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.