Multivariate hyperdensity functional theory for inhomogeneous equilibrium fluids: From first principles to simulation-based machine learning

Abstract

Hyperdensity functional theory facilitates the investigation of the equilibrium behavior of a general order parameter or statistical mechanical observable in spatially inhomogeneous classical many-body systems. The approach is based on applying the exact Mermin-Evans classical density functional mapping to an extended ensemble. Here we present the multivariate generalization for investigating simultaneously the properties and interrelations of several different hyperobservables of choice. The resulting framework gives rise to a systematic characterization and prediction scheme for general many-body phenomena. All pertinent equilibrium averages, variances, and covariances constitute universal density functionals, as we demonstrate explicitly. Associated one-body hyperfluctuation profiles quantify the degree of correlation of the local density with first- and second-order combinations of hyperobservables. These multivariate hyperfluctuation profiles are accessible in many-body simulations and they satisfy exact hyper-Ornstein-Zernike equations, which we derive from the minimization principle in the extended multivariate ensemble. The formal structure of the theory integrates naturally with supervised machine learning, which renders all hyperdensity functionals accessible in practice via training of neural networks on simulation data. We demonstrate all salient techniques using the illustrative case of clustering in confined hard rod fluids, thereby choosing the total number of particles and the largest cluster size as the representative hyperobservables of interest. Our numerical methodology enables the efficient and successful prediction of all statistical quantities induced by the chosen hyperobservables, which we verify via comparison to test data and which we attribute to the tight interplay of first-principles and machine-learning concepts that our general approach combines.