Thermodynamic stability, critical points, and phase transitions in the theory of partial distribution functions

We consider the stability of the minimum of the thermodynamic potential treated as a functional of partial densities or correlation functions. We show that the loss of stability is related to critical points of thermodynamic functions. Curves or points of phase transitions of the first kind are determined by comparing the thermodynamic potentials of different phases, and the condition for loss of stability with respect to density fluctuations can be taken as the phase transition criterion only approximately. Phase transitions of the second kind are related to the loss of stability with respect to the pair correlation fluctuations.