On the thermodynamics of the black hole and hairy black hole transitions in the asymptotically flat spacetime with a box

We study the asymptotically flat quasi-local black hole/hairy black hole model with nonzero mass of the scalar field. We disclose effects of the scalar mass on transitions in a grand canonical ensemble with condensation behaviors of the parameter ψ₂, which is similar to approaches in holographic theories. We find that a more negative scalar mass makes the phase transition easier. We also obtain the analytical relation ψ₂ ∝ (T_c-T)^1/2 around the critical phase transition points, implying a second order phase transition. Besides the parameter ψ₂, we show that metric solutions can be used to disclose properties of the transitions. In this work, we observe that phase transitions in a box are strikingly similar to holographic transitions in AdS gravity and the similarity provides insights into holographic theories. (orig.)