Aspect of classical density functional theory: crystals and interfaces in hard-disk systems and the problem of constructing functionals using machine learning methods

Abstract

The theoretical studies reported in this thesis are mainly concerned with two topics in classical density functional theory (DFT). First, we investigate the crystal–fluid interface and phase transitions in hard–disk systems. Second, we propose a novel machine learning architecture, the Functional Equation Learner, to obtain an explicit free energy functional directly from equilibrium density distributions in the presence of different external potentials. For this purpose, we briefly introduce DFT and Fundamental Measure Theory (FMT). DFT is an approach for evaluating the free energy based on particle density distributions, and FMT is a particular instance of DFT for hard spheres in three, two, and one dimensions, which gives highly accurate (in two and three dimensions) or exact (in one dimension) descriptions of homogeneous and inhomogeneous systems. For two–dimensional hard spheres (hard disks), we use the free energy functional based on FMT proposed by Roth et al. [3], which has been previously reported to show accurate thermodynamic properties for a triangular crystalline structure and crystal–fluid coexistence densities compared to Monte–Carlo simulations. For one–component hard–disk systems, our result for the surface tension of a crystal–liquid interface is in good agreement with experiments, and the melting transition is investigated. As has been confirmed in past years, the melting transition for hard disks proceeds via the formation of a “hexatic” phase. In our numerical experiment, the characteristics of a hexatic phase, i.e. dislocations, are found. Furthermore, we model genuine hard–disk mixtures and mixtures of hard disks with non–additive polymers to obtain phase diagrams and surface tensions. For genuine hard–disk mixtures, the phase diagrams are qualitatively very similar to those of three–dimensional hard spheres, where the sequence of types of phase diagrams spindle → azeotropic → eutectic is observed upon lowering the small size ratio from 1 to 0.6. For non–additive mixtures, the free energy functional is linearized with respect to the polymer density, which is analogous to a known DFT approach to the three–dimensional Asakura–Oosawa model. In this case, the typical continuous widening of the coexistence gap between fluid and solid is observed upon the addition of the smaller species. For the surface tension, it shows that the addition of a second component leads in general to a substantial decrease for both genuine and non–additive hard–disk mixtures. Furthermore, despite that FMT provides highly accurate free energy functionals for hard particles, an FMT-like treatment for a non–vanishing interaction outside the hardcore is missing. In general, in such a case, the analytical form of the free energy functional is unknown; therefore we adopt the recently introduced equation learning network and propose the Functional Equation Learner (FEQL) for this task. With flexible combination rules, composite functions from the FEQL are built from a number of basis functions and a set of weighted densities. The training, i.e. tuning the parameters in functions, is automatically done by minimizing the Euclidean distance between predicted and exact/simulated density distributions. As a result, we find well approximated free energy functionals for the hard–rod fluid (exact functional is known) and the Lennard–Jones fluid (exact functional is unknown). In both cases, the density profiles, equation of states, and the direct correlation functions delivered by the learned functionals are in good agreement with exact/simulated results, even outside the training regions.