Projection method, moment free energy

We show the equivalence of the two approaches in Section HC. There, we first demonstrate that the general form of the entropy of mixing obtained by the combinatorial method can be transformed to the standard expression — Jdon(o) In n a). Second, we show that the moment free energies arrived at by the two methods are in fact equal, with the projection method being slightly more generally applicable. 

H. DERIVATION OF MOMENT FREE ENERGY A. Projection Method... 

This is identical to the projected entropy spr, Eq. (9), except for the last term. But by construction, the combinatorial entropy assumes that p0—the overall density—is among the moment densities retained in the moment free energy. The difference scomb — spt = —p0 In p is then linear in this density, and the combinatorial and projection methods therefore predict exactly the same phase behavior. 

In summary, we have shown that the projection and combinatorial methods for obtaining moment free energies give equivalent results. The only difference between the two approaches is that within the projection approach, one need not necessarily retain the zeroth moment, which is the overall density p0 = p, as one of the moment densities on which the moment free energy depends. If p0 does not appear in the excess free energy, this reduces the minimum number of independent variables of the moment free energy by one (see Section V for an example). 

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