Maxwell Equal Area Law Research Articles

Revisiting the phase transition of AdS-Maxwell–power-Yang–Mills black holes via AdS/CFT tools

In the present work we investigate the Van der Waals-like phase transition of AdS black hole solution in the Einstein–Maxwell–power-Yang–Mills gravity (EMPYM) via different approaches. After reconsidering this phase structure in the entropy-thermal plane, we recall the nonlocal observables such as holographic entanglement entropy and two point correlation function to show that the both observables exhibit a Van der Waals-like behavior as the case of the thermal entropy. By checking the Maxwell's equal area law and calculating the critical exponent for different values of charge C and nonlinearity parameter q we confirm that the first and the second order phases persist in the holographic framework. Also the validity of the Maxwell law is governed by the proximity to the critical point.

T $T$ – S $S$ criticality of black holes with power Maxwell invariant source

In this paper, we show that black holes with PMI source exhibit the $T$ – $S$ criticality and derive the relevant critical physical quantities analytically. The values of critical quantities for the case $s\neq1$ vary from those for the case $s=1$ , showing the effect of PMI field on the critical phenomena of black holes. When $q< q_{c}$ , the $T$ – $S$ curve can be divided into three branches by a maximum point and a minimum point of Hawking temperature. With the help of specific heat analysis, we further show that the small entropy branch and the large entropy branch are stable. However, the medium entropy branch is unstable and can be removed via the technique of free energy analysis. Moreover, we examine the Maxwell equal area law and find that the relative errors for all the cases are small enough. So the existence of the PMI field does not affect the Maxwell equal area law.

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.

Validity of Maxwell equal area law for black holes conformally coupled to scalar fields in text {AdS}_5 spacetime

We investigate the P{-}V criticality and the Maxwell equal area law for a five-dimensional spherically symmetric AdS black hole with a scalar hair in the absence of and in the presence of a Maxwell field, respectively. Especially in the charged case, we give the exact P{-}V critical values. More importantly, we analyze the validity and invalidity of the Maxwell equal area law for the AdS hairy black hole in the scenarios without and with charges, respectively. Within the scope of validity of the Maxwell equal area law, we point out that there exists a representative van der Waals-type oscillation in the P{-}V diagram. This oscillating part, which indicates the phase transition from a small black hole to a large one, can be replaced by an isobar. The small and large black holes have the same Gibbs free energy. We also give the distribution of the critical points in the parameter space both without and with charges, and we obtain for the uncharged case the fitting formula of the co-existence curve. Meanwhile, the latent heat is calculated, which gives the energy released or absorbed between the small and large black hole phases in the isothermal–isobaric procedure.

Critical phenomena of regular black holes in anti-de Sitter space-time

In General Relativity, addressing coupling to a non-linear electromagnetic field, together with a negative cosmological constant, we obtain the general static spherical symmetric black hole solution with magnetic charges, which is asymptotic to anti-de Sitter (AdS) space-times. In particular, for a degenerate case the solution becomes a Hayward–AdS black hole, which is regular everywhere in the full space-time. The existence of such a regular black hole solution preserves the weak energy condition, while the strong energy condition is violated. We then derive the first law and the Smarr formula of the black hole solution. We further discuss its thermodynamic properties and study the critical phenomena in the extended phase space where the cosmological constant is treated as a thermodynamic variable as well as the parameter associated with the non-linear electrodynamics. We obtain many interesting results such as: the Maxwell equal area law in the P{-}V (or S{-}T) diagram is violated and consequently the critical point (T_*,P_*) of the first order small–large black hole transition does not coincide with the inflection point (T_c,P_c) of the isotherms; the Clapeyron equation describing the coexistence curve of the Van der Waals (vdW) fluid is no longer valid; the heat capacity at constant pressure is finite at the critical point; the various exponents near the critical point are also different from those of the vdW fluid.

Thermal phase transitions of dimensionally continued AdS black holes

We study the thermal phase transitions of charged black holes in dimensionally continued gravity in anti--de Sitter space. We find the van der Waals--like phase transition in the temperature-entropy plane of the black holes with spherical horizons in even dimensions, and there is no such phase transition of the black holes with flat and hyperbolic geometries. Near the critical inflection point, the critical exponent is computed and its value does not depend on the dimension. The Maxwell equal area law is checked to be fulfilled in the temperature-entropy diagram for the first order phase transition. In odd dimensions, there are no thermal phase transitions for the black hole with any geometry of the horizon.

Q– $$\Phi $$ Φ criticality in the extended phase space of $$(n+1)$$ ( n + 1 ) -dimensional RN-AdS black holes

In order to achieve a deeper understanding of gravity theories, it is important to further investigate the thermodynamic properties of black hole at the critical point, besides the phase transition and critical behaviors. In this paper, by using Maxwell's equal area law, we choose $T,Q,\Phi$ as the state parameters and study the phase equilibrium problem of general $(n+1)$-dimensional RN-AdS black holes thermodynamic system. The boundary of the two-phase coexistence region and its isotherm and isopotential lines are presented, which may provide theoretical foundation for studying the phase transition and phase structure of black hole systems.

Holographic research on phase transitions for a five dimensional AdS black hole with conformally coupled scalar hair

Abstract In the framework of holography, we survey the phase structure for a higher dimensional hairy black hole including the effects of the scalar field hair. It is worth emphasizing that, not only black hole entropy, but also entanglement entropy and two point correlation function exhibit the Van der Waals-like phase transition in a fixed scalar charge ensemble. Furthermore, by making use of numerical computation, we show that the Maxwell's equal area law is valid for the first order phase transition. In addition, we also discuss how the hair parameter affects the black hole's phase transition.

New perspective for black hole thermodynamics in Gauss–Bonnet–Born–Infeld massive gravity

Following earlier study regarding Einstein-Gauss-Bonnet-massive black holes in the presence of Born-Infeld nonlinear electromagnetic field [S. H. Hendi, B. Eslam Panah and S. Panahiyan, arXiv:1510.00108], we study thermodynamical structure and critical behavior of these black holes through various methods in this paper. Geometrical thermodynamics is employed to give a picture regarding phase transition of these black holes. Next, a new method is used to derive critical pressure and horizon radius of these black holes. In addition, Maxwell equal area law is employed to study the Van der Waals like behavior of these black holes. Moreover, the critical exponents are calculated and by using Ehrenfest equations, the type of the phase transitions are determined.

A consistent and unified picture for critical phenomena of f(R) AdS black holes

A consistent and unified picture for critical phenomena of charged AdS black holes in f(R) gravity is drawn in this paper. Firstly, we investigate the phase transition in canonical ensemble. We derive the explicit solutions corresponding to the divergence of CQ. The two solutions merge into one when the condition Qc=√(−1/3R0) is satisfied. The curve of specific heat for Q < Qc has two divergent points and can be divided into three regions. Both the large radius region and the small radius region are thermodynamically stable with positive specific heat while the medium radius region is unstable with negative specific heat. However, when Q > Qc, the specific heat is always positive, implying the black holes are locally stable and no phase transition will take place. Secondly, both the T−r+ curve and T−S curve f(R) AdS black holes are investigated and they exhibit Van der Vaals like behavior as the P−v curve in the former research. Critical physical quantities are obtained and they are consistent with those derived from the specific heat analysis. We carry out numerical check of Maxwell equal area law for the cases Q=0.2Qc, 0.4Qc, 0.6Qc, 0.8Qc. The relative errors are amazingly small and can be negligible. So the Maxwell equal area law holds for T−S curve of f(R) black holes. Thirdly, we establish geometrothermodynamics for f(R) AdS black hole to examine the phase structure. It is shown that the Legendre invariant scalar curvature ℜ would diverge exactly where the specific heat diverges. To summarize, the above three perspectives are consistent with each other, thus providing a unified picture which deepens the understanding of critical phenomena of f(R) AdS black holes.

Thermodynamics of noncommutative high-dimensional AdS black holes with non-Gaussian smeared matter distributions

Considering non-Gaussian smeared matter distributions, we investigate thermodynamic behaviors of the noncommutative high-dimensional Schwarzschild-Tangherlini anti-de Sitter black hole, and obtain the condition for the existence of extreme black holes. We indicate that the Gaussian smeared matter distribution, which is a special case of non-Gaussian smeared matter distributions, is not applicable for the 6- and higher-dimensional black holes due to the hoop conjecture. In particular, the phase transition is analyzed in detail. Moreover, we point out that the Maxwell equal area law maintains for the noncommutative black hole whose Hawking temperature is within a specific range, but fails for that whose the Hawking temperature is beyond this range.

Two-Phase Equilibrium Properties in Charged Topological Dilaton AdS Black Holes

We discuss phase transition of the charged topological dilaton AdS black holes by Maxwell equal area law. The two phases involved in the phase transition could coexist and we depict the coexistence region inP-vdiagrams. The two-phase equilibrium curves inP-Tdiagrams are plotted, the Clapeyron equation for the black hole is derived, and the latent heat of isothermal phase transition is investigated. We also analyze the parameters of the black hole that could have an effect on the two-phase coexistence. The results show that the black holes may go through a small-large phase transition similar to that of a usual nongravity thermodynamic system.

An equal area law for holographic entanglement entropy of the AdS-RN black hole

The Anti-de Sitter-Reissner-Nordstrom (AdS-RN) black hole in the canonical ensemble undergoes a phase transition similar to the liquid-gas phase transition, i.e. the isocharges on the entropy-temperature plane develop an unstable branch when the charge is smaller than a critical value. It was later discovered that the isocharges on the entanglement entropy-temperature plane also exhibit the same van der Waals-like structure, for spherical entangling regions. In this paper, we present numerical results which sharpen this similarity between entanglement entropy and black hole entropy, by showing that both of these entropies obey Maxwell's equal area law to an accuracy of around 1 %. Moreover, we checked this for a wide range of size of the spherical entangling region, and the equal area law holds independently of the size. We also checked the equal area law for AdS-RN in 4 and 5 dimensions, so the conclusion is not specific to a particular dimension. Finally, we repeated the same procedure for a similar, van der Waals-like transition of the dyonic black hole in AdS in a mixed ensemble (fixed electric potential and fixed magnetic charge), and showed that the equal area law is not valid in this case. Thus the equal area law for entanglement entropy seems to be specific to the AdS-RN background.

Phase transition and Clapeyron equation of black holes in higher dimensional AdS spacetime

Using Maxwell's equal area law, we study the phase transition in higher dimensional anti-de Sitter (AdS) Reissner–Nordström black holes and Kerr black holes in this paper. The coexisting region of the two phases involved in the phase transition is found and some coexisting curves are shown in figures. We also analytically investigate the parameters that affect the phase transition. To better compare with a general thermodynamic system, the Clapeyron equation is derived for the higher dimensional AdS black holes.

Clapeyron equations and fitting formula of the coexistence curve in the extended phase space of charged AdS black holes

In this paper, we first review the equal area laws and Clapeyron equations in the extended phase space of the charged anti-de Sitter black holes. With different fixed parameters, the Maxwell's equal area law holds not only in the pressure-thermodynamic volume oscillatory line, but also in the charge-electric potential and temperature-entropy oscillatory lines. The conventional Clapeyron equation is generalized and two extra equations are found. Moreover, we show that the coexistence curve of the small and large charged black holes is charge independent in the reduced parameter space for any dimension of spacetime. The highly accurate fitting formula for the coexistence curve is also presented. Using this fitting formula of the coexistence curve, we find that the Clapeyron equations are highly consistent with the calculated values. The fitting formula is also very useful for further study on the thermodynamic property of the system varying along the coexistence curve.

Maxwell’s equal-area law for Gauss–Bonnet–Anti-de Sitter black holes

Interpreting the cosmological constant \Lambda as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we study the Maxwell's equal area law of higher dimensional Gauss-Bonnet-AdS black holes in extended phase space. These black hole solutions critically behave like Van der Waals systems. It has been realized that below the critical temperature T_c the stable equilibrium is violated. We show through calculations that the critical behaviors for the uncharged black holes only appear in d=5. For the charged case, we analyse solutions in d = 5 and d = 6 separately and find that, up to some constrains, the critical behaviors only appear in the spherical topology. Using the Maxwell's construction, we also find the isobar line for which the liquid-gas-like phases coexist.

Maxwellʼs equal-area law for charged Anti-de Sitter black holes

In this Letter we present the construction of the Maxwell equal-area law in the Hawking temperature graph for a charged black hole in Anti-de Sitter background. We are able to find exact solution for the corresponding isotherm and entropies for “gaseous” (large) black holes and “liquid” (near-extremal) black holes. Isothermal construction removes the unphysical, negative heat capacity, regions. Furthermore, extremal black holes turn out to be dual to un-shrinkable molecules of Van der Waals real fluid, which may explain their thermodynamical stability.