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. \(\partial F/\partial h|_{h \to 0}\) and \(\partial F/\partial T|_{h \to 0}\) have discontinuities in a T range; (2) Continuous phase-transition. \(\partial F/\partial h|_{h \to 0}\) and \(\partial F/\partial T|_{h \to 0}\) are continuous with T, while \(\partial^{2} F/\partial h\partial T|_{h \to 0}\) and \(\partial^{2} F/\partial T^{2} |_{h \to 0}\) have discontinuities at a T point; and (3) Diffuse phase-transition. \(\partial^{3} F/\partial h\partial^{2} T|_{h \to 0}\) and \(\partial^{3} F/\partial T^{3} |_{h \to 0}\) 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 \(T_{s} - T_{d}\) and the diffuse-degree \(\left( {T_{s} - T_{d} } \right)/T_{d} \times {1}00\%\) , naturally giving the relation of the phase-transition to the specific-heat peak.