Thermodynamic Products in Extended Phase Space

TL;DR

This paper investigates the thermodynamic properties of various charged AdS black holes in extended phase space, focusing on entropy and volume products, phase transitions, and universality of critical relations across horizons.

Contribution

It introduces the calculation of entropy and volume products in extended phase space for different charged AdS black holes, revealing their universal nature and analyzing phase transition conditions.

Findings

Entropy and volume products are universal quantities for spherically symmetric charged-AdS black holes.

Second order phase transitions are indicated by divergences in specific heat.

Universal relations between critical values of pressure, volume, and temperature are found across horizons.

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.