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Fundamental Measure Theory (FMT) is a successful and versatile approach for describing the properties of the hard-sphere fluid and hard-sphere mixtures within the framework of classical density functional theory (DFT). Lutsko [Phys. Rev. E 102, 062137 (2020)] introduced a version of FMT containing two free parameters, to be fixed by additional physical constraints. Whereas Lutsko focused on the stability of crystalline phases, we introduce and employ two statistical mechanical sum rules pertinent for the fluid phase, that are not automatically satisfied by FMT. By minimizing the relative deviation between different routes to calculate the excess chemical potential and the isothermal compressibility we determine the two free parameters of the theory. Our results indicate that requiring consistency with these sum rules can improve the quality of predictions of FMT for properties of the hard-sphere fluid phase. We suggest that employing these (test particle) sum rules, which apply for any interparticle pair-potential, might provide a means of testing the performance and accuracy of general DFT approximations.

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., Comment: 21 pages, 9 figures

Measurements of local density fluctuations are crucial to characterizing the interfacial properties of equilibrium fluids. A specific case that has been well-explored involves the heightened compressibility of water near hydrophobic entities. Commonly, a spatial profile of local fluctuation strength is constructed from measurements of the mean and variance of solvent particle number fluctuations in a set of contiguous sub-volumes of the system adjacent to the solvo/hydrophobe. An alternative measure proposed by Evans and Stewart (J. Phys.: Condens. Matter 27, 194111 (2015)) defines a local compressibility profile in terms of the chemical potential derivative of the spatial number density profile. Using Grand Canonical Monte Carlo simulation, we compare and contrast the efficacy of these two approaches for a Lennard-Jones solvent at spherical and planar solvophobic interfaces, and SPC/E water at a hydrophobic spherical solute. Our principal findings are that: (i) the local compressibility profile $\chi({\bf r})$ of Evans and Stewart is considerably more sensitive to variations in the strength of local density fluctuations than the spatial fluctuation profile $F({\bf r})$ and can resolve much more detailed structure; (ii) while the local compressibility profile is essentially independent of the choice of spatial discretization used to construct the profile, the spatial fluctuation profile exhibits strong systematic dependence on the size of the subvolumes on which the profile is defined. We clarify the origin and nature of this finite-size effect., Comment: 12 pages, 6 figures

Let $A/\mathbb{Q}$ be a Jacobian variety and let $F$ be a totally real, tamely ramified, abelian number field. Given a character $\psi$ of $F/\mathbb{Q}$, Deligne's Period Conjecture asserts the algebraicity of the suitably normalised value $\mathcal{L}(A,\psi,1)$ at $z=1$ of the Hasse-Weil-Artin $L$-function of the $\psi$-twist of $A$. We formulate a conjecture regarding the integrality properties of the family of normalised $L$-values $(\mathcal{L}(A,\psi,1))_{\psi}$, and its relation to the Tate-Shafarevich group of $A$ over $F$. We numerically investigate our conjecture through $p$-adic congruence relations between these values., Comment: 24 pages

Simulations of water near extended hydrophobic spherical solutes have revealed the presence of a region of depleted density and accompanying enhanced density fluctuations.The physical origin of both phenomena has remained somewhat obscure. We investigate these effects employing a mesoscopic binding potential analysis, classical density functional theory (DFT) calculations for a simple Lennard-Jones (LJ) solvent and Grand Canonical Monte Carlo (GCMC) simulations of a monatomic water (mw) model. We argue that the density depletion and enhanced fluctuations are near-critical phenomena. Specifically, we show that they can be viewed as remnants of the critical drying surface phase transition that occurs at bulk liquid-vapor coexistence in the macroscopic planar limit, i.e.~as the solute radius $R_s\to\infty$. Focusing on the radial density profile $\rho(r)$ and a sensitive spatial measure of fluctuations, the local compressibility profile $\chi(r)$, our binding potential analysis provides explicit predictions for the manner in which the key features of $\rho(r)$ and $\chi(r)$ scale with $R_s$, the strength of solute-water attraction $\varepsilon_{sf}$, and the deviation from liquid-vapor coexistence of the chemical potential, $\delta\mu$. These scaling predictions are confirmed by our DFT calculations and GCMC simulations. As such our theory provides a firm basis for understanding the physics of hydrophobic solvation., Comment: 18 pages

We investigate density fluctuations in a liquid close to a solvophobic substrate at which a surface critical drying transition occurs. Using classical density functional theory, we determine three measures of the spatial extent and strength of the fluctuations, i.e., the local compressibility $\chi_{\mu}(z)$, the local thermal susceptibility $\chi_T(z)$ and the reduced density $\chi_*(z)$; $z$ is the distance from the substrate. Whilst the first measure is frequently used, the second and third were introduced very recently by Eckert et.al., Phys. Rev. Lett. 125, 268004 (2020). For state points in the critical drying regime, all three measures, each scaled by its bulk value, exhibit very similar forms and the ratio of $\chi_T(z)$ to $\chi_{\mu}(z)$, for fixed $z$ in the vapour-liquid transition region, is constant. Using a scaling treatment of surface thermodynamics we show that such behaviour is to be expected on general grounds., Comment: 7 pages, 3 figures. Accepted by Physical Review E

We investigate the origin of the density depletion and enhanced density fluctuations that occur in water in the vicinity of an extended hydrophobic solute. We argue that both phenomena are remnants of the critical drying surface phase transition that occurs at liquid-vapor coexistence in the macroscopic planar limit, ie. as the solute radius $R_s\to\infty$. Focusing on the density profile $\rho(r)$ and a sensitive spatial measure of fluctuations, the local compressibility profile $\chi(r)$, we develop a scaling theory which expresses the extent of the density depletion and enhancement in compressibility in terms of $R_s$, the strength of solute-water attraction $\varepsilon_s$, and the deviation from liquid-vapor coexistence $\delta\mu$. Testing the predictions against results of classical density functional theory for a simple solvent and Grand Canonical Monte Carlo simulations of a popular water model, we find that the theory provides a firm physical basis for understanding how water behaves at a hydrophobe., Comment: 5 pages, 4 figures. To appear in Phys. Rev. Lett