SECOND ORDER HIGH DENSITY EXPANSION FOR FREE ENERGY AND ORDER PARAMETER

1991 ◽  
Vol 05 (18) ◽  
pp. 2935-2949
Author(s):  
M. BARTKOWIAK ◽  
K.A. CHAO
Keyword(s):  
Free Energy ◽  
Order Parameter ◽  
Order Approximation ◽  
Local Approximation ◽  
Second Order ◽  
High Density ◽  
First Order ◽  
First Order Approximation ◽  
Consistent Equation ◽  
Density Expansion

The self-consistently renormalized high-density expansion (SHDE) is first used to determine temperature dependence of order parameter. Free energy and magnetization of the Ising model has been calculated to the second order. It is shown that the unphysical discontinuity of the order parameter as a function of temperature, which appears in the first-order approximation, still remains in the second-order calculation. Based on the 1/d expansion, we then construct a method to select (1/z)i contributions from the high density expansion terms. This method is applied to the first and second-order self-consistent equation for magnetization. Selection of the first order in 1/z contributions within the first order of the SHDE leads to considerable improvement of the behavior of magnetization as a function of temperature, and application of the local approximation to the second order of the SHDE term gives an acceptable single-value behavior of the order parameter.

scholarly journals On the justification of Gaussian heuristic method in the theory of random vibration

1979 ◽  
Vol 1 (3-4) ◽  
pp. 1-11
Author(s):  
Nguyen Cao Menh
Keyword(s):  
Small Parameter ◽  
Random Vibration ◽  
Order Approximation ◽  
Heuristic Method ◽  
Second Order ◽  
Input Process ◽  
Mean Values ◽  
First Order ◽  
Gaussian Input ◽  
First Order Approximation

Recently in the problems of random vibration, the heuristic method, in which output process is supposed to be Gaussian when Gaussian input process is given, is applied [1, 2]. This method is called the “Gaussian heuristic method”. This paper deals with the justification of “Gaussian heuristic method”, form that two following important conclusions are proved: - “Gaussian heuristic method” gives density function of probability with the first order approximation with respect to the small parameter ε. - Applying this method we get mean values and second order correlation functions in second order approximation with respect to the small parameter ε.

Effective attenuation anisotropy of thin-layered media

Geophysics ◽  
2007 ◽  
Vol 72 (5) ◽  
pp. D93-D106 ◽  
Author(s):  
Yaping Zhu ◽  
Ilya Tsvankin ◽  
Ivan Vasconcelos
Keyword(s):  
Stiffness Matrix ◽  
Order Approximation ◽  
Seismic Attenuation ◽  
Second Order ◽  
The Real ◽  
First Order ◽  
Intrinsic Attenuation ◽  
Effective Velocity ◽  
First Order Approximation ◽  
Attenuation Anisotropy

One of the well-known factors responsible for the anisotropy of seismic attenuation is interbedding of thin attenuative layers with different properties. Here, we apply Backus averaging to obtain the complex stiffness matrix of an effective medium formed by an arbitrary number of anisotropic, attenuative constituents. Unless the intrinsic attenuation is uncommonly strong, the effective velocity function is controlled by the real-valued stiffnesses (i.e., independent of attenuation) and can be determined from the known equations for purely elastic media. Attenuation analysis is more complicated because the attenuation parameters are influenced by the coupling between the real and imaginary parts of the stiffness matrix. The main focus of this work is on effective transversely isotropic models with a vertical symmetry axis (VTI) that include isotropic and VTI constituents. Assuming that the stiffness contrasts, as well as the intrinsic velocity and attenuation anisotropy, are weak, we develop explicit first-order (linear) and second-order (quadratic) approximations for the attenuation-anisotropy parameters [Formula: see text], [Formula: see text], and [Formula: see text]. Whereas the first-order approximation for each parameter isgiven sim-ply by the volume-weighted average of its interval values, the second-order terms include coupling between various factors related to both heterogeneity and intrinsic anisotropy. Interestingly, the effective attenuation for P- and SV-waves is anisotropic even for a medium composed of isotropic layers with identical attenuation, provided there is a velocity variation among the constituent layers. Contrasts in the intrinsic attenuation, however, do not create attenuation anisotropy, unless they are accompanied by velocity contrasts. Extensive numerical testing shows that the second-order approximation for [Formula: see text], [Formula: see text], and [Formula: see text] is close to the exact solution for most plausible subsurface models. The accuracy of the first-order approximation depends on the magnitude of the quadratic terms, which is largely governed by the strength of the velocity (rather than attenuation) anisotropy and velocity contrasts. The effective attenuation parameters for multiconstituent VTI models vary within a wider range than do the velocity parameters, with almost equal probability of positive and negative values. If some of the constituents are azimuthally anisotropic with misaligned vertical symmetry planes, the effective velocity and attenuation functions may have different principal azimuthal directions or even different symmetries.

Calculation of the Ion Optical Properties of Inhomogeneous Magnetic Sector Fields, including the Second Order Aberrations in the Median Plane

1959 ◽  
Vol 14 (2) ◽  
pp. 121-129 ◽  
Author(s):  
H. A. Tasman ◽  
A. J. H. Boerboom
Keyword(s):  
Optical Properties ◽  
Order Approximation ◽  
Median Plane ◽  
Second Order ◽  
Central Path ◽  
Resolving Power ◽  
Lagrange Equations ◽  
Magnetic Sector ◽  
First Order ◽  
First Order Approximation

Investigation is made of the ion optical properties of inhomogeneous magnetic sector fields. In first order approximation the field is assumed to vary proportional to r—n (0 ≦ n < 1); the term in the magnetic field expansion which determines the second order aberrations is chosen independent of n, which makes the elimination possible of e. g. the second order angular aberration. From the EULER— LAGRANGE equations the second order approximation of the ion trajectories in the median plane and the first order approximation outside the median plane are derived for the case of normal incidence and exit of the central path in the sector field. An equation is presented giving the shape of the pole faces required to produce the desired field. The influence of stray fields is neglected. The object ana image distances are derived, as well as the mass dispersion, the angular, lateral and axial magnification, the resolving power, and the inclination of the plane of focus of the mass spectrum. The maximum transmitted angle in the z-direction is calculated. The resolving power proves to be proportional to (1—n) -1 whereas the length of the central path is proportional to (1—n) -½. An actual example is given of a 180° sector field with n=0.91, where the mass resolving power is increased by a factor 11 as compared with a homogeneous sector field of the same radius and slit widths.

Letter Reading as a Function of Approximation to English and French

1971 ◽  
Vol 33 (3_suppl) ◽  
pp. 1139-1142 ◽  
Author(s):  
Renaud S. Le Blanc ◽  
J. Gerard Muise
Keyword(s):  
French Text ◽  
English Language ◽  
Order Approximation ◽  
Second Order ◽  
Differential Sensitivity ◽  
First Order ◽  
French And English ◽  
First Order Approximation

French Ss were required to read letter strings which approximated French and English texts. Ss performed similarly at the zero and first order approximation but read faster on the French text at the second order. The results may be due to the greater uncertainty of the English language or to a differential sensitivity to the statistical constraints of both languages.

Molecular Transition Moment Calculations Using the Intermediate Operator Quasidegenerate Perturbation Theory

1991 ◽  
Vol 02 (01) ◽  
pp. 546-548
Author(s):  
A.V. ZAITSEVSKII ◽  
A.I. DEMENT’EV
Keyword(s):  
Perturbation Theory ◽  
Order Approximation ◽  
Effective Property ◽  
Second Order ◽  
Transition Moment ◽  
Electron Property ◽  
First Order ◽  
Molecular Transition ◽  
Simple Molecules ◽  
First Order Approximation

We developed a procedure for molecular transition one-electron property calculations based on the simple second-order QDPT approximation for the intermediate Hamiltonian and corresponding first-order approximation for intermediate effective property operators. To test its abilities, a series of transition moment calculations for simple molecules was performed and the results were compared with CI results.

Unified Second-Order Stochastic Averaging Approach

1997 ◽  
Vol 64 (2) ◽  
pp. 281-291 ◽  
Author(s):  
M. Hijawi ◽  
N. Moschuk ◽  
R. A. Ibrahim
Keyword(s):  
Order Approximation ◽  
Stochastic Averaging ◽  
Nonlinear Damping ◽  
Second Order ◽  
Unified Approach ◽  
Order Analysis ◽  
First Order ◽  
Different Types ◽  
First Order Approximation ◽  
Good Agreement

First-order stochastic averaging has proven very useful in predicting the response statistics and stability of dynamic systems with nonlinear damping forces. However, the influence of system stiffness or inertia nonlinearities is lost during the averaging process. These nonlinearities can be recaptured only if one extends the stochastic averaging to second-order analysis. This paper presents a systematic and unified approach of second-order stochastic averaging based on the Stratonovich-Khasminskii limit theorem. Response statistics, stochastic stability, phase transition (known as noise-induced transition), and stabilization by multiplicative noise are examined in one treatment. A MACSYMA symbolic manipulation subroutine has been developed to perform the averaging processes for any type of nonlinearity. The method is implemented to analyze the response statistics of a second-order oscillator with three different types of nonlinearities, excited by both additive and multiplicative random processes. The second averaging results are in good agreement with those estimated by Monte Carlo simulation. For a special nonlinear oscillator, whose exact stationary solution is known, the second-order averaging results are identical to the exact solution up to first-order approximation.

Improved first-order approximation of eigenvalues and eigenvectors

AIAA Journal ◽  
1998 ◽  
Vol 36 ◽  
pp. 1721-1727
Author(s):  
Prasanth B. Nair ◽  
Andrew J. Keane ◽  
Robin S. Langley
Keyword(s):  
Order Approximation ◽  
Eigenvalues And Eigenvectors ◽  
First Order ◽  
First Order Approximation

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

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