Solid/Liquid Phase Transition and Heat Engine in Asymptotically Flat Schwarzschild Black Hole via the R\'enyi Extended Phase Space Approach

TL;DR

This paper explores phase transitions and heat engine efficiency of Schwarzschild black holes using Rényi statistics, revealing a solid/liquid phase transition analogous to Hawking-Page transition and confirming the generalized second law in this framework.

Contribution

It introduces a novel analysis of Schwarzschild black holes in Rényi extended phase space, identifying a first-order phase transition and evaluating black hole heat engine efficiency.

Findings

Identifies a solid/liquid phase transition in Schwarzschild black holes with Rényi statistics.

Shows the latent heat of fusion scales as 1/√λ, absent in standard statistics.

Confirms the generalized second law of thermodynamics in the Rényi framework.

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 reservior at a fixed temperature when its event horizon radius is larger than the characteristic length scale $L_\lambda=1/\sqrt{\pi \lambda}$, where $\lambda$ 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 $\sim 1/\sqrt{\lambda}$; this is evident of the absence of the HP phase transition in the case of asymptotically flat Schwarzschild black hole from the GB statistics ($\lambda=0$). 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 heat engine. Interestingly, an efficiency $\eta$ of the black hole in a Carnot cycle takes the form $\eta_c=1-T_\text{C}/T_\text{H}$. This confirms the validity of the GSL in the R\'enyi extended phase space.