# Parametric Solution of a Small-Large Black Hole Coexistence Curve

## TL;DR

This paper derives a parametric analytical description of the coexistence curve for small and large charged anti-de Sitter black holes, revealing unique thermodynamic properties and critical behaviors distinct from classical fluids.

## Contribution

It introduces a parametric solution for the black hole coexistence curve, characterizing thermodynamic quantities as functions of a single parameter and analyzing phase transition properties.

## Key findings

- Reduced volumes depend only on parameter ω.
- Thermodynamic functions are piecewise analytic with a discontinuous derivative at the demarcation point.
- Critical exponents and amplitudes are obtained for phase transition analysis.

## Abstract

We consider the first-order phase transition of a charged anti-de Sitter black hole, and find that the equation of state with the conditions of the two coexisting phases, leads to the two coupled equations about the thermodynamic volumes of small black hole and large black hole. By solving the equations, it is found that each reduced volume is only a function of the parameter $\omega$ . All properties of the coexistence curve can be studied from the two volume functions. In particular, each thermodynamic quantity is described by a piecewise analytic function. The demarcation point is located at $\omega_{d}=12(2\sqrt{3}-3)$. The thermodynamic function but not its derivative, is continuous at the point. This property is completely different from that of the ven der Waals fluid. Moreover, the thermodynamic behaviors as $\omega\rightarrow0$ are discussed. From which one can easily obtain some critical exponents and amplitudes for small-large black hole phase transitions.