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 filed. We disclose effects of the scalar mass on transitions in a grand canonical ensemble with condensation behaviors of a parameter ${\psi }_2$, which is similar to approaches in holographic theories. We find that more negative scalar mass makes the phase transition easier to happen. We also obtain an analytical relation ${\psi }_2\propto (T_c-T{)}^{1\mathrm{/}2}$ around the critical phase transition points implying a second order phase transition. Besides the parameter ${\psi }_2$, we show that metric solutions can be used to disclose properties of transitions. In this work, we observe that phase transitions in a box are strikingly similar to holographic transitions in the AdS gravity and the similarity provides insights into holographic theories.