Neural density functional theory of liquid-gas phase coexistence

We use supervised machine learning together with the concepts of classical density functional theory to investigate the effects of interparticle attraction on the pair structure, thermodynamics, 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 attractive density functional across a broad spectrum of physical behaviour associated with liquid-gas phase coexistence in bulk and at interfaces. We analyse the bulk radial distribution function $g(r)$ obtained from automatic differentiation and the Ornstein-Zernike route and determine i) the Fisher-Widom line, i.e.\ the crossover of the asymptotic (large distance) decay of $g(r)$ from monotonic to oscillatory, ii) the (Widom) line of maximal 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 density functional minimization and the corresponding surface tension from functional line integration. We also show that the neural functional describes accurately the phenomena of drying at a hard wall and of capillary evaporation for a liquid confined in a slit pore. Our neural framework yields results that improve significantly upon standard mean-field treatments of interparticle attraction. Comparison with independent simulation results demonstrates a consistent picture of phase separation even when restricting the training to supercritical states only.