Why hyperdensity functionals describe any equilibrium observable.

Area of Science:

• Statistical Mechanics
• Soft Matter Physics
• Computational Physics

Background:

• Classical density functional theory (DFT) is a powerful tool for studying many-body systems.
• Understanding the equilibrium statistical mechanics of inhomogeneous soft matter systems remains a challenge.
• Existing methods often struggle with complex correlations and thermal observables.

Purpose of the Study:

• Introduce the recent hyperdensity functional theory (HDFT) for soft matter systems.
• Provide a framework for calculating arbitrary thermal observables in inhomogeneous equilibrium systems.
• Demonstrate the accuracy and efficiency of HDFT using neural network representations.

Main Methods:

• Application of classical DFT to an extended ensemble.
• Development and utilization of generalized Mermin-Evans functional relationships.
• Representation of functionals using neural networks trained via simulation-based supervised machine learning.
• Employing automatic differentiation and numerical integration for functional calculus.

Main Results:

• HDFT provides access to thermal observables in inhomogeneous many-body systems.
• Neural functionals accurately represent the generalized Mermin-Evans relationships.
• Exact sum rules, including hard wall contact theorems and hyperfluctuation Ornstein-Zernike equations, are established.
• Connections to hyperforce correlation sum rules and statistical mechanical gauge invariance are shown.

Conclusions:

• HDFT offers a robust theoretical framework for soft matter systems.
• Neural network integration enables efficient and accurate functional calculus.
• The theory provides quantitative measures of collective self-organization and insight into structuring mechanisms.