Hyper-density functional theory of soft matter

We present a scheme for investigating arbitrary thermal observables in spatially inhomogeneous many-body systems. Extending the equilibrium ensemble yields any given observable as an explicit hyper-density functional. Associated local fluctuation profiles follow from an exact hyper-Ornstein-Zernike equation. Simulation-based supervised machine learning trains neural networks that act as hyper-direct correlation functionals which facilitate efficient and accurate predictions. We exemplify the approach for the cluster statistics of hard rods and square well particles. The theory provides access to complex order parameters, as is impossible in standard density functional theory.