Thermodynamics of Reissner-Nordstrom-anti-de Sitter black holes in the grand canonical ensemble

TL;DR

This work analyzes Reissner–Nordström–anti-de Sitter black holes in York's grand canonical ensemble by enclosing the system in a finite-radius cavity and fixing the boundary temperature and electrostatic potential. The authors derive the reduced action, compute thermodynamic quantities, and map equilibrium by solving a dimensionless 5th-degree equation for the horizon radius, uncovering regimes with one or two black-hole branches. Local stability aligns with standard thermodynamic stability, with the higher-horizon branch stable and the lower one unstable; global stability depends on the boundary data, yielding Hawking–Page-type transitions in the large-boundary limit. In the infinity limit, different boundary prescriptions alter the energy while leaving entropy unchanged, illustrating how boundary conditions shape global thermodynamic behavior in AdS black holes.

Abstract

The thermodynamical properties of the Reissner-Nordström-anti-de Sitter black hole in the grand canonical ensemble are investigated using York's formalism. The black hole is enclosed in a cavity with finite radius where the temperature and electrostatic potential are fixed. The boundary conditions allow us to compute the relevant thermodynamical quantities, e.g. thermal energy, entropy and charge. The stability conditions imply that there are thermodynamically stable black hole solutions, under certain conditions. Instantons with negative heat capacity are also found.

Figures (9)

• Figure 1: Solutions of equation (\ref{['x^5']}) for $\overline \alpha=0$ (Reissner-Nordström) as a function of the electrostatic potential at the boundary $\phi$ for fixed values of $\overline \beta=0.1,0.3,0.6,0.9,3,9$. The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of $\overline \beta$ and $\phi$, only the solution with higher value of $x$ is stable.
• Figure 2: Solutions of equation (\ref{['x^5']}) for $\overline \alpha=0.5$ as a function of the electrostatic potential at the boundary $\phi$ for fixed values of $\overline \beta=0.1,0.3,0.6,0.9,3,9$. Notice that $\phi \lesssim .89$, as imposed by condition (\ref{['phi<']}). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of $\overline \beta$ and $\phi$, only the solution with higher value of $x$ is stable.
• Figure 3: Solutions of equation (\ref{['x^5']}) for $\overline \alpha=1$ as a function of the electrostatic potential at the boundary $\phi$ for fixed values of $\overline \beta=0.1,0.3,0.6,0.9,3,9$. The maximum value of $\phi$ for which there are solutions, $\phi_{\rm max} \simeq 0.76$ is imposed by condition (\ref{['phi<']}). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of $\overline \beta$ and $\phi$, only the solution with higher value of $x$ is stable.
• Figure 4: Solutions of equation (\ref{['x^5']}) for $\overline \alpha=5$ as a function of the electrostatic potential at the boundary $\phi$ for fixed values of $\overline \beta=0.1,0.3,0.6,0.9,3,9$. The maximum value of $\phi$ for which there are solutions, $\phi_{\rm max} \simeq 0.71$, is imposed by condition (\ref{['phi<']}). The stable solutions correspond to the upper branch of the curves. This means that when there are 2 solutions for given values of $\overline \beta$ and $\phi$, only the solution with higher value of $x$ is stable.
• Figure 5: Number of solutions of equation (\ref{['x^5']}) with $\overline \beta \to 0$, in the space $\phi \times a$, where $a$ is given by equation (\ref{['aRNADS']}). There is one black hole solution in the confined region and also for $a=1$ (i.e. infinite cosmological constant), for $\phi < \sqrt{0.5}$. There are two solutions for $\phi=0$, i.e. for the Schwarzschild-anti-de Sitter black hole.