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Universality classes, Thermodynamics of Group Entropies, and Black Holes

Henrik Jeldtoft Jensen, Petr Jizba, Piergiulio Tempesta

Abstract

Conventional Boltzmann--Gibbs statistical mechanics successfully describes systems with weak to moderate correlations, where the number of accessible configurations $W(N)$ grows exponentially with the number of degrees of freedom~$N$. However, this framework breaks down for systems with strong correlations or long-range interactions, for which the configuration space exhibits non-exponential growth. While numerous generalized entropies have been proposed to address this limitation, a coherent link to classical thermodynamic laws has remained elusive. Here, we propose group entropies as a unifying framework, defining universality classes of entropies through the asymptotic scaling of $W(N)$, each yielding an extensive entropy. We show that this approach provides the basis for a consistent thermodynamic formulation beyond the Boltzmann--Gibbs paradigm. In particular, by expressing these entropies in terms of thermodynamic state variables and taking the thermodynamic limit, we demonstrate their consistency with classical thermodynamics, in close analogy to the emergence of the Clausius entropy from the Boltzmann--Gibbs formalism. Focusing on the zeroth thermodynamic law, we identify the empirical temperature and, by using Carathéodory's formulation of the second law, we derive the associated absolute temperature. As an application of the thermodynamic framework obtained, we analyze black-hole thermodynamics using the group entropy class corresponding to stretched-exponential behavior of $W(N)$. In particular, we show that a hallmark property of black holes -- their negative specific heat -- emerges naturally within this framework while the entropy remains extensive. This result holds for the stretched-exponential entropies associated with both the Bekenstein--Hawking and Barrow entropy scalings.

Universality classes, Thermodynamics of Group Entropies, and Black Holes

Abstract

Conventional Boltzmann--Gibbs statistical mechanics successfully describes systems with weak to moderate correlations, where the number of accessible configurations grows exponentially with the number of degrees of freedom~. However, this framework breaks down for systems with strong correlations or long-range interactions, for which the configuration space exhibits non-exponential growth. While numerous generalized entropies have been proposed to address this limitation, a coherent link to classical thermodynamic laws has remained elusive. Here, we propose group entropies as a unifying framework, defining universality classes of entropies through the asymptotic scaling of , each yielding an extensive entropy. We show that this approach provides the basis for a consistent thermodynamic formulation beyond the Boltzmann--Gibbs paradigm. In particular, by expressing these entropies in terms of thermodynamic state variables and taking the thermodynamic limit, we demonstrate their consistency with classical thermodynamics, in close analogy to the emergence of the Clausius entropy from the Boltzmann--Gibbs formalism. Focusing on the zeroth thermodynamic law, we identify the empirical temperature and, by using Carathéodory's formulation of the second law, we derive the associated absolute temperature. As an application of the thermodynamic framework obtained, we analyze black-hole thermodynamics using the group entropy class corresponding to stretched-exponential behavior of . In particular, we show that a hallmark property of black holes -- their negative specific heat -- emerges naturally within this framework while the entropy remains extensive. This result holds for the stretched-exponential entropies associated with both the Bekenstein--Hawking and Barrow entropy scalings.
Paper Structure (20 sections, 164 equations)

This paper contains 20 sections, 164 equations.

Table of Contents

  1. Introduction
  2. Group entropies
  3. Zeroth law of thermodynamics: empirical temperature and pressure
  4. Carathéodory's principle and absolute temperature
  5. Empirical vs. absolute temperature in non-additive thermodynamics
  6. Thermodynamics from the micro-canonical ensemble of the group entropies
  7. Deriving the equilibrium condition using the micro-canonical group entropy
  8. First law of thermodynamics
  9. The stretched exponential entropy and the black-hole thermodynamics
  10. Micro-canonical heat capacity
  11. Black-body radiation in a black-hole spacetime
  12. Stefan--Boltzmann law
  13. Some remarks
  14. Conclusions
  15. Formal group laws
  16. ...and 5 more sections

Theorems & Definitions (3)

  • Definition 1
  • Definition 2
  • Definition 3