A unified picture of phase transition: from liquid-vapour systems to AdS black holes

Based on fundamental concepts of thermodynamics we examine phase transitions in black holes defined in Anti-de Sitter (AdS) spaces. The method is in line with that used a long ago to understand the liquid-vapour phase transition where the first order derivatives of Gibbs potential are discontinuous and Clausius-Clapeyron equation is satisfied. The idea here is to consider the AdS black holes as grand-canonical ensembles and study phase transition defined by the discontinuity of second order derivatives of Gibbs potential. We analytically check that this phase transition between the `smaller' and `larger' mass black holes obey Ehrenfest relations defined at the critical point and hence confirm a second order phase transition. This include both the rotating and charged black holes in Einstein gravity.