Hyper-density functional theory of soft matter

TL;DR

This paper introduces a hyper-density functional framework for analyzing inhomogeneous soft matter systems, enabling the study of complex observables through analytic methods and machine learning, surpassing standard density functional theory.

Contribution

It develops a novel hyper-density functional approach that extends traditional methods, incorporating machine learning to handle complex observables in soft matter systems.

Findings

Neural networks accurately predict cluster statistics for various particles.

The framework can treat complex observables beyond standard density functional theory.

Analytic solutions are available for simple observables like local compressibility.

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.