On Rényi Microstructural Aspects of Asymptotically Flat Charged Black Holes: A Novel Duality

Abstract: We explore the microstructure of asymptotically flat charged black holes through the lens of nonextensive R\'enyi statistics. A modified form of the R\'enyi entropy is proposed to incorporate compressibility effects, and geometrothermodynamics is applied within such a framework. Besides, by quantizing the black hole horizon area, we derive a microscopic description in terms of discrete degrees of freedom, with R\'enyi entropy providing a nonextensive generalization of the Bekenstein-Hawking entropy. We compute the R\'enyi partition function and probability distribution in the canonical ensemble, highlighting significant deviations from Boltzmann-Gibbs behavior at finite temperature due to nonextensive effects. The black hole undergoes a first-order Van der Waals-like phase transition between small and large configurations, characterized by discontinuities in entropy and latent heat, both of which are shown to depend on the number of area quanta. A critical threshold emerges beyond which the transition becomes athermal. Additionally, we establish a thermodynamic duality between charged-AdS black holes in Boltzmann-Gibbs statistics and charged-flat R\'enyi black holes, expressed via a conformal transformation of state variables. This duality extends naturally to higher dimensions. Our results provide new insights into the microstructure of black holes.