Emergent Phase, Thermodynamic Geometry and Criticality of Charged Black Holes from R\'enyi Statistics

Recently, a novel emergent phase can occur from thermodynamic consideration of the asymptotically flat Reissner-Nordstr\"om black hole (RN-AF) using R\'enyi statistics. We present an analysis of the thermodynamical and mechanical stabilities of the RN-AF in both the Gibbs-Boltzmann (GB) and the alternative R\'enyi statistics when charge $q$ and electrostatic potential $\phi$ are treated as pressure and volume, respectively. Interestingly, the emergent phase of the RN-AF can be both thermodynamically and mechanically stable in some range of parameters in the framework of R\'enyi thermodynamics. With the construction of the Maxwell equal area law in $q-\phi$ plane, the coexistence line between the near-extremal black hole phase and the emergent phase can be found in some values of charge which can be associated as the vapor pressure at which the liquid and gas phases coexist. In the aspect of thermodynamic geometry, the microscopic interaction between the black hole microstructures can be repulsive in the R\'enyi description. This implies that a novel correlation between the microstates of a self-gravitating system could be emerged via the nonextensive nature of long-range interaction systems. Finally, we also investigate the critical phenomena of the RN-AF in R\'enyi statistics compared to that of the van der Waals (vdW) fluid and find that the critical exponents of the relevant physical quantities of both systems are identical. This implies that both systems are in the same universality class of the phase transition.