时间无卷积广义主方程中核函数的精确解和高阶微扰展开:在自旋-玻色模型和激发态能量转移中的应用

The time-convolutionless (TCL) quantum master equation provides a powerful tool to simulate reduced dynamics of a quantum system coupled to a bath. The key quantity in the TCL master equation is the so-called kernel or generator, which describes effects of the bath degrees of freedom. Since the exact TCL generators are usually hard to calculate analytically, most applications of the TCL generalized master equation have relied on approximate generators using second and fourth order perturbative expansions. By using the hierarchical equation of motion (HEOM) and extended HEOM methods, we present a new approach to calculating the exact TCL generator and its high order perturbative expansions. The new approach is applied to the spin-boson model with different sets of parameters, to investigate the convergence of the high order expansions of the TCL generator. We also discuss circumstances where the exact TCL generator becomes singular for the spin-boson model, and a model of excitation energy transfer in the Fenna-Matthews-Olson complex.     

Finite-field calculations of molecular polarizabilities using field-induced polarization functions: second- and third-order perturbation correlation corrections to the coupled Hartree-Fock polarizability of H2O

Ordinary Rayleigh-Schrödinger perturbation theory with Møller-Plesset (RSMP) partitioning is used to calculate second- and third-order correlation corrections to the CHF polarizability and dipole moment of the water molecule by a finite-field procedure. [2/1] Padé approximants are found to be useful in accelerating the convergence of the property perturbation expansions. Field-induced polarization functions suitable for polarizability calculations are determined. The average polarizability calculated, neglecting vibrational averaging, with Dunning's (9s5p/4s-4s2p/2s) contracted GTO basis set augmented by field-induced 1s1p2d/1p polarization functions is within 3 per cent of the experimental result. Correlation corrections to the dipole moment and polarizability of the water molecule calculated by the finite-field RSMP and single + double excitation CI(SDCI) methods for the same basis set are found to be in close agreement. The RSMP approach has the advantages of being size-consistent and of being capable of greater efficiency than the SCDI method. Comparative calculations carried out using Epstein-Nesbet partitioning show that through third order RSEN correlation perturbation expansions for the dipole moment and polarizability are less rapidly convergent than RSMP expansions. However, reasonable accord with RSMP results can be achieved by using [2/1] Padé approximants to accelerate the convergence of RSEN energy perturbation expansions. The convergence of RSMP property correlation expansions based on the zeroth-order uncoupled-Hartree-Fock (UCHF) and coupled-Hartree-Fock (CHF) approximations are compared through third order. Whereas the CHF + RSMP expansions are for practical purposes fully converged, the UCHF + RSMP expansions are not adequately converged.     

Series expansions of rotating two and three dimensional sound fields

The cylindrical and spherical harmonic expansions of oscillating sound fields rotating at a constant rate are derived. These expansions are a generalized form of the stationary sound field expansions. The derivations are based on the representation of interior and exterior sound fields using the simple source approach and determination of the simple source solutions with uniform rotation. Numerical simulations of rotating sound fields are presented to verify the theory.     

Exponential asymptotic expansions and approximations of the unstable and stable manifolds of singularly perturbed systems with the Henon map as an example

The subject of this paper is the construction of the exponential asymptotic expansions of the unstable and stable manifolds of the area-preserving Henon map. The approach that is taken enables one to capture the exponentially small effects that result from what is known as the Stokes phenomenon in the analytic theory of equations with irregular singular points. The exponential asymptotic expansions were then used to obtain explicit functional approximations for the stable and unstable manifolds. These approximations are compared with numerical simulations and the agreement is excellent. Several of the main results of the paper have been previously announced in A. Tovbis, M. Tsuchiya, and C. Jaffe ["Chaos-integrability transition in nonlinear dynamical systems: exponential asymptotic approach," Differential Equations and Applications to Biology and to Industry, edited by M. Martelli, K. Cooke, E. Cumberbatch, B. Tang, and H. Thieme (World Scientific, Singapore, 1996), pp. 495-507, and A. Tovbis, M. Tsuchiya, and C. Jaffe, "Exponential asymptotic expansions and approximations of the unstable and stable manifolds of the Henon map," preprint, 1994]. (c) 1998 American Institute of Physics.     

Green Function and Perturbation Method for Dissipative Systems Based on Biorthogonal Basis

Based on the approach of biorthogonal basis, we carry out the quasinormal modes （QNMs） expansions for a class of open systems described by the wave equation with outgoing wave boundary conditions. For such a non-Hermitian system, the eigenfunction perturbation expansions and Green function method, which are based on the orthogonal eigenvectors of the Hermitian Hamiltonian for the dosed quantum system, can be generalized in terms of the biorthogonal basis, the two sets of eigenfunctions of H and its adjointness H . The time-independent perturbation theory for the complex frequencies can be also developed.     

An efficient and stable spectral method for electromagnetic scattering from a layered periodic structure

The scattering of acoustic and electromagnetic waves by periodic structures plays an important role in a wide range of problems of scientific and technological interest. This contribution focuses upon the stable and high-order numerical simulation of the interaction of time-harmonic electromagnetic waves incident upon a periodic doubly layered dielectric media with sharp, irregular interface. We describe a boundary perturbation method for this problem which avoids not only the need for specialized quadrature rules but also the dense linear systems characteristic of boundary integral/element methods. Additionally, it is a provably stable algorithm as opposed to other boundary perturbation approaches such as Bruno and Reitich’s “method of field expansions” or Milder’s “method of operator expansions”. Our spectrally accurate approach is a natural extension of the “method of transformed field expansions” originally described by Nicholls and Reitich (and later refined to other geometries by the authors) in the single-layer case.     

Ternary Interaction Parameters in Calphad Solution Models

For random, diluted, multicomponent solutions, the excess chemical potentials can be expanded in power series of the composition, with coefficients that are pressure- and temperature-dependent. For a binary system, this approach is equivalent to using polynomial truncated expansions, such as the Redlich-Kister series for describing integral thermodynamic quantities. For ternary systems, an equivalent expansion of the excess chemical potentials clearly justifies the inclusion of ternary interaction parameters, which arise naturally in the form of correction terms in higher-order power expansions. To demonstrate this, we carry out truncated polynomial expansions of the excess chemical potential up to the sixth power of the composition variables.     

Analytical representations of the Mössbauer recoilless fraction,Debye-Waller factors,lattice energies,and heat capacities

Simple analytical expansions are given for the recoilless fraction in Mössbauer spectroscopy, the Debye-Waller factor in X-ray scattering, and the lattice energy and heat capacity of solids. While this problem has been discussed in an earlier paper [1], computer technology has now advanced to the point that direct evaluations of the simple expansions of these quantities are useful for quick curve fitting to experimental data at any desired temperature, and these expansions are easier to evaluate than using graphs to estimate recoilless fractions and Debye temperatures. We compare this approach with a polynomial expansion in terms of Bernoulli numbers, which has only a limited domain of convergence. We explicitly evaluate the convergence of these Debye integral expansions as a function of the number of terms used and the time required.This work was prepared with the support of the U.S. Department of Energy, Grant No. DE-FG02-85 ER 45199.     

Asymptotic expansions for the coupled wavenumbers in an infinite orthotropic flexible fluid-filled cylindrical shell

Analytical expressions are found for the coupled wavenumbers in flexible, fluid-filled, circular cylindrical orthotropic shells using the asymptotic methods. These expressions are valid for arbitrary circumferential orders. The Donnell-Mushtari shell theory is used to model the shell and the effect of the fluid is introduced through the fluid-loading parameter μ. The orthotropic problem is posed as a perturbation on the corresponding isotropic problem by defining a suitable orthotropy parameter ε, which is a measure of the degree of orthotropy. For the first study, an isotropic shell is considered (by setting ε=0) and expansions are found for the coupled wavenumbers using a regular perturbation approach. In the second study, asymptotic expansions are found for the coupled wavenumbers in the limit of small orthotropy (ε?1). For each study, isotropy and orthotropy, expansions are found for small and large values of the fluid-loading parameter μ. All the asymptotic solutions are compared with numerical solutions to the coupled dispersion relation and the match is seen to be good. The differences between the isotropic and orthotropic solutions are discussed. The main contribution of this work lies in extending the existing literature beyond in vacuo studies to the case of fluid-filled shells (isotropic and orthotropic).     

Acoustic field induced in spheres with inhomogeneous density by external sources

Acoustic or electromagnetic fields induced in the interior of inhomogeneous penetrable bodies by external sources can be evaluated via well-known volume integral equations. For bodies of arbitrary shape and/or composition, for which separation of variables fails, a direct attack for the solution of these integral equations is the only available approach. In a previous paper by the same authors the scalar (acoustic) field in inhomogeneous spheres of arbitrary compressibility, but with constant density, was considered. In the present one the direct hybrid (analytical-numerical) method applied to the much simpler integral equation for spheres with constant density is generalized to densities that vary with r, theta, or even psi. This extension is by no means trivial, owing to the appearance of the derivatives of both the density and the unknown function in the volume integral, a fact necessitating a more subtle and accuracy-sensitive approach. Again, the spherical shape allows use of the orthogonal spherical harmonics and of Dini's expansions of a general type for the radial functions. The convergence of the latter, shown to be superior to other possible sets of orthogonal expansions, can be further optimized by the proper selection of a crucial parameter in their eigenvalue equation.     

Asymptotic expansions of wave functions of three identical particels for small hyperradius andS-wave potentials

In the framework of the integro-differential approach asymptotic expansions of the wave functions of three identical particles are constructed as series in powers of the hyperradius, its logarithm powers and unknown functions of the hyperspherical angles. For calculations of these functions a recurrence chain of ordinary second-order differential equations is obtained. The dependence of the asymptotic expansions on the total angular momentum and the behaviour of the potentials at small distances are investigated.     

Asymptotics with respect to a small geometric parameter for solutions of three-dimensional Lamé equations

Our three-dimensional approach enables us to find the principal orders with respect to a small parameter to be imposed (for the problem in question) on the components of the displacement vector and ensuring the existence and uniqueness of the principal terms of the asymptotic expansions inside a domain of a thin solid. Financially supported by RFBR under grants nos. 08-01-00231, 08-01-00251, and 08-01-00353.     

New ways to solve the Schroedinger equation

We discuss a new approach to solve the low lying states of the Schroedinger equation. For a fairly large class of problems, this new approach leads to convergent iterative solutions, in contrast to perturbative series expansions. These convergent solutions include the long standing difficult problem of a quartic potential with either symmetric or asymmetric minima.     

Quasi-geometrical optics methods in the theory of wave propagation through a randomly inhomogeneous layer

Some possibilities of using asymptotic methods, such as the Maslov method and the interference-integral method, in problems of wave propagation in randomly inhomogeneous media are considered. It is shown that using the Maslov method and the interference-integral method in a small-angle approximation, and neglecting amplitude fluctuations of partial waves, provides results of heuristic spectral methods that utilize expansions in terms of incoming or outgoing waves. The combined use of these asymptotic methods gives a third heuristic method, a mixed-integral representation obtained earlier by applying the method of two-scale expansions. It is pointed out that results of a mixed-integral representation change to those of the method of smooth disturbances and of the phase-screen method. Unlike the method of two-scale expansions, the proposed approach based on the combined use of asymptotic methods facilitates the process of taking into account the heterogeneity of a background medium.     

Causes and cures for the infinities in slow-motion expansions in general relativity

The causes of the divergent integrals arising in slow-motion expansions of the general relativistic field equations are studied and a remedy for them is suggested. This is done within the context of a model problem involving a coupled nonlinear scalar field and isotropic oscillator. The model is shown to give rise to divergent integrals directly attributable to the nonlinearity when the field is assumed to be analytic in a slowness parameter. Application of a nonregular perturbation approach which includes the method of matched asymptotic expansions is shown to eliminate the infinite contributions.Supported in part by The National Science Foundation under grant no. PH 79-15.     

Equilibrium ensemble approach to disordered systems I: general theory,exact results

An outline of Morita? equilibrium ensemble approach to disordered systems is given, and hitherto unnoticed relations to other, more conventional approaches in the theory of disordered systems are pointed out. It is demonstrated to constitute a generalization of the idea of grand ensembles and to be intimately related also to conventional low-concentration expansions as well as to perturbation expansions about ordered reference systems. Moreover, we draw attention to the variational content of the equilibrium ensemble formulation. A number of exact results are presented, among them general solutions for site- and bond- diluted systems in one dimension, both for uncorelated, and for correlated disorder.     

On Analysis of Spectro-Correlation Characteristics of One-Dimensional Brownian Motion

We propose an analytical–numerical approach to finding the correlation function of one-dimensional Brownian motion in a one-mode potential profile described by a low-order polynomial. The approach is based on solving chains of differential equations for the statistical moment functions of particle coordinate fluctuations, which are broken in a certain manner. Two methods of such breaking are considered. One method is based upon quasi-linear expansions of the moment functions, and another one, on cumulantless expansions. Spectro-correlation characteristics of Brownian motion in biquadratic potential profiles of two types are studied.     

A Simple Inductive Approach to the Problem of Convergence of Cluster Expansions of Polymer Models

We explain a simple inductive method for the analysis of the convergence of cluster expansions (Taylor expansions, Mayer expansions) for the partition functions of polymer models. We give a very simple proof of the Dobrushin–Kotecký–Preiss criterion and formulate a generalization usable for situations where a successive expansion of the partition function has to be used.     

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

In this paper, we consider the relation between Evans-function-based approaches to the stability of periodic travelling waves and other theories based on long-wavelength asymptotics together with Bloch wave expansions. In previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the linearized Whitham averaged system accurately describes the spectral stability to long-wavelength perturbations. To clarify the connection between Bloch-wave-based expansions and Evans-function-based approaches, we reproduce this result without reference to the Evans function by using direct Bloch expansion methods and spectral perturbation analysis. One of the novelties of this approach is that we are able to calculate the relevant Bloch waves explicitly for arbitrary finite-amplitude solutions. Furthermore, this approach has the advantage of being applicable in the more general multi-periodic setting where no conveniently computable Evans function has yet been devised.     

Three-dimensional coupled double-distribution-function lattice Boltzmann models for compressible Navier–Stokes equations

Two three-dimensional (3D) lattice Boltzmann models in the framework of coupled double-distribution-function approach for compressible flows, in which specific-heat ratio and Prandtl number can be adjustable, are developed in this paper. The main differences between the two models are discrete equilibrium density and total energy distribution function. One is the D3Q25 model obtained from spherical function, and the other is the D3Q27 standard lattice model obtained from Hermite expansions of the corresponding continuous equilibrium distribution functions. The two models are tested by numerical simulations of some typical compressible flows, and their numerical stability and precision are also analysed. The results indicate that the two models are capable for supersonic flows, while the one from Hermite expansions is not suitable for compressible flows with shock waves.