) and of the mass distribution equation (11) (b) of the collapsed shell.
(a) (b)The mass distribution of this model,M(r), thus has the form(11)Mr=∫0rρr~4πr~2dr~=25Mr6l03+5r32,which, like (2), can also be understood as a kind of step functions with continuity and goes to the step function under the limit l0→0. By solving g00=0, we get the total mass M as the function of the horizon radius rH:(12)M=b2+rH2l03+5rH3250b2rH5,which is plotted in Figure 4. As done for the first model, one can also determine the allowed ranges of the ratios for the extreme configuration in this model: b/l0: [3.24, ∞), r0/l0: [0.943, 1), and M/l0: (0.720, 0.784], respectively, when the hoop conjecture is considered. Moreover, the data lead to the same result as in the first model; that is, a smaller extreme black hole is heavier than a greater one.Figure 4
Plot of the relation ofM with respect to rH. Each vertical line lies at the extremal radius of the curve with the same color.


## 3. Thermodynamics

### 3.1. Thermodynamics of Hydrogen-Atom-Like Black Holes

In this subsection we are going to analyze the thermodynamic properties of the hydrogen-atom-like black holes, especially new phase transitions associated with the minimal length.


#### 3.1.1. Equation of State and Entropy

We start the discussion of the thermodynamics by calculating the Hawking temperature:(13)TH≡κH2π=14π-g00g11dg00drr=r+,where κH is the surface gravity and r+ is the outer horizon radius. Using (3) and (2), we obtain(14)TH=14π1r++3r+b2+r+22a31+r+2b21+r+a+r+22a2-er+/a-1,which is plotted in Figure 5. These curves show one common property that the temperature of the extreme black hole vanishes, which can be strictly proved by substituting (8) into (14). This property means that the evaporation of black holes behaves well; that is, it has no divergency due to the appearance of the minimal length [7]. In addition, the curves display different characteristics when the vacuum pressure increases, which implies various phase transitions that will be discussed in the next subsection. The phase transitions can be classified into four types: (i) for the zero pressure case (b→∞, see, e.g., the green curve of Figure 5), there exists one maximum of temperature, and the temperature approaches zero when r+→∞; (ii) for the relatively low pressure (19.46a<b<∞, see, e.g., the cyan curve of Figure 5), the temperature has one additional minimum following one maximum, and then it increases when r+ increases; (iii) for the critical pressure Pc (b=19.46a, see, e.g., the dashed purple curve of Figure 5), the maximum and minimum merge into one inflexion; and (iv) for the relatively high pressure (9.996a<b<19.46a, see, e.g., the blue curve of Figure 5), the temperature rises smoothly as the horizon radius increases.Figure 5
Plot of the Hawking temperatureTH with respect to r+ under different vacuum pressure. The blue, cyan, and green solid curves correspond to the cases of b/a=10, 30, and ∞, respectively, and the extremal radii in the three cases take r0/a=3.000, 3.325, and 3.384, respectively. The dashed purple curve corresponds to b/a=19.46 and r0/a=3.253, that is, the critical case where the maximum and minimum of temperature coalesce.In fact, (14) is the equation of state and Figure 5 is the diagram of isobar in the temperature-volume plane.3 One can rewrite (14) in a familiar way by using (4),(15)P=-18πr+2+TH2r++r+4/3a31-er+/a+r+/a+r+2/2a2-1-r+4π6a31-er+/a+r+/a+r+2/2a2+r+3,and plot (15) as the diagram of isotherm in the pressure-volume plane shown in Figure 6. The diagram reveals the similarity to that of the van der Waals fluid, implying the occurrence of phase transitions in the range from zero temperature to the critical temperature Tc=0.014/a.Figure 6
Plot of the equation of state. The solid curves correspond to the cases ofaTH=0.018 (blue), 0.012 (cyan), and 0.0096 (green) from top to bottom, respectively. The orange curve corresponds to aTH=0.008, but the dotted part, where the pressure is negative, does not actually exist. The dashed purple curve corresponds to the critical temperature aTc=0.014.A convincing demonstration was given in [17], where the black hole mass M is regarded as the enthalpy. Alternatively, an intuitive explanation was provided in [18]. That is, the negative cosmological constant corresponds to the positive pressure and thus to the negative vacuum energy density ϵ, where ϵ+P=0. Therefore, the total energy is supposed to include the vacuum energy; that is, E=M+ϵV=M-PV, and hence M is the enthalpy H rather than the internal energy of black holes. As a result, the first law associated with the AdS spacetime is modified to be(16)dM=THdS+VdP.It is evident that the second term in (16) vanishes for the fixed cosmological constant or constant pressure. Thus we can derive the entropy from (6) and (14),(17)S=∫r0r+dMTH=∫r0r+2πr1-e-r/a1+r/a+r2/2a2dr,where the integration is performed from r0 in order to guarantee the vanishing entropy at zero temperature [10]. It is hard to integrate (17) analytically but it is much easier to plot it. As shown in Figure 7, for the near-extreme configuration, the entropy has an obvious deviation from the area law, but with the horizon radius increasing, the deviation is getting smaller and smaller and finally approaches a constant. Note that such a constant is varying for different pressure or for different b parameters because the lower bound of the integration r0, that is, the horizon radius of extreme black holes, depends on b parameter; see Figure 7(b).Figure 7
Plots of the entropyS (a) and of ΔS≡S-πr+2 (b) with respect to r+. The red curve in the left diagram corresponds to the entropy πr+2 of the ordinary Schwarzschild-AdS situation; the blue and green curves in both diagrams correspond to the cases of b/a=10 and b=∞, respectively.
(a) (b)


#### 3.1.2. Phase Transition

We discuss the phase transition of the hydrogen-atom-like model in this subsection. The heat capacity at constant pressure takes the form(18)CP≡∂H∂THP=∂M∂THP=∂M∂r+∂TH∂r+-1.It diverges at the extremal points of temperature where its sign changes from CP>0 to CP<0, or vice versa, which leads to phase transitions, as shown in Figure 8. Compared with the ordinary Schwarzschild-AdS black hole with a finite b parameter that has only one first-order phase transition, the self-regular Schwarzschild-AdS black hole has a complicated situation with one second-order, two first-order, and even no phase transitions,4 depending on the values of vacuum pressure; see the bottom left diagram, the top right diagram, and the bottom right diagram, respectively, in Figure 8. Even for the case without the AdS background, that is, b→∞, there is one first-order phase transition in the self-regular black hole, but no phase transition in the ordinary Schwarzschild black hole, see the top left diagram in Figure 8. In particular, one can see that the difference between the ordinary and self-regular black holes just exists in the near-extremal region, and such a difference disappears at a large horizon radius.Figure 8
Plots of the heat capacityCP with respect to r+ under different pressure. They correspond to b/a→∞ (a), b/a=30 (b), b/a=19.46 (the critical vacuum pressure (c)), and b/a=10 (d), respectively. The solid curves correspond to (18), and the dashed red curves correspond to the relationship CP=2πr+2(b2+3r+2)/3r+2-b2 of the ordinary Schwarzschild-AdS situation.
(a) (b) (c) (d)To analyze the phase transitions in detail, we should also investigate the thermodynamic potential. Since we are discussing an isothermal-isobaric ensemble, the thermodynamic potential should be the Gibbs free energy defined as(19)G≡H-THS=M-THS,which is plotted in Figure 9. By differentiating G with respect to r+, one obtains(20)∂G∂r+=-S∂TH∂r+,(21)∂2G∂r+2=-CPTH∂TH∂r+2-S∂2TH∂r+2,which implies that the curve of G(r+) has the same distributions of the extremal points and inflexion points as that of TH(r+) (see Figure 5), apart from the extreme configuration with zero entropy. This property is quite helpful in our following analyses on phase transitions.Figure 9
Plot of the Gibbs free energyG with respect to r+ under different pressure. The green, cyan, and blue solid curves correspond to the different cases of (b/a, G(r0)/a): (∞, 2.575), (30, 2.607), and (10, 2.835), respectively. The dashed purple curve corresponds to the critical vacuum pressure with (19.46, 2.649).If we consider the case without the AdS background (b→∞), we can obtain from the green curve of Figure 5 that there are two phases of black holes with horizon radii r1 and r2,5 the small black hole r1<rmax and the large black hole r2>rmax in the region of temperature 0<T<Tmax, where rmax=6.544a and Tmax=0.0096/a, and no minimum of temperature in this case. For the large black hole with r2, the negative heat capacity, see Figure 8(a), implies that it is unstable. Further, our calculation shows that a large black hole has a higher Gibbs free energy than a small one. For instance, for a small black hole with r1=4.89a, the Gibbs free energy is 2.42a, but for a large black hole with r2=9.66a, the Gibbs free energy equals 2.64a at the temperature T=0.0080/a. Thus, a large black hole would decay into a small one, i.e., the stable configuration with the positive capacity and lower energy.In the case with the relatively low pressure (19.46<b/a<∞), there are three regions divided by two vertical asymptotes in the CP-r+ plane (see, e.g., Figure 8(b)), corresponding to three types of black holes, or three phases, the small one with radius r1<rmax in the temperature range T<Tmax, the medium one with rmax<r2<rmin in the range Tmin<T<Tmax, and the large one r3>rmin in the range T>Tmin. Now we discuss different phases in different ranges of temperature. Due to the rich phase structures, this case is separated into the following three subcases.(i)
Takingb/a=30, for instance, we have Tmax=0.0113/a and Tmin=0.0092/a, and their corresponding radii rmax=7.070a and Tmin=17.31a, respectively. For T<Tmin, there is only a small black hole with r1<4.93a that is at least locally stable because of the positive heat capacity. Similarly, for T>Tmax, only a large locally stable black hole with r3>33.67a exists. As to the temperature range Tmin<T<Tmax, the three types of black holes probably exist. Configurations of the small black hole with r1 and of the large black hole with r3 are at least locally stable and one of them with a lower Gibbs free energy is more stable; when the two configurations have the same Gibbs free energy, they can coexist and convert into each other, corresponding to a first-order phase transition since there is the entropy change during the process. The configuration of the medium black hole with the radius r2 is unstable and probably decays into either the small black hole with r1 or the large black hole with r3, the more stable one with the lower Gibbs free energy.(ii)
When considering the pure thermal radiation with the vanishing Gibbs free energy [19], we can infer that the small black hole with r1 in the temperature range T<Tmax is still locally stable since its Gibbs free energy is positive and that the large black hole with r3 can coexist with the pure thermal radiation at the Hawking-Page temperature THP=0.0107/a. Moreover, if T<THP, the small and large black holes with r1 and r3, respectively, in the range of temperature Tmin<T<THP and the only small black hole with r1 in the range of temperature T<Tmin have positive Gibbs free energy, so the pure thermal radiation is dominant. If T>THP, in the range of THP<T<Tmax, the large black hole with r3 has negative free energy and thus dominates, and in the range of T>Tmax, the single large black hole is globally stable. Note that no black holes, even with positive Gibbs free energy, can decay into the pure thermal radiation because they cannot totally evaporate off, meaning that their radius decreases and finally reaches r0 (the radius of extreme configurations) at which both the temperature and the heat capacity vanish. That is, the extreme black hole stops evaporating and its Gibbs free energy admits a local maximum,6 as argued in [20]. It should be pointed out that the analyses for this and the above subcase only apply to the range of pressure b/a>27.53 which leads to Tmax>THP.(iii)
If the range of pressure is19.46<b/a<27.53 which leads to Tmax<THP, for the three types of black holes in the range of temperature Tmin<T<Tmax, the pure thermal radiation is dominant as all configurations of black holes have positive Gibbs free energy. Moreover, in the range of temperature Tmax<T<THP, the pure thermal radiation is still dominant because the single large black hole with horizon radius r3 has positive Gibbs free energy. At last, if T>THP, the single large black hole is globally stable.For the critical pressure case (b/a=19.46), the two vertical asymptotes in CP-r+ plane merge into one and the second region with negative heat capacity disappears (see Figure 8(c)). Only one locally stable small black hole (r+<rc=9.47a) or one at least locally stable large black hole (r+>rc=9.47a) exists. The phase transition at the inflexion is second order because the entropy is continuous in the two phases. Here are also three situations: (i) if r+<rHP, where rHP=19.87a is the horizon radius when the black hole is at the Hawking-Page temperature, the black hole with positive free energy stays in a locally stable configuration; (ii) if r+=rHP, the black hole with vanishing free energy coexists with the pure thermal radiation; and (iii) if r+>rHP, the black hole with negative free energy is globally stable.If the ensemble is in the relatively high pressure (b/a<19.46), the heat capacity changes continuously with respect to the radius without divergence (see Figure 8(d)). As a result, there is no phase transition between black holes. The analysis about the Hawking-Page phase transition resembles that of the case at the critical pressure.We would like to point out that a recent work by Frassino et al. reveals the phase structure of the holographic metric [21]. This phase structure is similar to that of our model, where the reason will be given in Section 4. In addition, the minimal length is treated as a variable in [20], where the increase of the minimal length leads to the same tendency of phase transitions as that of the decrease of the AdS radius we have just discussed above. We note that such comparability coincides with our analysis in Section 2 where the minimal length is related to the equivalent dS negative pressure; see (5).


### 3.2. Thermodynamics of the Collapsed Shell Model

We at first derive the Hawking temperature of the collapsed shell from (13):(22)TH=5b2r+3-l03+3r+25r+3-l034πb2r+l03+5r+3,which is plotted in Figure 10. Then we calculate the entropy(23)S=πr+2-r02-45πl031r+-1r0-150πl061r+4-1r04,where r0, the horizon radius of the extreme black hole, is the solution of dM/drH=0 (cf. (12)). The first term which equals one-fourth of the horizon area satisfies the area law of the ordinary black hole thermodynamics, and the other terms are quantum corrections which are induced by the minimal length. We notice that Figure 10 presents similar features to that in Figure 5, which implies that the thermodynamic properties of the collapsed shell model resemble that of the hydrogen-atom-like model due to the relations between the temperature and the heat capacity and between the temperature and the Gibbs free energy; see (18) and (21). From the physical point of view, the equilibrium state is described by the equation of state plotted in Figure 10 with isobars in the temperature-horizon plane, which displays the thermodynamic similarities for the two models (cf. Figures 10 and 5). In addition, in order to consider the pure thermal radiation of the AdS background, we have to investigate the Gibbs free energy. To this end, the G-r+ plane is plotted in Figure 11, which shows that the collapsed shell model indeed has similar phase structure to that of the hydrogen-atom-like model; see Figure 9. Thus, the remaining discussions are not necessary to be repeated.Figure 10
Plot of the Hawking temperatureTH with respect to r+ under different vacuum pressure. Particularly, the dashed purple curve corresponds to the case of the critical pressure.Figure 11
Plot of the Gibbs free energyG with respect to r+ under different pressure. Particularly, the dashed purple curve corresponds to the case of the critical pressure.


## 3.1. Thermodynamics of Hydrogen-Atom-Like Black Holes

In this subsection we are going to analyze the thermodynamic properties of the hydrogen-atom-like black holes, especially new phase transitions associated with the minimal length.


### 3.1.1. Equation of State and Entropy

We start the discussion of the thermodynamics by calculating the Hawking temperature:(13)TH≡κH2π=14π-g00g11dg00drr=r+,where κH is the surface gravity and r+ is the outer horizon radius. Using (3) and (2), we obtain(14)TH=14π1r++3r+b2+r+22a31+r+2b21+r+a+r+22a2-er+/a-1,which is plotted in Figure 5. These curves show one common property that the temperature of the extreme black hole vanishes, which can be strictly proved by substituting (8) into (14). This property means that the evaporation of black holes behaves well; that is, it has no divergency due to the appearance of the minimal length [7]. In addition, the curves display different characteristics when the vacuum pressure increases, which implies various phase transitions that will be discussed in the next subsection. The phase transitions can be classified into four types: (i) for the zero pressure case (b→∞, see, e.g., the green curve of Figure 5), there exists one maximum of temperature, and the temperature approaches zero when r+→∞; (ii) for the relatively low pressure (19.46a<b<∞, see, e.g., the cyan curve of Figure 5), the temperature has one additional minimum following one maximum, and then it increases when r+ increases; (iii) for the critical pressure Pc (b=19.46a, see, e.g., the dashed purple curve of Figure 5), the maximum and minimum merge into one inflexion; and (iv) for the relatively high pressure (9.996a<b<19.46a, see, e.g., the blue curve of Figure 5), the temperature rises smoothly as the horizon radius increases.Figure 5
Plot of the Hawking temperatureTH with respect to r+ under different vacuum pressure. The blue, cyan, and green solid curves correspond to the cases of b/a=10, 30, and ∞, respectively, and the extremal radii in the three cases take r0/a=3.000, 3.325, and 3.384, respectively. The dashed purple curve corresponds to b/a=19.46 and r0/a=3.253, that is, the critical case where the maximum and minimum of temperature coalesce.In fact, (14) is the equation of state and Figure 5 is the diagram of isobar in the temperature-volume plane.3 One can rewrite (14) in a familiar way by using (4),(15)P=-18πr+2+TH2r++r+4/3a31-er+/a+r+/a+r+2/2a2-1-r+4π6a31-er+/a+r+/a+r+2/2a2+r+3,and plot (15) as the diagram of isotherm in the pressure-volume plane shown in Figure 6. The diagram reveals the similarity to that of the van der Waals fluid, implying the occurrence of phase transitions in the range from zero temperature to the critical temperature Tc=0.014/a.Figure 6
Plot of the equation of state. The solid curves correspond to the cases ofaTH=0.018 (blue), 0.012 (cyan), and 0.0096 (green) from top to bottom, respectively. The orange curve corresponds to aTH=0.008, but the dotted part, where the pressure is negative, does not actually exist. The dashed purple curve corresponds to the critical temperature aTc=0.014.A convincing demonstration was given in [17], where the black hole mass M is regarded as the enthalpy. Alternatively, an intuitive explanation was provided in [18]. That is, the negative cosmological constant corresponds to the positive pressure and thus to the negative vacuum energy density ϵ, where ϵ+P=0. Therefore, the total energy is supposed to include the vacuum energy; that is, E=M+ϵV=M-PV, and hence M is the enthalpy H rather than the internal energy of black holes. As a result, the first law associated with the AdS spacetime is modified to be(16)dM=THdS+VdP.It is evident that the second term in (16) vanishes for the fixed cosmological constant or constant pressure. Thus we can derive the entropy from (6) and (14),(17)S=∫r0r+dMTH=∫r0r+2πr1-e-r/a1+r/a+r2/2a2dr,where the integration is performed from r0 in order to guarantee the vanishing entropy at zero temperature [10]. It is hard to integrate (17) analytically but it is much easier to plot it. As shown in Figure 7, for the near-extreme configuration, the entropy has an obvious deviation from the area law, but with the horizon radius increasing, the deviation is getting smaller and smaller and finally approaches a constant. Note that such a constant is varying for different pressure or for different b parameters because the lower bound of the integration r0, that is, the horizon radius of extreme black holes, depends on b parameter; see Figure 7(b).Figure 7
Plots of the entropyS (a) and of ΔS≡S-πr+2 (b) with respect to r+. The red curve in the left diagram corresponds to the entropy πr+2 of the ordinary Schwarzschild-AdS situation; the blue and green curves in both diagrams correspond to the cases of b/a=10 and b=∞, respectively.
(a) (b)


### 3.1.2. Phase Transition

We discuss the phase transition of the hydrogen-atom-like model in this subsection. The heat capacity at constant pressure takes the form(18)CP≡∂H∂THP=∂M∂THP=∂M∂r+∂TH∂r+-1.It diverges at the extremal points of temperature where its sign changes from CP>0 to CP<0, or vice versa, which leads to phase transitions, as shown in Figure 8. Compared with the ordinary Schwarzschild-AdS black hole with a finite b parameter that has only one first-order phase transition, the self-regular Schwarzschild-AdS black hole has a complicated situation with one second-order, two first-order, and even no phase transitions,4 depending on the values of vacuum pressure; see the bottom left diagram, the top right diagram, and the bottom right diagram, respectively, in Figure 8. Even for the case without the AdS background, that is, b→∞, there is one first-order phase transition in the self-regular black hole, but no phase transition in the ordinary Schwarzschild black hole, see the top left diagram in Figure 8. In particular, one can see that the difference between the ordinary and self-regular black holes just exists in the near-extremal region, and such a difference disappears at a large horizon radius.Figure 8
Plots of the heat capacityCP with respect to r+ under different pressure. They correspond to b/a→∞ (a), b/a=30 (b), b/a=19.46 (the critical vacuum pressure (c)), and b/a=10 (d), respectively. The solid curves correspond to (18), and the dashed red curves correspond to the relationship CP=2πr+2(b2+3r+2)/3r+2-b2 of the ordinary Schwarzschild-AdS situation.
(a) (b) (c) (d)To analyze the phase transitions in detail, we should also investigate the thermodynamic potential. Since we are discussing an isothermal-isobaric ensemble, the thermodynamic potential should be the Gibbs free energy defined as(19)G≡H-THS=M-THS,which is plotted in Figure 9. By differentiating G with respect to r+, one obtains(20)∂G∂r+=-S∂TH∂r+,(21)∂2G∂r+2=-CPTH∂TH∂r+2-S∂2TH∂r+2,which implies that the curve of G(r+) has the same distributions of the extremal points and inflexion points as that of TH(r+) (see Figure 5), apart from the extreme configuration with zero entropy. This property is quite helpful in our following analyses on phase transitions.Figure 9
Plot of the Gibbs free energyG with respect to r+ under different pressure. The green, cyan, and blue solid curves correspond to the different cases of (b/a, G(r0)/a): (∞, 2.575), (30, 2.607), and (10, 2.835), respectively. The dashed purple curve corresponds to the critical vacuum pressure with (19.46, 2.649).If we consider the case without the AdS background (b→∞), we can obtain from the green curve of Figure 5 that there are two phases of black holes with horizon radii r1 and r2,5 the small black hole r1<rmax and the large black hole r2>rmax in the region of temperature 0<T<Tmax, where rmax=6.544a and Tmax=0.0096/a, and no minimum of temperature in this case. For the large black hole with r2, the negative heat capacity, see Figure 8(a), implies that it is unstable. Further, our calculation shows that a large black hole has a higher Gibbs free energy than a small one. For instance, for a small black hole with r1=4.89a, the Gibbs free energy is 2.42a, but for a large black hole with r2=9.66a, the Gibbs free energy equals 2.64a at the temperature T=0.0080/a. Thus, a large black hole would decay into a small one, i.e., the stable configuration with the positive capacity and lower energy.In the case with the relatively low pressure (19.46<b/a<∞), there are three regions divided by two vertical asymptotes in the CP-r+ plane (see, e.g., Figure 8(b)), corresponding to three types of black holes, or three phases, the small one with radius r1<rmax in the temperature range T<Tmax, the medium one with rmax<r2<rmin in the range Tmin<T<Tmax, and the large one r3>rmin in the range T>Tmin. Now we discuss different phases in different ranges of temperature. Due to the rich phase structures, this case is separated into the following three subcases.(i)
Takingb/a=30, for instance, we have Tmax=0.0113/a and Tmin=0.0092/a, and their corresponding radii rmax=7.070a and Tmin=17.31a, respectively. For T<Tmin, there is only a small black hole with r1<4.93a that is at least locally stable because of the positive heat capacity. Similarly, for T>Tmax, only a large locally stable black hole with r3>33.67a exists. As to the temperature range Tmin<T<Tmax, the three types of black holes probably exist. Configurations of the small black hole with r1 and of the large black hole with r3 are at least locally stable and one of them with a lower Gibbs free energy is more stable; when the two configurations have the same Gibbs free energy, they can coexist and convert into each other, corresponding to a first-order phase transition since there is the entropy change during the process. The configuration of the medium black hole with the radius r2 is unstable and probably decays into either the small black hole with r1 or the large black hole with r3, the more stable one with the lower Gibbs free energy.(ii)
When considering the pure thermal radiation with the vanishing Gibbs free energy [19], we can infer that the small black hole with r1 in the temperature range T<Tmax is still locally stable since its Gibbs free energy is positive and that the large black hole with r3 can coexist with the pure thermal radiation at the Hawking-Page temperature THP=0.0107/a. Moreover, if T<THP, the small and large black holes with r1 and r3, respectively, in the range of temperature Tmin<T<THP and the only small black hole with r1 in the range of temperature T<Tmin have positive Gibbs free energy, so the pure thermal radiation is dominant. If T>THP, in the range of THP<T<Tmax, the large black hole with r3 has negative free energy and thus dominates, and in the range of T>Tmax, the single large black hole is globally stable. Note that no black holes, even with positive Gibbs free energy, can decay into the pure thermal radiation because they cannot totally evaporate off, meaning that their radius decreases and finally reaches r0 (the radius of extreme configurations) at which both the temperature and the heat capacity vanish. That is, the extreme black hole stops evaporating and its Gibbs free energy admits a local maximum,6 as argued in [20]. It should be pointed out that the analyses for this and the above subcase only apply to the range of pressure b/a>27.53 which leads to Tmax>THP.(iii)
If the range of pressure is19.46<b/a<27.53 which leads to Tmax<THP, for the three types of black holes in the range of temperature Tmin<T<Tmax, the pure thermal radiation is dominant as all configurations of black holes have positive Gibbs free energy. Moreover, in the range of temperature Tmax<T<THP, the pure thermal radiation is still dominant because the single large black hole with horizon radius r3 has positive Gibbs free energy. At last, if T>THP, the single large black hole is globally stable.For the critical pressure case (b/a=19.46), the two vertical asymptotes in CP-r+ plane merge into one and the second region with negative heat capacity disappears (see Figure 8(c)). Only one locally stable small black hole (r+<rc=9.47a) or one at least locally stable large black hole (r+>rc=9.47a) exists. The phase transition at the inflexion is second order because the entropy is continuous in the two phases. Here are also three situations: (i) if r+<rHP, where rHP=19.87a is the horizon radius when the black hole is at the Hawking-Page temperature, the black hole with positive free energy stays in a locally stable configuration; (ii) if r+=rHP, the black hole with vanishing free energy coexists with the pure thermal radiation; and (iii) if r+>rHP, the black hole with negative free energy is globally stable.If the ensemble is in the relatively high pressure (b/a<19.46), the heat capacity changes continuously with respect to the radius without divergence (see Figure 8(d)). As a result, there is no phase transition between black holes. The analysis about the Hawking-Page phase transition resembles that of the case at the critical pressure.We would like to point out that a recent work by Frassino et al. reveals the phase structure of the holographic metric [21]. This phase structure is similar to that of our model, where the reason will be given in Section 4. In addition, the minimal length is treated as a variable in [20], where the increase of the minimal length leads to the same tendency of phase transitions as that of the decrease of the AdS radius we have just discussed above. We note that such comparability coincides with our analysis in Section 2 where the minimal length is related to the equivalent dS negative pressure; see (5).


## 3.1.1. Equation of State and Entropy

We start the discussion of the thermodynamics by calculating the Hawking temperature:(13)TH≡κH2π=14π-g00g11dg00drr=r+,where κH is the surface gravity and r+ is the outer horizon radius. Using (3) and (2), we obtain(14)TH=14π1r++3r+b2+r+22a31+r+2b21+r+a+r+22a2-er+/a-1,which is plotted in Figure 5. These curves show one common property that the temperature of the extreme black hole vanishes, which can be strictly proved by substituting (8) into (14). This property means that the evaporation of black holes behaves well; that is, it has no divergency due to the appearance of the minimal length [7]. In addition, the curves display different characteristics when the vacuum pressure increases, which implies various phase transitions that will be discussed in the next subsection. The phase transitions can be classified into four types: (i) for the zero pressure case (b→∞, see, e.g., the green curve of Figure 5), there exists one maximum of temperature, and the temperature approaches zero when r+→∞; (ii) for the relatively low pressure (19.46a<b<∞, see, e.g., the cyan curve of Figure 5), the temperature has one additional minimum following one maximum, and then it increases when r+ increases; (iii) for the critical pressure Pc (b=19.46a, see, e.g., the dashed purple curve of Figure 5), the maximum and minimum merge into one inflexion; and (iv) for the relatively high pressure (9.996a<b<19.46a, see, e.g., the blue curve of Figure 5), the temperature rises smoothly as the horizon radius increases.Figure 5
Plot of the Hawking temperatureTH with respect to r+ under different vacuum pressure. The blue, cyan, and green solid curves correspond to the cases of b/a=10, 30, and ∞, respectively, and the extremal radii in the three cases take r0/a=3.000, 3.325, and 3.384, respectively. The dashed purple curve corresponds to b/a=19.46 and r0/a=3.253, that is, the critical case where the maximum and minimum of temperature coalesce.In fact, (14) is the equation of state and Figure 5 is the diagram of isobar in the temperature-volume plane.3 One can rewrite (14) in a familiar way by using (4),(15)P=-18πr+2+TH2r++r+4/3a31-er+/a+r+/a+r+2/2a2-1-r+4π6a31-er+/a+r+/a+r+2/2a2+r+3,and plot (15) as the diagram of isotherm in the pressure-volume plane shown in Figure 6. The diagram reveals the similarity to that of the van der Waals fluid, implying the occurrence of phase transitions in the range from zero temperature to the critical temperature Tc=0.014/a.Figure 6
Plot of the equation of state. The solid curves correspond to the cases ofaTH=0.018 (blue), 0.012 (cyan), and 0.0096 (green) from top to bottom, respectively. The orange curve corresponds to aTH=0.008, but the dotted part, where the pressure is negative, does not actually exist. The dashed purple curve corresponds to the critical temperature aTc=0.014.A convincing demonstration was given in [17], where the black hole mass M is regarded as the enthalpy. Alternatively, an intuitive explanation was provided in [18]. That is, the negative cosmological constant corresponds to the positive pressure and thus to the negative vacuum energy density ϵ, where ϵ+P=0. Therefore, the total energy is supposed to include the vacuum energy; that is, E=M+ϵV=M-PV, and hence M is the enthalpy H rather than the internal energy of black holes. As a result, the first law associated with the AdS spacetime is modified to be(16)dM=THdS+VdP.It is evident that the second term in (16) vanishes for the fixed cosmological constant or constant pressure. Thus we can derive the entropy from (6) and (14),(17)S=∫r0r+dMTH=∫r0r+2πr1-e-r/a1+r/a+r2/2a2dr,where the integration is performed from r0 in order to guarantee the vanishing entropy at zero temperature [10]. It is hard to integrate (17) analytically but it is much easier to plot it. As shown in Figure 7, for the near-extreme configuration, the entropy has an obvious deviation from the area law, but with the horizon radius increasing, the deviation is getting smaller and smaller and finally approaches a constant. Note that such a constant is varying for different pressure or for different b parameters because the lower bound of the integration r0, that is, the horizon radius of extreme black holes, depends on b parameter; see Figure 7(b).Figure 7
Plots of the entropyS (a) and of ΔS≡S-πr+2 (b) with respect to r+. The red curve in the left diagram corresponds to the entropy πr+2 of the ordinary Schwarzschild-AdS situation; the blue and green curves in both diagrams correspond to the cases of b/a=10 and b=∞, respectively.
(a) (b)


## 3.1.2. Phase Transition

We discuss the phase transition of the hydrogen-atom-like model in this subsection. The heat capacity at constant pressure takes the form(18)CP≡∂H∂THP=∂M∂THP=∂M∂r+∂TH∂r+-1.It diverges at the extremal points of temperature where its sign changes from CP>0 to CP<0, or vice versa, which leads to phase transitions, as shown in Figure 8. Compared with the ordinary Schwarzschild-AdS black hole with a finite b parameter that has only one first-order phase transition, the self-regular Schwarzschild-AdS black hole has a complicated situation with one second-order, two first-order, and even no phase transitions,4 depending on the values of vacuum pressure; see the bottom left diagram, the top right diagram, and the bottom right diagram, respectively, in Figure 8. Even for the case without the AdS background, that is, b→∞, there is one first-order phase transition in the self-regular black hole, but no phase transition in the ordinary Schwarzschild black hole, see the top left diagram in Figure 8. In particular, one can see that the difference between the ordinary and self-regular black holes just exists in the near-extremal region, and such a difference disappears at a large horizon radius.Figure 8
Plots of the heat capacityCP with respect to r+ under different pressure. They correspond to b/a→∞ (a), b/a=30 (b), b/a=19.46 (the critical vacuum pressure (c)), and b/a=10 (d), respectively. The solid curves correspond to (18), and the dashed red curves correspond to the relationship CP=2πr+2(b2+3r+2)/3r+2-b2 of the ordinary Schwarzschild-AdS situation.
(a) (b) (c) (d)To analyze the phase transitions in detail, we should also investigate the thermodynamic potential. Since we are discussing an isothermal-isobaric ensemble, the thermodynamic potential should be the Gibbs free energy defined as(19)G≡H-THS=M-THS,which is plotted in Figure 9. By differentiating G with respect to r+, one obtains(20)∂G∂r+=-S∂TH∂r+,(21)∂2G∂r+2=-CPTH∂TH∂r+2-S∂2TH∂r+2,which implies that the curve of G(r+) has the same distributions of the extremal points and inflexion points as that of TH(r+) (see Figure 5), apart from the extreme configuration with zero entropy. This property is quite helpful in our following analyses on phase transitions.Figure 9
Plot of the Gibbs free energyG with respect to r+ under different pressure. The green, cyan, and blue solid curves correspond to the different cases of (b/a, G(r0)/a): (∞, 2.575), (30, 2.607), and (10, 2.835), respectively. The dashed purple curve corresponds to the critical vacuum pressure with (19.46, 2.649).If we consider the case without the AdS background (b→∞), we can obtain from the green curve of Figure 5 that there are two phases of black holes with horizon radii r1 and r2,5 the small black hole r1<rmax and the large black hole r2>rmax in the region of temperature 0<T<Tmax, where rmax=6.544a and Tmax=0.0096/a, and no minimum of temperature in this case. For the large black hole with r2, the negative heat capacity, see Figure 8(a), implies that it is unstable. Further, our calculation shows that a large black hole has a higher Gibbs free energy than a small one. For instance, for a small black hole with r1=4.89a, the Gibbs free energy is 2.42a, but for a large black hole with r2=9.66a, the Gibbs free energy equals 2.64a at the temperature T=0.0080/a. Thus, a large black hole would decay into a small one, i.e., the stable configuration with the positive capacity and lower energy.In the case with the relatively low pressure (19.46<b/a<∞), there are three regions divided by two vertical asymptotes in the CP-r+ plane (see, e.g., Figure 8(b)), corresponding to three types of black holes, or three phases, the small one with radius r1<rmax in the temperature range T<Tmax, the medium one with rmax<r2<rmin in the range Tmin<T<Tmax, and the large one r3>rmin in the range T>Tmin. Now we discuss different phases in different ranges of temperature. Due to the rich phase structures, this case is separated into the following three subcases.(i)
Takingb/a=30, for instance, we have Tmax=0.0113/a and Tmin=0.0092/a, and their corresponding radii rmax=7.070a and Tmin=17.31a, respectively. For T<Tmin, there is only a small black hole with r1<4.93a that is at least locally stable because of the positive heat capacity. Similarly, for T>Tmax, only a large locally stable black hole with r3>33.67a exists. As to the temperature range Tmin<T<Tmax, the three types of black holes probably exist. Configurations of the small black hole with r1 and of the large black hole with r3 are at least locally stable and one of them with a lower Gibbs free energy is more stable; when the two configurations have the same Gibbs free energy, they can coexist and convert into each other, corresponding to a first-order phase transition since there is the entropy change during the process. The configuration of the medium black hole with the radius r2 is unstable and probably decays into either the small black hole with r1 or the large black hole with r3, the more stable one with the lower Gibbs free energy.(ii)
When considering the pure thermal radiation with the vanishing Gibbs free energy [19], we can infer that the small black hole with r1 in the temperature range T<Tmax is still locally stable since its Gibbs free energy is positive and that the large black hole with r3 can coexist with the pure thermal radiation at the Hawking-Page temperature THP=0.0107/a. Moreover, if T<THP, the small and large black holes with r1 and r3, respectively, in the range of temperature Tmin<T<THP and the only small black hole with r1 in the range of temperature T<Tmin have positive Gibbs free energy, so the pure thermal radiation is dominant. If T>THP, in the range of THP<T<Tmax, the large black hole with r3 has negative free energy and thus dominates, and in the range of T>Tmax, the single large black hole is globally stable. Note that no black holes, even with positive Gibbs free energy, can decay into the pure thermal radiation because they cannot totally evaporate off, meaning that their radius decreases and finally reaches r0 (the radius of extreme configurations) at which both the temperature and the heat capacity vanish. That is, the extreme black hole stops evaporating and its Gibbs free energy admits a local maximum,6 as argued in [20]. It should be pointed out that the analyses for this and the above subcase only apply to the range of pressure b/a>27.53 which leads to Tmax>THP.(iii)
If the range of pressure is19.46<b/a<27.53 which leads to Tmax<THP, for the three types of black holes in the range of temperature Tmin<T<Tmax, the pure thermal radiation is dominant as all configurations of black holes have positive Gibbs free energy. Moreover, in the range of temperature Tmax<T<THP, the pure thermal radiation is still dominant because the single large black hole with horizon radius r3 has positive Gibbs free energy. At last, if T>THP, the single large black hole is globally stable.For the critical pressure case (b/a=19.46), the two vertical asymptotes in CP-r+ plane merge into one and the second region with negative heat capacity disappears (see Figure 8(c)). Only one locally stable small black hole (r+<rc=9.47a) or one at least locally stable large black hole (r+>rc=9.47a) exists. The phase transition at the inflexion is second order because the entropy is continuous in the two phases. Here are also three situations: (i) if r+<rHP, where rHP=19.87a is the horizon radius when the black hole is at the Hawking-Page temperature, the black hole with positive free energy stays in a locally stable configuration; (ii) if r+=rHP, the black hole with vanishing free energy coexists with the pure thermal radiation; and (iii) if r+>rHP, the black hole with negative free energy is globally stable.If the ensemble is in the relatively high pressure (b/a<19.46), the heat capacity changes continuously with respect to the radius without divergence (see Figure 8(d)). As a result, there is no phase transition between black holes. The analysis about the Hawking-Page phase transition resembles that of the case at the critical pressure.We would like to point out that a recent work by Frassino et al. reveals the phase structure of the holographic metric [21]. This phase structure is similar to that of our model, where the reason will be given in Section 4. In addition, the minimal length is treated as a variable in [20], where the increase of the minimal length leads to the same tendency of phase transitions as that of the decrease of the AdS radius we have just discussed above. We note that such comparability coincides with our analysis in Section 2 where the minimal length is related to the equivalent dS negative pressure; see (5).


## 3.2. Thermodynamics of the Collapsed Shell Model

We at first derive the Hawking temperature of the collapsed shell from (13):(22)TH=5b2r+3-l03+3r+25r+3-l034πb2r+l03+5r+3,which is plotted in Figure 10. Then we calculate the entropy(23)S=πr+2-r02-45πl031r+-1r0-150πl061r+4-1r04,where r0, the horizon radius of the extreme black hole, is the solution of dM/drH=0 (cf. (12)). The first term which equals one-fourth of the horizon area satisfies the area law of the ordinary black hole thermodynamics, and the other terms are quantum corrections which are induced by the minimal length. We notice that Figure 10 presents similar features to that in Figure 5, which implies that the thermodynamic properties of the collapsed shell model resemble that of the hydrogen-atom-like model due to the relations between the temperature and the heat capacity and between the temperature and the Gibbs free energy; see (18) and (21). From the physical point of view, the equilibrium state is described by the equation of state plotted in Figure 10 with isobars in the temperature-horizon plane, which displays the thermodynamic similarities for the two models (cf. Figures 10 and 5). In addition, in order to consider the pure thermal radiation of the AdS background, we have to investigate the Gibbs free energy. To this end, the G-r+ plane is plotted in Figure 11, which shows that the collapsed shell model indeed has similar phase structure to that of the hydrogen-atom-like model; see Figure 9. Thus, the remaining discussions are not necessary to be repeated.Figure 10
Plot of the Hawking temperatureTH with respect to r+ under different vacuum pressure. Particularly, the dashed purple curve corresponds to the case of the critical pressure.Figure 11
Plot of the Gibbs free energyG with respect to r+ under different pressure. Particularly, the dashed purple curve corresponds to the case of the critical pressure.


## 4. General Analysis

Our discussions in the above two sections are based on two specific models which show the similar thermodynamic properties to those of the other models constructed in [10, 15, 16], where those mass distributions can also be understood as a kind of step functions with continuity. In this section, we demonstrate that the models with such a mass distribution have similar thermodynamic properties in a general way.When the mass density of black holes takes some analytic form of theδ-function with the minimal length as a parameter, the metric (3) can be extended to a general self-regular Schwarzschild-AdS solution of the modified Einstein equations [7]:(24)ds2=-1-2frMr+r2b2dt2+1-2frMr+r2b2-1dr2+r2dΩ2,where f(r), the original M(r)/M in (3), is unfixed but is supposed to be a kind of step functions with continuity; see the red curve in Figure 12 for illustration. Specifically, as ρ(r)=(M/4πr2)(df/dr) is finite everywhere, f(r) is a third-order infinitesimal with respect to r for the “collapsed core” (see, e.g., our first model) or a higher-order one for the “collapsed shell” (see, e.g., our second model) near the origin in order to get rid of the singularity of the metric, and it approaches one at a large r.7 The relationship between the total mass M and the horizon radius rH can be obtained as usual:(25)M=rH2frH1+rH2b2.Since M approaches the positive infinity both at rH→0 and at rH→∞, there is at least one minimum point. Based on the equation,(26)dMdrHrH=r0=frHb2+3rH2-f′rHrHb2+rH22frH2b2rH=r0=0,we obtain the extremal point r0 satisfying f(r0)=f′(r0)r01-2r02/b2+3r02. Figure 12 illustrates that there is only one intersection for f(rH) and f′(rH)rH1-2rH2/b2+3rH2, and thus there exists only one extreme black hole with the minimal mass and the minimal radius.Figure 12
Sketches off(r) (red), f′(r)r (purple), and f′(r)r1-2r2/(b2+3r2) (green). The curve of f′(r), the dashed blue curve with f′′(r∗)=0, is drawn for reference.The existence of extreme black holes for the metric solution equation (24) can be shown by the following analysis. For the case without the AdS background (i.e., b→∞), we obtain8dM/dr=fr-f′rr/2f(r)2. The sketches of f(r) and f′(r)r are drawn in Figure 12. For a small r, as f(r) is at least a third-order infinitesimal with respect to r, we obtain f′(r)r>fr-f0/rr=f(r); that is, the curve of f′(r)r is over that of f(r). Next, in order to analyze the increasing tendency of the two functions, we calculate their derivatives, f′rr′=f′′(r)r+f′(r) and f′(r), and deduce f′rr′>f′(r) if r<r∗; see the caption of Figure 12 for the definition of r∗. As a result, only if r>r∗ can one obtain f′rr′<f′(r), which leads to intersection of the two curves. After the two curves intersect at the horizon radius9r=1, f′′(r)<0 renders f′rr′<f′(r), which implies that r=1 is the single point where f(r)-f′(r)r vanishes, that is, the only one extreme configuration of black holes. Moreover, for the case with the AdS background, that is, the parameter b is finite and satisfies the hoop conjecture, the intersection of f(r) and f′(r)r1-2r2/b2+3r2 will have a shift to the left if compared with the case without the AdS background; that is, the intersecting point is smaller than 1 because f′(r)r1-2r2/b2+3r2 is smaller than f′(r)r.The thermodynamic properties of black holes merely depend on the equation of state, so we just need to analyze the relationship between the Hawking temperatureTH and the specific volume or simply between the Hawking temperature and the outer horizon radius r+; see Endnote 3. From (13), we calculate the temperature,(27)TH=14πfr+b2+3r+2-f′r+b2r++r+3fr+b2r+=fr+2πr+dMdr+,which vanishes at the extremal radius r0, that is, the solution of dM/dr+=0, and is always positive at other radii.


### 4.1. Features of Temperature of Black Holes without the AdS Background

In the absence of the AdS background (b→∞), the Hawking temperature and its derivative with respect to r+ take the forms(28)TH=14π1r+-f′r+fr+,(29)dTHdr+=-14πfr+2-f′r+2r+2+f′′r+fr+r+2fr+2r+2,and the temperature vanishes at r+→∞. As to the derivative of the temperature, it is positive due to fr0=f′(r0)r0 and f′′(r0)<0 at the extremal radius r0 and close to zero at infinity. As a result, there is at least one local maximum temperature T1 whose corresponding radius r1 satisfies the condition from the vanishing of (29) at r+=r1:(30)f′′r1fr1r12-f′r12r12+fr12=0.In the following we prove that T1 is the global maximum and thus indicate the similar TH-r+ relation to that of the above two models depicted by Figures 5 and 10, which ensures that the thermodynamic properties of the general case are similar to that of the other specific models mentioned.Set functiong(r) satisfying the following second-order differential equation,(31)g′′rgrr2-g′r2r2+gr2=0,we have a general solution, gr=c1rexp⁡(c2r), where c1 and c2 are two parameters to be determined. By choosing a positive parameter c1=fr1/r1exp⁡-f′r1r1+fr1 and a negative parameter c2=f′r1r1-fr1/r1, we obtain the particular solution that ensures g(r1)=f(r1) and g′(r1)=f′(r1). Again using g(r1)=f(r1), g′(r1)=f′(r1) and (30) and (31), we deduce g′′(r1)=f′′(r1). Since it is difficult to depict the equality of the second derivatives, g′′(r1)=f′′(r1), on the plot of curves of f(r) and g(r), we choose to draw the curves of f′(r) and g′(r) instead; see Figure 13. One can infer that the two curves are tangent at r1 due to g′(r1)=f′(r1) and g′′(r1)=f′′(r1). In addition, we note that as a result of g(r)=∫0rg′r′dr′+g(0) and g(0)=0, g(r) is equal to the integral area surrounded by the curve g′(r) and the transverse axis from the origin to r, and the same analysis is applicable for f(r). Consequently, g(r1)=f(r1) indicates that the integral areas surrounded by the curve g′(r) and the transverse interval [0,r1] and by the curve f′(r) and the transverse interval [0,r1] are the same; see the two solid curves in Figure 13. That is to say, the equality of the two integral areas implies the existence of the maximum of temperature in the TH-r+ plane.Figure 13
Sketches off′(r) (blue) and g′(r)=c11+c2rexp⁡(c2r) (purple). The dashed red curve and the dashed green curve represent g~′(r)=c~11+c~2rexp⁡(c~2r) with c~1>c1 and c~1<c1, respectively.Assuming that there still exists one local minimum of temperatureT~1 at r~1 on the curve of TH that meets f′′r~1fr~1r~12-f′(r~1)2r~12+f(r~1)2=0, we can certainly find out another particular solution g~r=c~1rexp⁡(c~2r) with c~1=fr~1/r~1exp⁡-f′r~1r~1+fr~1 and c~2=f′r~1r~1-fr~1/r~1 to the differential equation g~′′(r)g~(r)r2-g~′(r)2r2+g~(r)2=0. This particular solution satisfies g~(r~1)=f(r~1), g~′(r~1)=f′(r~1), and g~′′(r~1)=f′′(r~1). Using (29), we have the maximum of temperature at r1:(32)THr1=14π1r1-f′r1fr1=14π1r1-g′r1gr1=-14πc2>0,which means that TH(r1) is only related to the negative parameter c2. Considering TH(r1)>TH(r~1), one can infer that c~2>c2. Next we compare g′(r) with g~′(r)=c~11+c~2rexp⁡(c~2r) by considering the following two cases in order to determine that the local minimum of temperature T~1 conjectured above does not exist.(i)
For the casec~1>c1, we can get g~′(r)>g′(r), which means that the curve of g~′(r) is always above that of g′(r). So the integral area surrounded by g~′(r) and the interval [0,r~1] is not the same as that surrounded by f′(r) and the interval [0,r~1]. In fact, we always have g~(r~1)>f(r~1) in this case; see, for instance, the sketches in Figure 13.(ii)
For the casec~1<c1, we can first get g~′(0)=c~1<c1=g′(0); then if the two curves of g~′(r) and f′(r) are tangent at r~1, we can infer g~(r~1)=∫0r~1g~′(r)dr<∫0r~1f′rdr=f(r~1), which also leads to the inequality of the integral areas; see the sketches in Figure 13.In a word, no suitable parametersc~1 and c~2 can be found to simultaneously meet g~(r~1)=f(r~1), g~′(r~1)=f′(r~1), and g~′′(r~1)=f′′(r~1), or in other words, to meet TH′(r~1)=0. As a result, the curve of TH without the AdS background has one and only one extremal point, that is, the maximum point, which certainly leads to the conclusion that the TH-r+ relation in this case is similar to the green curves of Figures 5 and 10.


### 4.2. Features of Temperature of Black Holes with the AdS Background

For the case with the AdS background, we rewrite the Hawking temperature (27) and calculate the corresponding derivative as follows:(33)TH=14πfr+-f′r+r+fr+r++3fr+r+-f′r+r+2fr+b2,(34)dTHdr+=14π-fr+2-f′r+2r+2+f′′r+fr+r+2fr+2r+2+3fr+2+f′r+2r+2-2fr+f′r+r+-fr+f′′r+r+2b2fr+2.The Hawking temperature is still zero for extreme black holes, but because of the finiteb parameter, it grows to infinity as the horizon radius increases. As to the derivative of the temperature, TH′, it is positive according to (27) at the extremal radius r0 and also positive at infinity because it goes to 3/(4πb2). For a relatively large b, the zero point of TH′, where the main part of TH′ is the first term of (34), will slightly shift to the right compared with the case of b→∞, and TH′ can also be negative at some radius. As a result, there exist at least one maximum and one following minimum on the curve of TH when the horizon radius increases. In this case, the two points can be proven to be the only pair of extremal points of TH where the numerator of TH′ vanishes. In fact, to the differential equation,10(35)hrh′′rr2b2+r2-h′r2r2b2+r2+2hrh′rr3-hr23r2-b2=0,we can find a general solution,(36)hr=λrr2+b2exp⁡βbtan-1⁡rb,where λ and β are two parameters to be determined, and give its derivative as follows:(37)h′r=λ3r2+βr+b2exp⁡βbtan-1⁡rb.We can obtain two particular solutions,h1(r)=λ1rr2+b2exp⁡β1/btan-1⁡(r/b) and h2(r)=λ2rr2+b2exp⁡β2/btan-1⁡r/b, where the two pairs of parameters take the forms11(38)λ1=frIrIrI2+b2exp⁡-β1btan-1⁡rIb>0,β1=f′rIrIrI2+b2-frI3rI2+b2frIrI<0;λ2=frIIrIIrII2+b2exp⁡-β2btan-1⁡rIIb>0,β2=f′rIIrIIrII2+b2-frII3rII2+b2frIIrII<0,and rI and rII are the maximum and minimum points, respectively, and meet the inequality rI<rII. The particular solutions satisfy the conditions, h1(rI)=f(rI), h1′(rI)=f′rI, and h1′′(rI)=f′′(rI); h2(rII)=f(rII), h2′(rII)=f′(rII), and h2′′(rII)=f′′rII. As a result, the curve of f′(r) is tangent to that of h1′(r) at rI and also tangent to that of h2′(r) at rII. Again considering h(r)=∫0rh′(r~)dr~ and h(0)=0, we deduce that the integral area surrounded by the curve h1′(r) and the transverse interval [0,rI] equals that by the curve f′(r) and the transverse interval [0,rI], and the same happens for the integral areas surrounded by the curve h2′(r) and transverse interval [0,rII] and by the curve f′(r) and the transverse interval [0,rII].Using (33), we have the temperature at the extremal (maximum and minimum) points re,(39)THre=14πhre-h′rerehrere+3hrere-h′rere2hreb2=-β4πb2,which is only related to the negative parameter β. Due to TH(rI)>TH(rII), we obtain β1<β2. As to the relationship between λ1 and λ2, if λ1<λ2 is taken, the curve of h1(r) is always below that of h2(r), which cannot lead to h1(rI)=f(rI) and h2(rII)=f(rII) simultaneously. As a result, we get λ1>λ2.If another maximum pointr~I following rII (r~I>rII) is conjectured on the curve of TH, one can seek out the corresponding particular solution, h~(r)=λ~rr2+b2exp⁡β~/btan-1⁡r/b, where the parameters take the forms(40)λ~=fr~Ir~Ir~I2+b2exp⁡-β~1btan-1⁡r~Ib>0,β~=f′r~Ir~I3+b2r~I-fr~I3r~I2+b2fr~Ir~I<0,and meet the conditions h~(r~I)=f(r~I), h~′(r~I)=f′(r~I) and h~′′(r~I)=f′′(r~I). From TH(r~I)>TH(rII), we deduce that β~<β2. With this constraint and the assumption λ~>λ2 together we first consider the following four different situations which are drawn in Figures 14 and 15.(i)
In the range of parametersβ1<β~<β2 and λ2<λ~<λ1, h~′(0)=λ~b2 is between h1′(0)=λ1b2 and h2′(0)=λ2b2, so the point of tangency r~I, if it exists, should be between rI and rII in order to make the integral area surrounded by h~(r~I) and [0,r~I] be equal to the integral area surrounded by f(r~I) and [0,r~I], rather than being larger than rII as supposed; see Figure 14(a).(ii)
In the range of parametersβ1<β~<β2 and λ2<λ1<λ~, the curve of h~′(r) lies above that of h1′(r), so there is no r~I to satisfy the equality of integral areas about h~′(r) and f′(r); see Figure 14(b).(iii)
In the range of parametersβ~<β1<β2 and λ2<λ~<λ1, the curve of h~′(r) lies below that of h1′(r), so we find no r~I to satisfy the equality of integral areas about h~′(r) and f′(r); see Figure 15(a).(iv)
In the range of parametersβ~<β1<β2 and λ2<λ1<λ~, since h~′(0)=λ~ is larger than h1′(0)=λ1, in order to ensure the equality of the integral areas, the curve of h~′(r) must intersect with h1′(r) before h1′(r) is tangent to f′(r), which makes the point of tangency of h~′(r) and f′(r) smaller than rI or even nonexistent; see Figure 15(b).Figure 14
Sketches off′(r) (blue), h1′(r)=λ13r2+β1r+b2exp⁡β1/btan-1⁡r/b (green), and h2′(r)=λ23r2+β2r+b2exp⁡β2/btan-1r/b (red). The dashed purple (a) and dashed pink (b) curves represent h~′(r)=λ~3r2+β~r+b2exp⁡(β~/b)tan-1⁡(r/b) with the different ranges of parameters, (β1<β~<β2, λ2<λ~<λ1) and (β1<β~<β2, λ2<λ1<λ~), respectively.
(a) (b)Figure 15
Sketches off′(r) (blue), h1′(r)=λ13r2+β1r+b2exp⁡β1/btan-1⁡r/b (green), and h2′(r)=λ23r2+β2r+b2exp⁡β2/btan-1⁡r/b (red). The dashed brown (a) and dashed black (b) curves represent h~′(r)=λ~3r2+β~r+b2exp⁡β~/btan-1⁡r/b with the different ranges of parameters, (β~<β1<β2, λ2<λ~<λ1) and (β~<β1<β2, λ2<λ1<λ~), respectively.
(a) (b)In addition, for the situationλ~<λ2 where the curve of h~′(r) is below that of h2′(r), we find no points of tangency to meet h~(r~I)=f(r~I).As analyzed in the above for bothλ~>λ2 (four possible situations) and λ~<λ2, we can conclude that no other maximum of temperature exists. Moreover, no other minimum of temperature exists, either, since no other minimum points are available after the minimum point rII. Otherwise, the asymptotic behavior of the temperature at a large horizon radius would be violated. As a result, when b is relatively large, there are only one maximum point and one minimum point on the curve of TH, which is similar to the cyan curves of Figures 5 and 10.Equation (33) shows that TH increases as b decreases at fixed r+. Equation (34) indicates that TH′(r) is always positive for r<r1, and that for r>r1, it can also be positive for a small b. When the curve is considered as a whole to be varying smoothly with respect to b, there exist four types of situations: one single maximum, one maximum and one following minimum, one inflexion, and no extremal points on the curve. They appear in the order as the AdS radius b decreases from infinity to zero, which explains the similarity of thermodynamic properties in the models with a general mass distribution.


### 4.3. Entropy

From (16) and (27), the entropy at constant pressure is derived to be(41)S=∫r0r+dMTH=∫r0r+2πrfrdr,revealing that the entropy depends on the mass distribution f(r). Incidentally, it returns naturally to the ordinary Schwarzschild and Schwarzschild-AdS black holes if one sets f(r)=1.When we make a further extension, starting from the nonsingularity of metric equation (24) while neglecting the finiteness of density, the discussions above can be generalized to the case in which f(r) is infinitesimal higher than first order at the origin. Specifically, when one takes f(r) in the form (42)fr=1-l02r2+l02,the entropy turns out to be(43)S=πr+2-r02+2πl02ln⁡r+2r02,where the correction is logarithmic, as calculated in the holographic metric [21]. We thus include the result of [21] as our special case.


## 4.1. Features of Temperature of Black Holes without the AdS Background

In the absence of the AdS background (b→∞), the Hawking temperature and its derivative with respect to r+ take the forms(28)TH=14π1r+-f′r+fr+,(29)dTHdr+=-14πfr+2-f′r+2r+2+f′′r+fr+r+2fr+2r+2,and the temperature vanishes at r+→∞. As to the derivative of the temperature, it is positive due to fr0=f′(r0)r0 and f′′(r0)<0 at the extremal radius r0 and close to zero at infinity. As a result, there is at least one local maximum temperature T1 whose corresponding radius r1 satisfies the condition from the vanishing of (29) at r+=r1:(30)f′′r1fr1r12-f′r12r12+fr12=0.In the following we prove that T1 is the global maximum and thus indicate the similar TH-r+ relation to that of the above two models depicted by Figures 5 and 10, which ensures that the thermodynamic properties of the general case are similar to that of the other specific models mentioned.Set functiong(r) satisfying the following second-order differential equation,(31)g′′rgrr2-g′r2r2+gr2=0,we have a general solution, gr=c1rexp⁡(c2r), where c1 and c2 are two parameters to be determined. By choosing a positive parameter c1=fr1/r1exp⁡-f′r1r1+fr1 and a negative parameter c2=f′r1r1-fr1/r1, we obtain the particular solution that ensures g(r1)=f(r1) and g′(r1)=f′(r1). Again using g(r1)=f(r1), g′(r1)=f′(r1) and (30) and (31), we deduce g′′(r1)=f′′(r1). Since it is difficult to depict the equality of the second derivatives, g′′(r1)=f′′(r1), on the plot of curves of f(r) and g(r), we choose to draw the curves of f′(r) and g′(r) instead; see Figure 13. One can infer that the two curves are tangent at r1 due to g′(r1)=f′(r1) and g′′(r1)=f′′(r1). In addition, we note that as a result of g(r)=∫0rg′r′dr′+g(0) and g(0)=0, g(r) is equal to the integral area surrounded by the curve g′(r) and the transverse axis from the origin to r, and the same analysis is applicable for f(r). Consequently, g(r1)=f(r1) indicates that the integral areas surrounded by the curve g′(r) and the transverse interval [0,r1] and by the curve f′(r) and the transverse interval [0,r1] are the same; see the two solid curves in Figure 13. That is to say, the equality of the two integral areas implies the existence of the maximum of temperature in the TH-r+ plane.Figure 13
Sketches off′(r) (blue) and g′(r)=c11+c2rexp⁡(c2r) (purple). The dashed red curve and the dashed green curve represent g~′(r)=c~11+c~2rexp⁡(c~2r) with c~1>c1 and c~1<c1, respectively.Assuming that there still exists one local minimum of temperatureT~1 at r~1 on the curve of TH that meets f′′r~1fr~1r~12-f′(r~1)2r~12+f(r~1)2=0, we can certainly find out another particular solution g~r=c~1rexp⁡(c~2r) with c~1=fr~1/r~1exp⁡-f′r~1r~1+fr~1 and c~2=f′r~1r~1-fr~1/r~1 to the differential equation g~′′(r)g~(r)r2-g~′(r)2r2+g~(r)2=0. This particular solution satisfies g~(r~1)=f(r~1), g~′(r~1)=f′(r~1), and g~′′(r~1)=f′′(r~1). Using (29), we have the maximum of temperature at r1:(32)THr1=14π1r1-f′r1fr1=14π1r1-g′r1gr1=-14πc2>0,which means that TH(r1) is only related to the negative parameter c2. Considering TH(r1)>TH(r~1), one can infer that c~2>c2. Next we compare g′(r) with g~′(r)=c~11+c~2rexp⁡(c~2r) by considering the following two cases in order to determine that the local minimum of temperature T~1 conjectured above does not exist.(i)
For the casec~1>c1, we can get g~′(r)>g′(r), which means that the curve of g~′(r) is always above that of g′(r). So the integral area surrounded by g~′(r) and the interval [0,r~1] is not the same as that surrounded by f′(r) and the interval [0,r~1]. In fact, we always have g~(r~1)>f(r~1) in this case; see, for instance, the sketches in Figure 13.(ii)
For the casec~1<c1, we can first get g~′(0)=c~1<c1=g′(0); then if the two curves of g~′(r) and f′(r) are tangent at r~1, we can infer g~(r~1)=∫0r~1g~′(r)dr<∫0r~1f′rdr=f(r~1), which also leads to the inequality of the integral areas; see the sketches in Figure 13.In a word, no suitable parametersc~1 and c~2 can be found to simultaneously meet g~(r~1)=f(r~1), g~′(r~1)=f′(r~1), and g~′′(r~1)=f′′(r~1), or in other words, to meet TH′(r~1)=0. As a result, the curve of TH without the AdS background has one and only one extremal point, that is, the maximum point, which certainly leads to the conclusion that the TH-r+ relation in this case is similar to the green curves of Figures 5 and 10.


## 4.2. Features of Temperature of Black Holes with the AdS Background

For the case with the AdS background, we rewrite the Hawking temperature (27) and calculate the corresponding derivative as follows:(33)TH=14πfr+-f′r+r+fr+r++3fr+r+-f′r+r+2fr+b2,(34)dTHdr+=14π-fr+2-f′r+2r+2+f′′r+fr+r+2fr+2r+2+3fr+2+f′r+2r+2-2fr+f′r+r+-fr+f′′r+r+2b2fr+2.The Hawking temperature is still zero for extreme black holes, but because of the finiteb parameter, it grows to infinity as the horizon radius increases. As to the derivative of the temperature, TH′, it is positive according to (27) at the extremal radius r0 and also positive at infinity because it goes to 3/(4πb2). For a relatively large b, the zero point of TH′, where the main part of TH′ is the first term of (34), will slightly shift to the right compared with the case of b→∞, and TH′ can also be negative at some radius. As a result, there exist at least one maximum and one following minimum on the curve of TH when the horizon radius increases. In this case, the two points can be proven to be the only pair of extremal points of TH where the numerator of TH′ vanishes. In fact, to the differential equation,10(35)hrh′′rr2b2+r2-h′r2r2b2+r2+2hrh′rr3-hr23r2-b2=0,we can find a general solution,(36)hr=λrr2+b2exp⁡βbtan-1⁡rb,where λ and β are two parameters to be determined, and give its derivative as follows:(37)h′r=λ3r2+βr+b2exp⁡βbtan-1⁡rb.We can obtain two particular solutions,h1(r)=λ1rr2+b2exp⁡β1/btan-1⁡(r/b) and h2(r)=λ2rr2+b2exp⁡β2/btan-1⁡r/b, where the two pairs of parameters take the forms11(38)λ1=frIrIrI2+b2exp⁡-β1btan-1⁡rIb>0,β1=f′rIrIrI2+b2-frI3rI2+b2frIrI<0;λ2=frIIrIIrII2+b2exp⁡-β2btan-1⁡rIIb>0,β2=f′rIIrIIrII2+b2-frII3rII2+b2frIIrII<0,and rI and rII are the maximum and minimum points, respectively, and meet the inequality rI<rII. The particular solutions satisfy the conditions, h1(rI)=f(rI), h1′(rI)=f′rI, and h1′′(rI)=f′′(rI); h2(rII)=f(rII), h2′(rII)=f′(rII), and h2′′(rII)=f′′rII. As a result, the curve of f′(r) is tangent to that of h1′(r) at rI and also tangent to that of h2′(r) at rII. Again considering h(r)=∫0rh′(r~)dr~ and h(0)=0, we deduce that the integral area surrounded by the curve h1′(r) and the transverse interval [0,rI] equals that by the curve f′(r) and the transverse interval [0,rI], and the same happens for the integral areas surrounded by the curve h2′(r) and transverse interval [0,rII] and by the curve f′(r) and the transverse interval [0,rII].Using (33), we have the temperature at the extremal (maximum and minimum) points re,(39)THre=14πhre-h′rerehrere+3hrere-h′rere2hreb2=-β4πb2,which is only related to the negative parameter β. Due to TH(rI)>TH(rII), we obtain β1<β2. As to the relationship between λ1 and λ2, if λ1<λ2 is taken, the curve of h1(r) is always below that of h2(r), which cannot lead to h1(rI)=f(rI) and h2(rII)=f(rII) simultaneously. As a result, we get λ1>λ2.If another maximum pointr~I following rII (r~I>rII) is conjectured on the curve of TH, one can seek out the corresponding particular solution, h~(r)=λ~rr2+b2exp⁡β~/btan-1⁡r/b, where the parameters take the forms(40)λ~=fr~Ir~Ir~I2+b2exp⁡-β~1btan-1⁡r~Ib>0,β~=f′r~Ir~I3+b2r~I-fr~I3r~I2+b2fr~Ir~I<0,and meet the conditions h~(r~I)=f(r~I), h~′(r~I)=f′(r~I) and h~′′(r~I)=f′′(r~I). From TH(r~I)>TH(rII), we deduce that β~<β2. With this constraint and the assumption λ~>λ2 together we first consider the following four different situations which are drawn in Figures 14 and 15.(i)
In the range of parametersβ1<β~<β2 and λ2<λ~<λ1, h~′(0)=λ~b2 is between h1′(0)=λ1b2 and h2′(0)=λ2b2, so the point of tangency r~I, if it exists, should be between rI and rII in order to make the integral area surrounded by h~(r~I) and [0,r~I] be equal to the integral area surrounded by f(r~I) and [0,r~I], rather than being larger than rII as supposed; see Figure 14(a).(ii)
In the range of parametersβ1<β~<β2 and λ2<λ1<λ~, the curve of h~′(r) lies above that of h1′(r), so there is no r~I to satisfy the equality of integral areas about h~′(r) and f′(r); see Figure 14(b).(iii)
In the range of parametersβ~<β1<β2 and λ2<λ~<λ1, the curve of h~′(r) lies below that of h1′(r), so we find no r~I to satisfy the equality of integral areas about h~′(r) and f′(r); see Figure 15(a).(iv)
In the range of parametersβ~<β1<β2 and λ2<λ1<λ~, since h~′(0)=λ~ is larger than h1′(0)=λ1, in order to ensure the equality of the integral areas, the curve of h~′(r) must intersect with h1′(r) before h1′(r) is tangent to f′(r), which makes the point of tangency of h~′(r) and f′(r) smaller than rI or even nonexistent; see Figure 15(b).Figure 14
Sketches off′(r) (blue), h1′(r)=λ13r2+β1r+b2exp⁡β1/btan-1⁡r/b (green), and h2′(r)=λ23r2+β2r+b2exp⁡β2/btan-1r/b (red). The dashed purple (a) and dashed pink (b) curves represent h~′(r)=λ~3r2+β~r+b2exp⁡(β~/b)tan-1⁡(r/b) with the different ranges of parameters, (β1<β~<β2, λ2<λ~<λ1) and (β1<β~<β2, λ2<λ1<λ~), respectively.
(a) (b)Figure 15
Sketches off′(r) (blue), h1′(r)=λ13r2+β1r+b2exp⁡β1/btan-1⁡r/b (green), and h2′(r)=λ23r2+β2r+b2exp⁡β2/btan-1⁡r/b (red). The dashed brown (a) and dashed black (b) curves represent h~′(r)=λ~3r2+β~r+b2exp⁡β~/btan-1⁡r/b with the different ranges of parameters, (β~<β1<β2, λ2<λ~<λ1) and (β~<β1<β2, λ2<λ1<λ~), respectively.
(a) (b)In addition, for the situationλ~<λ2 where the curve of h~′(r) is below that of h2′(r), we find no points of tangency to meet h~(r~I)=f(r~I).As analyzed in the above for bothλ~>λ2 (four possible situations) and λ~<λ2, we can conclude that no other maximum of temperature exists. Moreover, no other minimum of temperature exists, either, since no other minimum points are available after the minimum point rII. Otherwise, the asymptotic behavior of the temperature at a large horizon radius would be violated. As a result, when b is relatively large, there are only one maximum point and one minimum point on the curve of TH, which is similar to the cyan curves of Figures 5 and 10.Equation (33) shows that TH increases as b decreases at fixed r+. Equation (34) indicates that TH′(r) is always positive for r<r1, and that for r>r1, it can also be positive for a small b. When the curve is considered as a whole to be varying smoothly with respect to b, there exist four types of situations: one single maximum, one maximum and one following minimum, one inflexion, and no extremal points on the curve. They appear in the order as the AdS radius b decreases from infinity to zero, which explains the similarity of thermodynamic properties in the models with a general mass distribution.


## 4.3. Entropy

From (16) and (27), the entropy at constant pressure is derived to be(41)S=∫r0r+dMTH=∫r0r+2πrfrdr,revealing that the entropy depends on the mass distribution f(r). Incidentally, it returns naturally to the ordinary Schwarzschild and Schwarzschild-AdS black holes if one sets f(r)=1.When we make a further extension, starting from the nonsingularity of metric equation (24) while neglecting the finiteness of density, the discussions above can be generalized to the case in which f(r) is infinitesimal higher than first order at the origin. Specifically, when one takes f(r) in the form (42)fr=1-l02r2+l02,the entropy turns out to be(43)S=πr+2-r02+2πl02ln⁡r+2r02,where the correction is logarithmic, as calculated in the holographic metric [21]. We thus include the result of [21] as our special case.


## 5. Summary

In this paper, we calculate the Hawking temperature and the entropy of black holes and analyze the phase transitions under different vacuum pressure for the self-regular Schwarzschild-AdS black hole with the hydrogen-atom-like and the collapsed shell mass distributions. In particular, we provide the fine phase structures that depend on the cosmological constant or the thermodynamic pressure. There are four types of phase structures that correspond to the zero, the relatively low, the critical, and the relatively high pressure, respectively, where the second type is further separated into three subtypes due to its complexity. Thanks to the existence of a minimal length, the models we studied are free of singularity, and their pathological outcome of evaporation can be cured. Specifically, when the temperature is smaller thanTHP, the black hole even with positive Gibbs free energy is unable to decay into the pure thermal radiation due to the existence of extreme configurations. In addition, the entropy satisfies the area law only at a large horizon radius but has an obvious deviation dependent on the mass distribution at the near-extremal scale. In order to give the reason that different mass-smeared schemes lead to similarities in thermodynamic properties [16], we investigate a general model whose mass density is based on an analytic expression of the δ-function and the mass distribution takes a kind of step functions with continuity. We find that all such models indeed have similar properties thermodynamically.



---

*Source: 1095217-2017-01-24.xml*