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

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 (rc, Tc, Pc). 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.