Abstract

Using supervised machine learning together with the rigorous concepts of classical density functional theory (DFT) we investigate the pair structure and thermodynamic properties, including bulk liquid-gas coexistence and associated interfacial phenomena, in many-body systems. Local learning of the one-body direct correlation functional is based on Monte Carlo simulations of inhomogeneous systems with randomized thermodynamic conditions, randomized planar shapes of the external potential, and randomized box sizes. Focusing on the prototypical Lennard-Jones system, we test predictions of the resulting neural DFT across a broad spectrum of physical behaviour. Specifically, we analyse the bulk radial distribution function g® obtained from automatic differentiation and the Ornstein-Zernike equation and determine : i) the Fisher-Widom line, i.e., the crossover of asymptotic (large distance r) decay of g® from monotonic to oscillatory, ii) the (Widom) line of maximal true correlation length, iii) the line of maximal isothermal compressibility and iv) the spinodal by calculating the poles of the structure factor in the complex plane. The bulk binodal and the density profile of the free liquid-gas interface are obtained from DFT minimization and the corresponding surface tension from functional line integration. Our neural DFT improves significantly upon standard mean-field treatments of interparticle attraction. It also describes accurately the phenomena of drying at a hard wall and capillary evaporation of a liquid confined in a slit pore. Comparison with independent simulation results demonstrates a consistent picture of phase separation even when restricting the training to supercritical states only. We argue that phase coexistence and its associated signatures can be discovered as emerging phenomena via functional mappings and educated extrapolation [1]. Extending the neural DFT to include functional dependence on the pair potential enables new and powerful inversion of structural data for liquids to discover the underlying interaction potential, important in soft matter inverse design [2].

[1] F. Sammueller, M. Schmidt & R. Evans PRX 15 , 011013-1-23 (2025).

[2] S.F. Kampa, F. Sammueller, M. Schmidt & R. Evans PRL (to appear).