Thermodynamic Products in Extended Phase Space

Abstract

We have examined the thermodynamic properties for a variety of spherically symmetric charged-AdS black hole (BH) solutions, including the charged AdS BH surrounded by quintessence dark energy and charged AdS BH in $f(R)$ gravity in \emph{extended phase-space}. This framework involves treating the cosmological constant as thermodynamic variable (for example: thermodynamic pressure and thermodynamic volume). Then they should behave as an analog of Van der Waal (VdW) like systems. In the extended phase space we have calculated the \emph{entropy product} and \emph{thermodynamic volume product} of all horizons. The mass (or enthalpy) independent nature of the said product signals they are \emph{universal} quantities. %Various type of phase diagram of the specific heat has been drawn. The divergence of the specific heat indicates that the second order phase transition occurs under certain condition. In the appendix-A, we have studied the thermodynamic volume products for axisymmetric spacetime and it is shown to be \emph{not universal} in nature. Finally, in appendix-B, we have studied the $P-V$ criticality of Cauchy horizon for charged-AdS BH and found to be an universal relation of critical values between two horizons as $P_{c}^{-} = P_{c}^{+}$, $v_{c}^{-}=v_{c}^{+}$, $T_{c}^{-} = -T_{c}^{+}$, $\rho_{c}^{-} = -\rho_{c}^{+}$. The symbols are defined in the main work.