Alternative approach towards critical behavior and microscopic structure of the higher dimensional Power-Maxwell black holes

Using an alternative approach, we investigate critical behavior and phase transition of higher dimensional charged black holes in an anti--de Sitter background and in the presence of conformally invariant Power-Maxwell electrodynamics. In this approach, we keep the cosmological constant (pressure) as a fixed thermodynamic quantity and instead allow the charge of the black hole to vary. We disclose that one can realize the critical behavior for the system in ${\mathbit{Q}}_{p}\text{\ensuremath{-}}\mathrm{\ensuremath{\Psi}}$ plane and deduce all the critical exponents of the system as well as calculate the critical point $({T}_{c},{\mathbit{Q}}_{{p}_{c}},{\mathrm{\ensuremath{\Psi}}}_{c})$, where ${\mathbit{Q}}_{p}={Q}^{2p/(2p\ensuremath{-}1)}$, $p$ is the power parameter of the Power-Maxwell Lagrangian, and $\mathrm{\ensuremath{\Psi}}$ is the conjugate of ${\mathbit{Q}}_{p}$. We observe that the critical exponents are independent of the details of the model and have the same values as Van der Waals liquid-gas system. We thus complete the analogy of these types of black holes with Van der Waals liquid-gas system. We also write down the equation of state as ${\mathbit{Q}}_{p}={\mathbit{Q}}_{p}(T,\mathrm{\ensuremath{\Psi}})$ and construct a Smarr relation based on this new phase space as $M=M(S,P,{\mathbit{Q}}_{p})$. We obtain the Gibbs free energy of the system and find a swallowtail behavior in Gibbs diagrams, which is a characteristic of the first-order phase transition. Finally, we explain the microscopic behavior of the black hole by using thermodynamic geometry. We observe a gap in the scalar curvature $R$ that occurs between small and large black holes. The maximum value of this gap increases with increasing the dimension of the spacetime. It is seen that the interaction among the internal constituents of the black hole, as a thermodynamical system, is intrinsically a strong repulsive interaction.