Extended Phase Space Research Articles

Critical phenomena and reentrant phase transition of asymptotically Reissner–Nordström black holes

By considering a small correction to the Maxwell field, we show that the resultant black hole solutions (also known as the asymptotically Reissner–Nordström black holes) undergo the reentrant phase transition and can have a novel phase behavior. We also show that such a small nonlinear correction of the Reissner–Nordström black holes has high effects on the phase structure of the solutions. It leads to a new classification in the canonical ensemble of extended phase space providing the values of the nonlinearity parameter α being α⋚4q2/7. We shall study these three classes and investigate deviations from those of the standard Reissner–Nordström solutions. Interestingly, we find that there is the reentrant phase transition for α<4q2/7, and for the case of α=4q2/7 there is no phase transition below (at) the critical point. For the last case, one finds that small and large black holes are thermodynamically distinguishable for temperatures and pressures higher than the critical ones.

The effect of dark matter on the weak cosmic censorship conjecture in the extended space

By analyzing the energy–momentum relationship of the absorbed fermions dropping into a Reissner–Nordstöm–anti-de Sitter black hole surrounded by dark matter, the laws of thermodynamic and weak cosmic censorship conjecture in the extended phase space are investigated. We find that the first law of thermodynamics is valid. However, the validity of the second law of thermodynamics depends on the density [Formula: see text] of the perfect fluid dark matter. In addition, we also find that when the fermions are absorbed, the structures of black hole surrounded by dark matter would not change. Therefore, weak cosmic censorship conjecture holds for the extreme black holes and the non-extreme black holes.

Divergence behavior of thermodynamic curvature scalar at critical point in the extended phase space of generic black holes

The P-V phase transition and critical behavior in the extended phase space of asymptotic Anti-de Sitter (AdS) black holes have been widely investigated, in which four critical exponents around critical point are found to be consistent with values in the mean field theory. Recently, another critical exponent ν related to divergent correlation length at critical point is proposed by using thermodynamic curvature scalar RN in the charged AdS black hole. In this paper, we develop a method to investigate the divergent behavior of RN at critical point, and find that the divergent behavior of RN around the critical point expresses a universal property in generic black holes. We further directly apply this method to investigate black holes in de Rham-Gabadadze-Tolley (dRGT) massive gravity to check this universality. Those results shed new lights on the microscopic properties of black holes.

Criticality of lower dimensional AdSd black holes

In lower dimensions, charged AdS black holes in an extended phase space, where the cosmological constant is interpreted as the thermodynamic pressure, are typically absent of liquid/gas phase transitions. We investigate the criticality of lower dimensional charged, dilatonic, asymptotically AdS (CDAdS$_d$) black holes generated from consistent truncations of RNAdS$_{d+2}$ black objects. We demonstrate that CDAdS$_{d}$ black holes in $d<4$ can exhibit rich van der Waals behavior and confirm that the associated critical exponents match those expected from mean field theory.

Solid-liquid phase transition and heat engine in an asymptotically flat Schwarzschild black hole via the Rényi extended phase space approach

Recently, it has been found that, with the R\'enyi statistics, the asymptotically flat Schwarzschild black hole can be in thermal equilibrium with infinite heat reservoir at a fixed temperature when its event horizon radius is larger than the characteristic length scale ${L}_{\ensuremath{\lambda}}=1/\sqrt{\ensuremath{\pi}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ is the nonextensivity parameter. In the R\'enyi extended phase space with the $PdV$ work term, an off shell free energy in the canonical ensemble with the thermodynamic volume as an order parameter is considered to identify a first-order Hawking-Page (HP) phase transition as a solid-liquid phase transition. It has the latent heat of fusion from solid (corresponding to thermal radiation) to liquid (corresponding to black hole) in the form of $\ensuremath{\sim}1/\sqrt{\ensuremath{\lambda}}$; this is evident in the absence of the HP phase transition in the case of an asymptotically flat Schwarzschild black hole from the Gibbs-Boltzmann statistics ($\ensuremath{\lambda}=0$). Moreover, we investigate the generalized second law of black hole thermodynamics (GSL) in R\'enyi statistics by considering the black hole as a working substance in a heat engine. Interestingly, an efficiency $\ensuremath{\eta}$ of the black hole in a Carnot cycle takes the form ${\ensuremath{\eta}}_{c}=1\ensuremath{-}{T}_{\mathrm{C}}/{T}_{\mathrm{H}}$. This confirms the validity of the GSL in the R\'enyi extended phase space.

Hawking–Page phase transitions of the black holes in a cavity

The Hawking–Page phase transitions of the Schwarzschild and charged black holes are investigated in an extended phase space, in which the black holes are enclosed in a spherical cavity of radius $$r_B$$ in asymptotically flat space. An effective thermodynamic volume $$V=4\pi r_B^3/3$$ is introduced for the black hole, and an effective pressure p is defined as the conjugate variable of V. The phase transition temperature $$T_{\mathrm{HP}}$$ and the Gibbs free energy G are systematically studied in a grand canonical ensemble with fixed electric potential $$\varPhi $$ , and $$T_{\mathrm{HP}}$$ is found to increase with p and decrease with $$\varPhi $$ . If the phase transition occurs, $$\varPhi $$ must have an upper bound, such that the cavity radius is always larger than the black hole horizon radius. These phase transition behaviors are further compared to those in the anti-de Sitter space, and the remarkable similarities and notable differences are also discussed in depth. Our work reveals the relationship of the thermodynamic properties of black holes and their specific boundary conditions in different extended phase spaces.

Exact 2+1 dimensional rotating black hole solution in f(R)-gravity and its thermodynamical properties

In this work, we obtain the exact rotating black hole solution in f ( R ) -gravity model( R is not considered as a constant here) and investigate its thermodynamics, especially phase transition and critical behavior in the anti-de-Sitter (AdS) extended phase space. One particular characteristic of this solution is that when we set J = 0 , the Massive BTZ black hole solution recovers. We also obtain the exact form of the f ( R ) model for this solution. In the thermodynamical analysis, we calculate the thermodynamical quantities like the temperature and entropy for this solution and we compare them with the BTZ corresponding quantities. Also, We show the validity of the first law of thermodynamics. After that, we investigate the stability(local and global) of this obtained solution. In the critical behavior analysis, we find that there is no evidence to show the existence of P − V criticality like the ordinary BTZ case.

Geometrical thermodynamics and P-V criticality of charged accelerating AdS black holes

The unusual asymptotic structure of the accelerating black holes led to ambiguity in their geometric characteristics and thermodynamic behavior. Motivated by the interesting properties of such black holes and the significant role of electric charge and string tension on their structure, we study the thermodynamic behavior of these black holes by two methods and examine the changes of free parameters on the thermal behavior of the black holes. First, we investigate phase transition and thermal stability of the system through the use of heat capacity in the non-extended phase space. We examine the effects of electric charge, string tension and the cosmological constant on the phase transition and stability of the system. We also find that to have a phase transition, we have to apply some constraints on the free parameters. Then, we employ the geometrical thermodynamic (GT) method to study phase transition and compare the obtained results with those of the heat capacity. Next, we work in the extended phase space by considering the cosmological constant as a dynamical pressure and evaluate the existence of van der Waals like phase transition. We obtain critical quantities and study the effective role of electric charge and string tension on these quantities. Finally, we make use of the GT method in the extended phase space and find that the results of the GT method, heat capacity and P−V diagram lead to a consistent conclusion.

Extended phase space thermodynamics for third-order Lovelock black holes with nonmaximally symmetric horizons

We study thermodynamics and critical behaviors of higher-dimensional Lovelock black holes with non-maximally symmetric horizons in the canonical ensemble of extended phase space. The effects from non-constancy of the horizon of the black hole via appearing two chargelike parameters in thermodynamic quantities of third-order Lovelock black holes are investigated. We find that Ricci flat black holes with nonconstant curvature horizon show critical behavior. This is an interesting feature that is not seen for any kind of black hole in Einstein or Lovelock gravity in the literature. We examine how various interesting thermodynamic phenomena such as standard first-order small-large black hole phase transition, a reentrant phase transition, or zeroth order phase transition happens for Ricci flat, spherical, or hyperbolic black holes with nonconstant curvature horizon depending on the values of Lovelock coefficient and chargelike parameters. While for a spherical black hole of third order Lovelock gravity with constant curvature horizon phase transition is observed only for $7\leq d \leq11$, for our solution criticality and phase transition exist in every dimension. With a proper choice of the free parameters, a large-small-large black hole phase transition occurs. This process is accompanied by a finite jump of the Gibbs free energy referred to as a zeroth-order phase transition. For the case $\kappa=-1$ a novel behavior is found for which three critical points could exist.

Nonlinearly charged black hole chemistry with massive gravitons in the grand canonical ensemble

In the context of Black Hole Chemistry (BHC), holographic phase transitions of asymptotically anti-de Sitter (AdS) charged topological black holes (TBHs) in massive gravity coupled to Power Maxwell Invariant (PMI) electrodynamics are discussed in the grand canonical (fixed $U(1)$ potential, $\Phi$) ensemble. Considering all higher-order graviton's self-interactions of (dRGT) massive gravitational field theory in arbitrary dimensions, exact TBH solutions are derived, the explicit form of the on-shell action is computed, and the associated thermodynamic quantities in the Grand Canonical Ensemble (GCE) are calculated. In addition, the validity of the first law of thermodynamics and Smarr relation are examined in the extended phase space. Regarding this model, it is shown that a van der Waals (vdW) behavior takes place in $d \ge 4$ dimensions, a typical reentrant phase transition is observed in $d \ge 5$, and both anomalous vdW and triple point phenomena are recognized in $d \ge 6$. So, the results are quite different from their counterparts in the GCE of Maxwell-massive gravity. Besides, we briefly study the critical behaviors of higher-dimensional TBHs with a conformally invariant Maxwell source as a special subgroup of our solutions for both Einstein and massive gravity theories in various ensembles.

Effects of thermal fluctuations on the Kerr–Newman–NUT–AdS black hole

This paper is devoted to studying the impact of thermal fluctuations on thermodynamics of rotating as well as charged anti-de Sitter black holes with the Newman–Unti–Tamburino (NUT) parameter. To this end, we derive the analytic expression of thermodynamic variables, namely the Hawking temperature, volume, angular velocity, and entropy within the limits of extended phase space. These variables meet the first law of thermodynamics as well as the Smarr relation in the presence of new NUT charge. To analyze the effects of quantum fluctuations, we derive the exact expression of corrected entropy, which yields modification in other thermodynamical equations of state. The local stability and phase transition of the considered black hole are also examined through specific heat. It is found that the NUT parameter increases the stability of small black holes, while the logarithmic corrections induce instability in the system.

Dyonic Born-Infeld black hole in four-dimensional Horndeski gravity

The action of four-dimensional Horndeski gravity coupled to Born-Infeld electromagnetic fields is given via the Kaluza-Klein process. Dyonic black hole solution of the theory is constructed. The metric is devoid of singularity at the origin independent of the parameter selections, this property is different from the one of Einstein-Born-Infeld black holes. Thermodynamics of the black hole is studied, thermodynamic quantities are calculated and the first law is checked to be satisfied. Thermodynamic phase transitions of the black holes are studied in extended phase space.

Van der Waals black hole as a heat engine

In this work, we attempt to study the thermodynamic behavior of static Van der Waals (VdW) black hole with an anti-de Sitter (AdS) background in the extended phase space. Treating the negative cosmological constant as thermodynamic pressure, we obtain the expressions for enthalpy, Gibb’s free energy and Helmholtz free energy. We study the stability and Joule–Thomson expansion for the black hole. Next, we construct a heat engine by considering the VdW black hole as the working substance. We investigate the maximum efficiency of the black hole heat engine for the Carnot cycle. We also discuss the work done and the efficiency of a new heat engine. Finally, we study the efficiency of the black hole heat engine for the Rankine cycle.

Geometry of AdS black hole thermodynamics in extended phase space

We consider the geometry of anti-de-Sitter (AdS) black hole thermodynamics in four dimensions, where the equation of state in the extended phase space formalism allows explicit comparison with normal fluid systems. We show that for the two-dimensional parameter manifolds considered here, the scalar curvature is proportional to the thermodynamic volume. This allows us to critically examine the applicability of geometric methods in black hole thermodynamics in extended phase space. We show how several standard features that are expected to hold in normal fluid systems impose severe restrictions on the black hole parameters, leading to the fact that several results in the current literature on the geometry of thermodynamics in extended phase space may be physically invalid. These are true for both charged and rotating AdS black holes. As a by-product of our analysis, we examine a conjecture regarding the equality of the correlation lengths of co-existing phases near criticality, in charged AdS black hole backgrounds, and find reasonable validity.

Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang--Mills theories. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang--Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang--Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.

Geometric particle-in-cell methods for the Vlasov–Maxwell equations with spin effects

We propose a numerical scheme to solve the semiclassical Vlasov–Maxwell equations for electrons with spin. The electron gas is described by a distribution function $f(t,{\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ that evolves in an extended 9-dimensional phase space $({\boldsymbol x},{{{\boldsymbol p}}}, {\boldsymbol s})$ , where $\boldsymbol s$ represents the spin vector. Using suitable approximations and symmetries, the extended phase space can be reduced to five dimensions: $(x,{{p_x}}, {\boldsymbol s})$ . It can be shown that the spin Vlasov–Maxwell equations enjoy a Hamiltonian structure that motivates the use of the recently developed geometric particle-in-cell (PIC) methods. Here, the geometric PIC approach is generalized to the case of electrons with spin. Total energy conservation is very well satisfied, with a relative error below $0.05\,\%$ . As a relevant example, we study the stimulated Raman scattering of an electromagnetic wave interacting with an underdense plasma, where the electrons are partially or fully spin polarized. It is shown that the Raman instability is very effective in destroying the electron polarization.

Hayward black hole heat engine efficiency in anti-de Sitter space

In this paper, we attempt to further study the heat engine efficiency for the regular black hole (BH) with an anti-de Sitter (AdS) background where the working material is the Hayward–AdS (HAdS) BH. In the extended phase space, we investigate the heat engine efficiency of the HAdS BH by defining the cosmological constant as the thermodynamic pressure P and deriving the mechanical work from the PdV terms. Then, we obtain the relation between the efficiency and the entropy/pressure and plot these function figures. Meanwhile, we compare the relation between the HAdS BH with that of the Bardeen–AdS (BAdS) BH, where it is found that the efficiency of the HAdS BH increases with increase in the magnetic charge q in contrast to that of the BAdS BH decrease with increase in the magnetic charge q. We found that the HAdS BH is more efficient than the BAdS BH, and guess that it is related to the BH structure.

Thermodynamics of z=4 Hořava-Lifshitz black holes

Thermodynamics of $z=4$ Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz black holes in $3+1$ dimensions is studied in extended phase space. By using the scaling argument we find the Smarr relation and the first law for the black hole solutions of $z=4$ Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity. We find that it is necessary to take into account the variation of dimensionful parameters of the theory in the first law. We find that the reverse isoperimetric inequality can be violated for spherical, flat, and hyperbolic horizons and in all such cases we have black holes for which the specific heat at constant pressure and volume are positive. This provides a counterexample to a recent conjecture stating that black holes violating the reverse isoperimetric inequality are thermodynamically unstable. We find for $z=4$ Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz black holes with hyperbolic horizons that there are two critical points: one showing Van der Waals behavior, the other reverse Van der Waals behavior.

Thermodynamics and heat engines of black holes with Born–Infeld-type electrodynamics

In this paper, we have studied electrically charged black holes in a new model of nonlinear electrodynamics introduced by Kruglov in Mod. Phys. Lett. A 32, 1750201 (2017). There are two parameters for the theory and the black hole could have up to two horizons. Thermodynamics is studied in the extended phase space where the pressure is proportional to the cosmological constant. First law and the Smarr formula are derived. There are phase transitions similar to the Van der Waals liquid-gas phase transitions. Black hole is also studied as a heat engine and we have discussed how the parameters in the nonlinear electrodynamics theory affect the efficiency of the heat engine.

Critical behavior of charged AdS black holes surrounded by quintessence via an alternative phase space

Considering the variable cosmological constant in the extended phase space has a significant background in the black hole physics. It has been shown that the thermodynamic behavior of charged anti--de Sitter black hole surrounded by the quintessence in the extended phase space is similar to the van der Waals fluid. In this paper, we indicate that such a black hole admits the same criticality and van der Waals--like behavior in the nonextended phase space. In other words, we keep the cosmological constant as a fixed parameter, and instead, we consider the normalization factor as a thermodynamic variable. We show that there is a first-order small/large black hole phase transition which is analogous to the liquid/gas phase transition in fluids. We introduce a new picture of the equation of state and then we calculate the corresponding critical quantities. Moreover, we obtain the critical exponents and show that they are the same values as of the van der Waals system. Finally, we study the photon sphere and the shadow observed by a distant observer and investigate how the shadow radius may be affected by the variation of black hole parameters. We also investigate the relations between shadow radius and phase transitions and calculate the critical shadow radius where the black hole undergoes a second-order phase transition.