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The correlation entropy as a functional of radial distribution function$g(r)$ (or the total correlation function $h(r)=g(r)-1$) in classical fluidshas been obtained from the second Legendre transform of the grand potential. Wefocus on the correlation entropy difference between the two typical functionalsin the mean spherical approximation (MSA) and the hypernetted chain (HNC)approximation. While the entropy functional difference between theseapproximations is of a simple form, the diagrammatic approaches in the liquidstate theory are quite different from each other. Here we clarified the gapbetween the MSA and HNC functionals by developing a field theoretic descriptionof the correlation functional theory that combines the variational principle oflower bound free energy, the conventional saddle-point approximation of areference system to be optimized based on the variational principle, and thehybrid treatment of the saddle-point approximation and the fugacity expansionfor modifying the primary optimization. Our formulation demonstrates that theMSA functional is reproduced by the first maximization of the variationalfunctional in the saddle-point approximation, and that the HNC functional isobtained from the improved maximization of the virial term due to the fugacityexpansion around the MSA functional. The virial term leads to the modificationof a reference system interacting via the direct correlation function, therebycreating the correlation entropy difference.

journal of physics. a, mathematical and theoretical

Bridging the gap between correlation entropy functionals in the mean  spherical and the hypernetted chain approximations: a field theoretic  description