Riemann Surfaces and Winding Numbers of R\'enyi Phase Structure of Charged-Flat Black Holes

TL;DR

This paper explores the thermodynamic phase transitions of charged-flat black holes using a nonextensive Rnyi entropy framework, linking complex analysis, Riemann surfaces, and black hole spacetime dimensions.

Contribution

It extends the analysis of black hole phase transitions to a nonextensive Rnyi entropy context, revealing new relations between Riemann surface sheets and spacetime dimensions.

Findings

Predicted phase portraits of charged-flat black holes in different ensembles.

Established relations between Riemann surface sheets and black hole dimensions.

Connected nonextensive Rnyi parameter with cosmological constant.

Abstract

It's widely recognized that the free energy landscape captures the essentials of thermodynamic phase transitions. In this work, we extend the findings of [1] by incorporating the nonextensive nature of black hole entropy. Specifically, the connection between black hole phase transitions and the winding number of Riemann surfaces derived through complex analysis is extended to the R\'enyi entropy framework. This new geometrical and non-extensive formalism is employed to predict the phase portraits of charged-flat black holes within both the canonical and grand canonical ensembles. Furthermore, we elucidate novel relations between the number of sheets comprising the Riemann surface of the Hawking-Page and Van der Waals transitions and the dimensionality of black hole spacetimes. Notably, these new numbers are consistent with those found for charged-AdS black holes in Gibbs-Boltzmann statistics, providing another significant example of the potential connection between the cosmological constant and the nonextensive R\'enyi parameter.