Integral Equation Theory Research Articles

Comparison of integral equation theories of the liquid state

Comparison of integral equation theories of the liquid state

Existence and uniqueness of the solution to initial and inverse problems for integro-differential heat equations with fractional load

This work is devoted to the unique solvability of the direct and inverse problems for a multidimensional heat equation with a fractional load in Holder spaces. In the problem under consideration, the loaded term is in the form of a fractional integral operator for the time variable. We prove the existence and uniqueness of the solution to these problems by the contraction mapping theorem and the theory of integral equations. For more information see https://ejde.math.txstate.edu/Volumes/2024/64/abstr.html

Solvent‐Controlled Separation of Integratively Self‐Sorted Pd2LA2LB2 Coordination Cages

The integrative implementation of multiple different components into metallosupramolecular self‐assemblies requires sophisticated strategies to avoid the formation of statistical mixtures. Previously, the key focus was set on thermodynamically driven reactions of simple homoleptic into complex heteroleptic structures. Using Pd2LA2LB2‐type coordination cages, we herein show that integrative self‐sorting can be reversed by a change of solvent (from DMSO to MeCN) to favor narcissistic re‐segregation into coexisting homoleptic species Pd2LA4 and Pd3LB6. Full separation (“unsorting”) back to a mixture of the homoleptic precursors was finally achieved by selective precipitation of Pd3LB6with anionic guest G1 from MeCN, keeping pure Pd2LA4 in solution. When a mixture of homoleptic Pd3LB6 and heteroleptic Pd2LA2LB2 is exposed to a combination of two different di‐anions (G1 and G2) in DMSO, selective guest uptake gives rise to two defined coexisting host‐guest complexes. A joint experimental and deep theoretical investigation via liquid‐state integral equation theory of the reaction thermodynamics on a molecular level accompanied by solvent distribution analysis hints at solvent expulsion from Pd2LA4 to favor the formation of Pd2LA2LB2 in DMSO as the key entropic factor for determining the solvent‐specific modulation of the cage conversion equilibrium.

Solvent-Controlled Separation of Integratively Self-Sorted Pd2LA2LB2 Coordination Cages.

The integrative implementation of multiple different components into metallosupramolecular self-assemblies requires sophisticated strategies to avoid the formation of statistical mixtures. Previously, the key focus was set on thermodynamically driven reactions of simple homoleptic into complex heteroleptic structures. Using Pd2LA2LB2-type coordination cages, we herein show that integrative self-sorting can be reversed by a change of solvent (from DMSO to MeCN) to favor narcissistic re-segregation into coexisting homoleptic species Pd2LA4and Pd3LB6. Full separation ("unsorting") back to a mixture of the homoleptic precursors was finally achieved by selective precipitation of Pd3LB6with anionic guestG1from MeCN, keeping pure Pd2LA4in solution. When a mixture of homoleptic Pd3LB6and heteroleptic Pd2LA2LB2is exposed to a combination of two different di-anions (G1andG2) in DMSO, selective guest uptake gives rise to two defined coexisting host-guest complexes.A joint experimental and deep theoretical investigation via liquid-state integral equation theory of the reaction thermodynamics on a molecular level accompanied by solvent distribution analysis hints at solvent expulsion from Pd2LA4to favor the formation of Pd2LA2LB2in DMSO as the key entropic factor for determining the solvent-specific modulation of the cage conversion equilibrium.

Effect of nonpolar solute on the local structure of aqueous solution: Orientation-dependent integral equation study of nonpolar solute in rose water model

The presence of solute in water, even in small quantities, changes the properties of water. The ability of water molecules to form hydrogen-bonds is decisive for the structural and thermodynamic properties. When the solute is added to water, the network of hydrogen bonds is disrupted, changing the structure and properties of the water. In this work, we have used a simple two-dimensional rose water model together with orientation-dependent integral equation theory (ODIET) to gain insight into the effects of nonpolar solutes on the local structure of aqueous solutions. The theory shows good agreement with the radial distribution functions and enthalpies of solvation obtained with simulations, while improving the results of the orientation-averaged version of the theory. The advantage of the two-dimensional water model together with ODIET is that angle-dependent properties can be calculated relatively quickly and easily visualised. Using the potential of the mean force derivatives, the effect of added nonpolar solute on the local structure of the solution was investigated. Depending on the size of the solute, two different regimes of the effect of the solute on the structure of the solution were identified. Small solutes increase the tendency of water to form hydrogen-bonds and at the same time decrease solvation and association of the solute. On the other hand, large solutes have the opposite effect - they decrease the HB tendency of water and increase solvation and association of the solute.

Theoretical analysis of the structure, thermodynamics, and shear elasticity of deeply metastable hard sphere fluids.

The structure, thermodynamics, and slow activated dynamics of the equilibrated metastable regime of glass-forming fluids remain a poorly understood problem of high theoretical and experimental interest. We apply a highly accurate microscopic equilibrium liquid state integral equation theory, in conjunction with naïve mode coupling theory of particle localization, to study in a unified manner the structural correlations, thermodynamic properties, and dynamic elastic shear modulus in deeply metastable hard sphere fluids. Distinctive behaviors are predicted including divergent inverse critical power laws for the contact value of the pair correlation function, pressure, and inverse dimensionless compressibility, and a splitting of the second peak and large suppression of interstitial configurations of the pair correlation function. The dynamic elastic modulus is predicted to exhibit two distinct exponential growth regimes with packing fraction that have strongly different slopes. These thermodynamic, structural, and elastic modulus results are consistent with simulations and experiments. Perhaps most unexpectedly, connections between the amplitude of long wavelength density fluctuations, dimensionless compressibility, local structure, and the dynamic elastic shear modulus have been theoretically elucidated. These connections are more broadly relevant to understanding the slow activated relaxation and mechanical response of colloidal suspensions in the ultradense metastable region and deeply supercooled thermal liquids in equilibrium.

IDENTIFICATION OF AN ARBITRARY MOVING AXISYMMETRIC LOAD ACTING ON A CYLINDRICAL SHELL

Various types of external non-stationary loads can act on various structural elements and cylindrical shells of finite length in particular distributed and concentrated, stationary and moving load. When using various external load identification methods, the type of external load is usually known. In prac- tice this is not always the case. The purpose of the study is to develop a method for identifying an arbitrary axisymmetric load acting on an elastic cy- lindrical shell of finite length, which can be used to identify a moving load. To model the non-stationary load of a cylindrical shell, a system of differ- ential equations of the refined Timoshenko theory for medium thickness shells was used. The solution to this system of differential equations is ob- tained by expanding the unknown functions into Fourier series and applying the integral Laplace transform. The solution of the inverse problem was obtained using the theory of integral equations and the Tikhonov regularization method. As a result of the study, a solution of the inverse problem of solid mechanics to identify an arbitrary axisymmetric non-stationary load was obtained. A numerical experiment was carried out to use the developed method to identify a moving non-stationary load acting on a simply supported cylindrical shell of medium thickness. The simulation results indicate a fairly accurate identification of both the change in time and the distribution along the shell of a non-stationary axisymmetric moving load. A method has been developed for identifying an external non-stationary load randomly distributed along a cylindrical shell, which makes it possible to identify a moving load without preliminary information about the type of this load. The developed method allows us to reproduce moving non-stationary loads, which are often encountered in practice, and expand it to other types of structural elements.

Martini 3D-OZ: A Theoretical Investigation of Solvation Shell Structures and Solvation Free Energies of Martini Coarse-Grained Proteins.

We investigate the properties of aqueous solutions using integral equation theories and molecular dynamics (MD) simulations within the framework of the MARTINI coarse-grained force field. The integral equation theory used in the present work is based on the Ornstein-Zernike equation coupled with the hypernetted chain (HNC) and Kovalenko-Hirata (KH) closures. Overall, the solvation shell structures and solvation thermodynamics in the HNC approximation are shown to be in better agreement with those from the MD simulation than the KH results. Especially, through the analysis of spatial distribution functions of water around a protein, we have demonstrated that the HNC approximation can provide the highly anisotropic structure of the solvation shell of the protein. On the other hand, the KH approximation works well for simple particle solutes, but the results for highly hydrated proteins deviate quite significantly from the MD results. We further explore in detail the reason underlying the deviation caused by the KH approximation. Lastly, a potential application of the integral equation theory with the MARTINI model is outlined.

Learning integral operators via neural integral equations

Nonlinear operators with long-distance spatiotemporal dependencies are fundamental in modelling complex systems across sciences; yet, learning these non-local operators remains challenging in machine learning. Integral equations, which model such non-local systems, have wide-ranging applications in physics, chemistry, biology and engineering. We introduce the neural integral equation, a method for learning unknown integral operators from data using an integral equation solver. To improve scalability and model capacity, we also present the attentional neural integral equation, which replaces the integral with self-attention. Both models are grounded in the theory of second-kind integral equations, where the indeterminate appears both inside and outside the integral operator. We provide a theoretical analysis showing how self-attention can approximate integral operators under mild regularity assumptions, further deepening previously reported connections between transformers and integration, as well as deriving corresponding approximation results for integral operators. Through numerical benchmarks on synthetic and real-world data, including Lotka–Volterra, Navier–Stokes and Burgers’ equations, as well as brain dynamics and integral equations, we showcase the models’ capabilities and their ability to derive interpretable dynamics embeddings. Our experiments demonstrate that attentional neural integral equations outperform existing methods, especially for longer time intervals and higher-dimensional problems. Our work addresses a critical gap in machine learning for non-local operators and offers a powerful tool for studying unknown complex systems with long-range dependencies.

Pair correlations and freezing transitions in a two-dimensional system of tilted axially symmetric quadrupoles

We study the pair structure and the transition from the liquid to the solid state in a system of axially symmetric charged particles confined to a two-dimensional plane and characterised by only one non-zero component of the electric quadrupole moment, which we assume to be inclined with respect to the perpendicular to the confining plane. The particles interact via anisotropic repulsive quadrupole–quadrupole interactions. The pair correlation functions (PCF) were calculated by solving the hypernetted chain integral equation (HNC) theory in the range of the tilt angle 75.0 ∘ ≤ 90 ∘ − ϕ ≤ 90 ∘ . NVT Monte Carlo (MC) simulations were also performed to test the accuracy of the PCFs. We find that the PCFs calculated with the HNC theory agree well with those of the MC simulation at smaller tilt angles and show anomalous behaviour with respect to the in-plane angular variation. The PCFs obtained with the HNC theory were used in the Ramakrishnan-Yussouff (RY) density functional theory (DFT) to investigate the transition between liquid and solid freezing. It was found that the RY DFT stabilises the triangular solid only for tilt angles ≥ 80 ∘ . When the tilt angle is decreased compared to the normal orientation of the quadrupoles, the transition density increases.

Liquid state theory of the structure of model polymerized ionic liquids.

We employ polymer integral equation theory to study a simplified model of semiflexible polymerized ionic liquids (PolyILs) that interact via hard core repulsions and short range screened Coulomb interactions. The multi-scale structure in real and Fourier space of PolyILs (ions chosen to mimic Li, Na, K, Br, PF6, and TFSI) are determined as a function of melt density, Coulomb interaction strength, and ion size. Comparisons with a homopolymer melt, a neutral polymer-solvent-like athermal mixture, and an atomic ionic liquid are carried out to elucidate the distinct manner that ions mediate changes of polymer packing, the role of excluded volume effects, and the influence of chain connectivity, respectively. The effect of Coulomb strength depends in a rich manner on ion size and density, reflecting the interplay of steric packing, ion adsorption, and charge layering. Ion-mediated bridging of monomers is found, which intensifies for larger ions. Intermediate range charge layering correlations are characterized by a many-body screening length that grows with PolyIL density, cooling, and Coulomb strength, in disagreement with Debye-Hückel theory, but in accord with experiments. Qualitative differences in the collective structure, including an ion-size-dependent bifurcation of the polymer structure factor peak and pair correlation function, are predicted. The monomer cage order parameter increases significantly, but its collective ion counterpart decreases, as ions become smaller. Such behaviors allow one to categorize PolyILs into two broad classes of small and large ions. Dynamical implications of the predicted structural results are qualitatively discussed.

Two Methods Based on Integral Equation Approaches in Analyzing Polyelectrolyte Solutions: Macrophase Separation.

To understand the phase behaviors of polyelectrolyte solutions, we provide two analytical methods to formulate a molecular equation of state for a system of fully charged polyanions (PAs) and polycations (PCs) in a monomeric neutral component, based on integral equation theories. The mixture is treated in a primitive and restricted manner. The first method utilizes Blum's approach to charged hard spheres, incorporating the chain connectivity contribution by charged spheres via Stell's cavity function method. The second method employs Wertheim's multi-density Ornstein-Zernike treatment of charged hard spheres with Baxter's adhesive potential. The pressures derived from these methods are compared to available molecular dynamics simulations data for a solution of PAs and monomeric counterions as a limiting case. Two-phase equilibrium for the system is calculated using both methods to evaluate the relative strength of phase segregation that leads to complex coacervation. Additionally, the scaling exponents for a selected solution near its critical point are examined.

Surface Tension of Simple Molten Salts: Insight from a Charged Hard Sphere Model.

Surface tension of molten salts is rooted in many important phenomena in physical chemistry. We develop an analytical theory for the surface tension of molten salts, where the molten salt is described by a restricted primitive model electrolyte consisting of charged hard spheres, and surface tension is related to the formation of a cavity in the electrolyte. The integral equation theory is applied to the restricted primitive model electrolyte to derive an analytical formula of the cavity formation energy. The scaling relation of the cavity formation energy is further combined with morphological thermodynamics theory to determine the formula for surface tension. According to our formula, surface tension consists of a positive hard sphere contribution and a negative electrostatic contribution. Using the molar mass, interionic distance, density, and temperature as the input, our theory leads to a good prediction of surface tension of more than 16 molten salts at their melting point without introducing any adjustable parameters.

Spin-Spin Coupling Constant Based on Reference Interaction Site Model Self-Consistent Field with Constrained Spatial Electron Density.

A method for computing spin-spin coupling constants (SSCCs) using the reference interaction site model self-consistent field with constrained spatial electron density (RISM-SCF-cSED) is proposed for the first time. Describing solvents using integral equation theory allows us to reflect solvent effects at atomic resolution in SSCCs while accounting for thermal fluctuations at a low computational cost. Applying the method to water, 1,1-difluoroethylene, and 1-methylaminomethylene-2-naphthalenone revealed that the solvent shift was evaluated to a greater extent than in the continuum solvent model. The origin of this phenomenon was analyzed in terms of the physical mechanisms underlying SSCCs.

Density functional theory of freezing transition in a binary mixture of spheres and soft-ellipsoids

ABSTRACT This study uses the classical density functional theory (DFT) of freezing to investigate phase transitions in ellipsoidal-spherical binary mixtures interacting via a generalised Gay-Berne (GB) intermolecular potential, with a focus on I-N, I-SmA and N-SmA transitions. The mixtures, consisting mainly of ellipsoids, contain spherical additives at concentrations of 5 % , 10 % and 15 % . The pair correlation functions, which are crucial for the DFT, are derived from Percus-Yevick integral equation theory. Our results show a significant influence of spherical additives on liquid crystalline phases formed by ellipsoids. At an ellipsoid length-to-width ratio of 3.0, only the I-N transition is observed, with the SmA phase remaining unstable under all conditions. Increasing the ellipsoidal length-to-width ratio to κ = 4.0 introduces the SmA phase, whose stability depends on the concentration and size of the spherical additives. The stability of the SmA phase decreases with increasing concentration and size of the spheres. Phase diagrams, which are shown in temperature-density and pressure-temperature planes, emphasise the influence of the size of the spherical dopants on the phase stability.

On a class of Hausdorff type operators on interval

The paper studies integral operators on the interval of real line which naturally arise in some problems in the theory of integral equations and mathematical physics. We study boundedness in weighted Lebesgue spaces: Sufficient and necessary conditions for boundedness are given. Also, special important particular cases of operators and spaces are considered as examples. We construct identity approximation within the framework of the class of operators under study, and two different approaches are given.

Study of the thermodynamic inconsistency of the potential of mean force calculated using the integral equation theory of molecular liquids

In this study, we investigated thermodynamic inconsistency in the potential of mean force (PMF) calculated using the Ornstein-Zernike (OZ), one-dimensional reference interaction site model (1D-RISM), and three-dimensional OZ (3D-OZ) theories. Two methods were used to obtain the PMF: converting the radial distribution function (RDF) into the PMF, and utilizing the solvation free energy (SFE) of the diatomic solute molecule with an added direct interaction between the two solute atoms. Both for the Lennard-Jones (LJ) and Coulomb systems, the hypernetted chain (HNC) closure showed the best performance among the closure relationships tested in this study. However, the performance of the HNC was not perfect and exhibited significant inconsistency in the PMF. The Kovalenko–Hirata and Kobryn–Gusarov–Kovalenko closures were also tested, with fair results when combined with the 1D-RISM equation but poor results when combined with the 3D-OZ equation. The Verlet-modified (VM) closure was found to be inapplicable to Coulomb systems. In contrast, the modified HNC (MHNC) closure showed relatively good results at high temperatures. Furthermore, we modified the Martynov–Sarkisov (MS) closure for Coulomb systems, resulting in a modified MS closure that yields relatively accurate PMF results. We also discuss the behavior of the SFE for Coulomb systems obtained from the 1D-RISM theory, a thorough analysis of both the RDF and the integrand in the Kirkwood charging formula.

Hyers–Ulam stability of integral equations with infinite delay

Integral equations with infinite delay are considered as functional equations in a Banach space. Two types of Hyers–Ulam stability criteria are established. First, it is shown that a linear autonomous equation is Hyers–Ulam stable if and only if it has no characteristic value with zero real part. Second, it is proved that the Hyers–Ulam stability of a linear autonomous equation is preserved under sufficiently small nonlinear perturbations. The proofs are based on a recently developed decomposition theory of linear integral equations with infinite delay.

Acoustic wave propagation and scattering in visco-acoustic medium: Integral equation representation and De Wolf approximation

A visco-acoustic medium is an approximate representation of the absorption and attenuation characteristics of the actual earth medium, which will cause energy attenuation and velocity dispersion of seismic waves. The propagation law of scattered waves in viscoelastic media can be revealed through seismic wave forward simulation. However, the numerical simulation of scattered waves in a visco-acoustic medium using the De Wolf approximation method based on the Lippmann-Schwingr (L-S) equation has not been studied. Therefore, based on the Futterman dispersion relation, the mathematical expression of the integral equation of acoustic wave propagation and scattering in the visco-acoustic medium is derived. We combine the De Wolf approximation with the quality factor Q. The thin slab approximation and screen approximation methods are two realizations of the De Wolf approximation. The thin slab approximation and the screen approximation continuation operator in the viscous-acoustic medium are constructed and applied to the numerical simulation of the scattered wave field. The research results of the concave model and complex model indicate that due to the influence of medium viscosity, the energy of the deep reflected waves is weaker in the forward modeling records of the thin slab approximation method and screen approximation method, and the dominant frequency of seismic waves moves towards the low-frequency direction; Compared with the finite difference results, they only calculate one reflection, and there are no multiple wave signals in the seismic records. Finally, We discuss the thin slab approximation and the screen approximation method for detail around the division of the thickness of the thin slab, the selection of the reference frequency, and the non-uniformity of the medium.

Potential method in the coupled theory of thermoelastic triple-porosity nanomaterials

In this paper, the coupled linear theory of thermoelasticity for nanomaterials with triple porosity is considered in which the combination of Darcy’s law and the volume fraction concept for three levels of pores (macro-, meso- and micropores) is provided. The 3D basic boundary value problems (BVPs) of steady vibrations of this theory are formulated and these BVPs are investigated using the potential method (boundary integral equation method) and the theory of singular integral equations. Namely, the formula of integral representation of regular vectors is obtained. The surface (single-layer and double-layer) and volume potentials are introduced and their basic properties are given. Some useful singular integral operators are defined for which Noether’s theorems are valid. The symbolic determinants and indexes of these operators are calculated. The BVPs of steady vibrations are reduced to the equivalent singular integral equations. Finally, with the help of the potential method, we prove the existence theorems for classical solutions of the aforementioned BVPs.