Ehrenfest scheme for $P-V$ criticality of higher dimensional charged   black holes, rotating black holes and Gauss-Bonnet AdS black holes
  To provide an analytic verification of the nature of phase transition at the critical point of $P-V$ criticality, the original expressions of Ehrenfest equations have been introduced directly. By treating the cosmological constant and its conjugate quantity as thermodynamic pressure and volume respectively, we carry out analytical check of classical Ehrenfest equations. To show that our approach is universal, we investigate not only higher-dimensional charged AdS black holes, but also rotating AdS black holes. Not only are the examples of Einstein gravity shown, but also the example of modified gravity is presented for Gauss-Bonnet AdS black holes. The specific heat at constant pressure $C_P$, the volume expansion coefficient $\alpha$ and the isothermal compressibility coefficient $\kappa_T$ are found to diverge exactly at the critical point. It has been verified that both Ehrenfest equations hold at the critical point of $P-V$ criticality in the extended phase spaces of AdS black holes. So the nature of the critical point of $P-V$ criticality of AdS black holes has been demonstrated analytically to be a second-order phase transition. These results are consistent with the nature of liquid-gas phase transition at the critical point. In this sense, our research would deepen the understanding of the relations of AdS black holes and liquid-gas systems. Moreover, our successful approaches to introduce the original expressions of Erhenfest equations directly into black hole phase transition research demonstrate again that black hole thermodynamics is closely related to classical thermodynamics, which allows us to borrow techniques from classical thermodynamics to investigate the thermodynamics of black holes. 
Jie-Xiong Mo and Wen-Biao Liu
gr-qc