Holographic Van der Waals phase transition for a hairy black hole

The Van der Waals(VdW) phase transition in a hairy black hole is investigated by analogizing its charge, temperature, and entropy as the temperature, pressure, and volume in the fluid respectively. The two point correlation function(TCF), which is dual to the geodesic length, is employed to probe this phase transition. We find the phase structure in the temperature$-$geodesic length plane resembles as that in the temperature$-$thermal entropy plane besides the scale of the horizontal coordinate. In addition, we find the equal area law(EAL) for the first order phase transition and critical exponent of the heat capacity for the second order phase transition in the temperature$-$geodesic length plane are consistent with that in temperature$-$thermal entropy plane, which implies that the TCF is a good probe to probe the phase structure of the back hole.