Abstract

Employing a new approach toward thermodynamic phase space, we investigate the phase transition, critical behavior and microscopic structure of higher dimensional black holes in an Anti-de Sitter (AdS) background and in the presence of Power-Maxwell field. In contrast to the usual extended $P-V$ phase space where the cosmological constant (pressure) is treated as a thermodynamic variable, we fix the cosmological constant and treat the charge of the black hole (or more precisely $Q^s$) as a thermodynamic variable. Based on this new standpoint, we develop the resemblance between higher dimensional nonlinear black hole and Van der Waals liquid-gas system. We write down the equation of state as $% Q^s=Q^s(T,\psi)$, where $\psi$ is the conjugate of $Q^s$, and construct a Smarr relation based on this new phase space as $M=M(S,P,Q^s)$, while $% s=2p/(2p-1)$ and $p$ is the power parameter of the Power-Maxwell Lagrangian. We obtain the Gibbs free energy of the system and find a swallowtail behaviour in Gibbs diagrams, which is a characteristic of first-order phase transition and express the analogy between our system and van der Waals fluid-gas system. Moreover, we calculate the critical exponents and show that they are independent of the model parameters and are the same as those of Van der Waals system which is predicted by the mean field theory. Finally , we successfully explain the microscopic behavior of the black hole by using thermodynamic geometry. We observe a gap in the scalar curvature $R$ occurs between small and large black hole. The maximum amount of the gap increases as the number of dimensions increases. We finally find that character of the interaction among the internal constituents of the black hole thermodynamic system is intrinsically a strong repulsive interaction.

Highlights

  • The study of black holes thermodynamics is one of the most important subject in gravitational physics, which was anticipated by Bekenstein in 1973 [1]

  • We examine the critical behavior of power Maxwell black hole in alternative phase space where the Anti-de Sitter (AdS) radius is fixed and the electric charge of black hole can vary

  • We have investigated the critical behavior of higher dimensional AdS black holes with power-Maxwell nonlinear electrodynamics via an alternative approach toward the phase space

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Summary

THERMODYNAMIC OF HIGHER DIMENSIONAL ADS BLACK HOLE WITH A POWER MAXWELL FIELD

The action of Einstein gravity in (n + 1)-dimensional spacetime coupled to a power Maxwell field can be written as [42]. The quantity q is an integration constant related to the charge Q of the black hole per unite volume ωn−1 and one can find it by applying the Gauss law. Using Brown-York method [48], the total mass of the black hole per unit volume ωn−1 can be read as follows. In the case p = 1, the metric function Eq (3) and the electric potential Eq (8) reduce to n + 1-dimensional Reissner-Nordstrom (RN)-AdS black holes [7]. One can obtain the entropy of the black hole per unit volume ωn−1 as. If we set n = 3 and p = 1, Eq(14) reduces to the well-known Smarr relation for the 4-dimensional Einstein-Maxwell black holes [7]. Which obtained it from (2p − n) / (2p − 1) < 0 [42]

Alternative phase space
1: The behavior of isothermal
EQUATION OF STATS AND CRITICAL POINT
2: The relation
Critical exponent
GIBBS FREE ENERGY
THERMODYNAMIC GEOMETRY AND MICROSCOPIC STRUCTURE
SUMMARY AND CONCLUSION
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