A note on Maxwell’s equal area law for black hole phase transition

Abstract

The state equation of the charged AdS black hole is reviewed in the T -r + plane. With a view on the the phase transition, the T -S, P-V , P-ν graphs are plotted and then the equal area law is used in the three cases to get the phase transition point (P, T ). The analytical phase transition point relations for P-T of a charged AdS black hole has been obtained successfully. By comparing the three results, we find that the equal area law possibly cannot be used directly for the P-ν plane. According to the T -S, P-V results, we plot the P-T -Q graph and find that for a highly charged black hole a very low temperature condition is required for the phase transition.

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

“…As shown in Tables 6 and 7, the relative errors for the P-r + graph are very large while those for the P-V graph are amazingly small, leading to the conclusion that the Maxwell equal area law holds for the P-V graph, while it fails for the P-r + graph. Our numerical results here for the GB-BI-massive black holes further corroborate the findings in previous research [77,78].…”

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

“…The first is study the thermodynamics and state space geometry of black holes [18,19,25,26,[37][38][39]44,[48][49][50], which found that the black hole phase transition point meets the requirements of thermodynamic second-order phase transition. The second approach is to turn to Maxwell's equal area law and discuss the critical behavior of AdS space-time black hole system [19,[51][52][53][54][55], which have also proved the existence of second-order phase transition at black hole phase transformation point.…”

“…In this region with partial negative pressure, pressure increases together with volume ∂ M/∂ P > 0 in the isotherm, which is similar to the result obtained from the P-V phase diagram of a Van der Waals-Maxwell liquid/gas. We have a possible solution to the famous Maxwell equal area law, which was extensively applied to the study of the AdS space-time black hole thermodynamic system [49,[51][52][53][54][55]. The T -P curve yielded for the system;s biphase equilibrium and the slope expression of the biphase equilibrium curve showed that the AdS space-time black hole has the secondorder phase transition and can be in a two-phase coexistence state.…”

Maxwell’s equal area law for Lovelock thermodynamics

“…In fact, the isobar is a constant temperature line in constant pressure, so the AdS background geometry remains the same in the phase transition [51]. There have been some papers studying the this idea for different AdS black holes [51][52][53][54][55][56]. The equal area law construction can be equally applied on the T − S plane.…”

“…On the other hand, the integral of P dv do not have the scaling of Gibbs free energy, so the resulting solutions of the black hole do not share the same Gibbs free energy. The equal area law cannot be used on the P − v or T − v plane [56].…”