Insight into black hole phase transition from parametric solutions

We consider the first-order phase transition of a charged anti–de Sitter black hole and introduce a new dimensionless parameter, ω=(ΔS/πQ2)2. The parametric solutions of the two reduced volumes are obtained. Each volume is described by a piecewise analytic function. The demarcation point is located at ωd=12(23−3). The volume function is smoothly connected at the point. We show that all properties of the coexistence curve can be studied from the two volume functions. In other words, an arbitrary reduced thermodynamic variable of the two coexisting phases is only a function of ω. Some phase diagrams are plotted by using parametric solutions. We find that, when the reduced pressure P^>P^A (of order 7.4×10−4), the first-order phase transition of the black hole is similar to the van der Waals fluid. However, the similarity disappears when P^≤P^A. At a van der Waals fluidlike stage, the values of the reduced Gibbs function and the reduced density average are equal. At a non–van der Waals fluid stage, the phase diagrams have extraordinarily rich structure. It is worth pointing out that the phase transition is very important for the low-pressure case since the pressure in essence is the cosmological constant, which is normally very small. Moreover, the thermodynamic behaviors as ω→0 are discussed, from which one can easily obtain some critical exponents and amplitudes for small-large black hole phase transitions.