Abstract
The formally exact framework of equilibrium Density Functional Theory (DFT) is capable of simultaneously and consistently describing thermodynamic and structural properties of interacting many-body systems in arbitrary external potentials. In practice, however, DFT hinges on approximate (free-)energy functionals from which density profiles (and hence the thermodynamic potential) follow via an Euler–Lagrange equation. Here, we explore a relatively simple Machine-Learning (ML) approach to improve the standard mean-field approximation of the excess Helmholtz free-energy functional of a 3D Lennard-Jones system at a supercritical temperature. The learning set consists of density profiles from grand-canonical Monte Carlo simulations of this system at varying chemical potentials and external potentials in a planar geometry only. Using the DFT formalism, we nevertheless can extract not only very accurate 3D bulk equations of state but also radial distribution functions using the Percus test-particle method. Unfortunately, our ML approach did not provide very reliable Ornstein–Zernike direct correlation functions for small distances.
Highlights
Given the massive present-day availability of computer power and data, the grown general interest in machine learning (ML) should not come as a big surprise
The additional physics that can be extracted beyond the learning set includes density profiles for external potentials outside the learning set and (i) thermodynamic bulk quantities and (ii) the two-body direct correlation function from which the radial distribution function follows
We will show that from a learning set in a planar geometry, a machine-learned functional can be constructed that is capable of predicting the 3D mechanical bulk equation of state of the homogeneous fluid, the 3D radially symmetric direct correlation function and the radial distribution function at any density, and Lennard-Jones density profiles in an arbitrary external potential in 3D
Summary
Given the massive present-day availability of computer power and data, the grown general interest in machine learning (ML) should not come as a big surprise. We will show that from a learning set in a planar geometry, a machine-learned functional can be constructed that is capable of predicting the 3D mechanical bulk equation of state of the homogeneous fluid (the pressure–density–chemical potential relations), the 3D radially symmetric direct correlation function and the radial distribution function at any density, and (in principle) Lennard-Jones density profiles in an arbitrary external potential in 3D. The agreement of these predictions against simulations varies from very good (the equation of state, radial distributions from the Percus test-particle method, and density profiles outside the learning set) to, admittedly, rather poor (direct correlation function).
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