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

TL;DR

This paper reviews the phase transition behavior of charged AdS black holes, deriving analytical P-T transition points using the equal area law in various thermodynamic planes and analyzing the conditions for phase transitions.

Contribution

It provides new analytical relations for the P-T phase transition points of charged AdS black holes and examines the applicability of the equal area law in different thermodynamic planes.

Findings

The equal area law may not be directly applicable in the P-ν plane.

Analytical P-T transition relations are successfully derived.

A low temperature condition is necessary for phase transition in highly charged black holes.

Abstract

The state equation of the charged AdS black hole is reviewed in the $T-r$ plane. Thinking of the phase transition, the $T-S$, $P-V$, $P-\nu$ 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 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 $P-\nu$ 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.