Spinodal criterion exact free energy

Appendix A Moment (Gibbs) Free Energy for Fixed Pressure Appendix B Moment Entropy of Mixing and Large Deviation Theory Appendix C Spinodal Criterion From Exact Free Energy Appendix D Determinant Form of Critical Point Criterion References... 

Here / = 1/7 in the standard notation. From our general statements in Section in. A, the spinodal criterion derived from the exact free energy (38) must be identical to this this is shown explicitly in Appendix C. Note that the spinodal condition depends only on the (first-order) moment densities p, and the second-order moment densities py of the distribution p(cr) [given by Eqs. (40) and (41)] it is independent of any other of its properties. This simplification, which has been pointed out by a number of authors [11, 12], is particularly useful for the case of power-law moments (defined by weight functions vt>f(excess free energy only depends on the moments of order 0, 1... K — 1 of the density distribution, the spinodal condition involves only 2K— moments [up to order 2(K — 1)]. 

As expected from the general discussion in Section III. A, the criterion (57) can also be derived from the exact free energy an alternative form involving the spinodal determinant Y is given in Appendix D. Equation (57) shows that the location of critical points depend only on the moment densities p[t py, and pijk [11, 46]. For a system with an excess free energy depending only on power-law moments up to order K - 1, the critical point condition thus involves power-law moments of the parent only up to order 3 (K — 1). 

APPENDIX C SPINODAL CRITERION FROM EXACT FREE ENERGY... 

In this appendix, we apply the spinodal criterion (50) to the exact free energy (38) and show that it can be expressed in a form identical to Eq. (55). This result has been given by a number of authors [11, 12, 44], but we include it here for the sake of completeness. 

The exactness statements in Section HI can also be directly translated to the constant pressure case. The arguments above imply directly that the onset of phase coexistence is found exactly from the moment Gibbs free energy All phases are in the family (A3), because one of them (the parent) is, and the requirement of equal chemical potentials is satisfied. Spinodals and (multi-) critical points are also found exactly. Arguing as in Section HI. A and using the vector notation of Eq. (53), the criterion for such points is found as... 

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