Why hyperdensity functionals describe any equilibrium observable

Title data

Sammüller, Florian ; Schmidt, Matthias:
Why hyperdensity functionals describe any equilibrium observable.
In: Journal of Physics: Condensed Matter. Vol. 37 (2024) Issue 8 . - 083001.
ISSN 1361-648X
DOI: https://doi.org/10.1088/1361-648X/ad98da

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Abstract in another language

We give an introductory account of the recent hyperdensity functional theory for the equilibrium statistical mechanics of soft matter systems (Sammüller et al 2024 Phys. Rev. Lett. 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 (Robitschko et al 2024 Commun. Phys. 7 103) that arise from statistical mechanical gauge invariance (Müller et al 2024 Phys. Rev. Lett. 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.