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A consistent and unified picture for critical phenomena of $f(R)$ AdS black holes

A consistent and unified picture for critical phenomena of charged AdS black holes in $f(R)$ gravity is drawn in this paper. Firstly, we investigate the phase transition in canonical ensemble. We derive the explicit solutions corresponding to the divergence of $C_Q$. The two solutions merge into one when the condition $Q_c=\sqrt{\frac{-1}{3R_0}}$ is satisfied. The curve of specific heat for $Q<Q_c$ has two divergent points and can be divided into three regions. Both the large radius region and the small radius region are thermodynamically stable with positive specific heat while the medium radius region is unstable with negative specific heat. However, when $Q>Q_c$, the specific heat is always positive, implying the black holes are locally stable and no phase transition will take place. Secondly, both the $T-r_+$ curve and $T-S$ curve $f(R)$ AdS black holes are investigated and they exhibit Van der Vaals like behavior as the $P-v$ curve in the former research. Critical physical quantities are obtained and they are consistent with those derived from the specific heat analysis. We carry out numerical check of Maxwell equal area law for the cases $Q=0.2Q_c, 0.4Q_c, 0.6Q_c, 0.8Q_c$. The relative errors are amazingly small and can be negligible. So the Maxwell equal area law holds for $T-S$ curve of $f(R)$ black holes. Thirdly, we establish geometrothermodynamics for $f(R)$ AdS black hole to examine the phase structure. It is shown that the Legendre invariant scalar curvature $\mathfrak{R}$ would diverge exactly where the specific heat diverges. To summarize, the above three perspectives are consistent with each other, thus providing a unified picture which deepens the understanding of critical phenomena of $f(R)$ AdS black holes.