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 (\(\mu\)) dual to the number of colours (\(N\)) of the boundary gauge theory, in the grand canonical ensemble. By appropriately defining \(\mu\) 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 \(\mu\) 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 \(\mu\) as in the Kerr-AdS case.