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This paper extends the Euclidean path integral formalism to account for nonextensive statistical mechanics. Concretely, we introduce a generalized Wick's rotation from real time $t$ to imaginary time $\tau$ such that, $t\rightarrow-i f_\alpha(\tau)$, where $f_\alpha$ a differentiable function and $\alpha$ is a parameter related to nonextensivity. The standard extensive formalism is recovered in the limit $\alpha\rightarrow0$ and $f_0(\tau)=\tau$. Furthermore, we apply this generalized Euclidean path integral to black hole thermodynamics and derive the generalized Wick's rotations given the nonextensive statistics. The proposed formulation enables the treatment of nonextensive statistics on the same footing as extensive Gibbs-Boltzmann statistics. Moreover, we define a universal measure, $\eta$, for the nonextensivity character of statistics. Lastly, based on the present formalism, we strengthen the equivalence between the AdS-Schwarzschild black hole in Gibbs-Boltzmann statistics and the flat-Schwarzschild black hole within R\'enyi statistics and suggest a potential reformulation of the $AdS_5$/$CFT_4$ duality., Comment: 1 figure, 20 pages

It's widely recognized that the free energy landscape captures the essentials of thermodynamic phase transitions. In this work, we extend the findings of [1] by incorporating the nonextensive nature of black hole entropy. Specifically, the connection between black hole phase transitions and the winding number of Riemann surfaces derived through complex analysis is extended to the R\'enyi entropy framework. This new geometrical and non-extensive formalism is employed to predict the phase portraits of charged-flat black holes within both the canonical and grand canonical ensembles. Furthermore, we elucidate novel relations between the number of sheets comprising the Riemann surface of the Hawking-Page and Van der Waals transitions and the dimensionality of black hole spacetimes. Notably, these new numbers are consistent with those found for charged-AdS black holes in Gibbs-Boltzmann statistics, providing another significant example of the potential connection between the cosmological constant and the nonextensive R\'enyi parameter., Comment: 19 figures, 39 pages

Charged-flat black holes in the R\'enyi extended phase space demonstrate phase structures akin to those of a van der Waals fluid in four-dimensional spacetime and mirror the behaviors of Reissner-Nordstrom-Anti-de-Sitter black holes within the standard Gibbs-Boltzmann extended phase space. This study delves into the dynamics of states initially positioned within the unstable spinodal region of the phase space associated with the charged-flat black hole when subjected to time-periodic thermal perturbations. Our analysis based on the Mel'nikov method reveals that chaos emerges when the $\delta$ parameter surpasses a critical threshold, $\delta_c$. This critical quantity is dependent on the black hole charge; notably, a larger value of $Q$ impedes the onset of chaos. Furthermore, we examine the effects of space-periodic thermal perturbations on its equilibrium state and find that chaos invariably occurs, irrespective of the perturbation amplitude. Hence, the chaotic dynamics observed in the analysis of charged-flat black holes under R\'enyi statistics exhibit resemblances to those of asymptotically AdS-charged black holes investigated via the Gibbs-Boltzmann formalism. This serves as yet another example of a potential and significant connection between the cosmological constant and the nonextensivity R\'enyi parameter., Comment: 23 pages, 9 figures

In this paper, we extend the proposed setup in [1,2] for finding the topological charges associated with the Hawking-Page and Van-der-Waals transition points as well as equilibrium phases to catch the nonextensive nature of the black hole entropy, Rigorously speaking we incorporate the R\'enyi statistics formalism in off-shell Bragg-Williams free energy landscape to examine topologically the Hawking-Page phase transition related to the uncharged/charged-flat black hole in grand canonical and the Van-der- Waals transition in the canonical ensemble and where a vortex/anti-vortex structure is found. For this purpose, we introduce three mappings, the $\psi$- and $\xi$-mapping, for phase transitions classification and the $\eta$-mapping for equilibrium phases classification. We found that Hawking-Page and Van-der-Waals phase transitions belong to different topological classes and exhibit an interplay of total charge values hinting to a possible new correspondence. Our topological study provides further substantiation for a possible conjecture positing a correspondence between the thermodynamic characteristics of black holes in asymptotically flat spacetime using R\'enyi statistics, and those in asymptotically Anti-de-Sitter spacetime employing Gibbs-Boltzmann statistics., Comment: 37 pages, 18 figures, and 3 tables

Motivated by the nonextensive nature of entropy in gravitational context and the Gauge/Gravity duality, black hole thermodynamics has been attracting intense emphasis in the literature. Along the present work, we investigate some features of the phase structure and critical phenomena of the 4-dimensional charged black holes in asymptotically flat spacetime within the formalism of R\'enyi statistics. First, we explore the extended phase space via the R\'enyi statistics approach. Concretely, based on the modified version of the Smarr formula, we recall the equal-area law to remove the oscillatory non-physical region in the $P_R-V$ and $T_R-S_R$ planes. Then, the coexistence curves are determined, as well as the latent heat of phase change. Moreover, we prove that the critical exponent describing the behavior of the order parameter near the critical point is $\frac{1}{2}$, which is consistent with Landau's theory of continuous phase transition. Lastly, we apply the Hamiltonian approach to R\'enyi thermodynamics which provides a new and solid mathematical basis for the extension of phase space and puts more insight into an expected and profound possible connection between the nonextensivity R\'enyi parameter $\lambda$ and the cosmological constant $ \Lambda$., Comment: 28 pages, 8 figures