Revisiting van der Waals like behavior of f(R) AdS black holes via the two point correlation function

Abstract

Van der Waals like behavior of $f(R)$ AdS black holes is revisited via two point correlation function, which is dual to the geodesic length in the bulk. The equation of motion constrained by the boundary condition is solved numerically and both the effect of boundary region size and $f(R)$ gravity are probed. Moreover, an analogous specific heat related to $δL$ is introduced. It is shown that the $T-δL$ graphs of $f(R)$ AdS black holes exhibit reverse van der Waals like behavior just as the $T-S$ graphs do. Free energy analysis is carried out to determine the first order phase transition temperature $T_*$ and the unstable branch in $T-δL$ curve is removed by a bar $T=T_*$. It is shown that the first order phase transition temperature is the same at least to the order of $10^{-10}$ for different choices of the parameter $b$ although the values of free energy vary with $b$. Our result further supports the former finding that charged $f(R)$ AdS black holes behave much like RN-AdS black holes. We also check the analogous equal area law numerically and find that the relative errors for both the cases $θ_0=0.1$ and $θ_0=0.2$ are small enough. The fitting functions between $ \log\mid T -T_c\mid$ and $\log\midδL-δL_c\mid $ for both cases are also obtained. It is shown that the slope is around 3, implying that the critical exponent is about $2/3$. This result is in accordance with those in former literatures of specific heat related to the thermal entropy or entanglement entropy.