Hyperdensity Functional Theory of Soft Matter.

Area of Science:

• Statistical Mechanics
• Computational Physics
• Condensed Matter Theory

Background:

• Investigating thermal observables in inhomogeneous many-body systems is challenging.
• Standard density functional theory has limitations for complex observables.

Purpose of the Study:

• To present a novel theoretical framework for analyzing arbitrary thermal observables.
• To extend the grand canonical ensemble for inhomogeneous systems.
• To enable the study of complex order parameters inaccessible to standard methods.

Main Methods:

• Extending the grand canonical ensemble to define observables as hyperdensity functionals.
• Utilizing an exact hyper-Ornstein-Zernike equation for local fluctuation profiles.
• Employing simulation-based supervised machine learning for neural hyperdirect correlation functionals.

Main Results:

• The framework provides explicit hyperdensity functionals for any observable.
• Local compressibility and simple observables allow analytic treatment.
• Machine learning accurately predicts cluster statistics for hard rods, square-well rods, and hard spheres.

Conclusions:

• The presented theory offers a powerful tool for investigating complex thermal observables in inhomogeneous systems.
• This method overcomes limitations of standard density functional theory.
• Machine learning integration provides efficient and accurate predictions for challenging systems.