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The generalized thermodynamic equation related to condensation and freezing processes was derived by introducing the condensation and freezing probability function into the dynamic framework based on the statistical thermodynamic fluctuation theory. As a result, the physical mechanism of some weather phenomena covered by using (0,1) can in turn be studied and uncovered. From the generalized dynamic equations, the tendency equation of the generalized potential vorticity (GPV) is derived. From the discussion of tendency equation of GPV, in some very thin transitional areas, GPV is found nonconserved because of the introduction of the condensation and freezing probability function, even in frictionless and moist adiabatic air motion.

Plain Language Summary The condensation and freezing probability function was introduced to derive the generalized thermodynamic equation that can continuously represent all states present in the condensation and freezing processes. As a result, the physical mechanism of some weather phenomena covered by using (0,1) can, in turn, be studied and uncovered. From the discussion of tendency equation of the generalized potential vorticity, in some very thin transitional areas, generalized potential vorticity is found nonconserved because of the introduction of the freezing probability function and condensation probability function, even in frictionless and moist adiabatic air motion.