Rational Function Approximation Method Research Articles

Second virial coefficient, Boyle temperature and equation of state of van Hove fluids with a downward concavity attractive parabolic-well

The second virial coefficient, the Boyle temperature and the equation of state of van Hove fluids with a relatively short ranged attractive parabolic-well of downward concavity are considered. The analytic second virial coefficient for this fluid is obtained explicitly and it is used to compute the Boyle temperature of the fluid as a function of the range of the potential. Further, an equation of state is derived using the second-order thermodynamic perturbation theory of Barker and Henderson in the macroscopic compressibility approximation, with the hard-sphere fluid being the reference fluid. For this latter we profit from the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation state, derived within the so-called rational function approximation method up to a range twice the size of the hard-core diameter. The results for the reduced pressure of the fluid as a function of the packing fraction and two values of the range of the potential well at different temperatures are compared with Monte Carlo simulation data. Estimates of the values of the critical temperature are also provided.

Structural and thermodynamic properties of fluids whose molecules interact via one-, two-, and three-step potentials

The structural and thermodynamic properties of fluids whose molecules interact via potentials with a hard-core plus a square well, a square shoulder, and a second square well, are considered. Those properties are derived by using a (semi-analytical) rational-function approximation method as a particular case of the more general formulation provided earlier involving potentials with a hard-core plus n piecewise constant sections. Comparison of the results with recent simulation data confirms the usefulness of the approach.

Vector Fitting–Cauchy Method for the Extraction of Complex Natural Resonances in Ground Penetrating Radar Operations

In this paper, we obtain the Complex Natural Resonances of an object from the backscattered response in the frequency domain with a novel rational function approximation method based on both Vector Fitting and Cauchy methods. We determine the system order and an initial set of poles, which are used as a basis for a rational function approximation. The results from the simulations and experiments show an improvement in the reconstructed signals and the accuracy of the CNRs calculated, with an increased tolerance to the critical Signal-to-Noise Ratio. This is being used in the problem of GPR landmine humanitarian detection in Colombia.

Structural properties of additive binary hard-sphere mixtures. III. Direct correlation functions.

An analysis of the direct correlation functions c_{ij}(r) of binary additive hard-sphere mixtures of diameters σ_{s} and σ_{b} (where the subscripts s and b refer to the "small" and "big" spheres, respectively), as obtained with the rational-function approximation method and the WM scheme introduced in previous work [S. Pieprzyk et al., Phys. Rev. E 101, 012117 (2020)2470-004510.1103/PhysRevE.101.012117], is performed. The results indicate that the functions c_{ss}(r<σ_{s}) and c_{bb}(r<σ_{b}) in both approaches are monotonic and can be well represented by a low-order polynomial, while the function c_{sb}(r<1/2(σ_{b}+σ_{s})) is not monotonic and exhibits a well-defined minimum near r=1/2(σ_{b}-σ_{s}), whose properties are studied in detail. Additionally, we show that the second derivative c_{sb}^{''}(r) presents a jump discontinuity at r=1/2(σ_{b}-σ_{s}) whose magnitude satisfies the same relationship with the contact values of the radial distribution function as in the Percus-Yevick theory.

Thermodynamic Properties of the Parabolic-Well Fluid

The thermodynamic properties of the parabolic-well fluid are considered. The intermolecular interaction potential of this model, which belongs to the class of the so-called van Hove potentials, shares with the square-well and the triangular well potentials the inclusion of a hard-core and an attractive well of relatively short range. The analytic second virial coefficient for this fluid is computed explicitly and an equation of state is derived with the aid of the second-order thermodynamic perturbation theory in the macroscopic compressibility approximation and taking the hard-sphere fluid as the reference system. For this latter, the fully analytical expression of the radial distribution function, consistent with the Carnahan-Starling equation of state as derived within the rational function approximation method, is employed. The results for the reduced pressure of the parabolic-well fluid as a function of the packing fraction and two values of the range of the parabolic-well potential at different temperatures are compared with Monte Carlo and Event‐driven molecular dynamics simulation data. Estimates of the values of the critical temperature are also provided.

Model-Based Vector-Fitting Method for Circuit Model Extraction of Coupled-Resonator Diplexers

In this paper, a novel rational function approximation method, namely, model-based vector fitting (MVF), is proposed for accurate extraction of the characteristic functions of a coupled-resonator diplexer with a resonant type of junction from noise-contaminated measurement data. MVF inherits all the merits of the vector-fitting (VF) method and can also stipulate the order of the numerator of the model. Thus, MVF is suitable for the high-order diplexer system identification problem against measurement noise. With the extracted characteristic functions, a three-port transversal coupling matrix of a diplexer can be synthesized. A matrix orthogonal transformation strategy is also proposed to transform the obtained transversal matrix to a target coupling matrix configuration, whose entries have one-to-one relationship with the physical tuning elements. The whole model extraction procedure is analytical and robust, and can be used in a computer-aided tuning (CAT) program for coupled-resonator diplexers. A practical tuning example of a diplexer with a common resonator is given in detail to demonstrate the effectiveness and the practical value of the proposed method.

Rational-function approximation for fluids interacting via piece-wise constant potentials

The structural properties of fluids whose molecules interact via potentials with a hard-core plus n piece-wise constant sections of different widths and heights are derived using a (semi-analytical) rational-function approximation method. The results are illustrated for the cases of a square-shoulder plus square-well potential and a shifted square-well potential and compared both with simulation data and with those that follow from the (numerical) solutions of the Percus-Yevick integral equation.

Exact solution of the Percus-Yevick integral equation for fluid mixtures of hard hyperspheres

Structural and thermodynamic properties of multicomponent hard-sphere fluids at odd dimensions have recently been derived in the framework of the rational function approximation (RFA) [Rohrmann and Santos, Phys. Rev. E 83, 011201 (2011)]. It is demonstrated here that the RFA technique yields the exact solution of the Percus-Yevick (PY) closure to the Ornstein-Zernike (OZ) equation for binary mixtures at arbitrary odd dimensions. The proof relies mainly on the Fourier transforms c(ij)(k) of the direct correlation functions defined by the OZ relation. From the analysis of the poles of c(ij)(k) we show that the direct correlation functions evaluated by the RFA method vanish outside the hard core, as required by the PY theory.

Structure of the square-shoulder fluid

The structural properties of square-shoulder fluids are derived from the use of the rational function approximation method. The computation of both the radial distribution function and the static structure factor involves mostly analytical steps, requiring only the numerical solution of a single transcendental equation. The comparison with available simulation data and with numerical solutions of the Percus–Yevick and hypernetted-chain integral equations shows that the present approximation represents an improvement over the Percus–Yevick theory for this system and a reasonable compromise between accuracy and simplicity.

Depletion potential in the infinite dilution limit

The depletion force and depletion potential between two in principle unequal "big" hard spheres embedded in a multicomponent mixture of "small" hard spheres are computed using the rational function approximation method for the structural properties of hard-sphere mixtures [S. B. Yuste, A. Santos, and M. Lopez de Haro, J. Chem. Phys. 108, 3683 (1998)]. The cases of equal solute particles and of one big particle and a hard planar wall in a background monodisperse hard-sphere fluid are explicitly analyzed. An improvement over the performance of the Percus-Yevick theory and good agreement with available simulation results are found.

Direct correlation function of the hard-sphere fluid

Two analytic approximations for the direct correlation function of a hard-sphere fluid are considered. The first follows from a generalization of the Percus–Yevick result in d dimensions, whereas the second arises in the Rational Function Approximation (RFA) method. Both approximations require the equation of state of the hard-sphere fluid as input. The results, derived after use of the Carnahan–Starling and the Padé 4,3 equations of state in both approaches, are compared with simulation data. The comparison shows that the first approximation is rather accurate in the region inside the core, but inherits the limitation of the Percus–Yevick theory for distances beyond the hard-sphere diameter. On the other hand, the results of the RFA method are also accurate inside the core and capture well the initial part of the tail beyond the hard-sphere diameter, but fail to account for the subsequent oscillations observed in the simulations. Other merits and limitations of the two approaches are reported.

Multicomponent fluid of hard spheres near a wall

The rational function approximation method, density functional theory, and NVT Monte Carlo simulation are used to obtain the density profiles of multicomponent hard-sphere mixtures near a planar hard wall. Binary mixtures with a size ratio 1:3 in which both components occupy a similar volume are specifically examined. The results indicate that the present version of density functional theory yields an excellent overall performance. A reasonably accurate behavior of the rational function approximation method is also observed, except in the vicinity of the first minimum, where it may even predict unphysical negative values.

Application of asymptotic waveform approximation technique to hybrid FE/BI method for 3D scattering

The asymptotic waveform evaluation (AWE) technique is a rational function approximation method in computational mathematics, which is used in many applications in computational electromagnetics. In this paper, the performance of the AWE technique in conjunction with hybrid finite element/boundary integral (FE/BI) method is firstly investigated. The formulation of the AWE applied in hybrid FE/BI method is given in detail. The characteristic implementation of the application of the AWE to the hybrid FE/BI method is discussed. Numerical results demonstrate that the AWE technique can greatly speed up the hybrid FE/BI method to acquire wide-band and wide-angle backscatter radar-cross-section (RCS) by complex targets.

Theoretical scheme for the shear viscosity of Lennard-Jones fluids

We use the shear viscosity expression from the Enskog theory of dense gases in a perturbative scheme for the Lennard-Jones (LJ) fluid. This perturbative scheme is formulated by combining the analytic rational function approximation method of Bravo Yuste and Santos [Phys. Rev. A 43, 5418 (1991)] for the radial distribution function of hard-sphere fluids and the well known Mansoori-Canfield/Rasaiah-Stell perturbation theory to determine an effective diameter for the LJ fluid. The scheme is reliable on a wide range of temperatures and densities, and is very accurate around the critical point. Using this information, we build an accurate empirical formula for the shear viscosity in the liquid phase, which fits the recent data [K. Meier et al., J. Chem. Phys. 121, 3671 (2004)] in the whole simulation range.

The structure factor and equation of state of hard-sphere fluids

In a paper by Verberg et al (2002 Dynamics: Models and Kinetic Methods for Non-Equilibrium ManyBody Systems ed J Karkheck (Dordrecht: Kluwer)) it was shown that the static structure factorS(k) of a hard-sphere fluid plays an important role in the computation of the transportproperties of colloidal suspensions. To perform actual calculations, those authors used asemiempirical form due to Grundke and Henderson which stems out of the Percus–Yevickapproximation. In this work we present calculations of the static structure factor of ahard-sphere fluid within the rational function approximation method (Bravo Yuste et al 1996 Phys. Rev. E 53 4820) which is wholly compatible with the equation ofstate used for the fluid. The Carnahan–Starling equation of state (Carnahanand Starling 1969 J. Chem. Phys. 51 635), the Padé (4, 3) of van Rensburg (1993J. Phys. A: Math. Gen. 26 4805) and Sánchez (1994 J. Chem. Phys. 101 7003), the equation ofstate of Malijevský and Veverka (1999 Phys. Chem. Chem. Phys. 1 4267) and a recent proposalby us (Robles and López de Haro 2003 Europhys. Lett. 62 56) will be explicitlyexamined.

Hybrid Rational Function Approximation and its accuracy analysis

We propose a rational function approximation method combining numeric and symbolic computations. Given functions or data are first interpolated by a rational function, i.e. the ratio of polynomials. Undesired poles appearing in the rational interpolant are removed by an approximate-GCD method. We call the rational approximation a Hybrid Rational Function Approximation and abbreviate it as HRFA. In this paper we give a short survey of the HRFA and then discuss its accuracy analysis by using the approximate-GCD proposed by Pan.

On the contact values of the derivatives of the hard-sphere radial distribution function

Approximate analytical expressions for the values of the derivatives of the radial distribution function at contact in a hard-sphere fluid are derived. These expressions are obtained within a rational function approximation (RFA) method recently developed. The results are compared with those based on the Percus-Yevick approximation, as well as with those obtained from expressions for the cavity function inside the core or through computer simulations. While the qualitative dependence on the packing fraction is similar for the values of the first two derivatives at contact in all cases, the one corresponding to the third derivative is definitely different in our approach from that previously reported. The integral moments of the total correlation function as given by the RFA method are also computed.