Third-order Corrections Research Articles

Relativistic third-order viscous corrections to the entropy four-current from kinetic theory

By employing a Chapman-Enskog like iterative solution of the Boltzmann equation in relaxation-time approximation, we derive a new expression for the entropy four-current up to third order in gradient expansion. We show that unlike second-order and third-order entropy four-current obtained using Grad's method, there is a nonvanishing entropy flux in the present third-order expression. We further quantify the effect of the higher-order entropy density in the case of boost-invariant one-dimensional longitudinal expansion of a system. We demonstrate that the results obtained using the third-order evolution equation for the shear stress tensor, derived by employing the method of Chapman-Enskog expansion, show better agreement with the exact solution of the Boltzmann equation as well as with the parton cascade bamps, as compared to those obtained using the third-order equations from the method of Grad's 14-moment approximation.

Singular limit of a dispersive Navier–Stokes system with an entropy structure

We prove a low Mach number limit for a dispersive fluid system [3] which contains third-order corrections to the compressible Navier–Stokes. We show that the classical solutions to this system in the whole space ℝn converge to classical solutions to ghost-effect systems [7]. Our analysis follows the framework in [4], which is built on the methodology developed by Métivier and Schochet [6] and Alazard [1] for systems up to the second order. The key new ingredient is the application of the entropy structure of the dispersive fluid system. This structure enables us to treat cases not covered in [4] and to simplify the analysis in [4].

A correction of the half-power bandwidth method for estimating damping

The half-power bandwidth method is commonly used to evaluate the system damping by using frequency response curves and assuming a small damping ratio. Preceding derivations obtain the third-order corrections to the classical formula but still show large errors when the damping ratio is high, especially for the acceleration case. In this note, new approximate formulas for damping ratio in terms of the half-power bandwidth are established using displacement and acceleration frequency response functions, respectively. The proposed formulas, third-order correction and classical approximations are compared by using relative errors of calculated damping ratios for both displacement and acceleration cases. The new formulas for damping ratio are brief and show excellent accuracy for small as well as high damping ratios.

Corrections to the eikonal approximation for nuclear scattering at medium energies

The upcoming Facility for Rare Isotope Beams (FRIB) at the National Superconducting Cyclotron Laboratory (NSCL) at Michigan State University has reemphasized the importance of accurate modeling of low energy nucleus-nucleus scattering. Such calculations have been simplified by using the eikonal approximation. As a high energy approximation, however, its accuracy suffers for the medium energy beams that are of current experimental interest. A prescription developed by Wallace \cite{Wallace:1971zz,Wallace:1973iu} that obtains the scattering propagator as an expansion around the eikonal propagator (Glauber approach) has the potential to extend the range of validity of the approximation to lower energies. Here we examine the properties of this expansion, and calculate the first-, second-, and third-order corrections for the scattering of a spinless particle off of a ${}^{40}$Ca nucleus, and for nuclear breakup reactions involving ${}^{11}$Be. We find that, including these corrections extends the lower bound of the range of validity of the down to energies of 40 MeV. At that energy the corrections provide as much as a 15\% correction to certain processes.

Design of the EMF-PF and Error Correction

This paper designs the electromagnetic flowmeter in partially filled pipes (EMF-PF), which sensor alternate duties, sensing both the fluid velocity and fill height signals in order to get the flow in the partially filled pipes. Because the errors caused by two parameters measured, different sensitivities and zero-shifts reduce the accuracy of the EMF-PF, therefore calibration is needed. The repeated tests show that the third-order correction of the errors achieved good results.

Comparison of pulse compression methods using only a pulse shaper

Spectral phase compensation is crucial in the production of ultrafast laser pulses. To produce very short pulses, it is necessary to correct higher-order phase terms beyond the typical second-order and third-order corrections. It is helpful to have a pulse shaper to perform the compression, and to use an integrated, deterministic pulse-shaper-only method for pulse compression and characterization. While several algorithms exist for this purpose, it is desirable to know which method is optimal. Using a computer simulation, we investigate the speed of several existing approaches in several typical pulse shaping cases. We also present and demonstrate experimentally a new method named Spectral Phase of Electric field by Analytic Reconstruction (SPEAR), a variant of the Chirp Reversal Technique (CRT). The chirp-scan method, CRT, and SPEAR are found to be the fastest methods to achieve a desired level of accuracy, for the cases investigated.

Electrostatics-based finite-size corrections for first-principles point defect calculations

Finite-size corrections for charged defect supercell calculations typically consist of image-charge and potential alignment corrections. A wide variety of schemes for both corrections have been proposed for decades. Regarding the image-charge correction, Freysoldt, Neugebauer, and Van de Walle (FNV) recently proposed a novel method that enables us to accurately estimate the correction energy a posteriori through alignment of the defect-induced potential to the model charge potential [Freysoldt et al., Phys. Rev. Lett. 102, 016402 (2009)]. This method, however, still has two issues in practice. Firstly, it uses planar-averaged potential for determining the potential offset, which cannot be readily applied to relaxed system. Secondly, the long-range Coulomb interaction is assumed to be screened by a macroscopic dielectric constant. This is valid only for cubic systems and can bring forth huge errors for defects in anisotropic materials. In this study, we use the atomic site electrostatic potential as a potential marker instead of the planar-averaged potential, and extend the FNV scheme by adopting the point charge model in an anisotropic medium for estimating long-range interactions. We also revisit the conventional potential alignment correction and show that it is fully included in the image-charge correction and therefore unnecessary. In addition, we show that the potential alignment corresponds to a part of first-order and full of third-order image-charge correction; thus the third-order image-charge contribution is absent after the potential alignment. Finally, a systematic assessment of the accuracy of the extended FNV correction scheme is performed for a wide range of material classes. The defect formation energies calculated using around 100-atom supercells are successfully corrected even after atomic relaxation within a few tenths of eV compared to those in the dilute limit.

Exactly Solvable Counting Statistics in Open Weakly Coupled Interacting Spin Systems

We study the full counting statistics for interacting quantum many-body spin systems weakly coupled to the environment. In the leading order in the system-bath coupling, we derive exact spin current statistics for a large class of parity symmetric spin-1/2 systems driven by a pair of Markovian baths with local coupling operators. Interestingly, in this class of systems the leading-order current statistics are universal and do not depend on details of the Hamiltonian. Furthermore, in the specific case of a symmetrically boundary driven anisotropic Heisenberg (XXZ) spin-1/2 chain, we explicitly derive the third-order nonlinear corrections to the current statistics.

Effective Hamiltonians for fastly driven tight-binding chains

We consider a single particle tunnelling in a tight-binding model with nearest-neighbour couplings, in the presence of a periodic high-frequency force. An effective Hamiltonian for the particle is derived using an averaging method resembling classical canonical perturbation theory. Three cases are considered: uniform lattice with periodic and open boundary conditions, and lattice with a parabolic potential. We find that in the latter case, interplay of the potential and driving leads to appearence of the effective next-nearest neighbour couplings. In the uniform case with periodic boundary conditions the second- and third-order corrections to the averaged Hamiltonian are completely absent, while in the case with open boundary conditions they have a very simple form, found before in some particular cases by S.Longhi [Phys. Rev. B 77, 195326 (2008)]. These general results may found applications in designing effective Hamiltonian models in experiments with ultracold atoms in optical lattices, e.g. for simulating solid-state phenomena.

On the quantum L-operator for the two-dimensional lattice Toda model

We consider the two-dimensional quantum lattice Toda model for affine and simple Lie algebras of type A. For its known L-operator, the second-order correction in lattice parameter e is found. It is proved that the equation determining the third-order correction in e has no solutions. Bibliography: 9 titles.

Termodynamika metalichnoho heliyu

The internal and free energies of liquid metallic helium are calculated for wide ranges of density and temperature, and the corresponding equation of state is obtained in the framework of perturbation theory. The electron-ion interaction potential is selected as a small parameter, and the calculations are carried out to the third order of smallness inclusive. Conduction electrons are considered in the random phase approximation with regard for the exchange interaction and correlations in the local field approximation. The hard-sphere model is used for the nuclear subsystem, the sphere diameter being the only parameter of the theory. The sphere diameter and the system density, at which helium transforms from the single- into double-ionized state are evaluated by analyzing the effective pair interaction between helium nuclei also in the third order of perturbation theory. The case of double-ionized helium atoms is considered. The third-order correction turns out substantial in all examined cases. The values obtained for thermodynamic parameters such as the density, temperature, and pressure fall within the ranges typical of the central regions of giant planets, which allows us to suppose the existence of metallic helium in the solar system.

Electrical conductivity and thermopower of metallic helium

The pair effective interionic interaction, electrical resistance, and thermopower of liquid metallic helium have been calculated over wide temperature and density ranges using the perturbation theory for the potential of electron-ion interaction. For conduction electrons, the random-phase approximation has been used taking into account the exchange interaction and correlations in the local-field approximation. The nuclear subsystem has been described by the hard-sphere model. The sphere diameter is the only parameter of the theory. The diameter and the system density at which helium is transformed from the singly ionized to doubly ionized state have been estimated based on an analysis of the pair effective interaction between helium nuclei. The case of doubly ionized helium atoms has been considered. The numerical calculations have been performed taking into account the perturbation theory in terms up to the third order. In all cases, the role of the third-order correction is significant. In the case of metallic helium, the values of the electrical resistance and its temperature dependence are characteristic of divalent simple liquid metals, as well as the dependences of the thermopower on the density and temperature.

Accuracy of the half-power bandwidth method with a third-order correction for estimating damping in multi-DOF systems

A third-order correction was recently suggested to improve the accuracy of the half-power bandwidth method in estimating the damping of single DOF systems. This paper analyzes the accuracy of the half-power bandwidth method with the third-order correction in damping estimation for multi-DOF linear systems. Damping ratios in a two-DOF linear system are estimated using its displacement and acceleration frequency response curves, respectively. A wide range of important parameters that characterize the shape of these response curves are taken into account. Results show that the third-order correction may greatly improve the accuracy of the half-power bandwidth method in estimating damping in a two-DOF system. In spite of this, the half-power bandwidth method may significantly overestimate the damping ratios of two-DOF systems in some cases.

The Effect of Third-Order Nonlinearities on the Statistical Distributions of Wave Heights, Crests and Troughs in Bimodal Crossing Seas

This paper investigates the effect of third-order nonlinearities on the statistical distributions of wave heights, crests, and troughs of waves mechanically generated in a deep-water basin and simulating two crossing systems characterized by bimodal spectra. The observed statistics exhibits various effects of third-order nonlinearities, in a manner dependent on both the distance from the wave-maker and the angle between the mean directions of the component wave systems. In order to isolate and demonstrate the effects of third-order nonlinearities by themselves, the vertically asymmetric distortions induced by second-order bound waves are removed from the observed time series. It appears then that the distributions of wave crests, troughs and heights extracted from the nonskewed records clearly deviate from the Rayleigh distribution, suggesting that the waves are characterized by nonlinear corrections of higher-order than the typical of second-order waves. Nonetheless, some models developed for weakly nonlinear second-order waves can still be used in describing wave heights, crests and troughs in mixed seas, provided that the relevant distribution parameters are modified, so as to reflect the effects of third-order corrections and some basic characteristics of the mixed seas.

Higgs boson production in the SM and MSSM to NNLO and beyond

In this contribution a brief review about the status of higher order corrections to Higgs boson production within the Minimal Supersymmetric Standard Model (MSSM) is given. Furthermore the first activities towards third-order corrections in the Standard Model (SM) are discussed.

Susceptibility phase imaging with improved image contrast using moving window phase gradient fitting and minimal filtering

To enhance image contrast in susceptibility phase imaging using a new method of background phase removal. A background phase removal method is proposed that uses the spatial gradient of the raw phase image to perform a moving window third-order local polynomial estimation and correction of the raw phase image followed by minimal high pass filtering. The method is demonstrated in simulation, 10 healthy volunteers, and 5 multiple sclerosis patients in comparison to a standard phase filtering approach. Compared to standard phase filtering, the new method increased phase contrast with local background tissue in subcortical gray matter, cortical gray matter, and multiple sclerosis lesions by 67% ± 33%, 13% ± 7%, and 48% ± 19%, respectively (95% confidence interval). In addition, the new method removed more phase wraps in areas of rapidly changing background phase. Local phase gradient fitting combined with minimal high pass filtering provides better tissue depiction and more accurate phase quantification than standard filtering.

Breakdown of the Single-Exchange Approximation in Third-Order Symmetry-Adapted Perturbation Theory

We report third-order symmetry-adapted perturbation theory (SAPT) calculations for several dimers whose intermolecular interactions are dominated by induction. We demonstrate that the single-exchange approximation (SEA) employed to derive the third-order exchange-induction correction (E(exch-ind)((30))) fails to quench the attractive nature of the third-order induction (E(ind)((30))), leading to one-dimensional potential curves that become attractive rather than repulsive at short intermolecular separations. A scaling equation for (E(exch-ind)((30))), based on an exact formula for the first-order exchange correction, is introduced to approximate exchange effects beyond the SEA, and qualitatively correct potential energy curves that include third-order induction are thereby obtained. For induction-dominated systems, our results indicate that a "hybrid" SAPT approach, in which a dimer Hartree-Fock calculation is performed in order to obtain a correction for higher-order induction, is necessary not only to obtain quantitative binding energies but also to obtain qualitatively correct potential energy surfaces. These results underscore the need to develop higher-order exchange-induction formulas that go beyond the SEA.

Large-scale first-principles molecular dynamics simulations: application to the microsolvation of biologically-relevant ions in aqueous clusters

First-principles molecular dynamics simulations are employed to investigate the solvation structure of the biologically-relevant paradigm guanidinium-ion in water clusters, Gdn+(H2O)n. In these simulations, the intermolecular interactions are evaluated along the trajectories with the self-consistent-charge density-functional tight-binding (SCC-DFTB) model with the on-site third-order correction and modified effective Coulomb interaction, an approximate model of density-functional theory. For Gdn+(H2O)21, the simulated probability distribution of the ion-to-water cluster centre-of-mass distances and water angular coordinates suggests that Gdn+ is primarily localised at the surface of the water cluster, with the surface of the ion relatively devoid of water molecules. Estimates of the computational resources required for first-principles molecular dynamics simulations of larger Gdn+(H2O)n (n≤100) indicate that, with a modern supercomputer, such simulations can readily be performed in a matter of days.

Third-Order Corrections and Mass-Shedding Limit of Rotating Neutron Stars Computed By a Complex-Plane Strategy

We implement the so-called “complex-plane strategy” for computing general-relativistic polytropic models of uniformly rotating neutron stars. This method manages the problem by performing all numerical integrations, required within the framework of Hartle’s perturbation method, in the complex plane. We give emphasis on computing corrections up to third order in the angular velocity, and the mass-shedding limit. We also compute the angular momentum, moment of inertia, rotational kinetic energy, and gravitational potential energy of the models considered.

Perceptually informed synthesis of bandlimited classical waveforms using integrated polynomial interpolation

Digital subtractive synthesis is a popular music synthesis method, which requires oscillators that are aliasing-free in a perceptual sense. It is a research challenge to find computationally efficient waveform generation algorithms that produce similar-sounding signals to analog music synthesizers but which are free from audible aliasing. A technique for approximately bandlimited waveform generation is considered that is based on a polynomial correction function, which is defined as the difference of a non-bandlimited step function and a polynomial approximation of the ideal bandlimited step function. It is shown that the ideal bandlimited step function is equivalent to the sine integral, and that integrated polynomial interpolation methods can successfully approximate it. Integrated Lagrange interpolation and B-spline basis functions are considered for polynomial approximation. The polynomial correction function can be added onto samples around each discontinuity in a non-bandlimited waveform to suppress aliasing. Comparison against previously known methods shows that the proposed technique yields the best tradeoff between computational cost and sound quality. The superior method amongst those considered in this study is the integrated third-order B-spline correction function, which offers perceptually aliasing-free sawtooth emulation up to the fundamental frequency of 7.8 kHz at the sample rate of 44.1 kHz.