Integral Equation Theory Research Articles

An equation of state for London dispersion interaction with thermodynamic inconsistent terms

One of the most important potential in the physical systems is London dispersion potential which is very significant in a lot of systems, for instance, the soft materials systems, some gaseous systems and some liquid systems. In the following work, we use the principles of the integral equations theory for derivation of an equation of state for the London dispersion potential where we apply the mean-spherical-approximation for this purpose with thermodynamic inconsistent principle. We take into account the virial expansion formula and the compressibility formula of the pressure. We found that the pressure is function to the temperature of the described systems and fractional volume of the components in the systems, in addition to the first ionization energy and the polarizablitiy of the components in the systems. We believe that the formula of the equation of state for London dispersion interaction which we derive in this study is general and suitable for compositions which interact via London dispersion effects.

Pricing European continuous-installment currency options with mean-reversion

In this paper, we consider European continuous-installment currency option under the mean-reversion environment. Specifically, we provide efficient pricing formula of installment currency put option via a partial differential equation (PDE) approach when the exchange rate follows the mean reverting lognormal model. Using the Mellin transform techniques, we derive the integral equation representation for the optimal stopping boundary from the PDE for pricing of the option. To verify the efficiency and accuracy of our approach, we provide computational results with the least square Monte Carlo method proposed by Longstaff and Schwartz (2001). We also present some numerical examples to examine the characteristics of the optimal boundaries and prices.

An efficient method for singular integral equations of non-normal type with two convolution kernels

This article deals with the solvability and explicit solutions of one class of singular integral equations with two convolution kernels in the non-normal type case. Via using techniques of complex analysis, such equations are transformed into Riemann–Hilbert boundary value problems (R-HPs) with the discontinuous property. We establish a regularity theory and existence of solutions, and obtain the general solutions and the conditions of Noether solvability. Especially, we investigate the asymptotic behaviours of solutions at nodes. Thus, this paper generalizes the theories of integral equations and Riemann–Hilbert boundary value problems.

The Study of Generalized Synchronization between Two Identical Neurons Based on the Laplace Transform Method

In this paper, a new method is proposed based on the auxiliary system approach to investigate generalized synchronization between two identical neurons with unidirectional coupling. Different from other studies, the synchronization error system between the response and auxiliary systems is converted into a set of Volterra integral equations according to the Laplace transform method and convolution theorem. By using the successive approximation method in the theory of integral equations, an analytical criterion for the detection of generalized synchronization between two identical neurons is obtained. It is found that there is a time difference between two signals of neurons when the generalized synchronization between them is achieved. Furthermore, the value of the time difference has no relation to the generalized synchronization condition but depends on the coupling function between two neurons. The study in this paper shows that one can construct a coupling function between two identical neurons using the current signal of the drive system to predict its future signal or make its past signal reappear.

Recent Developments of Computational Methods for pKa Prediction Based on Electronic Structure Theory with Solvation Models

The protonation/deprotonation reaction is one of the most fundamental processes in solutions and biological systems. Compounds with dissociative functional groups change their charge states by protonation/deprotonation. This change not only significantly alters the physical properties of a compound itself, but also has a profound effect on the surrounding molecules. In this paper, we review our recent developments of the methods for predicting the Ka, the equilibrium constant for protonation reactions or acid dissociation reactions. The pKa, which is a logarithm of Ka, is proportional to the reaction Gibbs energy of the protonation reaction, and the reaction free energy can be determined by electronic structure calculations with solvation models. The charge of the compound changes before and after protonation; therefore, the solvent effect plays an important role in determining the reaction Gibbs energy. Here, we review two solvation models: the continuum model, and the integral equation theory of molecular liquids. Furthermore, the reaction Gibbs energy calculations for the protonation reactions require special attention to the handling of dissociated protons. An efficient method for handling the free energy of dissociated protons will also be reviewed.

Thermodynamically consistent generalized hydrodynamic theory of thermal conduction and integral equations of thermal conductivity of simple fluids in electromagnetic fields

We have reviewed thermodynamically consistent evolution equations of fluids in an invited review article in the McGill University bicentennial memorial issue [B. C. Eu, Can. J. Chem. (in press) (2021)]. In this article, we apply the generalized hydrodynamic evolution equations of heat fluxes to derive integral equations for thermal conductivity coefficients of fluids in electromagnetic fields, which generalize the linear response theory formula of Green and Kubo to nonlinear heat conduction phenomena. The constitutive equation for the temperature field derived is compared with the temperature evolution equation obtained by Guo et al. [Physica D 342, 24 (2017)] on the basis of extended irreversible thermodynamics. They are considerably different.

Integral equation theory based dielectric scheme for strongly coupled electron liquids.

In a recent paper, Lucco Castello et al. (arXiv:2107.03537) provided an accurate parameterization of classical one-component plasma bridge functions that was embedded in a novel dielectric scheme for strongly coupled electron liquids. Here, this approach is rigorously formulated, its set of equations is formally derived, and its numerical algorithm is scrutinized. A systematic comparison with available and new path integral Monte Carlo simulations reveals a rather unprecedented agreement especially in terms of the interaction energy and the long wavelength limit of the static local field correction.

Horizon effect on optimal retirement decision

We study an optimal consumption, investment, life insurance, and retirement decision of an economic agent who has an option to retire early any time before the mandatory retirement date. We conduct a thorough theoretical analysis for the optimal retirement problem with general utility function in the presence of a mandatory retirement date, which leads to the optimal stopping problem in finite horizon. Furthermore, the different marginal utility of consumption before and after retirement is considered, which can provide an explanation for the retirement-consumption puzzle, while it makes the problem technically more challenging. Based on the theory of partial differential equation, we analyze the variational inequality arising from the dual problem and establish the duality theorem. We show that the optimal retirement decision is determined by the time-varying optimal retirement wealth boundary, and we provide an integral equation representation for the optimal retirement wealth boundary, which can be solved accurately and efficiently by using recursive integration method. As an extension, the case with stochastic labor income is also considered. The properties of the optimal strategies are provided with emphasis on the role of the mandatory retirement date and the impact of having a retirement option on the optimal financial decisions.

A Joint Venture of Ab Initio Molecular Dynamics, Coupled Cluster Electronic Structure Methods, and Liquid-State Theory to Compute Accurate Isotropic Hyperfine Constants of Nitroxide Probes in Water.

The isotropic hyperfine coupling constant (HFCC, Aiso) of a pH-sensitive spin probe in a solution, HMI (2,2,3,4,5,5-hexamethylimidazolidin-1-oxyl, C9H19N2O) in water, is computed using an ensemble of state-of-the-art computational techniques and is gauged against X-band continuous wave electron paramagnetic resonance (EPR) measurement spectra at room temperature. Fundamentally, the investigation aims to delineate the cutting edge of current first-principles-based calculations of EPR parameters in aqueous solutions based on using rigorous statistical mechanics combined with correlated electronic structure techniques. In particular, the impact of solvation is described by exploiting fully atomistic, RISM integral equation, and implicit solvation approaches as offered by ab initio molecular dynamics (AIMD) of the periodic bulk solution (using the spin-polarized revPBE0-D3 hybrid functional), embedded cluster reference interaction site model integral equation theory (EC-RISM), and polarizable continuum embedding (using CPCM) of microsolvated complexes, respectively. HFCCs are obtained from efficient coupled cluster calculations (using open-shell DLPNO-CCSD theory) as well as from hybrid density functional theory (using revPBE0-D3). Re-solvation of “vertically desolvated” spin probe configuration snapshots by EC-RISM embedding is shown to provide significantly improved results compared to CPCM since only the former captures the inherent structural heterogeneity of the solvent close to the spin probe. The average values of the Aiso parameter obtained based on configurational statistics using explicit water within AIMD and from EC-RISM solvation are found to be satisfactorily close. Using either such explicit or RISM solvation in conjunction with DLPNO-CCSD calculations of the HFCCs provides an average Aiso parameter for HMI in aqueous solution at 300 K and 1 bar that is in good agreement with the experimentally determined one. The developed computational strategy is general in the sense that it can be readily applied to other spin probes of similar molecular complexity, to aqueous solutions beyond ambient conditions, as well as to other solvents in the longer run.

Problems of steady vibrations in the coupled linear theory of double-porosity viscoelastic materials

In the present paper the coupled linear theory of double-porosity viscoelastic materials is considered and the basic boundary value problems (BVPs) of steady vibrations are investigated. Indeed, in the beginning, the systems of equations of motion and steady vibrations are presented. Then, Green’s identities are established and the uniqueness theorems for classical solutions of the BVPs of steady vibrations are proved. The fundamental solution of the system of steady vibration equations is constructed and the basic properties of the potentials (surface and volume) are given. Finally, the existence theorems for classical solutions of the above mentioned BVPs are proved by using the potential method (the boundary integral equations method) and the theory of singular integral equations.

Finite horizon portfolio selection problem with a drawdown constraint on consumption

In this paper we investigate the consumption and portfolio selection problem of a finitely-lived agent who faces drawdown constraint on consumption: the agent does not accept falling in her consumption below a fixed proportion of the historically highest level. We utilize the dual martingale method and study the dual problem, which can be transformed into an infinite series of optimal stopping problems. Based on a theory of partial differential equation (PDE), we analytically characterize a variational inequality arising from the optimal stopping problem. We provide a verification theorem that the value function for the original agent's problem is the Legendre-Fenchel transform of the integral of the value functions for the optimal stopping problems. Moreover, we derive integral equation representations for the optimal strategies and provide some numerical implications for the optimal strategies.

Numerical Solution of Fractional Elliptic Problems with Inhomogeneous Boundary Conditions

The numerical solution of fractional-order elliptic problems is investigated in bounded domains. According to real-life situations, we assumed inhomogeneous boundary terms, while the underlying equations contain the full-space fractional Laplacian operator. The basis of the convergence analysis for a lower-order boundary element approximation is the theory for the corresponding continuous problem. In particular, we need continuity results for Riesz potentials and the fractional-order extension of the theory for boundary integral equations with the Laplacian operator. Accordingly, the convergence is stated in fractional-order Sobolev norms. The results were confirmed in a numerical experiment.

Grid independent convergence using multilevel circulant preconditioning: Poisson\u2019s equation

We consider the iterative solution of the discrete Poisson’s equation with Dirichlet boundary conditions. The discrete domain is embedded into an extended domain and the resulting system of linear equations is solved using a fixed point iteration combined with a multilevel circulant preconditioner. Our numerical results show that the rate of convergence is independent of the grid’s step sizes and of the number of spatial dimensions, despite the fact that the iteration operator is not bounded as the grid is refined. The embedding technique and the preconditioner is derived with inspiration from theory of boundary integral equations. The same theory is used to explain the behaviour of the preconditioned iterative method.

Computer simulations and integral equation study of a two length scale core-softened fluid

Monte Carlo simulations, molecular dynamics and integral equation theory were used to study the thermodynamics and structure of particles interacting through the core softened interaction. Core-softened disks have two length scales of interaction, a hard core with one diameter and a soft corona with a larger diameter. We checked the possibility that a fluid with a core-softened potential reproduces anomalies of liquid water and attempted to determine the critical points which we did not observe nor with computer simulations nor with integral equations. We showed that some versions of the integral equation theory completely fail to predict structure and thermodynamics of such system, while others predict it quite well depending on the position in phase space.

Competing interactions in the depletion forces of ternary colloidal mixtures

Depletion interactions between colloidal particles surrounded by smaller depletants are typically characterized by a strong attraction at contact and a moderately repulsive barrier in front of it that extends at distances similar to the size of the depletants; the appearance and height of the barrier basically depend on the concentration and, therefore, the correlation between depletants. From a thermodynamic point of view, the former can drive the system to phase separation or toward non-equilibrium states, such as gel-like states, but its effects on both local and global properties may be controlled by the latter, which acts as a kind of entropic gate. However, the latter has not been entirely analyzed and understood within the context of colloidal mixtures mainly driven by entropy. In this contribution, we present a systematic study of depletion forces in ternary mixtures of hard spherical particles with two species of depletants, in two and three dimensions. We focus the discussion on how the composition of the depletants becomes the main physical parameter that drives the competition between the attractive well and the repulsive barrier. Our results are obtained by means of the integral equation theory of depletion forces and techniques of contraction of the description adapted to molecular dynamics computer simulations.

Pair-correlation functions and freezing transition in a 2D binary mixture of ultrasoft colloidal particles interacting via Hertzian potential

Density functional theory (DFT) of freezing and hypernetted chain (HNC) integral equation theory has been used to investigate the nature of pair-correlations and the freezing transitions in a binary mixture of ultrasoft colloidal particles confined to a two-dimensional (2D) plane. The particles of the system interact via Hertzian type model potential with varying softness. The accuracy of the pair-correlations obtained by HNC theory has been tested by performing NVT Monte Carlo (MC) simulation. The partial pair correlation functions of the mixture are found to exhibit non-monotonous behaviour whose nature depends on the softness of the potential and composition. Pair-correlations obtained by solving the HNC integral equation theory have been used in DFT to determine the freezing parameters of fluid-substitutionally disordered triangular solid. The constant pressure phase diagrams in the composition-temperature plane are found to be narrow spindle types.

Theoretical study on isomerization of α-acids: A DFT calculation

The α-acids contained in hops are one of the ingredients of beer. The isomerization of α-acids produces iso-α-acids, the main source of bitterness in beer. In this study, the isomerization mechanism of the α-acid, cohumulone, was elucidated by using density functional theory in conjunction with the polarizable continuum model or 3D-RISM integral equation theory of liquids. The calculated reaction diagram is consistent with experimental results; the activation free energy difference between the cis and trans isomers is in good agreement with the experimental estimate. The activation energy difference results from solvation energy. Additionally, a calculation of NMR chemical shifts showed that the proton position of isocohumulone is different from that proposed previously. The effect of Mg2+ cation on the isomerization was also investigated. Both the activation and reaction free energy are stabilized by the presence of Mg2+, which is consistent with experimental results. Water solvation reduces the activation free energy.

Two-dimensional core-softened model with water like properties: solvation of non-polar solute

ABSTRACT Monte Carlo simulations and integral equation theory were used to study the thermodynamics and structure of mixture of particles interacting through the smooth version of Stell–Hemmer interaction and non-polar particles interacting through Lennard–Jones interaction in two dimensions. The theoretical results are tested against the Monte Carlo computer simulations. Very good agreement between the integral equation theory results and exact computer simulations is obtained. We showed that some versions of the integral equation theory completely fail to predict structure of such system at some state points so caution has to be taken when applying them but in comparison with results for pure component results are better. We also checked how position of critical point changes with content of non-polar particles.

GalaxyWater-wKGB: Prediction of Water Positions on Protein Structure Using wKGB Statistical Potential.

Proteins fold and function in water, and protein-water interactions play important roles in protein structure and function. In computational studies on protein structure and interaction, the effect of water is considered either implicitly or explicitly. Implicit water models are frequently used in protein structure prediction and docking because they are computationally much more efficient than explicit water models, which are often employed in molecular dynamics (MD) simulations. However, implicit water models that treat water as a continuous solvent medium cannot account for specific atomistic protein-water interactions that are critical for structure formation and interactions with other molecules. Various methods for predicting water molecules that form specific atomistic interactions with proteins have been developed. Methods involving MD simulations or the integral equation theory tend to produce more accurate results at a higher computational cost than simple geometry- or energy-based methods. Here, we present a novel method for predicting water positions on a protein surface called GalaxyWater-wKGB, which is based on a statistical potential, a water knowledge-based potential based on the generalized Born model (wKGB). This method is accurate and rapid because it does not require conformational sampling or iterative computation owing to the effective statistical treatment employed to derive the potential. The statistical potential describes specific protein atom-water interactions more accurately than conventional potentials by considering the dependence on the degree of solvent accessibility of protein atoms as well as on protein atom-water distances and orientations. The introduction of solvent accessibility allows effective consideration of competing nonspecific protein-water and intraprotein interactions. When tested on high-resolution protein crystal structures, this method could recover similar or larger fractions of crystallographic water 180 times faster than the sophisticated integral equation theory, 3D-RISM. A web service of this water prediction method is freely available at http://galaxy.seoklab.org/wkgb.

Elastic cloaking for a periodic distribution of parallel finite cracks

We achieve elastic cloaking for a periodic distribution of an infinite number of parallel finite mode III cracks by means of the complex variable method and the theory of Cauchy singular integral equations. The cloaking bimaterial structure is composed of an undisturbed uniformly stressed left half-plane perfectly bonded via a wavy interface to the right half-plane which contains periodic cracks. The original design of the wavy interface and the positions of the periodic cracks are ultimately reduced to the solution of a Cauchy singular integral equation which can be solved numerically.