Thermodynamics of general flat space and boson star quasi-local systems

We study thermodynamics of flat space/boson star systems enclosed in a scalar reflecting box with St$\ddot{u}$ckelberg mechanism. We also disclose effects of model parameters on transitions and the properties appear to be qualitatively the same as those in holographic St$\ddot{u}$ckelberg transitions. Moreover, we obtain a relation $\bar{\zeta}\thickapprox 2 \tilde{\zeta}$ and the second order characteristic exponent, which also hold in holographic superconductor theories. The similarity between quasi-local thermodynamic transitions and holographic transitions provides additional evidences that holographic theories may also exist in the quasi-local space.