Asymptotic Form Research Articles

Correlation functions and polaron-molecule crossover in one-dimensional attractive Fermi gases

We study the correlation functions in a one-dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media. By means of the exact Bethe ansatz wave function with string solutions for the momentum, we manage to explicitly deduce the asymptotic form of the correlation functions in the infinite attractive limit and show that the system undergoes a polaron-molecule crossover when the attraction grows. The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation. These analytical results are checked by the numerical Monte Carlo methods for multiple integrals. The calculation is further extended to the excited states, i.e., the super-Tonks-Girardeau gas, and there is no peak or dip found in the vicinity of the correlation center. Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.

Resonantly interacting lump chains in the Mel'nikov equation

We constructed resonant lump chain solutions to the Mel'nikov equation by utilizing Hirota bilinear approach, mainly includes the resonance between lump chains, resonance between lump and lump chains. By calculating the asymptotic form of the solution, the resonant interaction of these chains results in infinite phase shifts, which is analogous to the line soliton interactions in the Kadomtsev-Petviashvili-II equation. The interactions are classified into two types, oblique and parallel cases, depending on the velocities of the individual lump chains. Our study highlights several cases of obliquely collided lump chains with a Y-shaped structure. Furthermore, the parallel resonance of lump chains can lead to the transmission, splitting, or absorption of another lump chain. The interactions between lump and lump chains are classified into two types, lump is semi-localized in time or completely localized in time.

Parametric Expansions of an Algebraic Variety near Its Singularities

Presently, there is a method based on Power Geometry that allows one to find asymptotic forms and asymptotic expansions of solutions to different kinds of non-linear equations near their singularities. The method contains three algorithms: (1) Reducing the equation to its normal form, (2) separating truncated equations, and (3) power transformations of coordinates. Here, we describe the method for the simplest case, a single algebraic equation, and apply it to an algebraic variety, as described by an algebraic equation of order 12 in three variables. The variety was considered in study of Einstein’s metrics and has several singular points and singular curves. Near some of them, we compute a local parametric expansion of the variety.

Near edge tractions for a rounded quarter-plane pressed into an elastically similar half-plane

Asymptotic forms are a useful way of representing the state of stress at a contact edge, allowing us to characterise the region in which cracks nucleate. The asymptotes must match the behaviour implied by the local geometry. In this paper, we study the behaviour of a flat contact with a circular arced edge (i.e. a flat and rounded contact); a geometry that has extensive applications. We show explicitly how the very convenient closed-form solution for this problem, derived from half-plane theory, may be collocated into the more realistic three-quarter plane far field solution, obtained from Williams’ solution. This provides a closed-form representation of the edge, correctly geared to the far-field solution, for the first time.

Estimation of Thickness and Radius of Hollow Metallic Spheres With a Simplified Forward Model

The measurement of metallic spherical objects in various industrial applications has a significant role in guaranteeing safety and product quality, which can be realized by the eddy-current testing technique without contact. In this study, for the measurement of thickness and radius of non-magnetic spherical shells, the original analytical model is simplified according to the asymptotic forms of the modified spherical Bessel functions. The simplified model contains only elementary functions, and can therefore accelerates the sphere property estimation by employing the optimization approach based on the Newton-Raphson method. In numerical simulation and experimental study, the proposed method can provide accurate spherical shell radii with less computational time compared with the original analytical model.

Large-time correlation functions in bosonic lattice field theories

Large-time correlation functions have a pivotal role in extracting particle masses from Euclidean lattice field theory calculations, however little is known about the statistical properties of these quantities. In this work, the asymptotic form of the distributions of the correlation functions at vanishing momentum is determined for bosonic interacting lattice field theories with a unique gapped vacuum. It is demonstrated that the deviations from the asymptotic form at large Euclidean times can be utilized to determine the spectrum of the theory.

Asymptotic Quantization of a Particle on a Sphere

Quantum systems whose states are tightly distributed among several invariant subspaces (variable spin systems) can be described in terms of distributions in a four-dimensional phase-space T∗S2 in the limit of large average angular momentum. The cotangent bundle T∗S2 is also the classical manifold for systems with E(3) symmetry group with appropriately fixed Casimir operators. This allows us to employ the asymptotic form of the star-product proper for variable (integer) spin systems to develop a deformation quantization scheme for a particle moving on the two-dimensional sphere, whose observables are elements of e(3) algebra and the corresponding phase-space is T∗S2. We show that the standard commutation relations of the e(3) algebra are recovered from the corresponding classical Poisson brackets and the explicit expressions for the eigenvalues and eigenfunctions of some quantized classical observables (such as the angular momentum operators and their squares) are obtained.

Fermion scattering on topological solitons in theCPN−1model

The scattering of Dirac fermions in the background fields of topological solitons of the $(2+1)$-dimensional $\mathbb{CP}^{N-1}$ model is studied using analytical and numerical methods. It is shown that the exact solutions for fermionic wave functions can be expressed in terms of the confluent Heun functions. The question of the existence of bound states for the fermion-soliton system is then investigated. General formulae describing fermion scattering are obtained, and a symmetry property for the partial phase shifts is derived. The amplitudes and cross-sections of the fermion-soliton scattering are obtained in an analytical form within the framework of the Born approximation, and the symmetry properties and asymptotic forms of the Born amplitudes are investigated. The dependences of the first few partial phase shifts on the fermion momentum are obtained by numerical methods, and some of their properties are investigated and discussed.

Pair trajectories of uncharged conducting spheres in an electric field

In this paper, we study the role of electrostatic forces on pair trajectories of two uncharged conducting spheres subject to an external electric field. We consider the hydrodynamic interactions between the spheres as they move relative to one another. Previous studies have shown that electric-field-induced forces on a two-sphere system are always attractive, except for the configuration when the line joining the centers is perpendicular to the external electric field. In the current study, we derive the asymptotic form of the interparticle force induced by the electric field in the lubrication limit for arbitrary size ratios. The attractive electric force diverges as the separation approaches zero. Thus, our calculation shows that the electric-field-induced forces can overcome the continuum lubrication resistance and allow finite time contact between the surfaces of two spherical conductors. We calculate the asymptotic variation of interparticle separation using the near-field asymptotic expressions for the electric-field-induced forces, exploring the role of hydrodynamic interactions in interparticle motion parallel and perpendicular to the electric field.

Universal asymptotic correlation functions for point group C_{6v} and an observation for triangular lattice Q-state Potts model.

We investigate a universal curve in asymptotic correlation functions of off-critical systems that possess C_{6v} symmetry following the argument for C_{4v} symmetry in our previous paper [Phys. Rev. E 102, 032141 (2020)2470-004510.1103/PhysRevE.102.032141]. Unlike the C_{4v} case, a minimal asymptotic form exists, which contains only two free parameters: the normalization constant and the modulus of the universal curve. We perform large-scale Monte Carlo simulations of the triangular lattice Q-state Potts model above the transition temperature. For Q=1, 2, 3, and 4, we successfully obtain numerical evidence that the minimal form gives the leading asymptotic behavior. We also discuss the possibility that the corrections to the minimal form are expressed using this form as a building block. From the minimal form with optimized parameters, we derive the equilibrium crystal shape of the honeycomb lattice Potts model, which is given by an algebraic curve of genus 1 and is universal among models with C_{6v}. Although the curve differs from those obtained in the C_{4v} case, the latter curves also have genus 1. We indicate that the birational equivalence concept can play an important role in comparing asymptotic forms for different point group symmetries, for example, C_{6v} and C_{4v}.

An asymptotic formula for sound radiation from plates

We present a new approach to the modelling of sound radiated from vibrating structures that will provide a platform for simulation of larger problems by the method of Dynamical Energy Analysis (DEA), which characterises sound using densities in a ray-dynamical phase space. We consider in particular the radiated acoustic power from plates with diffuse bending vibrations, which we characterise using a two-point correlation function of normal velocities. Such field-field correlation functions are transformed to ray-dynamical phase space densities using a Wigner transformation, which allows insertion into ray-based methods such as DEA. Using correlation functions to characterise plate vibrations also allows the method to cater for stochastic, noisy driving of such systems. The results for the radiation efficiency of a plate are presented in an asymptotic form, with leading contributions from the plate interior and its boundary and with corner corrections also being given for particular boundary conditions and right-angled corners. A notable feature of this analysis is that the bulk contribution vanishes below a critical frequency, and the asymptotic estimate of radiated power then leads with a boundary contribution. This is shown to agree well with a more traditional calculation based on modal analysis in the special case of a rectangular plate.

Equilibrium states of confined ions in two dimensions

ABSTRACT Ions that are trapped in two dimensions and are subject to a harmonic confining potential have widely varying stationary states that exhibit various asymptotic forms and bifurcations. We present a ‘birds-eye’ view of these structures for N = 2 to 5 ions, and the full range of anisotropy. These results may be interrogated in detail using the software provided here. Energy variations at bifurcation points and limits are also identified; for N = 5 these include blue-sky (or saddle-node) bifurcations. A limited attempt is also made to explore such features for a larger system of ions, i.e. N = 10.

Theory of Quantum Mechanical Scattering in Hyperbolic Space

The theory of quantum mechanical scattering in hyperbolic space is developed. General formulas based on usage of asymptotic form of the solution of the Shrödinger equation in hyperbolic space are derived. The concept of scattering length in hyperbolic space, a convenient measurable in describing low-energy nuclear interactions is introduced. It is shown that, in the limit of the flat space, i.e., when ρ→∞, the obtained expressions for quantum mechanical scattering in hyperbolic space transform to corresponding formulas in three-dimensional Euclidean space.

HD/FD Cooperative NOMA under UAV Deployment for a Novel Disaster-Management Model

This work presents a novel system model consisting of an unmanned aerial vehicle (UAV) equipped with a half/full-duplex relay (HDR/FDR) operating as a near-user in the downlink non-orthogonal multiple access (NOMA) systems. In a disaster situation, there is no direct connectivity of the active base station (BS) to the far user due to the out-of-coverage range. Therefore, UAV communication is established to aid the transmission from the same BS to the far user via the UAV. To quantify the effect, outage probability and throughput expressions in the exact and asymptotic forms were developed over the Weibull distribution (WD) fading channel. Additionally, the separation distance of the UAV from the base station is considered to quantify the effect. In particular, this paper helps to determine the optimal location of the UAV deployment from the BS at a fixed height from the ground to either maximize the far-user throughput or attain far-user throughput over different conditions of the WD fading channel. In addition, the performance results of the UAV-HDR/FDR-NOMA system are compared with those of a conventional downlink orthogonal multiple access (OMA) system. The comparison reveals that the UAV-HDR/FDR-NOMA systems outperform corresponding OMA systems in terms of outage probability and throughput over different values of the Weibull shaping index. The analytical results are then validated through numerical simulations on MATLAB.

Asymptotic Forms of Solutions to System of Nonlinear Partial Differential Equations

Here, we considerably develop the methods of power geometry for a system of partial differential equations and apply them to two different fluid dynamics problems: computing the boundary layer on a needle in the first approximation and computing the asymptotic forms of solutions to the problem of evolution of the turbulent flow. For each equation of the system, its Newton polyhedron and its hyperfaces with their normals and truncated equations are calculated. To simplify the truncated systems, power-logarithmic transformations are used and the truncated systems are further extracted. Here, we propose algorithms for computing unimodular matrices of power transformations for differential equations. Results: (1) the boundary layer on the needle is absent in liquid, while in gas it is described in the first approximation; (2) the solutions to the problem of evolution of turbulent flow have eight asymptotic forms, presented explicitly.

Open boundary conditions for the simulation of leaky modes

We propose an open-boundary method for the simulation of the modes of confining dielectric structures. The technique is inclusive of normal modes, but is especially advantageous for the simulation of quasi-normal, or leaky, modes. The central idea is to utilize the asymptotic form of targeted solutions to eliminate the outer part of the computational domain and bring the numerical boundary close to the simulated structure. While a similar approach was previously demonstrated for scalar quantum models, here we put forward a generalization for fully vectorial fields. Accuracy in this new context is validated using step-index and tube-type hollow core fiber geometries. The method has broad applicability, as quasi-bound modes are intrinsic to many systems of interest in optics and photonics.

Product form solutions of some queueing inventory problems in random environment

The public is often attracted by the offers given by the shopkeepers. These offers do make a crucial impact in the company's annual turnover. On a close watch it can be observed that these offers do change from time to time according to the seasonal variations. Here we investigate a single commodity Queueing-Inventory with 'offer' in random environment. The 'offer' can be anything given to the customers as incentives, who purchase the item. There are 'n' distinct environments. The offers in distinct environments are different. In some environments there may not be any offer. Here an environment dependent inventory control policy $ (s_i ,S) $ for $ 1\leq i \leq n $ is followed. Lead time, service and replenishment of the inventories are independent exponentially distributed random variables. It has been observed that when certain restrictions are imposed on the system, it admits asymptotic product form solution. Two such variants are discussed here. Long run behaviour of the system is analysed under the assumption of system stability. An optimization problem is constructed to compute the values of $ s_i $ and $ S $ that minimize the average system cost. The effect of environment in the optimum values of control variables, is also investigated.

The multiplicity and asymptotic forms of eigenvalues of vectorial diffusion equations with some certain assumptions

The main motivation point of this study is to obtain some novel results on the multiplicity of eigenvalues of diffusion equations. A diffusion equation with some boundary and jump conditions have been analyzed and integral equations have been obtained for the solution under certain initial conditions. Later, integral representations of these solutions have been provided. Finally, asymptotic formulas of eigenvalues with zeros of the characteristic have been considered. A brief conclusion has been given.

Fast Estimation of Metallic Pipe Properties Using Simplified Analytical Solution in Eddy-Current Testing

The eddy-current testing technique has been extensively explored for the measurement of metallic pipes in various industrial processes. In this study, a simplified analytical model of metallic pipes encircled by a coaxial sensor is derived according to the asymptotic forms of modified Bessel functions. The original Dodd and Deeds (DD) model is thus approximated by an elementary function describing the relationship between the measured inductance spectrum and the pipe properties. Due to the simple nature of the elementary function, the Jacobian matrix can be evaluated much more efficiently. Accordingly, a fast Newton–Raphson method-based optimization approach is developed. The proposed method has been evaluated by comparing it with the numerical simulation of the finite-element method (FEM) and experimental measurements which indicate its effectiveness for the measurement of metallic pipe properties.

Study of character of modulation instability in cyclotron resonance interaction of an electromagnetic wave with a counterpropagating rectilinear electron beam

In this paper, the interaction of a monochromatic electromagnetic wave with a counterpropagating electron beam moving in an axial magnetic field is considered. The purpose of this study is to investigate the conditions for occurrence of modulation instability (MI) in such a system and to determine at which parameters of the incident wave the MI is absolute or convective. Methods. Theoretical analysis of the MI character is carried out by studying the asymptotic form of unstable perturbations using the saddle-point analysis. The analytical results are verified by numerical simulations. Results. Theoretically, the boundary of change in the character of MI on the plane of input signal parameters (amplitude and detuning of the frequency from the cyclotron resonance) is determined. Numerical simulations confirm that as the signal frequency increases, the regime of self-modulation, which corresponds to the absolute MI, is replaced by the stationary single-frequency transmission corresponding to the convective MI. The numerical results coincide with the analytical ones for the system, which is matched at the end. The matching is implemented by smooth increasing of the guiding magnetic field in the region of electron beam injection. Conclusion. Determining the analytical conditions for the implementation of the absolute MI is of practical interest, since the emerging self-modulation can lead to the generation of trains of pulses with the spectrum in the form of frequency combs.