Cluster density functional theory for lattice models based on the theory of Mobius functions

TL;DR

This paper rigorously formulates Rosenfeld's fundamental measure theory for lattice models using Mobius functions, providing a unique cluster expansion and extending the theory to various interactions and dimensions.

Contribution

It introduces a Mobius function-based formulation of lattice model free-energy functionals, ensuring a unique expansion and consistent dimensional reduction.

Findings

Established a rigorous cluster expansion for lattice models.

Redefined basic clusters to ensure correct zero-density limits.

Proved functional dimensional reduction from higher to lower dimensions.

Abstract

Rosenfeld's fundamental measure theory for lattice models is given a rigorous formulation in terms of the theory of Mobius functions of partially ordered sets. The free-energy density functional is expressed as an expansion in a finite set of lattice clusters. This set is endowed a partial order, so that the coefficients of the cluster expansion are connected to its Mobius function. Because of this, it is rigorously proven that a unique such expansion exists for any lattice model. The low-density analysis of the free-energy functional motivates a redefinition of the basic clusters (zero-dimensional cavities) which guarantees a correct zero-density limit of the pair and triplet direct correlation functions. This new definition extends Rosenfeld's theory to lattice model with any kind of short-range interaction (repulsive or attractive, hard or soft, one- or multi-component...). Finally, a proof is given that these functionals have a consistent dimensional reduction, i.e. the functional for dimension d' can be obtained from that for dimension d (d'<d) if the latter is evaluated at a density profile confined to a d'-dimensional subset.