Order Correction Terms Research Articles

Casimir-Polder intermolecular forces in minimal length theories

Generalized uncertainty relations are known to provide a minimal length $\ensuremath{\hbar}\sqrt{\ensuremath{\beta}}$. The effect of such minimal length in the Casimir-Polder interactions between neutral atoms (molecules) is studied. The first order correction term in the minimal uncertainty parameter is derived and found to describe an attractive potential scaling as ${r}^{\ensuremath{-}9}$, as opposed to the well-known ${r}^{\ensuremath{-}7}$ long range retarded potential.

Continuum limit semiclassical initial value representation for dissipative systems

In this paper, we consider a dissipative system in which the system is coupled linearly to a harmonic bath. In the continuum limit, the bath is defined via a spectral density and the classical system dynamics is given in terms of a generalized Langevin equation. Using the path integral formulation and factorized initial conditions, it is well known that one can integrate out the harmonic bath, leaving only a path integral over the system degrees of freedom. However, the semiclassical initial value representation treatment of dissipative systems has usually been limited to a discretized treatment of the bath in terms of a finite number of bath oscillators. In this paper, the continuum limit of the semiclassical initial value representation is derived for dissipative systems. As in the path integral, the action is modified with an added nonlocal term, which expresses the influence of the bath on the dynamics. The first order correction term to the semiclassical initial value approximation is also derived in the continuum limit.

Numerical Simulations of femtosecond-laser-induced dynamic alignment of molecules in the high-frequency off-resonance regime

The motion equation for ϑ between the molecular axis and laser polarization direction in a high-frequency off-resonance femtosecond laser field is deduced while simultaneously examining the effects of a permanent dipole moment and field-induced polarizability and hyperpolarizability to molecular rotation. Femtosecond-laser-induced dynamic alignment of CO, N2, and Br2 molecules are investigated by numerically solving the obtained rotation equation for the angle ϑ. The effects of the molecular permanent dipole moment and the field-induced polarizability and hyperpolarizability on the degree of alignment are presented at different intensities. Our computational results show that the dynamic alignment of molecules is primarily determined by field-induced polarizability and the second hyperpolarizability for the laser intensity range from 5 × 1014 to 5 × 1016 W/cm2. The contributions of higher order correction terms to molecular alignment can usually be neglected. The polarizability-field interaction makes the angular distributions of a molecule have a maximum along the polarization axis and a minimum perpendicular to it. The role of the second hyperpolarizability keeps the molecular counts maximum along the laser polarization direction but minimum at an angle of 45° between the molecular axis and the polarization direction. There is also a second maximum of molecular counts perpendicular to the polarization axis. For CO, N2, and Br2 molecules, the dependences of laser-induced dynamic alignment on laser intensity exhibit completely different characteristics.

High order correction terms for the peak-peak correlation function in nearly-Gaussian models

One possible way to investigate the nature of the primordial power spectrum fluctuations is by investigating the statistical properties of the local maximum in the density fluctuation fields. In this work we present a study of the mean correlation function, $\xi_r$, and the correlation function for high amplitude fluctuations (the peak-peak correlation) in a slighlty non-Gaussian context. From the definition of the correlation excess, we compute the Gaussian two-point correlation function and, using an expansion in Generalized Hermite polynomials, we estimate the correlation of high density peaks in a non-Gaussian field with generic distribution and power spectrum. We also apply the results obtained to a scale-mixed distribution model, which correspond to a nearly Gaussian model. The results reveal that, even for a small deviation from Gaussianity, we can expect high density peaks to be much more correlated than in a Gaussian field with the same power spectrum. In addition, the calculations reveal how the amplitude of the peaks in the fluctuations field is related to the existing correlations. Our results may be used as an additional tool to investigate the behavior of the N-point correlation function, to understand how non-Gaussian correlations affect the peak-peak statistics and extract more information about the statistics of the density field.

Accounting for Triplet and Saturation Effects in FCS Measurements

Fluorescence correlation spectroscopy (FCS) is an increasingly important tool for determining low concentrations and dynamics of molecules in solution. Oftentimes triplet transitions give rise to fast blinking effects, which are accounted for by including an exponential term in the fitting of the autocorrelation function (ACF). In such cases, concomitant saturation effects also modify the amplitude and shape of the remaining parts of the ACF. We review studies of triplet and saturation effects in FCS and present a simple procedure to obtain more accurate results of particle concentrations and diffusional dynamics in experiments where triplet kinetics are evident, or where moderate laser powers approaching saturation levels are used, for example, to acquire sufficient photon numbers when observation times are limited. The procedure involves use of a modified function for curve-fitting the ACF, but there are no additional fitting parameters. Instead, a simple calibration of the total fluorescence count rate as a function of relative laser power is fit to a polynomial, and the non-linear components of this fit, together with the relative laser power used for the FCS measurement, are used to specify the magnitude of additional terms in the fitting function. Monte Carlo simulations and experiments using Alexa dyes and quantum dots, with continuous and pulsed laser excitation, demonstrate the application of the modified fitting procedure with first order correction terms, in the regime where distortions in the ACF due to photobleaching and detector dead time are small compared to those of fluorescence saturation and triplet photophysics.

Non-relativistic binding energies of heavy neutral atoms: Dependence of correlation energy on atomic number

As shown by Milne in the early days of quantum mechanics, the non-relativistic binding energy of a heavy neutral atom with nuclear charge Ze is proportional to Z 7/3 in the large Z limit. Correction terms of order Z 2 and Z 5/3 to Milne’s limiting large Z result were first calculated by Scott. Here, the question as to the power of Z at which correlation energy first appears in such a binding energy formula is addressed. It is shown that a recent approximation proposed by one of us (ÁN) supports an earlier extrapolation from large dimensionality D to D = 3 by Herschbach and co-workers, that correlation energy enters the large Z binding energy formula for the first time at o( Z 4/3).

A derivation of the isothermal quantum hydrodynamic equations using entropy minimization

AbstractDedicated to Prof. Herbert Gajewski in occasion of his 65th birthday Isothermal quantum hydrodynamic equations of order 𝒪($\hbar$2) using the quantum entropy minimization method recently developed by Degond and Ringhofer are derived. The equations have the form of the usual quantum hydrodynamic model including a correction term of order 𝒪($\hbar$2 which involves the vorticity. If the initial vorticity is of order 𝒪($\hbar$4, the standard model is obtained up to order . The derivation is based on a careful expansion of the quantum equilibrium obtained from the entropy minimization in powers of $\hbar$2.

Wave Equation Depth Migration with the Optimum Split-Step Fourier Method In 3-D VTI Media

A dispersion equation for transversely isotropic media with a vertical symmetry axis (VTI) is derived from the phase velocity expressed in terms of Thomsen parameters. Based on the vertical wavenumber solved from the dispersion equation, the optimum split-step Fourier 3-D wave equation depth migration method in isotropic media is extended to 3-D VTI media. The macro velocity model used for describing VTI media is consistent with the current velocity estimation method and can simultaneously accommodate the isotropy, weak anisotropy and strong anisotropy. The higher order correction terms added to the phase-shift method makes the proposed algorithm be able to model wider-angle wavefield with high accuracy in the presence of severe lateral velocity variations in 3-D VTI media. The accuracy of the proposed algorithm is demonstrated by 2-D and 3-D migration impulse responses.

Quasinormal modes of the charged black hole in Gauss-Bonnet gravity

The d-dimensional string generated gravity models lead to Einstein-Maxwell equations with quadratic order correction term called the Gauss-Bonnet term. We calculate the quasinormal modes for the d-dimensional charged black hole in the framework of this model. The quasinormal spectrum essentially depends upon the Gauss-Bonnet coupling parameter $\alpha$ which is related to the string scale, and is totally different from that for black holes derived from Einstein action. In particular, at large $\alpha$ the quasinormal modes are proportional to $\alpha$, while as $\alpha$ goes to zero the qusinormal modes approach their Schwarzschild values. In contrary to Einstein theory black hole behavior, the damping rate of the charged GB black hole as a function of charge does not contain a chracteristic maximum, but instead the monotonic falling down is observed. In addition, there have been obtained an asymptotic formula for large multipole numbers.

Approximate analytical solutions of the Forchheimer equation

In this paper we derive approximate analytical solutions for non-steady-state, non-linear flows through porous media, described by the Forchheimer equation. We demonstrate that one has to distinguish between two characteristic regimes. In early times, the hydraulic gradient is steep, and subsequently the inertial terms are dominant. One obtains the leading hydraulic behavior by neglecting linear terms describing the viscous dissipation mechanisms. In moderate times, as the disturbance introduced upstream propagates through the entire medium, the hydraulic gradient, and subsequently the inertial effects become less important: the leading behavior corresponds to the Darcy solution. The influence of the inertia mechanisms in this regime is taken into account by computing higher order correction terms, by means of perturbation analysis. The verification of the analytical solution is done by comparison with numerical results of a finite volume code. As application of the developed theory the water volume accumulated in an aquifer due to river flooding is presented.

Dynamic characteristics of electrons in high-order corrected fields of ultrashort laser pulses

Highorder correction to the fields of ultrashort, tightlyfocused laser pulses expressed in power series of ε=1/(ω0t0) and s=1/(k0w0)（ω0=ck0 the central oscillatory frequency, t0 the pulse duration, w0 the beam waist radius), are derived. Based on paraxial approximation, the firstorder correction terms to the Gaussian pulses are explicitly given. Their corrections of amplitude and phase, are found to be related to the variable of ε. Applying them to the study of the electron dynamics in the intense laser pulse field, we found that as long as ω0t0>20, the zerothorder approximation (long pulse approximation) is adequate for describing the interaction. For ω0t0<20, higherorder corrections have to be taken into account.

Surface effects in quantum spin chains

Chains of quantum spins with open ends and isotropic Heisenberg exchangeare studied. By diagonalizing the Hamiltonian for chains of finite lengthN and obtaining all the energy eigenvalues, the magnetic susceptibilityχ, the specificheat Cv, and thepartition function Z can be calculated exactly for these chains. The high-temperature series expansions of these are then evaluated.For χ and Cv it is found that the terms in the series consist of three parts. One is the normalhigh-T series already known in great detail for the ring (chain with periodic boundary conditions). The othertwo consist of a ‘surface’ term and a correction term of order(1/T)N. The surface term is found as a series up to and including(1/T)8 forspin S = 1/2 and 1. Simple Padé approximant formulae are given to extend the range of validity belowT = 1.

First-principles calculations for the electronic band structures of small diameter single-wall carbon nanotubes

We present first-principles calculations on the band structures of 40 different small diameter $(d)$ single-wall carbon nanotubes (SWCNTs), including 14 chiral ones, employing density functional theory (DFT) within the local density approximation (LDA), using the Vienna ab initio simulation package (VASP). The band gaps are calculated and discussed for all of the tubes. From small to large diameters, the gap of semiconducting zigzag tubes first increases, then reaches a maximum of about $1\phantom{\rule{0.3em}{0ex}}\mathrm{eV}$ for (11,0), after which it decreases, approximately as $1∕d$, showing a ``buckling'' around this average behavior. The smallest diameter zigzag tubes are all metallic, due to $\ensuremath{\sigma}\ensuremath{-}\ensuremath{\pi}$ mixing caused by high curvature. The Fermi wave-vector of armchair tubes shows a downshift from its ideal, zone folding expected value; this shift is proportional to $1∕{d}^{\phantom{\rule{0.1em}{0ex}}2}$. Eight of the ZF-metallic tubes (4 zigzags, 4 chirals) show a small gap in the band structure. For the zigzag tubes, this band gap is roughly given by the formula $\ensuremath{\Delta}=1.99∕{d}^{\phantom{\rule{0.1em}{0ex}}2}+140.9∕{d}^{4}$ ($\ensuremath{\Delta}$ in eV, $d$ in \AA{}). The appearance of the $1∕{d}^{4}$ higher order correction term is due to high curvature at small diameters, however it's apparently overestimated in our calculations. This overestimation can likely be eliminated by considering many-electron effects.

Absorption of a quantum in a D1/D5 system

In Nucl. Phys. B 615 (2001) 285 [hep-th/0107113], the wave equation for a minimally coupled scalar was studied in the geometry of a D1/D5 system with non-zero angular momentum. The probability for a quantum to enter the throat was computed by taking small a parameter γ which is associated with the value of the angular momentum. In the leading order in γ, the result was found to agree with the dual CFT result. In this Letter, we report on an observation that there are corrections of higher order in γ. Our results should be useful for determining the higher order correction terms that the dual CFT needs in order to incorporate the presence of the ‘capped’ geometry.

Phase equilibria in the Ni–Al–Cr–Ti system at 1000 and 1100 °C

Design of advanced materials in general calls for better understanding of phase transitions at service temperatures and relevant equilibrium phase diagrams. In the case of multicomponent alloys, it is often necessary to study phase transformations on model systems. Ni–Al–Cr–X quaternary systems (X = Co, W, Ti, Ta, etc.) are relatively simple yet sufficiently representative models for Ni–base superalloys. However, their complete thermodynamic description requires qualified knowledge of thermodynamic parameters for lower order systems as well as higher order correction terms. This work continues our previous studies on Ni–Al–Cr–(Co, W, Mo) systems. This time it is focused on titanium as the fourth element. Experimental alloys were prepared with respect to γ + γ′ phases presence and studied by means of analytical electron microscopy after annealing at typical service temperatures. Experimentally identified types of phases are compared with those obtained by means of thermodynamic calculations.

Reconstruction of Thin Conductivity Imperfections*

We consider the case of a uniform plane conductor containing a thin curve-like inhomogeneity of finite conductivity. In this article we prove that the imperfection can be uniquely determined from the boundary measurements of the first order correction term in the asymptotic expansion of the steady state voltage potential as the thickness goes to zero.

A Primal-Dual Interior-Point Method to Solve the Optimal Power Flow Dispatching Problem

This paper presents a primal-dual path-following interior-point method for the solution of the optimal power flow dispatching (OPFD) problem. The underlying idea of most path-following algorithms is relatively similar: starting from the Fiacco-McCormick barrier function, define the central path and loosely follow it to the optimum solution. Several primal-dual methods for OPF have been suggested, all of which are essentially direct extensions of primal-dual methods for linear programming. Nevertheless, there are substantial variations in some crucial details which include the formulation of the non-linear problem, the associated linear system, the linear algebraic procedure to solve this system, the line search, strategies for adjusting the centring parameter, estimating higher order correction terms for the homotopy path, and the treatment of indefiniteness. This paper discusses some of the approaches that were undertaken in implementing a specific primal-dual method for OPFD. A comparison is carried out with previous research on interior-point methods for OPF. Numerical tests on standard IEEE systems and on a realistic network are very encouraging and show that the new algorithm converges where other algorithms fail.

Light transmission from a twisted nematic liquid crystal: accurate methods to measure the azimuthal anchoring energy.

In this paper, we analyze the light transmission from a twisted nematic liquid crystal (NLC) and we propose two accurate and very direct optical methods to measure the azimuthal anchoring energy. In both of them, a monochromatic beam of wavelength lambda with a polarization vector that rotates at an angular frequency omega impinges on a twisted nematic liquid crystal. The intensity of the transmitted beam is modulated at angular frequency 2omega with a phase shift beta, which is related to the surface azimuthal director angle phi(1) at the investigated interface. It is shown that there exists a special geometry where the simple adiabatic relation phi(1)=beta/2 is satisfied up to second order in the small perturbative parameter alpha=lambda/(2piDelta(n)xi), where Delta(n) is the anisotropy of the refractive indices of the NLC and xi is the twist distortion length. Furthermore, the small residual higher order correction terms can be greatly reduced by choosing a proper geometry for the experiment. With this choice, the range of validity of the adiabatic theory is greatly extended. The perturbative theoretical results are fully confirmed by numerical calculations. The azimuthal anchoring energy coefficient can be obtained by measuring phase shift beta versus the intensity of an applied magnetic field. These methods greatly improve the accuracy of the previous transmitted light techniques and also provide accurate measurements of strong azimuthal anchoring energies.

Magnetoconvection in a rapidly rotating sphere: the weak–field case

The linear stability of magnetoconvection in a rapidly rotating sphere is investigated. The weak–field regime is studied where the Elsasser number Λ is ;O ( ;E 1/3) and ;E is the Ekman number, assumed to be small. In this regime, the magnetic field is strong enough to affect the critical Rayleigh number, frequency and preferred azimuthal wavenumber by order–one amounts, but is weak enough that the convection still has the form of columnar rolls. A global asymptotic theory is constructed that differs from previous local theories of the onset of convection at asymptotically small Ekman number, and it provides a consistent Wentzel–Kramers–Brillouin solution which takes account of the phase–mixing phenomenon. The asymptotic theory is developed to give the leading–order and first–order correction terms, including those from Hartmann boundary layers. Numerical solutions of the relevant partial differential equations have also been found, for values of the Ekman number down to 10–6.5, and these are compared with the asymptotic theory. Good agreement with the asymptotic theory is found.

The role of the frozen surface approximation in small wave-height perturbation theory for moving surfaces.

In the standard development of the small wave-height approximation (SWHA) perturbation theory for scattering from moving rough surfaces [e.g., E. Y. Harper and F. M . Labianca, J. Acoust. Soc. Am. 58, 349-364 (1975) and F. M. Labianca and E. Y. Harper, J. Acoust. Soc. Am. 62, 1144-1157 (1977)] the necessity for any sort of frozen surface approximation is avoided by the replacement of the rough boundary by a flat (and static) boundary. In this paper, this seemingly fortuitous byproduct of the small wave-height approximation is examined and found to fail to fully agree with an analysis based on the kinematics of the problem. Specifically, the first order correction term from the standard perturbation approach predicts a scattered amplitude that depends of the source wave number, whereas the kinematics point to a scattered amplitude that depends on the scattered wave number. It is shown that a perturbation approach in which an explicit frozen surface approximation is made before the SWHA is invoked predicts (first order) scattered amplitudes that are in agreement with the analysis based on the kinematics.