Hyper-density functional theory of soft matter

Abstract

We present a scheme for investigating arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. Extending the grand canonical ensemble yields any given observable as an explicit hyper-density functional. Associated local fluctuation profiles follow from an exact hyper-Ornstein-Zernike equation. While the local compressibility and simple observables permit analytic treatment, complex order parameters are accessible via simulation-based supervised machine learning of neural hyper-direct correlation functionals. We exemplify efficient and accurate neural predictions for the cluster statistics of hard rods, square well rods, and hard spheres. The theory allows to treat complex observables, as is impossible in standard density functional theory.