Reclassification of thermal equilibrium phase-transitions in thermodynamic limit systems

Abstract 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.