Perturbation Expansion Method Research Articles

Gravity-induced flocculation of Brownian particles in quiescent media under weak gravity

The main scope of the present paper is to study the rigorous problem of coupled gravity-induced/Brownian flocculation for the strong Brownian/weak gravity case, by solving the coupled Langevin type equations on determining the Brownian particles' collision trajectories. When comparing with the previous results, our numerical calculated flocculation rates are as good as those solutions obtained from using the singular perturbation expansion method to solve the steady state governing convective diffusion equation. Our solutions also confirm that the effect of Brownian diffusion can decrease the flocculation rates of colloidal particles under the condition of strong Brownian/weak gravity.

A novel perturbation expansion method for coupled system of acoustics and structures

We formulate a coupled vibration problem between a structure and an acoustic field in a mathematically rigorous fashion. A typical example of the structure is a car body that can be modeled by a cluster of thin plates. The problem leads to a nonstandard eigenvalue problem in a function space. Furthermore, we introduce a coupling strength parameter ε as a multiplier applied to the nondiagonal coupling terms. A natural interpretation of this parameter is given. We represent an eigenpair for the coupled system by a perturbation series with respect to ε which enable us to express the eigenpair for the coupled case by those for the decoupled case. We prove that the series consists only of even order terms of ε. In some practical applications, by using this perturbation series, it would become unnecessary to perform time consuming computations to get coupled eigenvalues, and hence the present results obtained by the perturbation analysis might have considerable engineering importance. We confirm the adequacy of this perturbation analysis by investigating some numerical examples. The results are given for a two-dimensional coupled problem. Without loss of generality, the same operator theoretical approach applies to the eigenvalue problems of the coupled vibration problem in a more general three-dimensional acoustic region with a plate for a part of its boundary.

Investigation of the age-at-onset heterogeneity in type 1 diabetes through mathematical modeling

The heterogeneity between young- and adult-onset type 1 diabetes (T1D) is well known, but not well understood. We approach this question through mathematical formulation and analysis of the dynamic interactions between the immune cells and the pancreatic islet β-cells that lead to the β-cell destruction. Utilizing the perturbation expansion method we investigate the dynamic stability of our system under fast and slow β-cell turnover limits. We find that if autoimmunity is initiated when the turnover is slow (adult age), a stable steady state can exist with reduced number of β-cells, where the β-cell regeneration balances the ongoing autoimmune destruction. This implies that a slow disease process is possible. In contrast, if autoimmunity occurs when the β-cell turnover is rapid (young age), such a stable state will never be attained and the destruction will progress unabated, leading to an acute disease onset. The major findings of our model are consistent with clinical observations, and it offers an explanation for the dynamic and phenotypic heterogeneity between young- and adult-onset T1D. More importantly, the model analyses point out that pathways regulating β-cell turnover can be new targets to interfere with the disease process of T1D.

Nonlinear magnetic susceptibility of ferrofluids

In ferrofluids, external magnetic fields can produce a nonlinearity in the dependence of magnetic susceptibilities on the field strength, due to both normal saturation and abnormal saturation. We derive the nonlinear magnetic susceptibility by using a Langevin model. Then, we investigate the nonlinear ac responses of the magnetic susceptibility by means of a perturbation expansion method and an orthogonal numerical method. The two methods are numerically shown in excellent agreement. We find that the responses are sensitive to suspension structures and field frequencies. Thus, by detecting the ac responses, it seems possible to real-time-monitor the structure of ferrofluids.

Stability of an alternative solitary-wave solution of an ion-acoustic wave obtained from the MKdV–KdV–ZK equation in magnetized non-thermal plasma consisting of warm adiabatic ions

The Korteweg-de Varies-Zakharov-Kuznetsov (KdV-ZK) equation describes the behaviour of long-wavelength weakly nonlinear ion-acoustic waves propagating obliquely to an external magnetic field in a non-thermal plasma consisting of warm adiabatic ions. When the coefficient of the nonlinear term of this equation vanishes, the nonlinear behaviour of ion-acoustic wave is described by a modified KdV-ZK (MKdV-ZK) equation. A combined MKdV-KdV-ZK equation more efficiently describes the nonlinear behaviour of ion-acoustic waves at points in the neighbourhood of the curve in the parametric plane along which the coefficient of the nonlinear term of the KdV-ZK equation vanishes. This combined MKdV-KdV-ZK equation admits both double-layer and alternative solitary-wave solutions having profile different from sech 2 or sech. In this paper the three-dimensional stability of the alternative solitary-wave solution having profile different from sech 2 or sech has been investigated by the recently developed multiple-scale perturbation expansion method of Allen and Rowlands. The instability condition and the growth rate of instability have been derived at the lowest order. The correct expression of the growth rate of instability at the lowest order has been obtained for a limiting case and the stability analysis has been carried out numerically from our model as presented in this paper for arbitrary values of the parameters involved in the system.

Interaction of an isolated state with an infinite quantum system containing several one-parameter eigenvalue bands: II. time-dependent case

A new method for the exact solution of the interaction of an isolated state $$\left| \Theta \right\rangle $$ with an infinite dimensional quantum system § ∞ containing several one-parameter eigenvalue bands $$\lambda _\nu (k)\in I_\nu \equiv \left[ {a_\nu ,b_\nu } \right]$$ is developed. Unlike standard perturbation expansion approach, this method produces correct results however strong the interaction between the state $$\left| \Theta \right\rangle $$ and the system $$\S_\infty ^{\,b} $$ . It is shown that in the case of the weak interaction this method correctly reproduces standard results obtained within the formalism of the perturbation expansion method. In particular, due to the interaction with the system $$\S_\infty ^{\,b} $$ , eigenvalue E of the state $$\left| \Theta \right\rangle $$ shifts to a new position. In addition, if this eigenvalue is embedded inside the range $$D=\bigcup\nolimits_\nu {I_\nu } $$ of the unperturbed eigenvalues, this shifted eigenvalue broadens and spectral distribution of the state $$\left| \Theta \right\rangle $$ has the shape of the universal resonance curve. However, if the interaction is strong, one finds much more complex and much more complicated behavior

Ferromagnetic transition in ordered double perovskites and related alloys

We study the electronic and magnetic properties of ordered perovskites Sr 2FeMO6 (where M is a transition metal), among which some compounds, like Sr2FeMoO6 are half-metallic with high Curie temperature while Sr2FeWO6 is an antiferromagnetic insulator. Using a double exchange type model with an interaction between localized spins and conduction electrons together with a tight-binding Hamiltonian and the renormalized perturbation expansion method, we study the behavior of Tc as a function of the number of conduction electrons and also as a function of the Fe-M energy difference. We also consider the Sr2FeMoxW1-xO6 compounds which present a magnetic and metal-insulator transition as a function of doping.

Dynamics of three coupled van der Pol oscillators with application to circadian rhythms

In this work we study a system of three van der Pol oscillators. Two of the oscillators are identical, and are not directly coupled to each other, but rather are coupled via the third oscillator. We investigate the existence of the in-phase mode in which the two identical oscillators have the same behavior. To this end we use the two variable expansion perturbation method (also known as multiple scales) to obtain a slow flow, which we then analyze using the computer algebra system MACSYMA and the numerical bifurcation software AUTO. Our motivation for studying this system comes from the presence of circadian rhythms in the chemistry of the eyes. We model the circadian oscillator in each eye as a van der Pol oscillator. Although there is no direct connection between the two eyes, they are both connected to the brain, especially to the pineal gland, which is here represented by a third van der Pol oscillator.

Dissipation of System and Atom in Two-Photon Jaynes–Cummings Model with Degenerate Atomic Levels

The method of perturbative expansion of master equation is employed to study the dissipative properties of system and of atom in the two-photon Jaynes-Cummings model (JCM) with degenerate atomic levels. The numerical results show that the degeneracy of atomic levels prolongs the period of entanglement between the atom and the field. The asymptotic value of atomic linear entropy is apparently increased by the degeneration. The amplitude of local entanglement and disentanglement is suppressed. The better the initial coherence property of the degenerate atom, the larger the coherence loss.

Few particles correlation in a one‐dimensional quasiperiodic lattice

AbstractDuring the last two decades, two of the most important discoveries in condensed matter physics have been the discovery of quasicrystals and the discovery of high‐Tc ceramic superconductors. These topics have generated a large number of experimental and theoretical studies in the physics of low dimension. They have also modified some of the concepts in solid state physics. For instance, it was believed that the five‐fold symmetry was incompatible with a long‐range order and it was not expected that ceramic materials with a high‐Tc and a short coherence length exhibit superconductivity. Therefore, it is important to revise both the spatial symmetry and the electronic correlation to identify how they affect the physical properties of materials. The study of these subjects is complex since we cannot use the reciprocal space to study quasicrystals and the electronic correlation in many‐body systems has not entirely been solved. Even in one‐dimensional quasiperiodic structures, the interactions between electrons have often been neglected and only few results have been obtained. In this work, we solved the cases of two and three interacting particles in a Fibonacci lattice using a real‐space method, the Green function technique, the renormalized perturbation expansion method and the Hubbard model. For the case of two interacting particles an analytical solution for the pairing phase diagram was obtained using the extended Hubbard Hamiltonian. For the case of three interacting particles the binding energy was numerically calculated. The results present here are compared with those obtained for the periodic and binary lattices. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Nonlinear Alternating Current Responses of Electrorheological Solids

When a composite containing nonlinear dielectric particles suspended in a host medium is subjected to a sinusoidal alternating current (ac) electric field, the dielectric response of the composite will generally consist of ac fields at frequencies of higher order harmonics. For an electrorheological (ER) solid under structure transformations due to external fields, we apply the Ewald-Kornfeld formulation to derive the local electric fields and induced dipole moments explicitly, and then we perform the perturbation expansion method to extract their fundamental and third-order harmonics analytically. It is shown that the degree of anisotropy of the ER solid can affect these harmonics significantly. Our results are well understood in the spectral representation theory. Thus, it seems possible to perform a real-time monitoring of the structure transformation by measuring the nonlinear ac responses of ER solids.

Landau Theory of Dynamic Phase Transitions and Systematic Perturbation Expansion Method for Getting Phase Diagrams

We propose a systematic perturbation expansion method for getting phase diagrams on dynamical symmetry breaking which is caused by varying the amplitude h or frequency Ω of the periodic external force h cos (Ω t ) in systems under the Landau type potentials. The method is based on the Fourier expansion and the Floquet theorem. We formulate the method by utilizing an introductory example, bistable system driven by the periodic external force. An order estimation criterion based on the magnitude of the Fourier coefficient for the fundamental frequency Ω plays an essential role in order to construct the systematic expansion method. According to the criterion, analytical expressions for the transition point and weak nonlinear expansions (Landau type expansions) near the transition point are derived in desired orders of approximation. We also show two applications of the method: the dynamical symmetry breaking in the XY-spin system with uniaxial anisotropy and tristable system at the presence of the forcing. T...

Comment on “Depth‐integrated modeling of suspended sediment transport” by M. Bolla Pittaluga and G. Seminara

Comment on “Depth‐integrated modeling of suspended sediment transport” by M. Bolla Pittaluga and G. Seminara

Nonlinear alternating current responses of graded materials.

When a composite of nonlinear particles suspended in a host medium is subjected to a sinusoidal electric field, the electrical response in the composite will generally consist of alternating current (ac) fields at frequencies of higher-order harmonics. The situation becomes more interesting when the suspended particles are graded, with a spatial variation in the dielectric properties. The local electric field inside the graded particles can be calculated by the differential effective dipole approximation, which agrees very well with a first-principles approach. In this work, a nonlinear differential effective dipole approximation and a perturbation expansion method have been employed to investigate the effect of gradation on the nonlinear ac responses of these composites. The results showed that the fundamental and third-harmonic ac responses are sensitive to the dielectric-constant and/or nonlinear-susceptibility gradation profiles within the particles. Thus, by measuring the ac responses of the graded composites, it is possible to perform a real-time monitoring of the fabrication process of the gradation profiles within the graded particles.

Spectral representation theory for higher order nonlinear responses in random composites with arbitrary nonlinearity

AbstractThe higher order nonlinear responses of random two‐phase composite media are investigated within spectral representation theory. The first component with the volume fraction p is assumed to have the nonlinear displacement (D)‐electric field (E) relation of the form D = ε1E+ χ1 |E|βE, where ε1 is the linear dielectric constant and χ1 is the (β + 1)th order nonlinear response; the second component is linear with the volume fraction 1–p and the dielectric constant ε2. The effective nonlinear responses of the composite media can be defined as 〈D〉 = εeE0 + χe|E0|βE0 + ηe|E0|2βE0, where εe, χe and ηe are the effective linear dielectric constant, the effective (β + 1)th order nonlinear response and (2β + 1)th order nonlinear response respectively. Based on spectral representation theory and the perturbative expansion method, general expressions for χe and ηe are derived as a function of the material parameters and the spectral density function m(x), which are indeed valid for arbitrary β and for any complicated microstructures. For the dilute limit and the Maxwell‐Garnett microstructures, our expressions yield the same formulae as those in previous works. Moreover, the theoretical results for Bruggeman effective medium microstructures are compared with numerical simulations, good agreement is found. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A numerical wave tank model for cavitating hydrofoils

In this paper, an iterative boundary element method (IBEM) for both 2-D and 3-D cavitating hydrofoils moving steadily inside a numerical wave tank (NWT) is presented and some extensive numerical results are given. The cavitating hydrofoil part, the free surface part and the wall parts of NWT are solved separately, with the effects of one on the others being accounted for in an iterative manner. The cavitating hydrofoil surface, the free surface, the bottom surface and the side walls are modelled by a low-order potential based panel method using constant strength dipole and source panels. Second-order correction on the free surface in 2-D are included into the calculations by the method of small perturbation expansion both for potential and for wave elevation. The source strengths on the free surface are expressed in terms of perturbation potential by applying first-order (linearized) and second-order free surface conditions. The IBEM is applied to a 2-D (NACA16006 and NACA0012) and a 3-D rectangular cavitating hydrofoil and the effect of number of iterations, the effect of the depth of the hydrofoil from finite bottom and the effect of the walls of NWT, on the results are discussed.

Fourth‐order approximation of fluid flow through rough‐walled rock fracture

A stochastic approach for estimating the permeability between rough‐walled rock fracture surfaces is presented. In the present analysis, the Reynolds lubrication equation, which is adopted for analyzing fluid flow through a rough surface of rock fracture, is solved as a stochastic partial differential equation using perturbative expansion method. The results are compared with the numerical deterministic analysis for randomly generated aperture distributions. The analytical results are in a good agreement with numerical results obtained by the Monte Carlo approach. Through the present analysis, a fourth‐order accurate approximation to the effective permeability of a single joint is given in terms of the mean, the standard deviation, and the spatial correlation, so that the effect of higher‐order terms is clarified more accurately than the classical second‐order approximation.

GABITOV–TURITSYN EQUATION FOR SOLITONS IN OPTICAL FIBERS

The dynamics of optical solitons with dispersion-management propagating through optical fibers with damping and amplification is studied. A multiple-scale perturbation expansion method is used to analyze the nonlinear Schrödinger's equation that governs the propagation of such solitons. In this paper we have considered both polarization preserving as well as birefringent fibers. Finally, the case of dense wavelength-division-multiplexed soliton system is also considered.

Effective nonlinear response in random mixture of coated granular cylinders

AbstractEffective dielectric response of granular composites consisting of coated cylindrical inclusions embedded in a host medium is investigated in both the dilute and nondilute cases. We assume that the shell and the host are linear with the dielectric function ϵs0 and ϵh0, while the core has the field‐dependent dielectric function ϵc = ϵc0 + χc|Ec|β, where ϵc0 and χc are the linear dielectric function and nonlinear susceptibility, respectively. General expressions for the effective linear dielectric function and third‐order nonlinear susceptibility of the three‐phase system are analytically obtained with perturbative expansion method. The partially resonant circumstances, in which the dielectric response of the components satisfies ϵs0 + ϵh0 or ϵs0 + ϵc0 = 0, and the conductance case in which the coated inclusions possess superconducting shells (ϵs0 → ∞) of zero thickness (t → 0), are examined in detail. We show that under the partially resonant conditions the coated cylindrical inclusions are equivalent to solid cylinders with larger nonlinear susceptibility and volume fractions than these of original cylindrical cores, and thus the effective nonlinear susceptibility χe of the composite system may be greatly enhanced. For the conductance case, we find that the coated cylinders behave as solid cylinders with a larger linear dielectric function than ϵc0.

GABITOV-TURITSYN EQUATION FOR SOLITONS IN MULTIPLE CHANNELS

The dynamics of optical solitons, with dispersion-management, propagating through dense wavelength-division-multiplexed system with damping and amplification is studied. A multiple-scale perturbation expansion method is used to analyze the nonlinear Schrodinger's equation that governs the propagation of such solitons.