Phase transitions in four-dimensional AdS black holes with a nonlinear electrodynamics source

In this work we consider black hole solutions to Einstein's theory coupled to a nonlinear power-law electromagnetic field with a fixed exponent value. We study the extended phase space thermodynamics in canonical and grand canonical ensembles, where the varying cosmological constant plays the role of an effective thermodynamic pressure. We examine thermodynamical phase transitions in such black holes and find that both first- and second-order phase transitions can occur in the canonical ensemble while, for the grand canonical ensemble, Hawking–Page and second-order phase transitions are allowed.     

QED effects on phase transition and Ruppeiner geometry of Euler-Heisenberg-AdS black holes

Considering the quantum electrodynamics (QED) effect, we study the phase transition and Ruppeiner geometry of Euler-Heisenberg anti-de Sitter black holes in the extended phase space. For negative and small positive QED parameters, we observe a small/large black hole phase transition and reentrant phase transition, respectively, whereas a large positive value of the QED parameter ruins the phase transition. Phase diagrams for each case are explicitly shown. Then, we construct the Ruppeiner geometry in thermodynamic parameter space. Different features of the corresponding scalar curvature are shown for both the small/large black hole phase transition and reentrant phase transition cases. Of particular interest is the additional region of positive scalar curvature, indicating a dominant repulsive interaction among black hole microstructures, for the black hole with a small positive QED parameter. Furthermore, universal critical phenomena are observed for the scalar curvature of Ruppeiner geometry. These results indicate that the QED parameter has a crucial influence on the black hole phase transition and microstructure.     

Ruppeiner Geometry of （2 ＋ 1）-Dimensional Spinning Dilaton Black Hole

In this paper, we study the geometrothermodynamics of (2+1)-dimensional spinning dilaton black hole. We show that the Ruppeiner curvature vanishes, which implies that there exist no phase transitions and thermodynamic interactions. However when the thermodynamics fluctuation is included, the geometry structure is reconsidered. The non-vanishing Ruppeiner curvature is obtained, which means the phase space is non-flat. We also study the phasetransitions and show that it can indeed take place at some points.     

Extended Phase Space in the Framework of Holography

We consider a holographic extended phase space in the presence of Reissner-Nordstrom-Anti-de Sitter(RNAdS) and Born-Infeld-Anti-de Sitter(BI-AdS) black holes in the bulk. In this extended phase space the cosmological constant is investigated as pressure and volume is defined as the codimension one-time slice in the bulk geometry enclosed by the minimal area appearing in the computation of the holographic entanglement entropy. These thermodynamics quantities can serve as probes of the underlying phase transition dictated by black hole thermodynamics, but do not describe different structures. We find that the isocharges on the pressure-volume plane exhibit a Van der Waals-like structure, for RN-AdS black holes in the background. For BI-AdS black holes, we observe the analogy with a Van der Waals liquid-gas system for βQ 1/2 and Reentrant phase transition for βQ 1/2 in the holographic extended phase space. The same holographic thermodynamic behavior is observed when we use the fidelity susceptibility as the volume and the cosmological constant as the pressure for RN-AdS black hole in the background.     

Thermodynamics of a class of non-asymptotically flat black holes in Einstein–Maxwell–Dilaton theory

We analyse in detail the thermodynamics in the canonical and grand canonical ensembles of a class of non-asymptotically flat black holes of the Einstein-(anti) Maxwell-(anti) Dilaton theory in 4D with spherical symmetry. We present the first law of thermodynamics, the thermodynamic analysis of the system through the geometrothermodynamics methods, Weinhold, Ruppeiner, Liu–Lu–Luo–Shao and the most common, that made by the specific heat. The geometric methods show a curvature scalar identically zero, which is incompatible with the results of the analysis made by the non null specific heat, which shows that the system is thermodynamically interacting, does not possess extreme case nor phase transition. We also analyse the local and global stability of the thermodynamic system, and obtain a local and global stability for the normal case for $0<\gamma <1$ and for other values of $\gamma $ , an unstable system. The solution where $\gamma =0$ separates the class of locally and globally stable solutions from the unstable ones.     

Thermodynamics,phase transitions and Ruppeiner geometry for Einstein–dilaton–Lifshitz black holes in the presence of Maxwell and Born–Infeld electrodynamics

In this paper, we first obtain the higher-dimen-sional dilaton–Lifshitz black hole solutions in the presence of Born–Infeld (BI) electrodynamics. We find that there are two different solutions for the cases of \(z=n+1\) and \(z\ne n+1\) where z is the dynamical critical exponent and n is the number of spatial dimensions. Calculating the conserved and thermodynamical quantities, we show that the first law of thermodynamics is satisfied for both cases. Then we turn to the study of different phase transitions for our Lifshitz black holes. We start with the Hawking–Page phase transition and explore the effects of different parameters of our model on it for both linearly and BI charged cases. After that, we discuss the phase transitions inside the black holes. We present the improved Davies quantities and prove that the phase transition points shown by them are coincident with the Ruppeiner ones. We show that the zero temperature phase transitions are transitions in the radiance properties of black holes by using the Landau–Lifshitz theory of thermodynamic fluctuations. Next, we turn to the study of the Ruppeiner geometry (thermodynamic geometry) for our solutions. We investigate thermal stability, interaction type of possible black hole molecules and phase transitions of our solutions for linearly and BI charged cases separately. For the linearly charged case, we show that there are no phase transitions at finite temperature for the case \( z\ge 2\). For \(z<2\), it is found that the number of finite temperature phase transition points depends on the value of the black hole charge and there are not more than two. When we have two finite temperature phase transition points, there is no thermally stable black hole between these two points and we have discontinuous small/large black hole phase transitions. As expected, for small black holes, we observe finite magnitude for the Ruppeiner invariant, which shows the finite correlation between possible black hole molecules, while for large black holes, the correlation is very small. Finally, we study the Ruppeiner geometry and thermal stability of BI charged Lifshtiz black holes for different values of z. We observe that small black holes are thermally unstable in some situations. Also, the behavior of the correlation between possible black hole molecules for large black holes is the same as for the linearly charged case. In both the linearly and the BI charged cases, for some choices of the parameters, the black hole system behaves like a Van der Waals gas near the transition point.     

The microstructure and Ruppeiner geometry of charged anti-de Sitter black holes in Gauss–Bonnet gravity: from the critical point to the triple point

Ruppeiner geometry has been successfully applied in the study of the black hole microstructure by combining with the small–large black hole phase transition, and the potential interactions among the molecular-like constituent degrees of freedom are uncovered. In this paper, we will extend the study to the triple point, where three black hole phases coexist acting as a typical feature of black hole systems quite different from the small–large black hole phase transition. For the six-dimensional charged Gauss–Bonnet anti-de Sitter black hole, we thoroughly investigate the swallow tail behaviors of the Gibbs free energy and the equal area laws. After obtaining the black hole triple point in a complete parameter space, we exhibit its phase structures both in the pressure–temperature and temperature–horizon radius diagrams. Quite different from the liquid–vapor phase transition, a double peak behavior is present in the temperature–horizon radius phase diagram. Then we construct the Ruppeiner geometry and calculate the corresponding normalized curvature scalar. Near the triple point, we observe multiple negatively divergent behaviors. Positive curvature scalar is observed for the small black hole with high temperature, which indicates that the repulsive interaction dominates among the microstructure. Furthermore, we consider the variation of the curvature scalar along the coexisting intermediate and large black hole curves. Combining with the observation for different fluids, the result suggests that this black hole system behaves more like the argon or methane. Our study provides a first and preliminary step towards understanding black hole microstructure near the triple point, as well as uncovering the particular properties of the Gauss–Bonnet gravity.     

Fractional phase transitions of RN-AdS black holes at their Davies points

In this study, we investigate the phase transitions of the RN-AdS black hole at its Davies points according to the generalized Ehrenfest classification of phase transition established based on fractional derivatives. Notably, Davies points label the positions at which the heat capacity diverges. According to the usual Ehrenfest classification, second-order phase transitions occur at these points. For the RN-AdS black hole, the Davies points can be classified into two types. The first type corresponds to extreme values of the temperature, and the second type corresponds to the infection point (namely the critical point) of temperature. Employing the generalized Ehrenfest classification, we determine that the orders of phase transition at the two types of Davies points are different, that is, we note an order of 3/2 for the first type and 4/3 for the second type. Thus, this finer-grained classification can discriminate between phase transitions that are expected to lie in the same category, providing new insights leading toward a better understanding of black hole thermodynamics.     

The Thermodynamic Geometry and Phase Transition of the Plane Symmetric Black Hole

In this paper, we investigate the thermodynamic properties and the thermodynamic geometry of the plane symmetric black hole. We obtain the thermodynamic curvature based on the Weinhold geometry curvature, Ruppeiner geometry curvature and the Quevedo curvature. We find the Weinhold curvature always equals to zero and there is a phase transition point for the Ruppeiner curvature. The Quevedo curvature produces a same phase structure as the heat capacity.

     

Maxwell’s equal area law for black holes in power Maxwell invariant

In this paper, we consider the phase transition of black hole in power Maxwell invariant by means of Maxwell’s equal area law. First, we review and study the analogy of nonlinear charged black hole solutions with the Van der Waals gas–liquid system in the extended phase space, and obtain isothermal P-v diagram. Then, using the Maxwell’s equal area law we study the phase transition of AdS black hole with different temperatures. Finally, we extend the method to the black hole in the canonical (grand canonical) ensemble in which charge (potential) is fixed at infinity. Interestingly, we find the phase transition occurs in the both ensembles. We also study the effect of the parameters of the black hole on the two-phase coexistence. The results show that the black hole may go through a small-large phase transition similar to those of usual non-gravity thermodynamic systems.     

Homothetic motions and Newtonian cosmology

We study the thermodynamics and the different thermodynamic geometric methods of Reissener-Nordström-de Sitter black hole and its extremal case, which is similar to the de Sitter black hole coupled to a scalar field, rather called an MTZ black hole. While studying the thermodynamics of the systems, we could find some abnormalities. In both cases, the thermodynamic geometric methods could give the correct explanation for all abnormal thermodynamic behaviors in the system.     

Validity of thermodynamic laws and weak cosmic censorship for AdS black holes and black holes in a cavity

By throwing a test charged particle into a Reissner-Nordstrom (RN) black hole, we test the validity of the first and second laws of thermodynamics and the weak cosmic censorship conjecture (WCCC) with two types of boundary conditions: the asymptotically anti-de Sitter (AdS) space and a Dirichlet cavity wall placed in an asymptotically flat space. For the RN-AdS black hole, the second law of thermodynamics is satisfied, and the WCCC is violated for both extremal and near-extremal black holes. For the RN black hole in a cavity, the entropy can either increase or decrease depending on the change in the charge, and the WCCC is satisfied/violated for the extremal/near-extremal black hole. Our results indicate that there may be a connection between the black hole thermodynamics and the boundary condition imposed on the black hole.     

Charged BTZ-like black holes in higher dimensions

Motivated by many worthwhile papers about (2+1)-dimensional BTZ black hole solutions, we generalize them to (n+1)-dimensional solutions, the so-called BTZ-like solutions. We show that the electric field of BTZ-like solutions is the same as that of (2+1)-dimensional BTZ black holes, and also their lapse functions are approximately the same, too. By these similarities, it is also interesting to investigate the geometric and thermodynamics properties of the BTZ-like solutions. We find that, depending on the metric parameters, the BTZ-like solutions may be interpreted as black hole solutions with inner (Cauchy) and outer (event) horizons, an extreme black hole or naked singularity. Then, we obtain the conserved and thermodynamic quantities, and we show that they satisfy the first law of thermodynamics. Next, we perform a thermodynamic stability analysis in the canonical ensemble and find that the BTZ-like solutions are stable in the whole phase space.     

Thermodynamics of a two-dimensional charged black hole in the geometric framework

In this paper, we study the thermodynamic features of a two-dimensional charged black hole. Weinhold curvature and Ruppeiner curvature are explored as information geometry, respectively. Moreover, based on the Legendre invariant proposed by Hernando Quevedo, the geometro-thermodynamics behavior of this black hole is investigated.     

Temperature and thermodynamic geometry of the Kerr-Sen black hole

This paper studies the thermodynamic properties of the Kerr-Sen black hole from the viewpoint of geometry.It calculates the temperature and heat capacity of the black hole,Weinhold metric and Ruppeiner metric are also obtained respectively.It finds that they are both curved and the curvature scalar of Weinhold curvature implies no information about the phase transition while the Ruppeiner one does.But they both carry no information about the second-order phase transition point reproduced from the capacity.Besides,the Legendre invariant metric of the Kerr-Sen black hole has been discussed and its scalar curvature gives the information about the second-order phase transition point.     

Geometro-thermodynamics of tidal charged black holes

Tidal charged spherically symmetric vacuum brane black holes are characterized by their mass m and tidal charge q, an imprint of the five-dimensional Weyl curvature. For q>0 they are formally identical to the Reissner–Nordström black hole of general relativity. We study the thermodynamics and thermodynamic geometries of tidal charged black holes and discuss similarities and differences as compared to the Reissner–Nordströ m black hole. As a similarity, we show that (for q>0) the heat capacity of the tidal charged black hole diverges on a set of measure zero of the parameter space, nevertheless both the regularity of the Ruppeiner metric and a Poincaré stability analysis show no phase transition at those points. The thermodynamic state spaces being different indicates that the underlying statistical models could be different. We find that the q<0 parameter range, which enhances the localization of gravity on the brane, is thermodynamically preferred. Finally we constrain for the first time the possible range of the tidal charge from the thermodynamic limit on gravitational radiation efficiency at black hole mergers.     

Geometrothermodynamics of black holes

The thermodynamics of black holes is reformulated within the context of the recently developed formalism of geometrothermodynamics. This reformulation is shown to be invariant with respect to Legendre transformations, and to allow several equivalent representations. Legendre invariance allows us to explain a series of contradictory results known in the literature from the use of Weinhold’s and Ruppeiner’s thermodynamic metrics for black holes. For the Reissner–Nordström black hole the geometry of the space of equilibrium states is curved, showing a non trivial thermodynamic interaction, and the curvature contains information about critical points and phase transitions. On the contrary, for the Kerr black hole the geometry is flat and does not explain its phase transition structure.     

(2+1)-Dimensional AdS Black Hole in Grand Canonical Ensemble

We investigate thermodynamics of the (2+1)-dimensional AdS black hole in grand canonical ensemble. In the York's formalism, the black hole is enclosed in a “box” with a finite radius and the boundary temperature, radius and potential are fixed in the grand canonical ensemble. We investigate the thermodynamical properties such as action, entropy, temperature, etc. We only find the stable solution for (2+1)-dimensional AdS black hole and do not find the instanton with the negative heat capacity.     

THERMODYNAMICS OF THE SLOWLY ROTATING KERR-NEWMAN BLACK HOLE IN THE GRAND CANONICAL ENSEMBLE

We investigate the thermodynamics of the slowly rotating Kerr-Newman (K-N) black hole in the grand canonical ensemble with York's formalism. Some thermodynamical properties, such as the thermodynamical action, entropy, thermodynamical energy and heat capacity are studied, and solutions of the slowly rotating K-N black hole with different boundary conditions are analysed. We find stable solutions and instantons under certain boundary conditions.