  Modern statistical thermodynamics retains the concepts employed by Landau of
the order parameter and a functional depending on it, now called the
Hamiltonian. The present paper investigates the limits of validity for the use
of the functional to describe the statistical correlations of a thermodynamic
phase, particularly in connection with the experimentally accessible scattering
of X-rays, electrons and neutrons. Guggenheim's definition for the functional
is applied to a generalized system and the associated paradoxes are analyzed.
In agreement with Landau's original hypothesis, it is demonstrated that the
minimum is equal to the thermodynamic free energy, requiring no fluctuation
correction term. Although the fluctuation amplitude becomes large in the
vicinity of a second-order phase transition in low dimensionalities, it does
not diverge and the equilibrium order parameter remains well defined.
