Abstract
We take a bottom-up approach to find the interaction potential between the $AdS$ black hole molecules under mean-field approximation. We start with the equation of state of dyonic $AdS$ black holes in fixed charge ensemble and use the method of classical cluster expansion to find the mean-field potential. We show that the Lennard-Jones (LJ) potential is a feasible choice to describe the equation of state. The LJ potential describes a two-body interaction. There exists a critical distance $r_0$ such that two interacting particles repel (attract) each other for $r < r_0$ ($r > r_0$). We compute the value of $r_0$ for dyonic $AdS$ black holes and compare the result obtained from the Ruppeiner scalar curvature. Our analysis shows how the electric (and magnetic) charge effects the interaction between black hole molecules.
Highlights
AND SUMMARYBlack holes are believed to be thermal objects [1]
The area of the event horizon, the ADM mass and the surface gravity of a black hole are identified with entropy, total energy and temperature respectively [3,4] and their differential changes satisfy a relation which is similar to the first law of thermodynamics
We try to shed some light towards the understanding of anti–de Sitter (AdS) black hole microstructure
Summary
Black holes are believed to be thermal objects [1]. Different macroscopic variables of black holes are identified with those of standard thermodynamics and the relations between them resemble the laws of thermodynamics [2]. Following the earlier works [11,12,13,14], Kubiznak and Mann considered the negative cosmological constant to be thermodynamic pressure of the AdS black hole and volume covered by the event horizon to be thermodynamic volume conjugate to pressure [15,16,17,18] This gives a one to one mapping between charges AdS black holes and van der Waals fluid. Liu and Mann constructed the Ruppeiner scalar for charged AdS black holes in a temperature and volume plane and showed that the characteristic curves in the ðT; VÞ plane are similar to those of van der Waals fluid except for some interesting corners. V. (v) In Appendix A we present the construction of Ruppenier curvature scalar for dyonic AdS black hole and discuss the relation between the signature of Ruppenier scalar and the nature of interaction between black hole molecules. (vi) In Appendix B we compute the first few cluster integrals required in our computation
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