Order Perturbation Approximation Research Articles

Generalization of the Menegozzi and Lamb maser algorithm to the transient superradiance regime

ABSTRACT We investigate the application of the conventional quasi-steady state maser modelling algorithm of Menegozzi & Lamb (ML) to the high field transient regime of the one-dimensional Maxwell-Bloch (MB) equations for a velocity distribution of atoms or molecules. We quantify the performance of a first order perturbation approximation available within the ML framework when modelling regions of increasing electric field strength, and we show that the ML algorithm is unable to accurately describe the key transient features of R. H. Dicke’s superradiance (SR). We extend the existing approximation to one of variable fidelity, and we derive a generalization of the ML algorithm convergent in the transient SR regime by performing an integration on the MB equations prior to their Fourier representation. We obtain a manifestly unique integral Fourier representation of the MB equations which is $\mathcal {O}\left(N\right)$ complex in the number of velocity channels N and which is capable of simulating transient SR processes at varying degrees of fidelity. As a proof of operation, we demonstrate our algorithm’s accuracy against reference time domain simulations of the MB equations for transient SR responses to the sudden inversion of a sample possessing a velocity distribution of moderate width. We investigate the performance of our algorithm at varying degrees of approximation fidelity, and we prescribe fidelity requirements for future work simulating SR processes across wider velocity distributions.

High accuracy CO2 Fourier transform measurements in the range 6000–7000 cm−1

A Bruker IFS 125HR Fourier-transform spectrometer has been used to measure pure carbon dioxide transmittance spectra in the spectral range 6000–7000 cm−1, including the bands 30011-00001, 30012-00001, 30013-00001, 30014-00001, and 00031-00001. A total of 10 measurements with absorption path lengths between 14.6 and 59.4 m and sample pressures from 3 to 80 mbar were performed at 295 K. A multispectrum fitting approach was used applying the Hartmann-Tran profile extended to account for line mixing in the Rosenkranz first order perturbation approximation. Line positions, self-shifts, intensities, self-broadened widths, their speed dependence and in some cases line mixing were adjusted for fitting the measurements. A rigorous error analysis has been performed. The primary goal of this work was to investigate whether Fourier transform spectroscopy can deliver line intensity accuracy down to the 0.1% level. In this work, a combined systematic standard uncertainty of 0.15% has been achieved, further improvements down to the 0.1% level are feasible. The achieved level of combined standard uncertainty has, to our knowledge, not been reached by Fourier-transform spectroscopy before. Comparisons with most recent work are presented for line positions, line intensities, self-broadening and its speed dependence, and self-shifts. Measurement and line parameter databases are provided on Zenodo (doi:10.5281/zenodo.4525272).

Lowest order perturbative approximation to vibrational coupled cluster method in bosonic representation

We propose a perturbative approximation to the vibrational coupled cluster method in bosonic representation to reduce the cost of calculating the cluster matrix elements by considering only the first order of S and $$\sigma $$ for the construction of the effective Hamiltonian $$e^\sigma e^{-S}He^Se^{-\sigma }$$ . With the systematic analysis of the results of two molecules, $$\hbox {H}_2\hbox {O}$$ and 1,1-difluoroethylene, we find that the accuracy of the transition energies with such low order approximation is comparable to the fully converged VCCM. The lowest order perturbative approximation is used to calculate the cluster matrix elements of the vibrational coupled cluster method in bosonic representation. The computational scaling for cluster matric elements is reduced from iterative N6 to non-iterative N4.

Approximation of enzyme kinetics for high enzyme concentration by a first order perturbation approach

This contribution presents an approximate solution of the enzyme kinetics problem for the case of excess of an enzyme over the substrate. A first order perturbation approach is adopted where the perturbation parameter is the relation of the substrate concentration to the total amount of enzyme. As a generalization over existing solutions for the same problem, the presented approximation allows for nonzero initial conditions for the substrate and the enzyme concentrations as well as for nonzero initial complex concentration. Nevertheless, the approximate solution is obtained in analytical form involving only elementary functions like exponentials and logarithms. The presentation discusses all steps of the procedure, starting from amplitude and time scaling for a non-dimensional representation and for the identification of the perturbation parameter. Suitable time constants lead to the short term and long term behaviour, also known as the inner and outer solution. Special attention is paid to the matching process by the definition of a suitable intermediate layer. The results are presented in concise form as a summary of the required calculations. An extended example compares the zero order and first order perturbation approximations for the short term and long term solution as well as the uniform solution. A comparison to the numerical solution of the initial set of nonlinear ordinary differential equations demonstrates the achievable accuracy.

Correlation effects beyond coupled cluster singles and doubles approximation through Fock matrix dressing.

We present an accurate single reference coupled cluster theory in which the conventional Fock operator matrix is suitably dressed to simulate the effect of triple and higher excitations within a singles and doubles framework. The dressing thus invoked originates from a second-order perturbative approximation of a similarity transformed Hamiltonian and induces higher rank excitations through local renormalization of individual occupied and unoccupied orbital lines. Such a dressing is able to recover a significant amount of correlation effects beyond singles and doubles approximation, but only with an economic n5 additional cost. Due to the inclusion of higher rank excitations via the Fock matrix dressing, this method is a natural improvement over conventional coupled cluster theory with singles and doubles approximation, and this method would be demonstrated via applications on some challenging systems. This highly promising scheme has a conceptually simple structure which is also easily generalizable to a multi-reference coupled cluster scheme for treating strong degeneracy. We shall demonstrate that this method is a natural lowest order perturbative approximation to the recently developed iterative n-body excitation inclusive coupled cluster singles and doubles scheme [R. Maitra et al., J. Chem. Phys. 147, 074103 (2017)].

Measurement of positions, intensities and self-broadening line shape parameters of H2O lines in the spectral ranges 1850–2280 cm−1 and 2390–4000 cm−1

A Bruker IFS 125 HR Fourier transform spectrometer was used to measure several pure water transmittance spectra in the spectral range 1850–4000cm−1. A total of 15 measurements with absorption path lengths between 24.9cm and 174.6m and sample gas pressures from 0.1 to 20mbar were performed at 296K. The transmittance spectra were corrected for various error sources and were analyzed in the spectral ranges 1850–2280cm−1 and 2390–4000cm−1 for the majority of intensities between 3×10–26 and 3×10–19cm molecule−1. A multispectrum fitting approach was used applying a quadratic speed-dependent Voigt model extended to account for line mixing in the Rosenkranz first order perturbation approximation. Line positions, intensities, self-broadening widths, their speed-dependence and in some cases line mixing had to be adjusted for fitting the measurements to noise level. An extensive error estimation calculation was performed propagating several instrumental and measurement errors into individual parameter inaccuracies. The determined parameters are compared to HITRAN12 and independent experimental values while intensities are compared to recent ab initio calculations performed at UCL. The overall agreement between ab initio calculations and experimental values is remarkable and below 1% in most cases. The determined line parameters are provided as a supplement to this publication.

Dilepton production rate in a hot and magnetized quark–gluon plasma

The differential multiplicity of dileptons in a hot and magnetized quark–gluon plasma, ΔB≡dNB/d4xd4q, is derived from first principles. The constant magnetic field B is assumed to be aligned in a fixed spatial direction. It is shown that the anisotropy induced by the B field is mainly reflected in the general structure of photon spectral density function. This is related to the imaginary part of the vacuum polarization tensor, Im[Πμν], which is derived in a first order perturbative approximation. As expected, the final analytical expression for ΔB includes a trace over the product of a photonic part, Im[Πμν], and a leptonic part, Lμν. It is shown that ΔB consists of two parts, ΔB∥ and ΔB⊥, arising from the components (μ,ν)=(∥,∥) and (μ,ν)=(⊥,⊥) of Im[Πμν] and Lμν. Here, the transverse and longitudinal directions are defined with respect to the direction of the B field. Combining ΔB∥ and ΔB⊥, a novel anisotropy factor νB is introduced. Using the final analytical expression of ΔB, the possible interplay between the temperature T and the magnetic field strength eB on the ratio ΔB/Δ0 and νB is numerically studied. Here, Δ0 is the Born approximated dilepton multiplicity in the absence of external magnetic fields. It is, in particular, shown that for each fixed T and B, in the vicinity of certain threshold energies of virtual photons, ΔB≫Δ0 and ΔB⊥≫ΔB∥. The latter anisotropy may be interpreted as one of the microscopic sources of the macroscopic anisotropies, reflecting themselves, e.g., in the elliptic asymmetry factor v2 of dileptons.

Fermi’s golden rule: its derivation and breakdown by an ideal model

Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from to , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time, even when the first order perturbation approximation itself is still valid.

Double K-shell ionization of Kr induced by swift Xe54+ ions

SynopsisKα satellite and hypersatellite lines of Kr induced by 52-, 94-, 146-, and 197-MeV/u Xe54+ ions were measured. Experimental values were compared with theoretical ones obtained within the independent electron model (IEM) employing single electron probabilities calculated with the semi-classical approach (SCA). This comparison suggests that first order perturbation approximation is consistent well with the experimental result at 197-MeV/u projectile collision while breaks down for the lower energies.

A perturbative formalism for electronic transitions through conical intersections in a fully quadratic vibronic model.

We consider a fully quadratic vibronic model Hamiltonian for studying photoinduced electronic transitions through conical intersections. Using a second order perturbative approximation for diabatic couplings, we derive an analytical expression for the time evolution of electronic populations at a given temperature. This formalism extends upon a previously developed perturbative technique for a linear vibronic coupling Hamiltonian. The advantage of the quadratic model Hamiltonian is that it allows one to use separate quadratic representations for potential energy surfaces of different electronic states and a more flexible representation of interstate couplings. We explore features introduced by the quadratic Hamiltonian in a series of 2D models, and then apply our formalism to the 2,6-bis(methylene) adamantyl cation and its dimethyl derivative. The Hamiltonian parameters for the molecular systems have been obtained from electronic structure calculations followed by a diabatization procedure. The evolution of electronic populations in the molecular systems using the perturbative formalism shows a good agreement with that from variational quantum dynamics.

Insights into the dependent-scattering contributions to the extinction coefficient in highly scattering suspensions

We study theoretically the extinction of collimated light in random systems of highly scattering particles embedded in nonabsorbing media. We aim to provide rough guidelines on the behavior of the extinction coefficient in the so-called dependent-scattering regime. We base our analysis on Keller's second order perturbative approximation to the effective propagation constant. To gain physical insight, we also analyze a simple model based on the physical notion that particles in a dense system scatter light in an effective medium.

Electron-Positron Pair Production in an Arbitrary Polarized Ultrastrong Laser Field

Electron-positron pair production in an arbitrary polarized ultrastrong laser field is investigated in the first order perturbation approximation in which the Volkov states are used for convenient calculation of scattering amplitude and cross section. It is found surprisingly that the optimal pair production depends strongly on the polarization. For some cases of field parameters, the optimal field is elliptically polarized or evenly circularly polarized one, rather than the usual linear polarization as indicated by previous works. Some insights into pair generation are given and some interesting unexpected features are also discussed briefly.

Practical insight through perturbation analysis

Though industrial processes are perceived to be dauntingly complex from a mathematical modelling perspective, simple mathematical models often provide remarkable insight. One of the reasons behind this apparent contradiction is that the mathematical models often involve small and/or large non-dimensional parameters and therefore are amenable to simplification through the use of perturbation analysis. In many cases, the leading order perturbation approximation provides the bulk of the information about the structure and behaviour of the solution and is often sufficient to obtain profound insight into the phenomenon under examination. This article illustrates such utility of the perturbation analysis by using examples in which leading order perturbation approximations provide substantial insight into the mode transition phenomena in the vibrational behaviour of curved beams and helices. The methodology described in this article can be used in a wide range of applications to reveal the simplest possible structure of the mathematical model that answers the question under examination. References R. S. Anderssen. Acceptance speech on the award of the George Szekeres Medal. October, 2004. C. M. Bender, and S. A. Orszag. Advanced Mathematical Methods for Scientists and Engineers. McGraw-Hill, 1978. F. R. de Hoog. Why are simple models often appropriate in industrial mathematics? In R.S. Anderssen et al., (editors), {18th World IMACS Congress and MODSIM09 International Congress on Modelling and Simulation}, pages 23--36, 2009, http://www.mssanz.org.au/modsim09/Plenary/deHoog.pdf T. Tarnopolskaya, F. R. de Hoog, N. H. Fletcher, and S. Thwaites. Asymptotic analysis of the free in-plane vibrations of beams with arbitrarily varying curvature and cross-section. Journal of Sound and Vibration, 196: 659--680, 1996, doi:10.1006/jsvi.1996.0507 T. Tarnopolskaya, F. R. de Hoog, A. Tarnopolsky, and N. H. Fletcher. Vibration of beam and helices with arbitrarily large uniform curvature. Journal of Sound and Vibration, 228(2): 305--332, 1999, doi:10.1006/jsvi.1999.2413 T. Tarnopolskaya, F. R. de Hoog, and N. H. Fletcher. Low-frequency mode transition in the free in-plane vibration of curved beams. Journal of Sound and Vibration, 228(1): 69--90, 1999, doi:10.1006/jsvi.1999.2400 N. H. Fletcher, T. Tarnopolskaya, and F. R. de Hoog. Wave propagation on helices and hyperhelices: a fractal regression. Proc. R. Soc. Lond. A, 457(1): 33-43, 2001, doi:10.1098/rspa.2000.0654 N. H. Fletcher. Hyperhelices: a classical analog for strings and hidden dimensions. Am. J. Phys., 72(5): 701--703, 2004, doi:10.1119/1.1652038 K. A. Seffen and E.Toews. Hyperhelical actuators: coils and coiled-coils. AIAA Structures, Structural Dynamics and Materials Conference, 2004, California. J. Ueda, T. W. Secord, and H. H. Asada. Large effective-strain piezoelectric actuators using nested cellular architecture with exponential strain amplification mechanisms. IEEE/ASME Transactions on Mechatronics, 15(5): 770--782, 2010, doi:10.1109/TMECH.2009.2034973

Analytical modeling of static behavior of electrostatically actuated nano/micromirrors considering van der Waals forces

In this paper, the effect of van der Waals (vdW) force on the pull-in behavior of electrostatically actuated nano/micromirrors is investigated. First, the minimum potential energy principle is utilized to find the equation governing the static behavior of nano/micromirror under electrostatic and vdW forces. Then, the stability of static equilibrium points is analyzed using the energy method. It is found that when there exist two equilibrium points, the smaller one is stable and the larger one is unstable. The effects of different design parameters on the mirror’s pull-in angle and pull-in voltage are studied and it is found that vdW force can considerably reduce the stability limit of the mirror. At the end, the nonlinear equilibrium equation is solved numerically and analytically using homotopy perturbation method (HPM). It is observed that a sixth order perturbation approximation can precisely model the mirror’s behavior. The results of this paper can be used for stable operation design and safe fabrication of torsional nano/micro actuators.

INFLUENCE OF VAN DER WAALS FORCE ON STATIC BEHAVIOR OF NANO/MICROMIRRORS UNDER CAPILLARY FORCE

In the current paper, the effect of van der Waals (vdW) force on the static behavior and pull-in characteristics of nano/micromirrors under capillary force is investigated. At first, the dimensionless equation governing the static behavior of nano/micromirrors is obtained. The dependence of the critical tilting angle on the physical and geometrical parameters of the nano/micromirror and its supporting torsional beams is investigated. It is found that the existence of vdW force can considerably reduce the stability limits of the nano/micromirror. It is also found that rotation angle of the mirror due to capillary force highly depends on the vdW force applied to the mirror. Finally, analytical tool Homotopy Perturbation Method (HPM) is utilized for prediction of the nano/micromirror behavior under combined capillary and vdW force. It is observed that a sixth order perturbation approximation accurately predicts the rotation angle and stability limits of the mirror. The results of this paper can be used for successful fabrication of nano/micromirrors using wet etching release process where capillary force plays a major role in the system.

Modeling of Pull-In Instability of Nano/Micromirrors Under the Combined Effect of Capillary and Casimir Forces

In the current article the effect of the Casimir force on the static behavior and pull-in characteristics of nano/micromirrors under capillary force is investigated. At the first, the dimensionless equation governing the static behavior of nano/micromirrors is obtained. The dependency of the critical tilting angle on the physical and geometrical parameters of the nano/micromirror and its supporting torsional beams is investigated. It is found that the Casimir effect can considerably reduce the pull-in instability limits of the nano/micromirror. It is also found that rotation angle of the mirror under capillary force highly depends on the Casimir force applied to the mirror. Finally the analytical tool Homotopy Perturbation Method (HPM) is utilized for prediction of the mirror's behavior under combined capillary and Casimir forces. It is observed that a sixth order perturbation approximation accurately predicts the rotation angle and stability limits of the mirror. Results of this article can be used for successful fabrication of nano/micromirrors using the wet etching process where capillary force plays a major role in the system.

An Efficient Method of Computing Higher-Order Bond Price Perturbation Approximations

This paper develops a fast method of computing arbitrary order perturbation approximations to bond prices in DSGE models. The procedure is implemented to third order where it can shorten the approximation process by more than one hundred times. In a consumption-based endowment model with habits, it is further shown that a third order perturbation solution is more accurate than the log-normal method and a procedure using consol bonds. In addition, we present MATLAB codes that implement the suggested method to third order.

On the effect of magnetic field on the low-l solar p-mode oscillations

Abstract One of the possible magnetic field effects on the stellar pulsations is known to be a splitting in the observed frequencies. Using this knowledge in the solar convection zone, there are two aims in this work Considering the Sun as an incompressible fluid, our first objective was to investigate the variation of the physical parameters in the 30% outermost convective solar layer, during a pulsation period. The second purpose was to calculate, by means of the spherical harmonics, the shifts on the low-l p-mode frequencies which could be caused by the presence of the magnetic field in the Sun. The first order perturbation approximation was used in order to calculate analytically the resulting frequency shifts and the small perturbations on the magnetic field, as well as the physical parameters, such as density, pressure and temperature, of a Standard Solar Model excluding both rotation and magnetic field ( Christensen-Dalsgaard et al., 1996 ) in the unperturbed equilibrium case.

Reverberation clutter from combined internal wave refraction and bottom backscatter.

A hypothesis is proposed that one mechanism for clutter observed in shallow water reverberation measurements is due to the combined effect of forward scatter and subsequent backscatter. The forward scatter refracts part of the sound from low grazing angle to high grazing angle, then being backscattered by bottom roughness. The refracted sound impinges onto the bottom at higher grazing angle, resulting in higher backscatter because of elevated bottom scattering cross section as compared to that at lower grazing angles. The resultant reverberation will stand out as a target-like clutter. An example is presented as a non-linear internal wave propagates in a shallow water channel, resembling conditions found on the New Jersey Shelf. The effect on reverberation of the internal wave as clutter is investigated using a time-domain numerical model. The model uses parabolic equation for the two-way propagation and first order perturbation approximation for bottom backscatter. [Work supported by ONR.]

On the effect of magnetic field on the low- l solar p-mode oscillations

One of the possible magnetic field effects on the stellar pulsations is known to be a splitting in the observed frequencies. Using this knowledge in the solar convection zone, there are two aims in this work Considering the Sun as an incompressible fluid, our first objective was to investigate the variation of the physical parameters in the 30% outermost convective solar layer, during a pulsation period. The second purpose was to calculate, by means of the spherical harmonics, the shifts on the low- l p-mode frequencies which could be caused by the presence of the magnetic field in the Sun. The first order perturbation approximation was used in order to calculate analytically the resulting frequency shifts and the small perturbations on the magnetic field, as well as the physical parameters, such as density, pressure and temperature, of a Standard Solar Model excluding both rotation and magnetic field ( Christensen-Dalsgaard et al., 1996) in the unperturbed equilibrium case.