Extensive Black Hole Thermodynamics

• The extensive black hole thermodynamic system is a framework where black holes are modeled as macroscopic objects possessing energy, entropy, temperature, pressure, and volume with modifications from Rényi entropy.
• The model introduces a nonextensivity parameter (λ) that adjusts traditional thermodynamic laws, enabling well-defined work terms and phase transitions analogous to classical systems.
• Analyses reveal that the framework supports a Hawking–Page phase transition and a stability mechanism based on a characteristic length scale, offering a generalized view of black hole thermodynamics.

An extensive black hole thermodynamic system is a macroscopic thermodynamic framework in which black holes are treated as thermodynamic objects with well-defined energy, entropy, temperature, and additional variables such as pressure and volume. In such systems, black holes obey generalized laws of thermodynamics that parallel those of classical macroscopic systems—often extending them into domains characterized by gravity, quantum corrections, and nonextensive statistical mechanics. The extensive property refers to scaling relations and additivity typical of classical thermodynamic systems, but in black hole contexts, extensivity can acquire modified or generalized forms due to gravity’s fundamentally long-range nature, the presence of horizons, and effects from quantum gravity.

1. Foundations: Rényi Extended Phase Space and the Role of Nonextensivity

In the Rényi extended phase space approach (Promsiri et al., 2021), the Bekenstein–Hawking area entropy $S_\mathrm{bh} = \pi r_h^2$ is replaced by the Rényi entropy,

$$S_{(R)} = \frac{1}{\lambda} \ln(1 + \lambda S_\mathrm{bh}),$$

where $\lambda$ is the nonextensivity parameter. This forms a basis for nonextensive thermodynamics in gravitational systems—deviating from the standard Boltzmann–Gibbs (GB) entropy, which is recovered as $\lambda \to 0$. The modified temperature is given by

$T_{(R)} = T_h (1 + \lambda S_\mathrm{bh}),$

introducing $\lambda$-dependent corrections to the thermal spectrum. The parameter $\lambda$ does not merely characterize statistical correlations but also enables a “black hole chemistry” structure: by treating $\lambda$ as a dynamical variable, one defines an effective thermodynamic pressure,

$P = \frac{3\lambda}{32},$

with associated thermodynamic volume,

$V = \frac{4\pi}{3} r_h^3.$

Consequently, the first law is generically extended to

$dU = T_{(R)}\,dS_{(R)} - P\,dV,$

or, more naturally, with enthalpy ($M$) replacing internal energy ($U$) due to the nontrivial work term $PdV$.

2. Phase Structure: Hawking–Page Transition and Solid/Liquid Analogy

In the canonical ensemble (with thermodynamic volume as the order parameter), the Rényi extended framework supports a first-order Hawking–Page (HP) phase transition, analogous to solid/liquid melting (Promsiri et al., 2021). The HP transition occurs between:

• a low-temperature “solid” phase (thermal radiation, no black hole)
• and a high-temperature “liquid” phase (stable large black hole).

The coexistence curve is characterized by a latent heat (analogous to heat of fusion in solids),

$L \sim \frac{1}{\sqrt{\lambda}},$

which diverges in the Boltzmann–Gibbs limit $\lambda \to 0$, thus precluding the phase transition in standard Schwarzschild black hole thermodynamics. The phase transition is identified in the free energy landscape, where the off-shell free energy exhibits two minima at a critical temperature (or pressure), separated by a barrier corresponding to a first-order transition. The thermodynamic volume $V$ serves as a clear order parameter, and the $P$–$V$ phase diagram displays the familiar features of two-phase coexistence.

3. Thermodynamic Stability and the Nonextensivity Length Scale

The Rényi parameter $\lambda$ introduces a characteristic length scale,

$L_\lambda = \frac{1}{\sqrt{\pi\lambda}},$

which separates stable from unstable phases:

• For $r_h < L_\lambda$, the Schwarzschild black hole in flat space has negative heat capacity and is thermodynamically unstable.
• For $r_h > L_\lambda$, the heat capacity becomes positive and the black hole can stably coexist with an infinite heat reservoir at fixed temperature (Promsiri et al., 2021).

This stabilization mechanism underscores the departure from the purely extensive GB framework, where Schwarzschild–like black holes exhibit unremitting instability due to negative heat capacity, blocking canonical ensemble constructions and phase transitions.

4. Generalized Second Law and Black Hole Heat Engines

The inclusion of the $PdV$ term enables a complete thermodynamic cycle analysis. Considering a Carnot cycle with the black hole as the working substance, the cycle efficiency is found to be

$\eta = 1 - \frac{T_C}{T_H},$

where $T_C$ and $T_H$ are the cold and hot reservoir temperatures, respectively. This recovers the standard Carnot efficiency and confirms that the generalized second law (GSL) holds in the Rényi extended framework: no heat engine based on black holes can surpass the Carnot bound, securing the consistency of the nonextensive extension (Promsiri et al., 2021).

5. Contrast with Boltzmann–Gibbs Statistics

The table below summarizes key differences between the Rényi and GB schemes:

Feature	Rényi ($\lambda > 0$)	Boltzmann–Gibbs ($\lambda=0$)
Thermodynamic stability	Stable for $r_h > L_\lambda$	Always unstable
Phase transition	Hawking–Page (solid/liquid) exists	HP absent
$PdV$ work term	Well-defined	Absent
Canonical ensemble	Well-defined for large $r_h$	Not well-defined
Latent heat	$L \sim 1/\sqrt{\lambda}$	Diverges ($L\to\infty$)

In GB statistics, the absence of a Hawking–Page-like transition in asymptotically flat Schwarzschild black holes is rooted in the lack of a stabilizing length scale and persistent negative heat capacity.

6. Physical Interpretation and Broader Implications

The Rényi extended phase space serves as an effective analog of AdS black hole thermodynamics, but for asymptotically flat spacetimes. It provides a thermodynamic scaffold where black holes show attributes (equilibrium with a reservoir, first order phase transitions, consistent work terms) previously believed to be exclusive to negative cosmological constant scenarios. The framework is suggestive of an underlying microscopic structure: nonextensive statistics are noted for encompassing systems with long-range correlations or interactions—mirroring expected quantum gravitational microphysics.

The identification of new extensive variables (such as $\lambda$ or the effective pressure) and the demonstration of the generalized first and second laws underscore the robustness and versatility of the extensive black hole thermodynamic system concept in modern gravitational thermodynamics.

7. Key Formulas and Definitions

• Rényi entropy: $$S_{(R)} = \dfrac{1}{\lambda} \ln(1 + \lambda \pi r_h^2)$$
• Modified temperature: $T_{(R)} = T_h (1 + \lambda \pi r_h^2)$
• Nonextensivity scale: $L_\lambda = 1/\sqrt{\pi\lambda}$
• First law (Rényi): $dU = T_{(R)}\,dS_{(R)} - P\,dV$, $P=(3\lambda)/32$, $V=(4\pi/3)r_h^3$
• Carnot efficiency: $\eta_c = 1 - T_C/T_H$

• Rényi extended phase-space formalism, stability analysis, phase structure, and Carnot cycle results: (Promsiri et al., 2021).
• Failure of the extensive (GB) formulation for Schwarzschild black holes, negative heat capacity: (Promsiri et al., 2021).
• Broader context of black hole chemistry and extensivity in AdS: (Karch et al., 2015).

This synthesis clarifies how the extensive black hole thermodynamic system, particularly in the Rényi framework, provides a self-consistent, generalized model for black hole thermodynamics. It incorporates phase transitions, stability structure, and heat engine efficiency—all in analogy with, yet distinctly generalized from, ordinary laboratory thermodynamics.