  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.

High Energy Physics - Theory