  A stationary cylindrical vessel containing a rotating plate near the bottle
surface is partially filled with liquid. With the bottom rotating, the shape of
the liquid surface would become polygon-like. This polygon vortex phenomenon is
an ideal system to demonstrate the Laminar-Turbulent transition process. Within
the framework of equilibrium statistical mechanics, a profound comparison with
Landau's phase transition theory was applied in the symmetry-breaking aspect to
derive the evolution equation of this system phenomenologically. A comparison
between theoretical prediction and experimental data is carried out. We
concluded a considerably highly matched result, while some exceptions are
viewed as the natural result that the experiment breaks through the up-limit of
using equilibrium mechanics as an effective theory, namely breaking through the
Arnold Tongue. Some extremely complex Non-equilibrium approaches were desired
to solve this problem thoroughly in the future. So our method could be viewed
as a linear approximation of this theoretical framework.
