Multicritical phenomena of Reissner-Nordström anti–de Sitter black holes

We present a study of the thermodynamic critical behavior of a Reissner-Nordstr\"om anti--de Sitter (RNAdS) black hole in the vicinity of certain critical points in the set of black hole parameters $(M,Q)$ at which the heat capacity at constant charge of the black hole becomes divergent, a characteristic which suggests that the second order phase transition may occur for a RNAdS black hole. Critical exponents of the relevant thermodynamical quantites are computed. Thermal fluctuation in the canonical ensemble near criticality is calculated and anomalous behavior is found for the mean square fluctuation of certain thermodynamical variables. Scaling symmetry for the free energy near a van der Waals--like critical point is also found from which scaling laws among the critical exponents are derived. The thermodynamic analogy of a RNAdS black hole with a van der Waals's liquid-gas system is then discussed and its possible relevance to our microscopic understanding of black hole physics is speculated.