Rényi Extended Phase Space Approach

• The extended phase space approach is a framework that replaces conventional Bekenstein–Hawking entropy with Rényi entropy to account for nonextensive gravitational effects in black hole thermodynamics.
• It reveals a first-order Hawking–Page phase transition by introducing a pressure-volume term, differentiating between stable and unstable black hole phases through a latent heat mechanism.
• This formulation enables modeling black hole heat engines under a Carnot cycle, ensuring that the generalized second law remains valid even with nonextensive effects.

The extended phase space approach provides a natural and physically motivated framework for the thermodynamics of asymptotically flat Schwarzschild black holes within the context of Rényi statistics (Promsiri et al., 2021). This perspective replaces the traditional Boltzmann–Gibbs extensivity with nonextensive effects characterized by a Rényi entropy, leading to new phase structures, a well-defined latent heat, and a direct connection between black hole thermodynamics and heat engines. Below, the central features of the Rényi extended phase space approach are systematically described.

1. Rényi Extended Phase Space: Formulation and Thermodynamics

The Rényi extended phase space arises by adopting the Rényi entropy in place of the conventional Bekenstein–Hawking (BH) or Boltzmann–Gibbs (GB) entropy. The entropy is defined as

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

where $S_{bh}$ is the standard BH area entropy and $\lambda$ is the nonextensivity parameter. As $\lambda \to 0$, $S_R$ recovers the usual extensive entropy. However, for $\lambda > 0$, $S_R$ remains well-behaved and incorporates the effects of long-range gravitational correlations.

Remarkably, in this framework, $\lambda$ admits an interpretation as a thermodynamic variable akin to the cosmological constant in AdS black hole chemistry. It is promoted to a thermodynamic pressure,

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

with its conjugate variable defined as the thermodynamic volume,

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

where $r_h$ is the event horizon radius.

The first law of black hole thermodynamics in the Rényi extended phase space thus includes the $PdV$ term, and the black hole mass is reinterpreted as an enthalpic potential $H(S_R, P)$ rather than an internal energy.

2. Phase Transitions: Analogy and Nonextensive Effects

Introducing the work term in the first law allows for constructing an equation of state in the $P$–$V$ plane analogous to conventional fluids. The asymptotically flat Schwarzschild black hole, when described with Rényi entropy, exhibits both small (unstable) and large (stable) black hole branches. The off-shell Gibbs free energy in the canonical ensemble is used as a function of the order parameter $V$ to identify phase transitions.

The Hawking–Page (HP) phase transition is realized as an analogue of a solid/liquid first-order transition. Thermal radiation corresponds to the "solid" phase, while the black hole corresponds to the "liquid" phase. The transition is characterized by a latent heat of fusion,

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

or, equivalently, since $P\sim\lambda$, $L \sim 1/\sqrt{P}$. This latent heat vanishes in the GB (extensive) limit, and therefore, the HP transition is absent for $\lambda = 0$. The emergence of the phase transition at $\lambda > 0$ underscores the central role of nonextensive effects.

The transition is identified by a discontinuous jump in the order parameter $V$ across the coexistence line, a haLLMark of a first-order transition, closely resembling the fusion of a solid to a liquid in classical thermodynamics.

3. Thermodynamic Volume as the Order Parameter

Within this extended phase space, the thermodynamic volume,

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

serves not only as the conjugate to the pressure in the thermodynamic sense, but also as the relevant order parameter for distinguishing between phases. In free energy plots, the off-shell Gibbs free energy $G_{R} = M - T_{R}S_{R}$ as a function of $V$ displays two distinct branches, with the phase transition demarcated by a discontinuous change in $V$.

This facilitates the precise identification of the solid/liquid (Hawking–Page) transition and allows unambiguous classification of the black hole and thermal phases in the canonical ensemble.

4. Generalized Second Law and Black Hole Heat Engines

The Rényi statistics framework enables rigorous examination of the generalized second law of thermodynamics (GSL) in the extended phase space. When the black hole is modeled as a working substance, the inclusion of the mechanical work term ($PdV$) enables construction of reversible thermodynamic cycles ("heat engines") involving the black hole.

Through suitable cyclic processes (e.g., Carnot cycles), the total change in entropy of the universe (black hole plus heat reservoir) is shown to satisfy the GSL. Even in processes where the black hole entropy decreases, the overall entropy (including the environment) remains non-decreasing, indicating that the GSL is respected in this nonextensive setting.

This thermodynamic consistency holds despite the use of Rényi entropy and the associated nonextensive corrections.

5. Carnot Efficiency in Rényi Extended Phase Space

The inclusion of the $PdV$ work term makes it possible to model the black hole as the working fluid of a classical heat engine—most fundamentally, in a Carnot cycle composed of two isothermal and two adiabatic legs in the $P$–$V$ space.

The efficiency of such a Carnot heat engine, denoted $\eta_c$, is derived directly using the extended phase space equation of state: $\eta_c = 1 - \frac{T_C}{T_H},$ where $T_H$ and $T_C$ are the temperatures of the hot and cold reservoirs, respectively. This is the universal classical Carnot efficiency.

Therefore, even within Rényi statistics and the extended phase space, the maximal heat engine efficiency remains bounded by the Carnot limit. This result robustly confirms the validity of the generalized second law and demonstrates that no violation arises from nonextensive effects or the modified entropy.

Summary Table

Feature	Rényi Extended Phase Space	Boltzmann–Gibbs (GB) Case
Entropy	$S_R = \frac{1}{\lambda} \ln(1 + \lambda S_{bh})$	$S_{bh}$
Pressure	$P = \frac{3\lambda}{32}$	$P = 0$
Phase Transition	1st order (solid/liquid–like HP) for $\lambda > 0$	Absent
Latent Heat ($L$)	$L \sim 1/\sqrt{\lambda}$ (nonzero for $\lambda > 0$)	$L = 0$
Thermodynamic Cycle (Carnot)	$\eta = 1 - T_C/T_H$	$\eta = 1 - T_C/T_H$
Generalized Second Law	Valid	Valid

Conclusion

The Rényi extended phase space approach advances the thermodynamic understanding of asymptotically flat Schwarzschild black holes by incorporating nonextensive entropy effects and formulating a thermodynamic pressure-volume framework. This approach predicts a first-order solid/liquid-like Hawking–Page phase transition with a nonzero latent heat that vanishes in the extensive GB limit, frames the thermodynamic volume as a true order parameter, and supports the construction of classical heat engines where Carnot efficiency and the generalized second law remain unviolated. These findings reinforce the deep connections between nonextensive statistical mechanics, black hole thermodynamics, and classical thermodynamic principles (Promsiri et al., 2021).