arXiv subjects

Alessandro Simon

Publications and source records attributed to Alessandro Simon.

6 recordsLinked to original sources

arXiv2026

Cubic-Equivariant Neural Density Functional Theory for Three-Dimensional Lattice Fluids

We construct a neural classical density functional that acts directly on unrestricted three-dimensional density profiles. As a computationally tractable test bed, we consider parallel hard cubes of side length three on a simple cubic lattice. A fully convolutional network learns the one-body direct-correlation functional $c^{(1)}[\rho]$ from data obtained with grand-canonical Monte Carlo simulations in randomized external potentials. Complete profiles are used during both training and inference; a stochastic Bernoulli mask on the output sites makes full-profile training effective without explicitly extracting and storing overlapping local density windows. Averaging the first-layer kernels over all 48 rotations and reflections of the cubic point group additionally imposes exact cubic equivariance without data augmentation. We compare the learned functional with independent simulation data and with the lattice fundamental-measure functional of Lafuente and Cuesta. The neural functional markedly improves the homogeneous equation of state and the density profile at a planar hard wall. For the anisotropic pair distribution around a fixed particle, both functionals reproduce the principal packing shells, with their relative accuracy depending on crystallographic direction. These results demonstrate neural density-functional calculations on complete three-dimensional profiles while also identifying accurate full-dimensional training data, thermodynamic consistency, and structural correlations as the central challenges for extensions to continuum fluids.

cond-mat.stat-mech↗

arXiv2026

Free-volume origin of diverging direct correlations in hard crystals: insights from an exact one-dimensional model

Second order direct correlation functions in three-dimensional hard-sphere crystals have much larger amplitudes (of the order of $1/n_{\mathrm{vac}}$ where $n_{\mathrm{vac}}$ is the vacancy concentration) and a more strongly structured spatial form of apparent shorter range than their liquid-state counterparts. We separate the two underlying questions---why the correlations are large and why they have the form they do---using the exact one-dimensional Percus functional for hard rods. A periodic crystal-like density is represented by Gaussian peaks with occupation probability $q=1-n_{\mathrm{vac}}$. The relevant thermodynamic quantity is the local insertion free volume $A(x)$. Its minimum can be written as $A_{\mathrm{min}} =\Delta_{\alpha a}+(1-\Delta_{\alpha a})n_{\mathrm{vac}}$, where $\Delta_{\alpha a}$ is the residual free volume at full occupation caused by finite localization (characterized by a width parameter $\alpha$) and lattice spacing $a$. The first and second direct correlations therefore contain the singular dependences $c^{(1)}\sim\ln A_{\mathrm{min}} $ and $c^{(2)} \sim-1/A_{\mathrm{min}} $. The observed $1/n_{\mathrm{vac}}$ dependence is the vacancy-dominated limit $n_{\mathrm{vac}}\gg\Delta_{\alpha a}$, rather than the most general result. The spatial form of $c^{(2)}$ has a separate geometrical origin: the hard-rod weight functions select configurations in which exclusion intervals and their boundaries intersect regions of small free volume. This produces plateaus, edges, and localized ridges tied to the underlying periodic density. A numerical solution of the inhomogeneous Ornstein--Zernike equation shows how these singular direct correlations are redistributed in the total and pair correlations. The model provides a minimal free-volume explanation for both the magnitude and the lattice-specific form of crystalline direct correlation functions.

cond-mat.stat-mech↗

arXiv2026

Density Profiles and Direct Correlation Functions from Density Functional Theory in Binary Hard-Sphere Crystals: Substitutional Solid and Interstitial Solid Solution

We determine the fully resolved equilibrium density profiles for two binary hard-sphere crystal structures using classical density functional theory through the White Bear II functional from fundamental measure theory. While for the substitutional crystal, in which some hard spheres are replaced by spheres of slightly smaller diameter, the density profiles are rather similar to the single-component case (narrow Gaussian peaks centered at fcc lattice sites), we observe a more complex behavior for the case of interstitial solid solutions, where the small species is fairly delocalized in the unit cell. Further, we compute the species-resolved inhomogeneous two-body direct correlation functions, depending on two three-dimensional vectors, for these two types of binary crystals. The large--large components are mainly determined by the vacancy concentration $n_\text{vac}$ and show a characteristic magnitude $\sim 1/n_\text{vac}$. Based on this observation, we propose a simple geometric picture. The components of the direct correlation function involving the small spheres substantially differ in interstitial solid solutions from those of the substitutional crystal.

cond-mat.stat-mech↗

arXiv2024

The orientational structure of a model patchy particle fluid: simulations, integral equations, density functional theory and machine learning

We investigate the orientational properties of a homogeneous and inhomogeneous tetrahedral 4-patch fluid (Kern--Frenkel model). Using integral equations, either (i) HNC or (ii) a modified HNC scheme with simulation input, the full orientational dependence of pair and direct correlation functions is determined. Density functionals for the inhomogeneous problem are constructed via two different methods. The first, molecular density functional theory, utilizes the full direct correlation function and an isotropic hard-sphere bridge functional. The second method, a machine learning approach, uses a decomposition of the functional into an isotropic reference part and a mean-field orientational part, where both parts are improved by machine learning techniques. Comparison to simulation data at hard walls and around hard tracers show a similar performance of the two functionals. Machine learning strategies are discussed to eliminate residual differences, with the goal of obtaining machine-learning enhanced functionals for the general anisotropic fluid.

cond-mat.soft↗

arXiv2024

Machine Learning approaches to classical density functional theory

In this chapter, we discuss recent advances and new opportunities through methods of machine learning for the field of classical density functional theory, dealing with the equilibrium properties of thermal nano- and micro-particle systems having classical interactions. Machine learning methods offer the great potential to construct and/or improve the free energy functional (the central object of density functional theory) from simulation data and thus they complement traditional physics- or intuition-based approaches to the free energy construction. We also give an outlook to machine learning efforts in related fields, such as liquid state theory, electron density functional theory and power functional theory as a functionally formulated approach to classical nonequilibrium systems.

cond-mat.stat-mech↗

arXiv2023

Machine learning of a density functional for anisotropic patchy particles

Anisotropic patchy particles have become an archetypical statistical model system for associating fluids. Here we formulate an approach to the Kern-Frenkel model via classical density functional theory to describe the positionally and orientationally resolved equilibrium density distributions in flat wall geometries. The density functional is split into a reference part for the orientationally averaged density and an orientational part in mean-field approximation. To bring the orientational part into a kernel form suitable for machine learning techniques, an expansion into orientational invariants and the proper incorporation of single-particle symmetries is formulated. The mean-field kernel is constructed via machine learning on the basis of hard wall simulation data. Results are compared to the well-known random-phase approximation which strongly underestimates the orientational correlations close to the wall. Successes and shortcomings of the mean-field treatment of the orientational part are highlighted and perspectives are given for attaining a full density functional via machine learning.

cond-mat.stat-mech↗