External Potential EP Theory Saddle Point Integration in

External Potential (EP) Theory Saddle Point Integration in U 

The fluctuations described by the two fields, U and W, are qualitatively different. This is already apparent from the fact that one, U, gives rise to a complex contribution to the field that acts on a chain, while the other, W, corresponds to a real one. The field U couples to the total density 0 + b and has been introduced to decouple the incompressibility constraint. Qualitatively, it con- 

Instead of enforcing the incompressibility constraint on each of the microscopic conformations, we thus only require that the single chain averages in the external field obey the constraint. We recall that the 4 a functionals of Wa = (iU + W)/2 and Wb = iU - W)/2 (cf. Eq. 28), thus Eq. 40 implicitly defines a functional U [W. Substituting the saddle point value into the free energy functional (Eq. 11), we obtain an approximate partition function 

At this point, one has two possible choices for calculating the thermodynamic averages of the microscopic composition [29]  

One can introduce a spatially varying exchange potential 5/x and calculate the thermodynamic averages with respect to the free energy functional M[W] of the EP theory. This leads to expressions similar to Eq. 27 and 

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