Hawking--Page phase transitions of the black holes in a cavity

The Hawking--Page phase transitions of the Schwarzschild and charged black holes are investigated in an extended phase space, in which the black holes are enclosed in a spherical cavity of radius $r_B$ in asymptotically flat space. An effective thermodynamic volume $V=4\pi r_B^3/3$ is introduced for the black hole, and an effective pressure $p$ is defined as the conjugate variable of $V$. The phase transition temperature $T_{\rm HP}$ and the Gibbs free energy $G$ are systematically studied in a grand canonical ensemble with fixed electric potential $\Phi$, and $T_{\rm HP}$ is found to increase with $p$ and decrease with $\Phi$. If the phase transition occurs, $\Phi$ must have an upper bound, such that the cavity radius is always larger than the black hole horizon radius. These phase transition behaviors are further compared to those in the anti-de Sitter space, and the remarkable similarities and notable differences are also discussed in depth. Our work reveals the relationship of the thermodynamic properties of black holes and their specific boundary conditions in different extended phase spaces.