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  • Featured in Physics
  • Open Access

Neural Density Functional Theory of Liquid-Gas Phase Coexistence

Florian Sammüller* and Matthias Schmidt†

Robert Evans

  • *Contact author: florian.sammueller@uni-bayreuth.de
  • †Contact author: matthias.schmidt@uni-bayreuth.de
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Phys. Rev. X 15, 011013 – Published 24 January, 2025

DOI: https://doi.org/10.1103/PhysRevX.15.011013

Abstract

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 behavior associated with liquid-gas phase coexistence in bulk and at interfaces. We analyze 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. We argue that phase coexistence and its associated signatures can be discovered as emerging phenomena via functional mappings and educated extrapolation.

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Machine Learning Predicts Liquid–Gas Transition

Published 24 January, 2025

Conventional theory has trouble predicting the conditions that will cause a liquid to boil, but a neural-network-based approach performs better.

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Popular Summary

Investigating the basic physics of complex many-body systems requires one to make approximations. Analytical approaches, which input information about underlying physical mechanisms, usually stand in contrast to brute-force numerical simulation methods, which enable quantitative predictions at the expense of increased computational demands and often physical insight. Focusing on the challenging case of liquids, we combine the virtues of simulations with the rigorous framework of classical density functional theory. Our approach yields an accurate and workable implementation of the statistical mechanics based on functional mappings represented by neural networks.

Using thermally trained neural density functionals, we investigate and connect an extensive spectrum of physical behavior arising from interparticle attraction in a many-body system. We cast new light on the entropy-versus-energy mechanism that drives bulk liquid-gas phase coexistence, the nature of pair correlation functions both away from and close to the critical point, and the subtle interfacial phenomena that occur for liquids at substrates and in capillaries. A key example is the drying transition, whereby a stable gas film intrudes between the bulk liquid and a repulsive wall. The framework that we develop provides access to a broad array of physical properties while remaining computationally efficient.

The numerical accuracy of the neural functional predictions that we have gleaned for a simple model liquid, and the ease with which these are obtained, promise exciting new perspectives for the discovery of emerging collective phenomena in soft matter.

Article Text

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Phys. Rev. X 15, 011013– Published 24 January, 2025
  • Received 28 August 2024
  • Accepted 17 December 2024
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