Closed and exact functions in the context of Ginzburg-Landau models

For a general vector field we exhibit two Hilbert spaces, namely the space of so called closed functions and the space of exact functions and we calculate the codimension of the space of exact functions inside the larger space of closed functions. In particular we provide a new approach for the known cases: the Glauber field and the second-order Ginzburg-Landau field, and for the case of the fourth-order Ginzburg-Landau field.