Thermodynamics and Van der Waals phase transition of charged black holes in flat spacetime via Rényi statistics

The phase structure and critical phenomena of the $3+1$ dimensional charged black holes in asymptotically flat spacetime are investigated within the R\'enyi statistics. As the nonextensive parameter $\ensuremath{\lambda}$ above zero, a charged black hole can be in thermodynamic equilibrium with surrounding thermal radiation and have a Hawking--Page phase transition. This gives more evidence supporting the proposed conjectured equivalence between the black hole thermodynamics in asymptotically flat spacetime via R\'enyi statistics and that in asymptotically anti--de Sitter (AdS) spacetime via Gibbs--Boltzmann statistics. The present work also provides another aspect of supporting evidence through exploring the extended phase space within the R\'enyi statistics. Working on a modified version of the Smarr formula, the thermodynamic pressure $P$ and volume $v$ of a charged black hole are found to be related to $\ensuremath{\lambda}$. The thermodynamics of asymptotically flat charged black holes via R\'enyi statistics has the Van der Waals phase structure, $P\ensuremath{-}v$ criticality and universal constant, in a similar way as that of asymptotically AdS charged black hole via Gibbs--Boltzmann statistics. This raises an interesting question of how $\ensuremath{\lambda}$ in the former system relates to $|\mathrm{\ensuremath{\Lambda}}|$ in the latter one.

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