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Non-extensive and quasi-homogeneous geometrothermodynamics

Hernando Quevedo, María N. Quevedo

TL;DR

The paper addresses the non-extensive nature of black hole entropy by adopting the Rényi entropy and enforcing quasi-homogeneity within geometrothermodynamics (GTD) in an extended thermodynamics setting. By analyzing the Schwarzschild black hole with the Rényi entropy, the authors show a curvature-driven second-order phase transition and the emergence of a stable phase for a finite range of the non-extensivity parameter $r$, with the Hawking temperature effectively rescaled by $e^{rS}$. The Rényi parameter is treated as an independent thermodynamic variable, and GTD provides invariant, representation-independent insights into stability and phase structure, suggesting a physically meaningful role for $r$ and guiding extensions to more general black holes.

Abstract

We study the thermodynamic properties of black holes, taking into account the non-extensive character of their entropy at the thermodynamic and statistical level. To this end, we assume that the Rényi entropy determines the fundamental thermodynamic equation of black holes and is represented by a quasi-homogeneous function. As a consequence, the Rényi parameter turns out to be an independent thermodynamic variable, which must be treated in the framework of extended thermodynamics. As a particular example, we use the formalism of geometrothermodynamics to show that the Schwarzschild black hole can become stable for certain values of the Rényi parameter.

Non-extensive and quasi-homogeneous geometrothermodynamics

TL;DR

The paper addresses the non-extensive nature of black hole entropy by adopting the Rényi entropy and enforcing quasi-homogeneity within geometrothermodynamics (GTD) in an extended thermodynamics setting. By analyzing the Schwarzschild black hole with the Rényi entropy, the authors show a curvature-driven second-order phase transition and the emergence of a stable phase for a finite range of the non-extensivity parameter , with the Hawking temperature effectively rescaled by . The Rényi parameter is treated as an independent thermodynamic variable, and GTD provides invariant, representation-independent insights into stability and phase structure, suggesting a physically meaningful role for and guiding extensions to more general black holes.

Abstract

We study the thermodynamic properties of black holes, taking into account the non-extensive character of their entropy at the thermodynamic and statistical level. To this end, we assume that the Rényi entropy determines the fundamental thermodynamic equation of black holes and is represented by a quasi-homogeneous function. As a consequence, the Rényi parameter turns out to be an independent thermodynamic variable, which must be treated in the framework of extended thermodynamics. As a particular example, we use the formalism of geometrothermodynamics to show that the Schwarzschild black hole can become stable for certain values of the Rényi parameter.

Paper Structure

This paper contains 6 sections, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-extensivity and quasi-homogeneity
  3. Quasi-homogeneous geometrothermodynamics
  4. The Schwarzschild black hole with Rényi entropy
  5. Interpretation of the results
  6. Conclusions

Figures (3)

  • Figure 1: The singularity conditions for $S =1$ as functions of the non-extensivity parameter $r$. Only the curve $CII$ crosses the zero axis, where the curvature of the metric $g^{II}$ diverges.
  • Figure 2: Mass $M$, temperature $T$, and dual non-extensivity $R$ of the Schwarzschild black hole with Rényi entropy as functions of the non-extensivity parameter $r$. The entropy is fixed as $S=1$. The doted lines represent the mass $M_0=\frac{1}{2} \sqrt{\frac{S}{\pi}}$ and temperature $T_0=\frac{1}{4\sqrt{\pi S}}$ for the Schwarzschild black hole with $r\rightarrow 0$.
  • Figure 3: Heat capacity of the Schwarzschild black hole with Rényi entropy for $S=1$ and different values of the non-extensivity parameter $r$.