Phase statistics and the Hamiltonian
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.