Order In Perturbative Expansion Research Articles

High-Informative Theorizing in Plasma Physics: Basic Principles and Importance for Researches on Beam-Plasma Heating, Plasma Confinement and Transport Phenomena

An analysis of existed concepts of basic turbulent plasma phenomena have shown that the most fundamental of the beginnings of plasma kinetic theory are defective. Basically, common methods of the theory yield equally rigorous justifications to incompatible versions of the same physical phenomenon. This general property stems from two inseparable reasons: the asymptotic convergence of intermediate iterative calculations and the common substitution of real plasmas by plasma ensembles. Via variations in the leading order of perturbation expansion, one generates a diversity of scenarios of the plasma physical evolution: The conditional limit of successive iterations depends on the theory leading order. Similarly, the laws of evolution of statistics of plasma ensemble cannot be independent on the ensemble content. Basic principles were formulated of gaining the informativeness of plasma theoretical deductions with account for above reasons. For a case of turbulent plasma, the technique was developed of reducing full plasma description to the most informative of possible final macrophysical scenarios. The importance of respective knowledge for researches on beam-plasma heating, plasma confinement and transport phenomena is discussed.

STRINGENT VERIFICATION OF THIRD ORDER + SECOND ORDER PERTURBATION DENSITY FUNCTION THEORY: BASED ON SHORT-RANGE SQUARE WELL POTENTIAL

A recently proposed third order + second order perturbation density functional theory (DFT) approach is made self-contained by using a virial pressure from the Ornstein–Zernike integral equation theory as input to determine the numerical value of an associated physical parameter. An exacting examination is formulated by applying the self-contained perturbation DFT approach to a short-range square well fluid of low temperatures subject to various external fields and comparing the theoretical results for density profiles to the corresponding grand canonical ensemble Monte Carlo simulation results. The comparison seems favorable for the third order + second order perturbation DFT approach as a self-contained and accurate predictive approach. It is surprisingly found that this self-contained third order + second order perturbation DFT approach is displayed outstandingly even if a deep SW perturbation term is being accounted for by a second order perturbation expansion. A discussion is presented about potential opportunity for this perturbation scheme.

Higher order perturbation expansion of waves in water of variable depth

In this work, we extended the application of “the modified reductive perturbation method” to long waves in water of variable depth and obtained a set of KdV equations as the governing equations. Seeking a localized travelling wave solution to these evolution equations we determine the scale function c 1 ( τ ) so as to remove the possible secularities that might occur. We showed that for waves in water of variable depth, the phase function is not linear anymore in the variables x and t . It is further shown that, due to the variable depth of the water, the speed of the propagation is also variable in the x coordinate.

Refactorizing NRQCD short-distance coefficients in exclusive quarkonium production

In a typical exclusive quarkonium production process, when the center-of-mass energy, s , is much greater than the heavy quark mass m, large kinematic logarithms of s / m 2 will unavoidably arise at each order of perturbative expansion in the short-distance coefficients of the nonrelativistic QCD (NRQCD) factorization formalism, which may potentially harm the perturbative expansion. This symptom reflects that the hard regime in NRQCD factorization is too coarse and should be further factorized. We suggest that this regime can be further separated into “hard” and “collinear” degrees of freedom, so that the familiar light-cone approach can be employed to reproduce the NRQCD matching coefficients at the zeroth order of m 2 / s and order by order in α s . Taking two simple processes, exclusive η b + γ production in e + e − annihilation and Higgs boson radiative decay into ϒ, as examples, we illustrate how the leading logarithms of s / m 2 in the NRQCD matching coefficients are identified and summed to all orders in α s with the aid of Brodsky–Lepage evolution equation.

Absolutely Local Excited Orbitals in the Higher Order Perturbation Expansion for the Molecular Interaction

Based on the locally projected molecular orbitals, the third and fourth order perturbation corrections for the molecular interaction within the single excitations are evaluated, and the calculated interaction energies are compared with the counterpoise (CP) corrected interaction energy of the Hartree-Fock level of theory. It is demonstrated that the third order calculation is a practical and powerful method to obtain the binding energy almost equal to the CP corrected energy. It requires only one more two-electron integral handling after the LP MO calculation, which is faster than a usual supermolecule HF calculation. For the perturbation expansion, the absolutely local excited orbitals are determined. For small basis sets, it is shown that the partial delocalization of the absolutely local excited orbitals is a compromizing technique to take into account the charge-transfer contribution without reintroducing a large basis set superposition error.

First order spectral perturbation theory of square singular matrix pencils

Let H(λ)=A0+λA1 be a square singular matrix pencil, and let λ0∈C be an eventually multiple eigenvalue of H(λ). It is known that arbitrarily small perturbations of H(λ) can move the eigenvalues of H(λ) anywhere in the complex plane, i.e., the eigenvalues are discontinuous functions of the entries of A0 and A1. Therefore, it is not possible to develop an eigenvalue perturbation theory for arbitrary perturbations of H(λ). However, if the perturbations are restricted to lie in an appropriate set then the eigenvalues change continuously. We prove that this set of perturbations is generic, i.e., it contains almost all pencils, and present sufficient conditions for a pencil to be in this set. In addition, for perturbations in this set, explicit first order perturbation expansions of λ0 are obtained in terms of the perturbation pencil and bases of the left and right null spaces of H(λ0), both for simple and multiple eigenvalues. Infinite eigenvalues are also considered. Finally, information on the eigenvectors of the generically regular perturbed pencil is presented. We obtain, as corollaries, results for regular pencils that are also new.

A Jacobi–Davidson like method for nonlinear eigenvalue problems based on singularity theory

AbstractWe present a Jacobi–Davidson like correction formula for left and right eigenvector approximations for non‐Hermitian nonlinear eigenvalue problems. It exploits techniques from singularity theory for characterizing singular points of nonlinear equations. Unlike standard nonlinear Jacobi‐Davidson, the correction formula does not contain derivative information and works with orthogonal projectors only. Moreover, the basic method is modified in that the new eigenvalue approximation is taken as a nonlinear Rayleigh functional obtained as root of a certain scalar nonlinear equation the existence of which – as well as a first order perturbation expansion – is shown. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Random-phase-approximation-based correlation energy functionals: Benchmark results for atoms

The random phase approximation for the correlation energy functional of the density functional theory has recently attracted renewed interest. Formulated in terms of the Kohn-Sham orbitals and eigenvalues, it promises to resolve some of the fundamental limitations of the local density and generalized gradient approximations, as, for instance, their inability to account for dispersion forces. First results for atoms, however, indicate that the random phase approximation overestimates correlation effects as much as the orbital-dependent functional obtained by a second order perturbation expansion on the basis of the Kohn-Sham Hamiltonian. In this contribution, three simple extensions of the random phase approximation are examined; (a) its augmentation by a local density approximation for short-range correlation, (b) its combination with the second order exchange term, and (c) its combination with a partial resummation of the perturbation series including the second order exchange. It is found that the ground state and correlation energies as well as the ionization potentials resulting from the extensions (a) and (c) for closed subshell atoms are clearly superior to those obtained with the unmodified random phase approximation. Quite some effort is made to ensure highly converged data, so that the results may serve as benchmark data. The numerical techniques developed in this context, in particular, for the inherent frequency integration, should also be useful for applications of random phase approximation-type functionals to more complex systems.

Crosstalk in Balanced Interconnections Used for Differential Signal Transmission

This paper introduces a two-stage model for assessing crosstalk in balanced interconnections used for differential signal transmission, such as multipair cables. The first stage considers the interconnection as uniform and uses a change of variables based on the symmetries inherent to balancing, for the definition of a set of parameters to be measured. The second stage of the model takes into account the nonuniformity related to the fluctuations of the characteristics of the interconnection, using a first order perturbation expansion and a probabilistic approach. This model is compatible with published results on crosstalk in multipair cables.

The single excitation perturbation expansion theory based on the locally projected molecular orbitals for molecular interaction: Comparison with the counterpoise corrected energy

The single excitation second order perturbation expansion (SPT) based on the locally projected self-consistent field molecular orbitals for molecular interaction is numerically tested for isomers of uracil–water and for 24 water clusters. For the extensive basis sets, the calculated binding energy BE SPT becomes close to the counterpoise corrected binding energy BE SCF CP . The required cpu time is less than that for a single supermolecule calculation. For smaller basis sets, by restricting the excited orbitals to the strictly monomer basis set (SMBS), BE SPT becomes approximate to BE SCF CP . The use of the SMBS for the excited orbitals has a clear advantage over the orthogonal but delocalized excited orbitals.

Influence of a magnetic field on a single-plate thermoacoustic system

In this paper, an analytical model for a single-plate thermoacoustic system in the presence of a magnetic field is proposed. The magnetic field acts perpendicular to the direction of the fluid oscillation. The governing momentum and energy equations are simplified to quasi-one-dimensional forms by approximations. Also these equations are used to derive the expressions for the fluctuating velocity ( u 1 ) , fluctuating temperature ( T 1 ) , Nusselt number ( Nu τ ) , and critical temperature gradient ( ∇ T cr ) by a first order perturbation expansion. After the effects of Hartmann number ( Ha δ ) , drive ratio ( DR ) , and temperature gradient ( ∇ T m ) on the flow and thermal fields are discussed, they are graphically presented. Finally, for the present problem, a wave equation is modeled by using the continuity, momentum, and energy equations. In addition, some possible solutions to this equation are presented.

Spin–orbit and tensor interactions in homogeneous matter of nucleons: accuracy of modern many-body theories

We study the energy per particle of symmetric nuclear matter and pure neutron matter using realistic nucleon–nucleon potentials having noncentral tensor and spin–orbit components, up to three times the empirical nuclear matter saturation density, ρ0=0.16 fm−3. The calculations are carried out within the frameworks of the Brueckner–Bethe–Goldstone (BBG) and correlated basis functions (CBF) formalisms, in order to ascertain the accuracy of the methods. The two hole-line approximation, with the continuous choice for the single particle auxiliary potential, is adopted for the BBG approach, whereas the variational Fermi hypernetted chain/single operator chain theory, corrected at the second order perturbative expansion level, is used in the CBF one. The energies are then compared with the available quantum and variational Monte Carlo results in neutron matter and with the BBG, up to the three hole-line diagrams. For neutron matter and potentials without spin–orbit components all methods, but perturbative CBF, are in reasonable agreement up to ρ∼3ρ0. After the inclusion of the LS interactions, we still find agreement around ρ0, whereas it is spoiled at larger densities. The spin–orbit potential lowers the energy of neutron matter at ρ0 by ∼3–4 MeV per nucleon. In symmetric nuclear matter, the BBG and the variational results are in agreement up to ∼1.5ρ0. Beyond this density, and in contrast with neutron matter, we find good agreement only for the potential having spin–orbit components.

An analytical solution and computer simulation for a multi-plate thermoacoustic system

In this paper, an analytical model for a multi-plate thermoacoustic system is developed. The governing momentum and energy equations are simplified to quasi-one-dimensional forms using reasonable approximations. The expressions for the fluctuating velocity (u1), fluctuating temperature (T1), Nusselt number (Nu), fluid friction irreversibility (NsFF), heat transfer irreversibility (NsHT), energy flux density (Ė) and transverse velocity (v1) are derived using a first order perturbation expansion of the governing momentum and energy equations. Finally, a wave equation is modelled for the present problem by using the continuity, momentum and energy equations.

BV Quantization of Topological Open Membranes

We study bulk-boundary correlators in topological open membranes. The basic example is the open membrane with a WZ coupling to a 3-form. We view the bulk interaction as a deformation of the boundary string theory. This boundary string has the structure of a homotopy Lie algebra, which can be viewed as a closed string field theory. We calculate the leading order perturbative expansion of this structure. For the 3-form field we find that the C-field induces a trilinear bracket, deforming the Lie algebra structure. This paper is the first step towards a formal universal quantization of general quasi-Lie bialgebroids.

Density matrix perturbation theory.

An orbital-free quantum perturbation theory is proposed. It gives the response of the density matrix upon variation of the Hamiltonian by quadratically convergent recursions based on perturbed projections. The technique allows treatment of embedded quantum subsystems with a computational cost scaling linearly with the size of the perturbed region, O(N(pert.)), and as O(1) with the total system size. The method allows efficient high order perturbation expansions, as demonstrated with an example involving a 10th order expansion. Density matrix analogs of Wigner's 2n+1 rule are also presented.

Nonequilibrium dynamics of optical-lattice-loaded Bose-Einstein-condensate atoms: Beyond the Hartree-Fock-Bogoliubov approximation

In this work a two-particle irreducible (2PI) closed-time-path (CTP) effective action is used to describe the nonequilibrium dynamics of a Bose Einstein condensate (BEC) selectively loaded into every third site of a one-dimensional optical lattice. The motivation of this work is the recent experimental realization of this system at National Institute of Standards and Technology (NIST) where the placement of atoms in an optical lattice is controlled by using an intermediate superlattice. Under the 2PI CTP scheme with this initial configuration, three different approximations are considered: a) the Hartree-Fock-Bogoliubov (HFB) approximation, b) the next-to-leading order 1/$\mathcal{N}$ expansion of the 2PI effective action up to second order in the interaction strength and c) a second order perturbative expansion in the interaction strength. We present detailed comparisons between these approximations and determine their range of validity by contrasting them with the exact many body solution for a moderate number of atoms and wells. As a general feature we observe that because the second order 2PI approximations include multi-particle scattering in a systematic way, they are able to capture damping effects exhibited in the exact solution that a mean field collisionless approach fails to produce. While the second order approximations show a clear improvement over the HFB approximation our numerical result shows that they do not work so well at late times, when interaction effects are significant.

Exciton in matrix formulation of quantum Hall effect

The quantum Hall effect is studied by introducing two different matrix variables for electrons and holes, having Chern–Simons type interactions. By generalizing the constraint condition proposed by Susskind to realize the Pauli's exclusion principle in this two component matrix model, the classical exciton solution having excitation of both quasi-electron and quasi-hole is obtained. The constraint condition is also solved quantum mechanically in the infinite-sized matrix case, giving the examples of the physical states. Using these quantum states, the corrections of the exciton energy, coming from the noncommutativity of space (Pauli principle) and from the quantum effects of the background state, are estimated in the lowest order perturbation expansion. As a result, the dispersion relation of exciton is obtained.

Perturbation theory for Lyapunov exponents of a toral map: Extension of a result of Shub and Wilkinson

Starting from a hyperbolic toral automorphism times a rotation of the circle, we obtain, for a small volume preserving perturbation, an exact and rigorous second order perturbation expansion of the Lyapunov exponents.

Higher Order Perturbation Expansion for Pairing Interaction in Repulsive Hubbard Model

Higher Order Perturbation Expansion for Pairing Interaction in Repulsive Hubbard Model

Higher Order Perturbation Expansion for Pairing Interaction in Repulsive Hubbard Model

We calculate the effective pairing interaction in the Hubbard model on the two-dimensional square lattice, by applying perturbation expansion up to the fourth order in the on-site repulsion U . We evaluate transition temperature as functions of the repulsion U . We consider only the following two typical cases (I) and (II): (I) The system is near the half-filled state, and the Fermi surface nesting yields the strong antiferromagnetic fluctuations. (II) The system is away from the half-filled state. This case is intended to reproduce the single main band of the spin–triplet superconductor Sr 2 RuO 4 . In the both cases, the Fermi level is set close to the Van Hove singularity. In the case (I), each order of the perturbation expansion for the pairing interaction gives a similar momentum dependence, and in addition the perturbation expansion for the pairing interaction converges relatively well even for moderately strong U . On the other hand, in the case (II), the perturbation expansion does not converge we...