  We investigate the non-commutative (NC) field theory approach to the vortex
liquid system restricted to the lowest Landau level (LLL) approximation. NC
field theory effectively takes care of the phase space reduction of the LLL
physics in a $\star$-product form and introduces a new gauge invariant form of
a quartic potential of the order parameter in the Ginzburg-Landau (GL) free
energy. This new quartic interaction coupling term has a non-trivial
equivalence relation with that obtained by Br\'ezin, Nelson and Thiaville in
the usual GL framework. The consequence of the equivalence is discussed.
