Why hyperdensity functionals describe any equilibrium observable

TL;DR

This paper introduces hyperdensity functionals, a novel approach in equilibrium statistical mechanics, enabling detailed analysis of arbitrary thermal observables in inhomogeneous soft matter systems using neural network-based functional representations.

Contribution

It presents the development of hyperdensity functionals, trained via machine learning, to accurately describe correlation functions and collective behaviors in complex many-body systems.

Findings

Neural functionals accurately represent generalized Mermin-Evans relationships.

Exact sum rules and hyperfluctuation profiles are derived and interconnected.

The approach provides deep insights into structuring mechanisms in soft matter systems.

Abstract

We give an introductory account of the recent hyperdensity functional theory for the equilibrium statistical mechanics of soft matter systems [F. Samm\"uller et al., Phys. Rev. Lett. 133, 098201 (2024); 10.1103/PhysRevLett.133.098201]. Hyperdensity functionals give access to the behaviour of arbitrary thermal observables in spatially inhomogeneous equilibrium many-body systems. The approach is based on classical density functional theory applied to an extended ensemble using standard functional techniques. The associated formally exact generalized Mermin-Evans functional relationships can be represented accurately by neural functionals. These neural networks are trained via simulation-based supervised machine learning and they allow one to carry out efficient functional calculus using automatic differentiation and numerical functional line integration. Exact sum rules, including hard wall contact theorems and hyperfluctuation Ornstein-Zernike equations, interrelate the different correlation functions. We lay out close connections to hyperforce correlation sum rules [S. Robitschko et al., Commun. Phys. 7, 103 (2024); 10.1038/s42005-024-01568-y] that arise from statistical mechanical gauge invariance [J. M\"uller et al., Phys. Rev. Lett. 133, 217101 (2024); 10.1103/PhysRevLett.133.217101]. Further quantitative measures of collective self-organization are provided by hyperdirect correlation functionals and spatially resolved hyperfluctuation profiles. The theory facilitates to gain deep insight into the inherent structuring mechanisms that govern the behaviour of both simple and complex order parameters in coupled many-body systems.