Perturbation Series Expansion Research Articles

A symplectic conservative perturbation series expansion method for nonlinear Hamiltonian equations with perturbation terms

A novel numerical method is proposed to solve the nonlinear Hamiltonian equation with consideration of perturbation terms in this study. Firstly, the original nonlinear Hamiltonian equation is formulated in the formal way and the corresponding conservation law is introduced. Considering the constant perturbation terms in the original nonlinear Hamiltonian equation, the perturbation series expansion method is proposed to obtain the response of the nonlinear Hamiltonian equation. Within the method, the solution to the original equation can be transformed into a perturbation series by introducing said small parameters. Based on the Taylor series expansion method, a series of modified nonlinear Hamiltonian equations for predicting the responses are derived. Moreover, the time history and phase diagram of the nonlinear Hamiltonian equation can be obtained based on the symplectic conservative algorithm. Additionally, the conservative law for the modified Hamiltonian equation is given. The performance of the proposed perturbation series expansion method is evaluated by using three numerical examples. Numerical examples indicate that the proposed method is effective and feasible for calculating the response of nonlinear Hamiltonian equation taking into account perturbation terms.

High order quark number susceptibilities in hot QCD from lattice electrostatic QCD

Building on the experience of [1], we develop a formalism to construct operators for higher derivatives of the pressure in hot QCD with respect to the quark chemical potential $\mu$. We provide formulae for the operators up to the sixth derivative, and obtain continuum-extrapolated results from lattice EQCD at zero and finite $\mu$ and at six different pairs of temperature T and number of massless quark flavors $n_f$. Our data is benchmarked against full-QCD lattice and perturbative results, allowing to judge the quality of the perturbative series expansion in EQCD and the dimensional reduction procedure as a whole.

Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure

We have recently proposed [1] a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with SU(N) gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values.First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots T[2,2k+1]. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank N and the representation R.

On the equations of warped disc dynamics

ABSTRACT The 1D evolution equations for warped discs come in two flavours: For very viscous discs, the internal torque vector $\boldsymbol {G}$ is uniquely determined by the local conditions in the disc, and warps tend to damp out rapidly if they are not continuously driven. For very inviscid discs, on the other hand, $\boldsymbol {G}$ becomes a dynamic quantity, and a warp will propagate through the disc as a wave. The equations governing both regimes are usually treated separately. A unified set of equations was postulated recently by Martin et al., but not yet derived from the underlying physics. The standard method for deriving these equations is based on a perturbation series expansion, which is a powerful, but somewhat abstract technique. A more straightforward method is to employ the warped shearing box framework of Ogilvie & Latter, which so far has not yet been used to derive the equations for the wave-like regime. The goal of this paper is to analyse the warped disc equations in both regimes using the warped shearing box framework, to derive a unified set of equations, valid for small warps, and to discuss how our results can be interpreted in terms of the affine tilted-slab approach of Ogilvie.

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

Entropy of fully packed hard rigid rods on d-dimensional hypercubic lattices.

We determine the asymptotic behavior of the entropy of full coverings of a L×M square lattice by rods of size k×1 and 1×k, in the limit of large k. We show that full coverage is possible only if at least one of L and M is a multiple of k, and that all allowed configurations can be reached from a standard configuration of all rods being parallel, using only basic flip moves that replace a k×k square of parallel horizontal rods by vertical rods, and vice versa. In the limit of large k, we show that the entropy per site S_{2}(k) tends to Ak^{-2}lnk, with A=1. We conjecture, based on a perturbative series expansion, that this large-k behavior of entropy per site is superuniversal and continues to hold on all d-dimensional hypercubic lattices, with d≥2.

Sugar Transport in a Merging Phloem Vessels: A Hydrodynamic Model

Green plants are the major tappers of the energy from the sun. The collected solar energy in the form of light is used to activate the chemical reaction occurring in matured leaves between carbon dioxide and water, leading to the synthesis of sugar (chemical energy). Two main transport processes are involved in the transport of mineral salt water from the soil through the roots, via the trunk and branches to the leaves where photosynthetic activity occurs, and the translocation of sugar from the leaves to where they are needed and possibly, stored. The xylem vessels bear the absorbed mineral salt water while the phloem vessels bear the manufactured sugar. In this study, neglecting the effects of occlusion and clogging of the phloem channels, we investigate the transport of sugars in the merging phloem vessels using the hydrodynamic approach. The model is designed using the Boussinesq approximation and solved semi-analytically using the regular perturbation series expansion solutions and Mathematica 11.2 computational software. Expressions for the concentration, temperature, and velocity are obtained and presented quantitatively and graphically. The results show among others, that increase in the merging angle causes a reduction in the concentration, temperature, and velocity profiles. However, there exists fluctuations in the concentration and temperature structures.

Dynamical formulation of low-energy scattering in one dimension

The transfer matrix M of a short-range potential may be expressed in terms of the time-evolution operator for an effective two-level quantum system with a time-dependent non-Hermitian Hamiltonian. This leads to a dynamical formulation of stationary scattering. We explore the utility of this formulation in the study of the low-energy behavior of the scattering data. In particular, for the exponentially decaying potentials, we devise a simple iterative scheme for computing terms of arbitrary order in the series expansion of M in powers of the wavenumber. The coefficients of this series are determined in terms of a pair of solutions of the zero-energy stationary Schrödinger equation. We introduce a transfer matrix for the latter equation, express it in terms of the time-evolution operator for an effective two-level quantum system, and use it to obtain a perturbative series expansion for the solutions of the zero-energy stationary Schrödinger equation. Our approach allows for identifying the zero-energy resonances for scattering potentials in both full line and half-line with zeros of the entries of the zero-energy transfer matrix of the potential or its trivial extension to the full line.

Understanding Phase Transitions in Supersymmetric Quantum Electrodynamics With Resurgence Theory

Using resurgence theory to describe phase transitions in quantum field theory shows that information on non-perturbative effects like phase transitions can be obtained from a perturbative series expansion.

Total cross sections in five methods for two-electron capture by alpha particles from helium: CDW-4B, BDW-4B, BCIS-4B, CDW-EIS-4B and CB1-4B

This work is on quantum-mechanical four-body distorted wave theories for double electron capture in collisions between fast heavy multiply charged ions and heliumlike atomic systems. The five widely used distorted wave methods of the first- and second-order in the perturbation series expansions are compared with the available experimental data on alpha –He collisions. These are the four-body boundary-corrected first Born (CB1-4B), the boundary-corrected continuum intermediate state (BCIS-4B), the Born distorted wave (BDW-4B), the continuum distorted wave (CDW-4B) and the continuum distorted wave-eikonal initial state (CDW-EIS-4B) methods.We address the complete breakdown of the CDW-EIS-4B method at all impact energies within its expected validity domain (100–10000 keV). Further, the relative performance is evaluated of the second-order theories with and without the eikonalization of the two-electron Coulomb wavefunctions for double continuum intermediate states. Finally, at all the considered intermediate and high energies, the practical aspects of the studied five methods are investigated by protracted evaluations of the convergence rates of total cross sections as a function of the number of quadrature points per axis in numerical computations of multi-dimensional (3D-5D) integrals.

Wandering bumps in a stochastic neural field: A variational approach

We develop a generalized variational method for analyzing wandering bumps in a stochastic neural field model defined on some domain U. For concreteness, we take U=S1 and consider a stochastic ring model. First, we decompose the stochastic neural field into a phase-shifted deterministic bump solution and a small error term, which is assumed to be valid up to some exponentially large stopping time. An exact, implicit stochastic differential equation (SDE) for the phase of the bump is derived by minimizing the error term with respect to a weighted L2(U,ρ) norm. The positive weight ρ is chosen so that the error term consists of fast transverse fluctuations of the bump profile. We then carry out a perturbation series expansion of the exact variational phase equation in powers of the noise strength ϵ to obtain an explicit nonlinear SDE for the phase that decouples from the error term. Solving the corresponding steady-state Fokker–Planck equation up to O(ϵ), we determine a leading-order expression for the long-time distribution of the position of the bump. Finally, we use the variational formulation to obtain rigorous exponential bounds on the error term, demonstrating that with very high probability the system stays in a small neighborhood of the bump for times of order exp(Cϵ−1).

Comparing optimized renormalization schemes for QCD observables

Based on the renormalization group summation method of McKeon ${\it et\; al.}$, it is shown that the renormalization group equation, while related to the radiatively mass scale $\mu$, would perform a summation over QCD perturbative terms. Employing the full QCD $\beta$-function within this summation, all logarithmic corrections can be presented as log-independent contributions. In another step of this approach, the renormalization scheme dependence of QCD observables, characterized by Stevenson, is required to be examined. In this regard, two choices of renormalization scheme would be exposed. In one of them, the QCD observable is expressed in terms of two powers of running coupling constant. In the other one, the perturbative series expansion is written as an infinite series in terms of the two-loop running coupling represented by the Lambert $W$-function. In both cases, the QCD observable involves parameters which are renormalization scheme invariant and a coupling constant which is independent of the renormalization scale. We then consider the approach of complete renormalization group improvement (CORGI). In this approach, using the self consistency principle, it is possible to reconstruct the conventional perturbative series in terms of scheme invariant quantities and the coupling constant as a function of Lambert $W$-function. One of the key differences between CORGI and the renormalization group summation method of McKeon ${\it et\; al.}$ is that the later treats renormalization scale and scheme independently while the former would encounter both dependencies together. In continuation, numerical results of the two approaches are compared for $R_{e^+e^-}$ ratio and the Higgs decay width to gluon-gluon.

On Modelling Wind-Farm Wake Turbulence Autospectra and Coherence from a Database

This study addresses the feasibility of modeling wind-farm wake-turbulence autospectra and coherences from a database: flow velocity points from experimental and computational fluid dynamics (CFD) investigations. Specifically, it first applies an earlier-exercised framework to construct the autospectral models from a database and then it adopts a recently proposed framework to construct the coherence models from a database. While this proposed framework has not been tested against a database, the methodology has been completely formulated with a theoretical basis. These models of autospectrum and coherence are interpretive, and in closed form. Both frameworks basically involve the perturbation series expansion of the autospectra and coherences. The framework for modeling autospectra is tested against a demanding database of wake turbulence inside a wind farm over a complex terrain from a full-scale test. The suitability of these autospectral models for simulation through white-noise driven filters is also demonstrated. Finally, coherence models are generated for assumed values of the perturbation series constants, and these coherence models are used to demonstrate how the coherence models of homogeneous isotropic turbulence deviate from the coherence models of non-homogeneous non-isotropic turbulence such as wind-farm wake turbulence. This feasibility of extracting both the one-point statistics of autospectral models and the two-point statistics of coherence models from a database represents a research avenue that is new and promising in the treatment of wind-farm wake turbulence. This paper also demonstrates the feasibility of fruitfully exploiting the wake treatment methods developed in other fields.

Unsteady MHD free convective chemically reacting fluid flow over a vertical plate with thermal radiation, Dufour, Soret and constant suction effects

Unsteady MHD free convective flow of a chemically reacting incompressible fluid over a vertical porous plate under the combined effects of thermal radiation, Dufour, Soret and constant suction is examined. The governing non-linear equations are solved semi analytically using the methods of similarity transformation and perturbation series expansions. Expressions for the velocity, temperature, concentration and Nusselt number for the flow are obtained and presented graphically. The analysis shows that Dufour number increases the temperature and velocity; Soret number increases the concentration and velocity; suction parameter decreases the temperature, concentration and velocity, but increases the Nusselt number; chemical reaction rate decreases the temperature but increases the concentration, velocity and Nusselt number; magnetic field force increases the velocity; Grashof numbers increase the velocity when their values are increasingly varied.

Factorization of the Jet Mass Distribution in the Small R Limit

We derive a factorization theorem for the jet mass distribution with a given $$p_T^J$$ for the inclusive production, where $$p_T^J$$ is a large jet transverse momentum. Considering the small jet radius limit (R ≪ 1), we factorize the scattering cross section into a partonic cross section, the fragmentation function to a jet, and the jet mass distribution function. The decoupled jet mass distributions for quark and gluon jets are well-normalized and scale invariant, and they can be extracted from the ratio of two scattering cross sections such as $$d\sigma/(dp_T^JdM_J^2)$$ and $$d\sigma/dp_T^J$$ . When $$M_J \sim p_T^JR$$ , the perturbative series expansion for the jet mass distributions works well. As the jet mass becomes small, large logarithms of $$M_J/(p_T^JR)$$ appear, and they can be systematically resummed through a more refined factorization theorem for the jet mass distribution.

Off-diagonal series expansion for quantum partition functions

We derive an integral-free thermodynamic perturbation series expansion for quantum partition functions which enables an analytical term-by-term calculation of the series. The expansion is carried out around the partition function of the classical component of the Hamiltonian with the expansion parameter being the strength of the off-diagonal, or quantum, portion. To demonstrate the usefulness of the technique we analytically compute to third order the partition functions of the 1D Ising model with longitudinal and transverse fields, and the quantum 1D Heisenberg model.

Time-dependent reliability assessment of fatigue crack growth modeling based on perturbation series expansions and interval mathematics

A novel method of the time-dependent reliability, which combines the perturbation series expansions with the interval mathematics, is presented in this study for life prediction of fatigue crack growth problems. Distinct from the treatment in statistical analysis, characteristic parameters that exist in structural crack propagation model are described as unknown-but-bounded (UBB) variables with perturbation terms owing to the fact of variability of geometric sizes, material properties, and loading conditions. Then, based on the perturbation principle and the Taylor extension approach, a new interval perturbation series expansion method (PSEM) to predict boundary rules of the fatigue crack length a(t) is derived. Meanwhile, the auto-correlation features between a(tk) and a(tk+1) are also confirmed by employing the interval process theory. Additionally, inspired by the classical out-crossing approach in random process issues, a non-probabilistic time-dependent reliability (NTR) index Rs(T), as a safety measure for in-service engineering structures with crack, is defined, and its solution algorithm is expounded in details. Numerical examples are eventually proposed to demonstrate the validity of the developed methodology of uncertainty quantification and reliability evaluation.

Large-scale structure perturbation theory without losing stream crossing

We suggest an approach to perturbative calculations of large-scale clustering in the Universe that includes from the start the stream crossing (multiple velocities for mass elements at a single position) that is lost in traditional calculations. Starting from a functional integral over displacement, the perturbative series expansion is in deviations from (truncated) Zel'dovich evolution, with terms that can be computed exactly even for stream-crossed displacements. We evaluate the one-loop formulas for displacement and density power spectra numerically in 1D, finding dramatic improvement in agreement with N-body simulations compared to the Zel'dovich power spectrum (which is exact in 1D up to stream crossing). Beyond 1D, our approach could represent an improvement over previous expansions even aside from the inclusion of stream crossing, but we have not investigated this numerically. In the process we show how to achieve effective-theory-like regulation of small-scale fluctuations without free parameters.

On the ultimate uncertainty of the top quark pole mass

We combine the known asymptotic behaviour of the QCD perturbation series expansion, which relates the pole mass of a heavy quark to the MS‾ mass, with the exact series coefficients up to the four-loop order to determine the ultimate uncertainty of the top-quark pole mass due to the renormalon divergence. We perform extensive tests of our procedure by varying the number of colours and flavours, as well as the scale of the strong coupling and the MS‾ mass. Including an estimate of the internal bottom and charm quark mass effect, we conclude that this uncertainty is around 110 MeV. We further estimate the additional contribution to the mass relation from the five-loop correction and beyond to be around 300 MeV.

Fatigue crack growth modeling and prediction with uncertainties via stochastic perturbation series expansion method

The fatigue crack growth process is characterized by uncertainties inherent to the variations in geometrical properties, material properties, and operating conditions. This paper proposes a novel stochastic perturbation series expansion method for predicting fatigue crack growth evolution which accounts for uncertain parameters. Unlike the deterministic method, the parameters of the crack growth model are considered to be stochastic with perturbation terms. The solution to the crack growth equation is then transformed into a perturbation series after introducing said small parameters. A series of modified differential equations for predicting the stochastic characteristics of the crack length history are derived via first-order Taylor series expansion method. The initial crack length under perturbation can be expressed via the artificial small parameter method per the measurement errors. Further, by substituting the initial conditions into the modified stochastic equations, variations in the mean value and standard deviation as well as the probabilistic region of crack length history can be successfully obtained. The performance of the proposed method is evaluated through two examples. Comparison against experimental and Monte-Carlo simulation data confirm that the proposed method is indeed feasible and effective for predicting fatigue crack growth evolution with consideration of stochastic uncertainties.