Coexistence curves and molecule number densities of AdS black holes in the reduced parameter space

In this paper, we investigate the coexistence curves and molecule number densities of $f(R)$ anti-de Sitter (AdS) black holes and Gauss–Bonnet AdS black holes. Specifically, we work with the reduced parameter space and derive the analytic expressions of the universal coexistence curves that are independent of theory parameters. Moreover, we obtain the explicit expressions of the physical quantity describing the difference of the number densities of black hole molecules between the small and large black holes. It is found that both the coexistence curve and the difference of the molecule number densities of $f(R)$ AdS black holes coincide with those of Reissner–Nordström–AdS black holes. It may be attributed to the same equation of state they share in the reduced parameter space. The difference of the molecule number densities between the small and large Gauss–Bonnet AdS black holes exhibit different behavior. This may be attributed to the fact that the charge of the Reissner–Nordström-AdS black hole is nontrivial. Our research will not only deepen the understanding of both the physics along the coexistence curve and the underlying microscopic freedom of AdS black holes but also highlight the importance of the law of the corresponding states.

Figure 1

(a) coexistence curve of $f(R)$ AdS black holes in the reduced parameter space and (b) the number density difference of the small and large $f(R)$ AdS black holes when $\tau \ne 1$.

Figure 2

(a) Coexistence curve of Gauss–Bonnet AdS black hole in the reduced parameter space and (b) the number density difference of the small and large Gauss–Bonnet AdS black holes when $\tau \ne 1$.