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We perform a study of phase transitions of RN-AdS black hole at its Davies points according to a generalized Ehrenfest classification of phase transition established on the basis of fractional derivatives. Davies points label the positions where heat capacity diverges. According to the usual Ehrenfest classification, second-order phase transitions occur there. For RN-AdS black hole, the Davies points can be classified into two types. The first type corresponds to the extreme values of temperature and the second type corresponds to the infection point(namely the critical point) of temperature. Employing the generalized Ehrenfest classification, we find that the orders of phase transition at the two types of Davies points are different. It is $3/2$-order for the first type and $4/3$-order for the second type. Thus this finer-grained classification can discriminate phase transitions that are supposed to be in the same category, which may provide some new insights toward a better understanding of black hole thermodynamics., Comment: 13 pages, 4 figures, to be published in Chinese Physics C

Following Barrow's idea of fractal black hole horizon, we re-derive black hole entropy of static spherically symmetric black holes. When a black hole absorbs matter its horizon area will increase. Given the spherically fractal structure, we conjecture that the minimal increase of the horizon area should be the area of the smallest bubble sphere. From this, we find the black hole entropy has a logarithmic form, which is similar to that of Boltzmann entropy if we consider $A/A_{pl}$ as the number of microscopic states. We further calculate temperatures and heat capacities of Schwarzschild, Reissner-Nordstr{\"o}m(RN), and RN-AdS black holes. It is found that their temperatures are all monotonically increasing and the heat capacities are all positive, which means these black holes are thermodynamically stable. Besides, for RN-AdS black hole we find its heat capacity has Schottky anomaly-like behavior, which may reflect the existence of the discrete energy level and restricted microscopical degree of freedom., Comment: 10 pages, 5 figures

Employing the fractional derivatives, we construct a more elaborate classification of phase transitions compared to the original Ehrenfest classification. In this way, a thermodynamic system can even undergo a fractional-order phase transition. We use this method to restudy the charged AdS black hole and Van der Waals fluids and find that at the critical point they both have a $4/3$-order phase transition, but not the previously recognized second-order one., Comment: 14 pages, 3 figures

We study the thermodynamic properties and critical behaviors of the topological charged black hole in AdS space under the consideration of the generalized uncertainty principle (GUP). It is found that only in the spherical horizon case there are Van der Waals-like first-order phase transitions and reentrant phase transitions. From the equation of state we find that the GUP-corrected black hole can have one, two and three apparent critical points under different conditions. However, it is verified by the Gibbs free energy that in either case there is at most one physical critical point., Comment: 13 pages, 9 figures, Advances in High Energy Physics, in press

On the basis of two requirements: the avoidance of the curvature singularity and the Maxwell theory as the weak field limit of the nonlinear electrodynamics, we find two restricted conditions on the metric function of $(2+1)$-dimensional regular black hole in general relativity coupled with nonlinear electrodynamics sources. By the use of the two conditions, we obtain a general approach to construct $(2+1)$-dimensional regular black holes. In this manner, we construct four $(2+1)$-dimensional regular black holes as examples. We also study the thermodynamic properties of the regular black holes and verify the first law of black hole thermodynamics., Comment: 13 pages, 4 figures. in press in PLB

We demonstrate the existence of $P-V$ criticality of the topological Ho\v{r}ava-Lifshitz(HL) black holes with a spherical horizon $(k=1)$ in the extended phase space. With the electric charge, we find that the critical behaviors of the HL black hole are nearly the same as those of van der Waals(VdW) system. For the uncharged case, the HL black hole has a peculiar $P-V$ criticality. The critical behavior is completely controlled by a parameter $\epsilon$, but not the temperature $T$. When $\epsilon$ is larger than a critical value $\epsilon_c$, no matter what the temperature is, there will be the first-order phase transition. Moreover, we find that there is an infinite number of critical points which form a "critical curve". As far as we know, this is the first time to find this kind of peculiar $P-V$ criticality., Comment: 16 pages, 8 figures. some typos are corrected. a note is added at the last of the paper

Under two different metric ansatzes, the noncommutative geometry inspired black holes (NCBH) in the framework of Rastall gravity are derived and analyzed. We consider the fluid-type matter with the Gaussian-distribution smeared mass density. Taking a Schwarzschild-like metric ansatz, it is shown that the noncommutative geometry inspired Schwarzschild black hole (NCSBH) in Rastall gravity, unlike its counterpart in general relativity (GR), is not a regular black hole. It has at most one event horizon. After showing a finite maximal temperature, the black hole will leave behind a point-like massive remnant at zero temperature. Considering a more general metric ansatz and a special equation of state of the matter, we also find a regular NCBH in Rastall gravity, which has a similar geometric structure and temperature to that of NCSBH in GR., Comment: 12 pages, 5 figures. to match the published version

In the framework of horizon thermodynamics, the field equations of Einstein gravity and some other second-order gravities can be rewritten as the thermodynamic identity: $dE=TdS-PdV$. However, in order to construct the horizon thermodynamics in higher-order gravity, we have to simplify the field equations firstly. In this paper, we study the fourth-order gravity and convert it to second-order gravity via a so-called " Legendre transformation " at the cost of introducing two other fields besides the metric field. With this simplified theory, we implement the conventional procedure in the construction of the horizon thermodynamics in 3 and 4 dimensional spacetime. We find that the field equations in the fourth-order gravity can also be written as the thermodynamic identity. Moreover, we can use this approach to derive the same black hole mass as that by other methods., Comment: 12 pages, no figure

Based on the consideration that the black hole horizon and the cosmological horizon of Kerr-de Sitter black hole are not independent each other, we conjecture the total entropy of the system should have an extra term contributed from the correlations between the two horizons, except for the sum of the two horizon entropies. By employing globally effective first law and effective thermodynamic quantities, we obtain the corrected total entropy and find that the region of stable state for kerr-de Sitter is related to the angular velocity parameter $a$, i.e., the region of stable state becomes bigger as the rotating parameters $a$ is increases., Comment: 12 pages, 5 figures. arXiv admin note: text overlap with arXiv:1610.09886

Based on the consideration that the black hole horizon and the cosmological horizon of Reissner-Nordstr\"om black hole in de Sitter space are not independent each other, we conjecture the total entropy of the system should have an extra term contributed from the entanglement between the two horizons, except for the sum of the two horizon entropies. Making use of the globally effective first law and the effective thermodynamic quantities, we derive the total entropy and find that it will diverge as the two horizons tends to coincide., Comment: 3 pages, 2 figures. In the published version, there are some errors in the equations (13-15). We correct them in this version. The conclusion is not affected

We study the thermodynamic stabilities of uncharged and charged black holes surrounded by quintessence (BHQ) by means of effective thermodynamic quantities. When the state parameter of quintessence $\omega_q$ is appropriately chosen, the structures of BHQ are something like that of black holes in de Sitter space. Constructing the effective first law of thermodynamics in two different ways, we can derive the effective thermodynamic quantities of BHQ. Especially, these effective thermodynamic quantities also satisfy Smarr-like formulae. It is found that the uncharged BHQ is always thermodynamically unstable due to negative heat capacity, while for the charged BHQ there are phase transitions of the second order. We also show that there is a great deal of difference on the thermodynamic properties and critical behaviors of BHQ between the two ways we employed., Comment: 18 pages, 13 figures. arXiv admin note: text overlap with arXiv:1605.08225

We study the thermodynamic properties of Schwarzschild-de Sitter (SdS) black hole and Reissner-Nordstr\"{o}m-de Sitter (RNdS) black hole in the view of global and effective thermodynamic quantities. Making use of the effective first law of thermodynamics, we can derive the effective thermodynamic quantities of de Sitter black holes. It is found that these effective thermodynamic quantities also satisfy Smarr-like formula. Especially, the effective temperatures are nonzero in the Nariai limit, which is consistent with the idea of Bousso and Hawking. By calculating heat capacity and Gibbs free energy, we find SdS black hole is always thermodynamically stable and RNdS black hole may undergoes phase transition at some points., Comment: 12 pages, 5 figures, accepted by Mod. Phys. Lett. A. arXiv admin note: text overlap with arXiv:1606.06070

On the basis of horizon thermodynamics, we study the thermodynamic stability and $P-V$ criticality of topological black holes constructed in Ho\v{r}ava-Lifshitz (HL) gravity without the detailed-balance condition (with general $\epsilon$). In the framework of horizon thermodynamics, we do not need the concrete black hole solution (the metric function) and the concrete matter fields. It is shown that the HL black hole for $k=0$ is always thermodynamically stable. For $k=1$, the thermodynamic behaviors and $P-V$ criticality of the HL black hole are similar to those of RN-AdS black hole for some $\epsilon$. For $k=-1$, the temperature is classified into six types by their different features. Among them, we mainly focus on the type with triply degenerate thermodynamic state. It is also shown that there is a "thermodynamic singularity" for the $k=-1$ HL black hole, where the temperature and Gibbs free energy both diverge apart from a special pressure $P_s$., Comment: 10 pages, 11 figures

On the basis of horizon thermodynamics we study the thermodynamic stability of black holes constructed in general relativity and Gauss-Bonnet gravity. In the framework of horizon thermodynamics there are only five thermodynamic variables $E,P,V,T,S$. It is not necessary to consider concrete matter fields, which may contribute to the pressure of black hole thermodynamic system. In non-vacuum cases, we can derive the equation of state, $P=P(V,T)$. According to the requirements of stable equilibrium in conventional thermodynamics, we start from these thermodynamic variables to calculate the heat capacity at constant pressure and Gibbs free energy and analyze the local and global thermodynamic stability of black holes. It is shown that $P>0$ is the necessary condition for black holes in general relativity to be thermodynamically stable, however this condition cannot be satisfied by many black holes in general relativity. For black hole in Gauss-Bonnet gravity negative pressure can be feasible, but only local stable black hole exists in this case., Comment: 6 pages, 7 figures

We obtain a magnetically charged regular black hole in general relativity. The source to the Einstein field equations is nonlinear electrodynamic field in a physically reasonable model of nonlinear electrodynamics (NED). "Physically" here means the NED model is constructed on the basis of three conditions: the Maxwell asymptotic in the weak electromagnetic field limit; the presence of vacuum birefringence phenomenon; and satisfying the weak energy condition (WEC). In addition, we analyze the thermodynamic properties of the regular black hole in two ways. According to the usual black hole thermodynamics, we calculate the heat capacity at constant charge, from which we know the smaller black hole is more stable. We also employ the horizon thermodynamics to discuss the thermodynamic quantities, especially the heat capacity at constant pressure., Comment: 12 pages, 5 figures

By Maxwell equal area law we study the phase transition in higher dimensional Anti-de Sitter (AdS) Reissner-Nordstr\"{o}m (RN) black hole and Kerr black hole in this paper. The coexisting region of the two phases involved in the phase transition is found and the coexisting curves are shown in $P-T$ figures. We also analytically investigate the parameters which affect the coexisting curves. To better compare with ordinary thermodynamic systems, the Clapeyon equation is derived for higher dimensional AdS black holes. This paper can help to deepen the understanding of the phase transition and phase structure of AdS black holes., Comment: 21 pages, 16 figures

We show by explicit computations that there is a superficial inconsistency between the conventional first law of black hole thermodynamics and Bekenstein-Hawking area law for three types of regular black holes. The corrected form of the first law for these regular black holes is given. The derivation relies on the general structure of the energy-momentum tensor of the matter fields. When the black hole mass parameter $M$ is included in the energy-momentum tensor, the conventional form of the first law should be modified with an extra factor. In this case, the black hole mass $M$ can no longer be considered as the internal energy of the regular black holes., Comment: 11 pages, no figures, to be published in Classical and Quantum Gravity

We study the phase transition of charged Gauss-Bonnet-de Sitter (GB-dS) black hole. For black holes in de Sitter spacetime, there is not only black hole horizon, but also the cosmological horizon. The thermodynamic quantities on the both horizons satisfy the first law of the black hole thermodynamics, respectively; moreover, there are additional connections between them. Using the effective temperature approach, we obtained the effective thermodynamic quantities of charged GB-dS black hole. According to Ehrenfest classification, we calculate some response functions and plot their figures, from which one can see that the spacetime undergoes a second-order phase transition at the critical point. It is shown that the critical values of effective temperature and pressure decrease with the increase of the value of GB parameter $\alpha$., Comment: 9 pages, 16 figures

We study the $P-V/r_{+}$ criticality and phase transition of quantum-corrected black hole in asymptotic safety (AS) gravity in the extended phase space. For the black hole, the cosmological constant is dependent on the momentum cutoff or energy scale, therefore one can naturally treat it as a variable and connect it with the thermodynamic pressure. We find that for the quantum-corrected black hole there is a similar first-order phase transition to that of the van der Waals liquid/gas system. We also analyze the types of the phase transition at the critical points according to Ehrenfest's classification. It is shown that they are second-order phase transition., Comment: 10 pages,7 figures

It is well known that there are black hole and the cosmological horizons for the Reissner-Nordstr\"{o}m-de Sitter spacetime. Although the thermodynamic quantities on the horizons are not irrelevant, they satisfy the laws of black hole thermodynamics respectively. In this paper by considering the relations between the two horizons we give the effective thermodynamic quantities in $(n+2)$-dimensional Reissner-Nordstr\"{o}m-de Sitter spacetime. The thermodynamic properties of these effective quantities are analyzed, moreover, the critical temperature, critical pressure and critical volume are obtained. We carry out an analytical check of Ehrenfest equations and prove that both Ehrenfest equations are satisfied. So the spacetime undergoes a second order phase transition at the critical point. This result is consistent with the nature of liquid--gas phase transition at the critical point, hence deepening the understanding of the analogy of charged dS spacetime and liquid--gas systems., Comment: 18 pages, 7 figures, 1 table

We study the phase transition and the critical behavior of the BTZ black hole with torsion obtained in $(1+2)$-dimensional Poincar\'{e} gauge theory. According to Ehrenfest's classification, when the parameters in the theory are arranged properly the BTZ black hole with torsion may posses the second order phase transition which is also a smaller mass/larger mass black hole phase transition. Nevertheless, the critical behavior is different from the one in the van der Waals liquid/gas system. We also calculated the critical exponents of the relevant thermodynamic quantities, which are the same as the ones obtained in the Ho\v{r}ava-Lifshitz black hole and the Born-Infeld black hole., Comment: 15 pages, 5 figures. arXiv admin note: text overlap with arXiv:1310.1491

After introducing the connection between the black hole horizon and the cosmological horizon, we discuss the thermodynamic properties of Reissner-Nordstrom-de Sitter (RN-dS) spacetime. We present the condition under which RN-dS black hole can exist. Employing Ehrenfest' classification we conclude that the phase transition of RN-dS black hole is the second-order one. The position of the phase transition point is irrelevant to the electric charge of the system. It only depends on the ratio of the black hole horizon and the cosmological horizon., Comment: 12 pages, 18 figures

In this paper, we study the phase transition and the entropy spectrum of BTZ black hole obtained in a model of three-dimensional gravity with torsion. By calculating the heat capacity we find that the BTZ black hole we considered will experience phase transition at some critical point. This indicates that the critical behaviors of black holes do not only depend on the spacetime metric, but have to do with the theory of gravity under consideration. In addition we derived the entropy spectrum of the BTZ black hole according to the quasinormal modes(QNMs) and the adiabatic invariance. It shows that the area or entropy spectrum will also rely on the concrete gravitational action., Comment: 17 pages, 5 figures

In this paper, we study the phase structure and equilibrium state space geometry of charged topological dilaton black holes in $(n+1)$-dimensional anti-de Sitter spacetime. By considering the pairs of parameters $(P\sim V)$ and $(Q\sim U)$ as variables, we analyze the phase structure and critical phenomena of black holes and discuss the relation between the two kinds of critical phenomena. We find that the phase structures and critical phenomena drastically depend on the cosmological constant $l$ (or the static electric charge $Q$ of the black holes), dimensionality $n$ and dilaton field $\Phi $., Comment: 18 pages, 7 figures. arXiv admin note: text overlap with arXiv:1209.1707, arXiv:1205.0559, arXiv:1208.6251 by other authors

Based on the idea of quantized space-time of Snyder, we derive new generalized uncertainty principle and new modified density of states. Accordingly we discuss the influences of the modified density of states on some physical quantities and laws. In addition we analyzed the exact solution of the harmonic oscillator in Snyder's quantized space-time., Comment: 5pages, 1 figure

In the model of de Sitter gauge theory of gravity, the empty homogenous and isotropic spacetimes with constant curvature scalar and nonvanishing homogenous and isotropic torsion must have de Sitter metrics. The static de Sitter spacetime with static, O(3)-symmetric, vector torsion is the only spherically symmetric, vacuum solution with the metric of the form $g_{\mu\nu}={\rm diag}(A^2(r),-B^2(r),-r^2,-r^2\sin^2\th)$. The expressions of the torsion for different de Sitter spacetimes are presented. They are different from one to another. The properties of different de Sitter spacetimes with torsion are also studied., Comment: 15 pages,4 figures,to appear in PRD

It is shown that all torsion-free vacuum solutions of the model of dS gauge theory of gravity are the vacuum solutions of Einstein field equations with the same positive cosmological constant. Furthermore, for the gravitational theories with more general quadratic gravitational Lagrangian ($F^2+T^2$), the torsion-free vacuum solutions are also the vacuum solutions of Einstein field equations., Comment: 4 pafes,0 figure