Reclassification of thermal equilibrium phase-transitions in thermodynamic limit systems

For relaxor-ferroelectrics and relaxor-ferromagnets, the initial Ehrenfest-classification gives no phase-transition that contradicts the measured order-parameter, while the classification according to order-parameter and its derivatives raises the question about the relationships between the phase-transition and the specific-heat peak above and near the transition temperature. Here, based on the free-energy (F) of the thermodynamic limit systems when the external-field (h) tends 0, thermal equilibrium phase-transitions of thermodynamic limit systems with temperature (T) are reclassified into: (1) Discontinuous phase-transition. [Formula: see text] and [Formula: see text] have discontinuities in a T range; (2) Continuous phase-transition. [Formula: see text] and [Formula: see text] are continuous with T, while [Formula: see text] and [Formula: see text] have discontinuities at a T point; and (3) Diffuse phase-transition. [Formula: see text] and [Formula: see text] are continuous with T, while they are respectively equal to 0 at the transition-temperature (T_d) and diffuse-temperature (T_s). The diffuse-region of the phase-transition is [Formula: see text] and the diffuse-degree [Formula: see text], naturally giving the relation of the phase-transition to the specific-heat peak.