Charged black hole in a canonical ensemble

We consider the thermodynamics of a charged black hole enclosed in a cavity. The charge in the cavity and the temperature at the walls are fixed, yielding a canonical ensemble. We derive the phase structure and stability of black hole equilibrium states. We compare our results to that of other work which uses asymptotically anti-de Sitter (AdS) boundary conditions to define the thermodynamics. The thermodynamic properties have extensive similarities which suggest that the idea of AdS holography is more dependent on the existence of the boundary than on the exact details of asymptotically AdS metrics.

Figure 1

Temperature versus $x$ for (a) $q=0.1$ and (b) $q=0.4$.

Figure 2

Phase diagram with charge $q$ on the horizontal axis and temperature $T$ on the vertical. The shaded region has three equilibria as opposed to only one outside.

Figure 3

Free energy versus temperature for (a) $q=0.1$, (b) $q=0.22\approx q_{\mathrm{crit}}$, and (c) $q=0.4$.

Figure 4

Free energy versus charge for (a) $\theta =2.33$, (b) $\theta =2.56$, and (c) $\theta =3.57$.