Extended phase space thermodynamics and critical behavior of charged AdS black holes surrounded by polytropic scalar field gas

Abstract

By looking at the notion of polytropic structures as a scalar field gas behaving like dark energy, we derive a static spherically symmetric (SSS) charged black hole (BH) solution in the context of General Relativity (GR). In this paper, the $P-v$ thermodynamic phase transition of charged anti-de Sitter (AdS) BH with a polytropic background is investigated in the perspective of extended phase space, where the negative cosmological constant operates as a pressure. It follows that the phase transition, viz., $P$-$v$, reveals that the small/large BH phase transition is identical to the van der Waals liquid/gas phase transition. Then, in $P-v$ critical behavior, we proceed to describe the BH equations of state $T-P$ and numerically assess the quantities in terms of the critical triplet $(r_c,T_c,P_c)$. Moreover, critical exponents are specified and the findings show the scaling behavior of thermodynamic quantities close to criticality in a global order. Moreover, the application of geometrothermodynamic (GT) tools is essential to describe the critical phase transition points. At this point, we consider a series of GT tools, namely Weinhold, Ruppeiner, Quevedo I and II types, Hendi, Panahiyan, Eslam Panah, and Momennia (HPEM). Results are thus nontrivial, since all the GT class metrics cover, at a minimum, the physical limitation or critical point of the phase transition. Furthermore, the sparsity of Hawking radiation is another task that served to ascertain the resemblance with the blackbody radiation properties.