Solid-liquid phase transition and heat engine in an asymptotically flat Schwarzschild black hole via the Rényi extended phase space approach

Abstract

Recently, it has been found that, with the R\'enyi statistics, the asymptotically flat Schwarzschild black hole can be in thermal equilibrium with infinite heat reservoir at a fixed temperature when its event horizon radius is larger than the characteristic length scale ${L}_{\ensuremath{\lambda}}=1/\sqrt{\ensuremath{\pi}\ensuremath{\lambda}}$, where $\ensuremath{\lambda}$ is the nonextensivity parameter. In the R\'enyi extended phase space with the $PdV$ work term, an off shell free energy in the canonical ensemble with the thermodynamic volume as an order parameter is considered to identify a first-order Hawking-Page (HP) phase transition as a solid-liquid phase transition. It has the latent heat of fusion from solid (corresponding to thermal radiation) to liquid (corresponding to black hole) in the form of $\ensuremath{\sim}1/\sqrt{\ensuremath{\lambda}}$; this is evident in the absence of the HP phase transition in the case of an asymptotically flat Schwarzschild black hole from the Gibbs-Boltzmann statistics ($\ensuremath{\lambda}=0$). Moreover, we investigate the generalized second law of black hole thermodynamics (GSL) in R\'enyi statistics by considering the black hole as a working substance in a heat engine. Interestingly, an efficiency $\ensuremath{\eta}$ of the black hole in a Carnot cycle takes the form ${\ensuremath{\eta}}_{c}=1\ensuremath{-}{T}_{\mathrm{C}}/{T}_{\mathrm{H}}$. This confirms the validity of the GSL in the R\'enyi extended phase space.