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.     

Phase transition and thermodynamic stability in extended phase space and charged Hořava–Lifshitz black holes

For charged black holes in Ho?ava–Lifshitz gravity, a second order phase transition takes place in extended phase space where the cosmological constant is taken as thermodynamic pressure. We relate the second order nature of phase transition to the fact that the phase transition occurs at a sharp temperature and not over a temperature interval. Once we know the continuity of the first derivatives of the Gibbs free energy, we show that all the Ehrenfest equations are readily satisfied. We study the effect of the perturbation of the cosmological constant as well as the perturbation of the electric charge on thermodynamic stability of Ho?ava–Lifshitz black hole. We also use thermodynamic geometry to study phase transition in extended phase space. We investigate the behavior of scalar curvature of Weinhold, Ruppeiner, and Quevedo metric in extended phase space of charged Ho?ava–Lifshitz black holes. It is checked if these curvatures could reproduce the result of specific heat for the phase transition.     

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.     

Thermodynamic geometry of the Garfinkle-Horowitz-Strominger dilaton black hole

This paper studies the thermodynamic properties of the Garfinkle-Horowitz-Strominger dilaton black hole from the viewpoint of geometry. It calculates the heat capacity and the temperature of the black hole, Weinhold metric and Ruppeiner metric are also obtained respectively. It finds that they are both curved and the scalar curvature of the Weinhold geometry consists with the first-order transition point reproduced from the capacity, while the Ruppeiner one is both in accordance with the first-order and the second-order phase transition points reproduced from the capacity.     

Thermodynamic and geometric framework of a （2＋ 1）-dimensional black hole with non-linear electrodynamics

The thermodynamic properties of a (2 + 1)-dimensional black hole with non-linear electrodynamics from the viewpoint of geometry is studied and some kinds of temperatures of the black hole have been obtained. Weinhold curvature and Ruppeiner curvature are explored as information geometry. Moreover, based on Quevedo's theory, the Legendre invariant geometry is investigated for the black hole. We also study the relationship between the scalar curvatures of the above several metrics and the phase transitions produced from the heat capacity.     

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.     

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.     

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.     

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.     

Thermodynamic Properties and Thermodynamic Geometries of Black p-Branes

The heat capacity and the electric capacitance of the black p-branes(BPB) are generally defined,then they are calculated for some special processes.It is found that the Ruppeiner thermodynamic geometry of BPB is flat.Finally,we give some discussions for the flatness of the Ruppeiner thermodynamic geometry of BPB and some black holes.     

Black holes and critical points in moduli space

We study the stabilization of scalars near a supersymmetric black hole horizon using the equation of motion of a particle moving in a potential and background metric. When the relevant 4-dimensional theory is described by special geometry, the generic properties of the critical points of this potential can be studied. We find that the extremal value of the central charge provides the minimal value of the BPS mass and of the potential under the condition that the moduli space metric is positive at the critical point. This is a property of a regular special geometry. We also study the critical points in all N 2 supersymmetric theories. We relate these ideas to the Weinhold and Ruppeiner metrics introduced in the geometric approach to thermodynamics and used for the study of critical phenomena.     

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.     

Geometry of Black Hole Thermodynamics

The Hessian of the entropy function can be thought of as a metric tensor on the state space. In the context of thermodynamical fluctuation theory Ruppeiner has argued that the Riemannian geometry of this metric gives insight into the underlying statistical mechanical system; the claim is supported by numerous examples. We study this geometry for some families of black holes. It is flat for the BTZ and Reissner–Nordström black holes, while curvature singularities occur for the Reissner–Nordström–anti–de Sitter and Kerr black holes.     

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.     

Fine micro-thermal structures for Reissner-Nordstr?m black hole

We solve the condundrum on whether the molecules of the Reissner-Nordstr?m black hole interact through the Ruppeiner thermodynamic geometry, basing our study on the concept of the black hole molecule proposed in [Phys. Rev. Lett. 115 (2015) 111302] and choosing the appropriate extensive variables. Our results show that the Reissner-Nordstr?m black hole is indeed an interaction system that may be dominated by repulsive interaction. More importantly, with the help of a novel quantity, namely the thermal-charge density, we describe the fine micro-thermal structures of the Reissner-Nordstr?m black hole in detail. Three different phases are presented, namely the free, interactive, and balanced phases. The thermal-charge density plays a role similar to the order parameter, and the back hole undergoes a new phase transition between the free phase and interactive phase. The competition between the free phase and interactive phase exists, which leads to extreme behavior of the temperature of the Reissner-Nordstr?m black hole. For the extreme Reissner-Nordstr?m black hole, the entire system is completely in the interactive phase. More importantly, we provide the thermodynamic micro-mechanism for the formation of the naked singularity of the Reissner-Nordstr?m black hole.     

Nucleation on a sphere: the roles of curvature,confinement and ensemble

ABSTRACT

By combining Monte Carlo simulations and analytical models, we demonstrate and explain how the gas-to-liquid phase transition of colloidal systems confined to a spherical surface depends on the curvature and size of the surface, and on the choice of thermodynamic ensemble. We find that the geometry of the surface affects the shape of the free energy profile and the size of the critical nucleus by altering the perimeter–area ratio of isotropic clusters. Confinement to a smaller spherical surface results in both a lower nucleation barrier and a smaller critical nucleus size. Furthermore, the liquid domain does not grow indefinitely on a sphere. Saturation of the liquid density in the grand canonical ensemble and the depletion of the gas phase in the canonical ensemble lead to a minimum in the free energy profile, with a sharp increase in free energy for additional growth beyond this minimum.     

Thermodynamic properties of asymptotically Reissner–Nordström black holes

Motivated by possible relation between Born–Infeld type nonlinear electrodynamics and an effective low-energy action of open string theory, asymptotically Reissner–Nordström black holes whose electric field is described by a nonlinear electrodynamics (NLED) are studied. We take into account a four dimensional topological static black hole ansatz and solve the field equations, exactly, in terms of the NLED as a matter field. The main goal of this paper is investigation of thermodynamic properties of the obtained black holes. Moreover, we calculate the heat capacity and find that the nonlinearity affects the minimum size of stable black holes. We also use Legendre-invariant metric proposed by Quevedo to obtain scalar curvature divergences. We find that the singularities of the Ricci scalar in Geometrothermodynamics (GTD) method take place at the Davies points.