RPST in 4D-EGB Black Hole Thermodynamics

• Restricted Phase Space Thermodynamics (RPST) is a framework that employs Rényi entropy to extend traditional black hole thermodynamics in 4D Einstein–Gauss–Bonnet spacetimes.
• It introduces a dual conjugate pair via the Rényi deformation parameter and its response potential, mirroring key aspects of holographic thermodynamics.
• The formalism uncovers Van der Waals–like phase transitions and leverages geometrothermodynamics to reveal structural similarities with AdS thermodynamic systems.

A restricted phase space–like (RPST-like) thermodynamic framework has been formulated for four-dimensional Einstein–Gauss–Bonnet (4D-EGB) black holes in asymptotically flat spacetime, using Rényi entropy as a non-extensive extension of the traditional Bekenstein–Hawking prescription. This formalism yields thermodynamic structures and critical phenomena that closely mimic those observed for AdS black holes under standard RPST. It achieves this by replacing the cosmological-constant/central charge machinery with a Rényi deformation parameter and conjugate response potential, facilitating a duality between non-extensive entropy and holographic thermodynamics. The approach reveals Van der Waals-like phase transitions, even for flat black holes, and employs geometrothermodynamics (GTD) to demonstrate a deep structural similarity with the AdS/RPST case (Bhattacharjee et al., 27 Jun 2025).

1. Rényi Entropy and the RPST-like Thermodynamic Structure

The RPST-like formalism developed for 4D-EGB black holes substitutes the standard entropy with the Rényi entropy: $S = \frac{1}{\lambda} \ln[1 + \lambda S_\text{BH}]$ where $S_\text{BH} = \frac{\pi r_+^2}{G}$ is the usual Bekenstein–Hawking entropy and $\lambda$ is the Rényi parameter governing deviations from extensivity. The inverse, $\beta = 1/\lambda$, is introduced for notational convenience. The horizon radius can be written as: $$r_+ = \sqrt{ \frac{G\beta}{\pi} \left[ e^{S/\beta} - 1 \right] }$$ Substitution of $r_+$ into the mass formula (which also incorporates the Gauss–Bonnet coupling $\alpha$ and electric charge $Q$) yields an explicit dependence of thermodynamic quantities on both Rényi entropy and the deformation parameter. For instance, the mass function becomes: $$M = \frac{ \beta ( e^{S/\beta} - 1) + \pi \alpha + \pi Q^2 }{2 \sqrt{ \pi G \beta ( e^{S/\beta} - 1 ) } }$$ Crucially, scaling arguments confirm that mass remains homogeneous of first order in all extensive variables, mirroring the extensively checked property of AdS/RPST systems.

2. Dual Conjugate Pair: Rényi Deformation and Response Potential

The first law of thermodynamics in this framework incorporates both the Rényi deformation and the Gauss–Bonnet coupling: $$dM = T\,dS + \Phi\,dQ + \mathcal{A}\,d\alpha + \zeta\,d\beta$$ Here, $T$ is the temperature, $\Phi$ the electric potential, and $\mathcal{A}$ the conjugate to $\alpha$. The response potential $\zeta = (\partial M/\partial \beta)$ forms a conjugate pair $(\beta, \zeta)$, which is directly analogous to the (central charge, chemical potential) pair $(C, \mu)$ in holographic (@@@@1@@@@) thermodynamics and RPST. The duality is both quantitative (in the structure of the first law) and qualitative (in governing the scaling response of the system): a large $\beta$ (small $\lambda$) mimics a large central charge.

3. Van der Waals–like Phase Transitions and Ensemble Analysis

An unexpected but robust result of this RPST-like formalism is the occurrence of Van der Waals–type first-order phase transitions for asymptotically flat black holes. This is established in both fixed charge ($\tilde{Q}$) and fixed potential ($\tilde{\Phi}$) ensembles via analysis of T–S (temperature–entropy) and F–T (Helmholtz free energy–temperature) curves:

Ensemble	Order Parameter	Critical Features
Fixed \$\tilde{Q}\$	Entropy/T/S	Swallowtail in F–T, non-monotonic T–S
Fixed \$\tilde{\Phi}\$	Entropy/T/S	Similar features; criticality depends on rescaled \$\tilde{\alpha}\$

The phase transition is identified via the standard inflection point conditions: $\frac{dT}{dS} = 0, \quad \frac{d^2T}{dS^2}=0$ Below the critical charge, these curves feature swallowtail structures—a hallmark of first-order transitions—whereas at the critical charge the system exhibits a second-order transition. This behavior is familiar in AdS/RPST frameworks but is highly nontrivial for asymptotically flat black holes.

4. Geometrothermodynamics and Structural Correspondence

The geometrothermodynamic (GTD) approach introduces a Legendre-invariant metric on the thermodynamic phase space, with the equilibrium-state subspace characterized by: $$g = S \left( \frac{\partial M}{\partial S}\right) \left[ -\frac{\partial^2 M}{\partial S^2} dS^2 + \frac{\partial^2 M}{\partial Q^2} dQ^2 \right]$$ The scalar curvature $R_\text{GTD}$ diverges precisely where the specific heat does, establishing an exact correspondence between geometry and phase transitions. Notably, GTD analysis demonstrates that the thermodynamic geometry of the Rényi-modified flat black hole closely mimics the geometry of AdS RPST black holes, with the inverse Rényi parameter $\beta$ playing a role quantitatively similar to the central charge $C$ in holographic thermodynamics.

5. Universality, Duality, and Implications

This RPST-like construction for flat black holes has several significant implications:

• The emergence of the $(\beta,\zeta)$ pair as an effective analog of $(C, \mu)$ highlights a thermodynamic duality: deviations from extensivity (Rényi parameter) are dual to the holographic central charge.
• The presence of Van der Waals–like first-order and critical second-order phase transitions in flat black holes, typical of AdS RPST systems, points to a deeper universality in black hole thermodynamics, unifying seemingly distinct settings via entropy deformation.
• GTD results show that non-extensive effects (Rényi entropy) can encode features traditionally associated with AdS holography, suggesting a structural similarity at both the thermodynamic and geometric level.

A plausible implication is that non-extensive entropy—represented here by the Rényi formulation—may act as a “bridge” between flat-space black hole thermodynamics and the AdS/CFT paradigm, allowing aspects of holographic duality to appear in classical and quantum gravity settings where no cosmological constant or conformal boundary exists.

6. Future Directions and Broader Impact

The RPST-like program for flat black holes, especially when extended to higher-curvature gravities and alternative entropy measures, opens new pathways for exploring:

• The microscopic origins of black hole entropy in non-AdS backgrounds.
• The utility of alternative statistical mechanics formalisms (beyond Bekenstein–Hawking) for capturing universal features of gravitational thermodynamics.
• Deeper connections between the geometric structure of thermodynamic phase space and critical phenomena, potentially informing the quantum gravity–statistical mechanics interface.

This framework suggests fruitful grounds for further exploration of dualities and universalities in black hole and gravitational thermodynamics, both in and out of holographic contexts (Bhattacharjee et al., 27 Jun 2025).