Black Hole Phase Transitions and the Chemical Potential

In the context of extended phase space thermodynamics and the AdS-CFT correspondence, we consider the chemical potential ($μ$) dual to the number of colours ($N$) of the boundary gauge theory, in the grand canonical ensemble. By appropriately defining $μ$ via densities of thermodynamic quantities, we show that it changes sign precisely at the Hawking-Page transition for AdS-Schwarzschild and RN-AdS black holes in five dimensions, signalling the onset of quantum effects at the transition point. Such behaviour is absent for non-rotating black holes in four dimensions. For Kerr-AdS black holes in four and five dimensions, our analysis points to the fact that $μ$ can change sign in the stable black hole region, i.e above the Hawking-Page transition temperature, for a range of angular frequencies. We also analyse AdS black holes in five dimensional Gauss-Bonnet gravity, and find similar features for $μ$ as in the Kerr-AdS case.

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