Validity of Maxwell Equal Area Law for Black Holes Conformally Coupled to Scalar Fields in $\text{AdS}_5$ Spacetime

TL;DR

This paper examines the phase transition behavior of five-dimensional AdS black holes with scalar hair, analyzing the validity of Maxwell's equal area law and providing critical point data, including for charged and uncharged cases.

Contribution

It offers the first detailed analysis of Maxwell's equal area law validity for AdS black holes with scalar hair, including exact critical values and phase transition characteristics.

Findings

Exact critical values for charged AdS black holes with scalar hair.

Validation of Maxwell's equal area law in certain regimes.

Distribution of critical points and co-existence curves.

Abstract

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 that 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 share the same Gibbs free energy. We also give the distribution of the critical points in the parameter space both without and with charges, and 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.