Van der Waals criticality in AdS black holes: a phenomenological study

AdS black holes exhibit van der Waals type phase transition. In the {\it extended} phase-space formalism, the critical exponents for any spacetime metric are identical to the standard ones. Motivated by this fact, we give a general expression for the Helmholtz free energy near the critical point which correctly reproduces these exponents. The idea is similar to the Landau model which gives a phenomenological description of the usual second order phase transition. Here two main inputs are taken into account for the analysis: (a) black holes should have van der Waals like isotherms and (b) free energy can be expressed solely as a function of thermodynamic volume and horizon temperature. Resulting analysis shows that the form of Helmholtz free energy correctly encapsulates the features of Landau function. We also discuss the {\it isolated critical point} accompanied by nonstandard values of critical exponents. The whole formalism is then extended to other two criticalities, namely $Y-X$ and $T-S$ (based on the standard; i.e. non-extended phase-space), where $X$ and $Y$ are generalized force and displacement, whereas $T$ and $S$ are horizon temperature and entropy. We observe that in the former case Gibbs free energy plays the role of Landau function, whereas in the later case that role is played by the internal energy (here it is black hole mass). Our analysis shows that, although the existence of van der Waals phase transition depends on the explicit form of the black hole metric, the values of the critical exponents are universal in nature.