Phase transition and thermodynamic geometry of $f(R)$ AdS black holes in the grand canonical ensemble

TL;DR

This paper studies phase transitions of charged AdS black holes in $f(R)$ gravity within the grand canonical ensemble, revealing unique characteristics and divergences in thermodynamic quantities not seen in canonical ensemble, using thermodynamic geometry tools.

Contribution

It introduces the analysis of $f(R)$ AdS black holes in the grand canonical ensemble, highlighting differences from canonical ensemble and deriving explicit thermodynamic geometric quantities.

Findings

No critical point for the $T-S$ curve in grand canonical ensemble.

Divergence of specific heat $C_\Phi$ when $0<\Phi<b$, no divergence for $\Phi>b$.

Thermodynamic scalar curvatures diverge at the same points as $C_\Phi$.

Abstract

The phase transition of four-dimensional charged AdS black hole solution in the $R+f(R)$ gravity with constant curvature is investigated in the grand canonical ensemble, where we find novel characteristics quite different from that in canonical ensemble. There exists no critical point for $T-S$ curve while in former research critical point was found for both the $T-S$ curve and $T-r_+$ curve when the electric charge of $f(R)$ black holes is kept fixed. Moreover, we derive the explicit expression for the specific heat, the analog of volume expansion coefficient and isothermal compressibility coefficient when the electric potential of $f(R)$ AdS black hole is fixed. The specific heat $C_\Phi$ encounters a divergence when $0<\Phi<b$ while there is no divergence for the case $\Phi>b$. This finding also differs from the result in the canonical ensemble, where there may be two, one or no divergence points for the specific heat $C_Q$. To examine the phase structure newly found in the grand canonical ensemble, we appeal to the well-known thermodynamic geometry tools and derive the analytic expressions for both the Weinhold scalar curvature and Ruppeiner scalar curvature. It is shown that they diverge exactly where the specific heat $C_\Phi$ diverges.