Alternative Derivation of the SCF Theory

The SCF equations are often derived in a slightly different way by additionally 

The starting point is the partition function, Eq. 6. Then, one proceeds to introduce collective densities fields and a pressure field U to 

The saddle point approximation of the partition functions yield the SCF theory. Let us first make the saddle point approximation in the fields. like before, we denote the values of the fields at the saddle point by an asterisk. 

Note that the saddle point values, ii2X and it2 are real. These equations correspond to Eq. 40. Inserting these saddle point values into the free energy functional, we obtain a fimctional that depends only on the composition a- 

This functional can be used to investigate the dynamics of collective composition fluctuations (cf. Sect. 5). If we proceed to make a saddle point approximation for the collective density we arrive at 

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