Method Of Collective Variables Research Articles

Critical Temperature Determination for Simple Fluids: an Analytical Approach Based on Collective Variables Method

An explicit equation for the liquid-vapor critical temperature of simple fluids is derived within an analytic approach – the method of collective variables with a reference system. This equation is applied to calculate the critical temperature values for several hard-core van der Waals fluids. The study also examines how the critical temperature depends on parameters of the interaction. Specifically, it is observed that, as the range of attractive interaction decreases, the critical temperature decreases as well.

Mechanical vs. agonist induced activation in the angiotensin II type 1 receptor.

The angiotensin II (AngII) type 1 (AT1) receptor is a member of the GPCR type A family whose function is directly linked to diseases such as hypertension and congestive heart failure. The natural agonist for the AT1 receptor is the vasoconstricting peptide AngII, yet it can also be mechanically activated by membrane stretch or shear in the absence of an agonist. Recent experiments have shown that the AT1 receptor adopts distinct active configurations depending on the chemical structure of bound agonist, which may result in different interactions with effectors such as G-proteins and beta-arrestin to produce various functional outcomes. Here, we present a systematic simulation study that characterizes the activation of the AT1 receptor under various membrane environments and mechanical stimuli. Through microsecond molecular dynamics (MD) simulations, we show that stability of the active state is highly sensitive to membrane properties such as thickness and tension. We compare membrane-mediated vs. agonist-induced activation to show that the AT1 receptor has distinct active conformations. Furthermore, we computed free energy surfaces for the ligand and membrane activate states using our previously developed locally distributed tension collective variable method. Comparison of the free energy maps shows that inactive ad various active states have unique energy landscapes. Our modeling results provide structural insights into the mechanical activation of the AT1 receptor and how it may produce different functional outcomes within the framework of biased agonism.

Kink dynamics of the sine-Gordon equation in a model with three identical attracting or repulsive impurities

Purpose of this work is to use analytical and numerical methods to consider the problem of the structure and dynamics of the kinks in the sine-Gordon model with “impurities” (or spatial inhomogeneity of the periodic potential). Methods. Using the method of collective variables for the case of three identical point impurities located at the same distance from each other, a system of differential equations is obtained. Resulting system of equations makes it possible to describe the dynamics of the kink taking into account the excitation of localized waves on impurities. To analyze the dynamics of the kink in the case of extended impurities, a numerical finite difference method with an explicit integration scheme was applied. Frequency analysis of kink oscillations and localized waves calculated numerically was performed using a discrete Fourier transform. Results. For the kink dynamics, taking into account the excitation of oscillations in modes, a system of equations for the coordinate of the kink center and the amplitudes of waves localized on impurities is obtained and investigated. Significant differences are observed in the dynamics of the kink when interacting with a repulsive and attractive impurity. The dynamics of the kink in a model with three identical extended impurities, taking into account possible resonant effects, was solved numerically. It is established that the found scenarios of kink dynamics for an extended rectangular impurity are qualitatively similar to the scenarios obtained for a point impurity described using a delta function. All possible scenarios of kink dynamics were determined and described taking into account resonant effects. Conclusion. The analysis of the influence of system parameters and initial conditions on possible scenarios of kink dynamics is carried out. Critical and resonant kink velocities are found as functions of the impurity parameters.

New computational optical solitons for generalized complex Ginzburg–Landau equation by collective variables

The purpose of the paper is to implement the collective variable method to investigate the generalized complex Ginzburg–Landau equation, which characterizes the kinetics of solitons in respect of pulse parameters for fiber optics. The statistical simulations of the interacting system of ordinary differential equations that reflect all the collective variables included in the pulse ansatz have been successfully carried out using a well-known numerical methodology, the fourth-order Runge–Kutta technique. The collective variable method is employed to plot the pulse variation characteristics as a function of propagation distance. The amplitude, temporal position, width, chirp, frequency, and phase of the pulse are all depicted against the propagated coordinate, where the width, phase of soliton, amplitude, and chirp all show a strong periodicity. The numerical dynamics of solitons have also been exhibited against varying values of pulse parameters to highlight differences in collective variables. Other key bits of the current investigation are also determined.

Collective variables approach to the vector-coupled system of Chen-Lee-Liu equation

The present manuscript employed a rare approach to tackle a vector-coupled system of Chen-Lee-Liu Equation (CLLE). This approach was based on the combination of the Collective Variable Method (CVM) and the higher-order Runge-Kutta Method (RKM) to examine the dynamics of pulse parameters and total energy on the evolution of a system of CLLE. Furthermore, to expatiate more on the pulse propagation associated with the system, the resulting dynamical equations of motions were graphically illustrated with the help of Maple and Matlab mathematical software. Lastly, the present approach was not only proved to be efficient in pulse characterization, but was also believed to effectively tackle high-order coupled systems of complex evolution equations with more wave functions (more dependent variables).

To the kinetic theory of dense gases and liquids. Calculation of quasi-equilibrium particle distribution functions by the method of collective variables

Based on a chain of BBGKI equations with a modified boundary condition that takes into account multiparticle correlations, kinetic equations in the approximate "pairs" collisions and in the polarization approximation, taking into account the interaction through the third particle, obtained. The specifics of the model representation of the pair potential of particle interaction through short-range and long-range parts were taken into account. In the case of the short-range potential in the form of the potential of solid spheres, the contribution of Enskog's revised theory to the complete integration of the collision of the kinetic equation is obtained. The collision integrals include paired quasi-equilibrium distribution functions that depend on the nonequilibrium mean values of the particle number density and the inverse temperature. The method of collective variables Yukhnovskii is applied for the calculation of pair quasi-equilibrium distribution function with an allocation of short-range and long-range parts in the potential of the interaction of particles. In this case, the system with short-range interaction is considered as a frame of reference.

New Optical Solitons with Variational Principle and Collective Variable Strategy for Cold Bosons in Zig‐Zag Optical Lattices

The theme of this piece of research is to investigate the collective variable (CV) as well as semi‐inverse techniques to explore a significant model of cold bosonic atoms in a zig‐zag optical lattice. The system is reduced to an important equation by utilizing the continuum approximation and explains the soliton’s dynamics in sense of pulse variables. These parameters are amplitude, temporal position, chirp, width, frequency, and phase which are termed as collective variables (CVs). The proposed methods are more straightforward, succinct, accurate, and simple to calculate. Furthermore, to employ the computational counterfeit on the system of six ordinary differential equations that denote all the CVs incorporated in the supposed ansatz, a well‐established computational method which is the Runge–Kutta scheme of order four is applied. The CV method is exerted to resolve the evolution of pulse parameters with the propagation distance and graphical illustrations which are also given. Moreover, figures reveal the fascinating periodic oscillations of frequency, width, amplitude, and chirp of soliton. Solitons and their numerical behavior to interpret fluctuations in CVs are presented for several values of super‐Gaussian pulse parameters. Also, the results for the semi‐inverse method which are bright solitons are provided in the form of 2D, 3D, and density plots for the distinct values of the fractal parameter to understand their physical significance. This scheme is effective in finding variational principles of various nonlinear evolution equations. Some compelling characteristics pertaining to the current scrutiny are also deduced.

Optical solitons for the Kaup–Newell equation by collective variables method

In this work, we present a collective variable (CV) approach to establish dispersive solitary wave solutions for the Kaup–Newell Equation (KNE). The full CV theory has been utilized to enunciate the soliton molecules through its ground-laying parameters including the power of each pulse, phase and center-of-mass. Additionally, the dynamics of an ultra short pulse has been analyzed by using CV. This work may be utilized for various dynamics of solitons as well as the influence the amplitude, temporal position, frequency, phase and chirp on the solitons’ nonlinear parameters. Moreover, the numerical simulations have been designed by means of appropriate parameter values to explain more on the obtained results.

Optical solitons via the collective variable method for the classical and perturbed Chen–Lee–Liu equations

Abstract In this article, the collective variable method to study two types of the Chen–Lee–Liu (CLL) equations, is employed. The CLL equation, which is also the second member of the derivative nonlinear Schrödinger equations, is known to have vast applications in optical fibers, in particular. More specifically, a consideration to the classical Chen–Lee–Liu (CCLL) and the perturbed Chen–Lee–Liu (PCLL) equations, is made. Certain graphical illustrations of the simulated numerical results that depict the pulse interactions in terms of the soliton parameters are provided. Also, the influential parameters in each model that characterize the evolution of pulse propagation in the media, are identified.

Computational Soliton solutions for the variable coefficient nonlinear Schrödinger equation by collective variable method

Computational Soliton solutions for the variable coefficient nonlinear Schrödinger equation by collective variable method

Magnetic vortex near the extended linear magnetic inhomogeneity

In this paper, the problem of a magnetic vortex motion in a field of extended linear inhomogeneity is solved theoretically. The motion parameters are calculated by the method of collective variables (Thiel’s equation) with the vortex effective mass and the third-order gyrovector. On the basis of the equation of motion, the influence of the core mass and the third-order gyrovector on the character of the vortex motion as a quasiparticle is analyzed. Using the method of magnetostatic charges, analytical expressions are obtained for the effective potentials where the vortex core is located: (a) near the edge of the magnet or at the boundary of different magnetic phases, (b) near the linear inhomogeneity of local anisotropy (bidirectional and unidirectional). The solution of the equation of motion made it possible to obtain the trajectories of the core in various physical situations. In addition, the paper discusses the features of the Hall effect for vortices/skyrmions, which are provided by the inertial properties and a third-order gyrovector. It is shown that when the core of a linear inhomogeneity crosses a unidirectional anisotropy or the boundary of magnetic phases, a curvature of the trajectory is observed, which is similar to the refraction of light at the boundary of optically dissimilar environment. It is important to note that the introduction of the mass and gyroscopic effect of the third order in the equation of motion showed that the motion of the vortex, even in a homogeneous potential, is not translational. In this case, the trajectory is an overlay of cycloids of different rotation frequencies (analytical expressions are obtained for the frequencies). It is shown that, the introduction of the mass and the gyroscopic term of the third order, the motion of the vortex core cannot be considered translational at least during the time of the transition mode until the stationary mode of motion takes place.

Optical Solitons via the Collective Variable Method for the Schrödinger–Hirota Equation

The collective variable (CV) method is an important technique that deals with both conservative and non-conservative systems by revealing set of equations of motion irrespective of their nonlinearities and dissipative terms. Thus, we employ in this paper the CV method for the complete investigation of the Schrodinger–Hirota equation, which plays a vital role in optical communication and fibers. The standard fourth-order Runge–Kutta technique for integration is further sought for the numerical simulation of the resulting system with an emphasis on the pulse width, amplitude, and frequency. Finally, graphical illustrations are supplied to support the results.

Optical solitons for the Lakshmanan-Porsezian-Daniel model by collective variable method

The present study introduces a Collective Variable (CV) approach to investigate an important type of Schrödinger equation called the Lakshmanan-Porsezian-Daniel (LPD) model which has great deals in optical materials. A state of numerical simulation via the application of the fourth-order Runge-Kutta method is further employed for the numerical treatment of the resultant system of dynamical equations of motion. The CV method gives the dynamics of the pulse parameters. Graphical representations of the pulse width, amplitude, temporal position, chirp, frequency, and phase of the pulse verses the propagation coordinate are displayed, correspondingly. More so, a significant periodicity is observed in the soliton’s amplitude, width and chirp. Other vital features with regards to the present study are also deduced.

Interaction of a Magnetic Vortex with Magnetic Anisotropy Nonuniformity

The problem of propagation of a magnetic inhomogeneity in the form of a magnetic vortex near a defect simulated by a crystallite with uniaxial anisotropy has been solved theoretically. The defect (crystallite) is implanted into a homogeneous 2D ferromagnetic matrix. Apart from the anisotropy energy, the term responsible for the existence of a centrosymmetric potential is included into the total energy. For calculations, we have used the method of collective variables (Thiele equation). We have considered the variants of bidirectional and unidirectional anisotropy of the crystallite. Analysis of the equations of motion for different directions of the anisotropy axis of the implanted defect has revealed the variety in the behavior of the vortex core as a quasiparticle. The vortex core can be trapped by the defect with equilibrium position of the vortex at rest directly on the crystallite or during its motion at a certain distance from it. It is shown that for a small damping parameter and in the case when the defect anisotropy axis lies in the plane of the magnet, the vortex moves so as if its core experiences the action of the repulsive axially symmetric potential.

Investigation of thermodynamic properties of single-layer two-component coatings

The results of studying the dynamics of disordered alloys by the method of collective variables are presented. The densities of phonon states of disordered alloys with BCC structure of the kxrb1-x system are calculated. The contribution of phonons to the free energy and entropy of the alloy at different temperatures is determined. It is shown that the contribution of phonons to the total free energy of the alloy decreases with decreasing temperature. The contribution of phonons to the entropy of alloys at 300 K is crucial.

Distribution of electropotential in the electrode area of a solid state ion electrolyte

A solid state electrolyte is considered as a system consisting of cations moving through the volume of a solid body and anions whose mobility can be neglected due to their large size compared to the size of the cations. Accordingly, in the homogeneous case local charge compensation takes place. Under the action of an external electric field, cations create in the near electrode area inhomogeneous redistribution of mobile charges and electric field. The model is used for the statistical­mechanical description of high­temperature ionic conductors and current sources. To obtain the free energy functional of the mobile charge subsystem depending on the distribution of their density, the cluster expansion scheme for the renormalized Mayer functions is used. The Hamiltonian of a system consisting of electric charges moving in the field of single­particle cell potentials of average forces is used as the basis one. The binary function of the host system is expressed in terms of the screened potentials and the potentials of the average forces based on the results of the method of collective variables. The internal energy of the system is calculated taking into account the short­ and long­range effects. The Gibbs – Duhem relation was used for calculating the free energy functional of the system. The distribution of the number density of moving particles and the electric potential in the near electrode region were found from the condition of extremality of the free energy. The potentials of average forces are obtained as a result of solving a system of integral equations in the lattice approximation, with accounting of the short­ and long­range effects. The transition from the correlative function to the correlation function allowed us to identify the correlated and uncorrelated parts of the electric potential. The linear contributions of the deviation of the charge concentrations from a uniform distribution to the chemical potential are considered. The calculations take into account the contribution of the correlation between the particles in the first three coordination spheres that leads to attraction of the first, repulsion of the second and third neighbors. The description is carried out using a linear differential equation of the fourth­order with complex values of the roots of the characteristic equation. The paper analyzes the results of the analytical solution.

Phase behaviour in ionic solutions: Restricted primitive model of ionic liquid in explicit neutral solvent

We study fluid-fluid equilibrium in the simplest model of ionic solutions where the solvent is explicitly included, i.e., a binary mixture consisting of a restricted primitive model (RPM) and neutral hard-spheres (RPM-HS mixture). First, using the collective variable method we find free energy, pressure and partial chemical potentials in the random phase approximation (RPA) for a rather general model that takes into consideration solvent-solvent and solvent-ion interactions beyond the hard core. In the special case of a RPM-HS mixture, we consider two regularizations of the Coulomb potential inside the hard core, i.e., the Weeks-Chandler-Andersen (WCA) regularization leading to the WCA approximation and the optimized regularization giving the optimized RPA (ORPA) or the mean spherical approximation (MSA). Furthermore, we calculate the phase coexistence using the associative mean spherical approximation (AMSA). In general, the three approximations produce qualitatively similar phase diagrams of the RPM-HS mixture, i.e., a fluid-fluid coexistence envelope with an upper critical solution point, a shift of the coexistence region towards higher total number densities and higher solvent concentrations with increasing pressure, and a small increase of the critical temperature with an increase of pressure. As for a pure RPM, the AMSA leads to the best agreement with the available simulation data when the association constant proposed by Olaussen and Stell is used. We also discuss the peculiarities of the phase diagrams in the WCA approximation.

Multisoliton Dynamics in the Sine-Gordon Model with Two Point Impurities

Collective variables method is used to derive a set of differential equations to describe the dynamics of a kink in the sine-Gordon model with two identical point impurities taking damping into account. It is shown that the scenarios of kink interaction with the waves localized on the impurities, found from the reduced model, are similar to those obtained earlier by numerical integration of the continuous sine-Gordon equation. For the case of the kink passage through the region with the impurities, the structure and properties of the arising on impurities long-lived four-kink multisolitons are analyzed. For the approximate analytical description of the two bound impurity-localized nonlinear waves, the system of differential equations for harmonic oscillators with elastic link is obtained. The analytical model qualitatively reproduces the results of the sine-Gordon equation numerical simulation. The cases of large and small distances between impurities are analyzed. The results of our study uncover new features of the kink-impurity interaction which is important for a number of applications where the sine-Gordon model is used.

The system of mobile ions in lattice models: Screening effects, thermodynamic and electrophysical properties

The lattice fluid model of the system with short range and long range Coulomb interactions is suggested. In the framework of the collective variables method, the screening of the Coulomb interactions in the bulk is considered. It is shown that the Debye length includes additional concentration dependence inversely proportional to the square root of the mean concentration of vacant sites like what was known at the plane boundary. The Coulomb interaction contribution to the free energy of the system is calculated in the approach close to the mean spherical approximation and is given in an analytical form.The influence of the variation of the crystal field near the system boundary on the structure and characteristics of the electric double layer is investigated. As compared to the system with equal crystal potentials at the lattice sites throughout the system the pronounced difference for the electrical capacitance appears at low absolute values of the surface potential and it is more pronounced for negative electric potentials. The capacitance diverges as the potential values at which the electric field tends to zero and attains negative values in regions of the surface potentials depending on their polarity and values of the surface crystal potential. Negative values of the capacitance may indicate the thermodynamic instability of the system that can result from neglecting the short range interaction contribution.

Vapor-liquid phase behavior of a size-asymmetric model of ionic fluids confined in a disordered matrix: The collective-variables-based approach.

We develop a theory based on the method of collective variables to study the vapor-liquid equilibrium of asymmetric ionic fluids confined in a disordered porous matrix. The approach allows us to formulate the perturbation theory using an extension of the scaled particle theory for a description of a reference system presented as a two-component hard-sphere fluid confined in a hard-sphere matrix. Treating an ionic fluid as a size- and charge-asymmetric primitive model (PM) we derive an explicit expression for the relevant chemical potential of a confined ionic system which takes into account the third-order correlations between ions. Using this expression, the phase diagrams for a size-asymmetric PM are calculated for different matrix porosities as well as for different sizes of matrix and fluid particles. It is observed that general trends of the coexistence curves with the matrix porosity are similar to those of simple fluids under disordered confinement, i.e., the coexistence region gets narrower with a decrease of porosity and, simultaneously, the reduced critical temperature T_{c}^{*} and the critical density ρ_{i,c}^{*} become lower. At the same time, our results suggest that an increase in size asymmetry of oppositely charged ions considerably affects the vapor-liquid diagrams leading to a faster decrease of T_{c}^{*} and ρ_{i,c}^{*} and even to a disappearance of the phase transition, especially for the case of small matrix particles.