Use Of Green Function Research Articles

Spectral Green's-function method in driven open quantum dynamics

A novel method based on spectral Green functions is presented for the simulation of driven open quantum dynamics that can be described by the Lindblad master equation in Liouville density operator space. The method extends the Hilbert space formalism and provides simple algebraic connections between the driven and non-driven dynamics in the spectral frequency domain. The formalism shows remarkable analogies to the use of Green functions in quantum field theory such as the elementary excitation energies and the Dyson self-energy equation. To demonstrate its potential, we apply the novel method to a coherently driven dissipative ensemble of 2-level systems comprising a single "active" subsystem interacting with $N$ "passive" subsystems -- a generic model with important applications in quantum optics and dynamic nuclear polarization. The novel method dramatically reduces computational cost compared with simulations based on solving the full master equation, thus making it possible to study and optimize many-body correlated states in the physically realistic limit of an arbitrarily large $N$.

Approximate calculation method for angular acoustic scattering by non-analytic objects with use of Green functions

Approximate calculation method for angular acoustic scattering by non-analytic objects with use of Green functions

Análisis de vigas sobre fundación flexible empleando funciones de Green

Beams on elastic foundation are basic elements within structural analysis, which are used to model foundation beams, foundation piles, retaining walls, and more complex structures that include some of these elements. For their analysis, the finite element method is usually used [1], which produces an approximate solution of the problem; and the Green's function stiffness method [2], which produces an exact solution. This article presents a methodology 100% based on the use of Green function's (response to a unit point force), to obtain the exact response of beams on elastic foundation. The main advantage of this formulation is its computational low cost compared to the aforementioned alternatives, and even for a large number of problems, it can be expressed only by means of sums and integrals, which can be easily performed numerically. Also, a great variety of Green function's for finite and infinite beams on elastic foundations with different boundary conditions are also presented, as well as some examples with the implementation of the proposed methodology.

Regularized integral formulation of mixed Dirichlet-Neumann problems

This paper presents a theoretical discussion as well as novel solution algorithms for problems of scattering on smooth two-dimensional domains under Zaremba boundary conditions, for which Dirichlet and Neumann conditions are specified on various portions of the domain boundary. The theoretical basis of the proposed numerical methods, which is provided for the first time in the present contribution, concerns detailed information about the singularity structure of solutions of the Helmholtz operator under boundary conditions of Zaremba type. The new numerical method is based on the use of Green functions and integral equations, and it relies on the Fourier continuation method for regularization of all smooth-domain Zaremba singularities as well as newly derived quadrature rules which give rise to high-order convergence, even around Zaremba singular points. As demonstrated in this paper, the resulting algorithms enjoy high-order convergence, and they can be used to efficiently solve challenging Helmholtz boundary value problems and Laplace eigenvalue problems with high-order accuracy.

On the SN 1993J Radio Shell Structure

SummaryAn accurate measurement of the expansion deceleration of SN 1993J depends on how well the shell size and its emission structure are known. With the goal of determining the emission structure of the shell, we have developed a new approach, which we call “Green Function Deconvolution” (GFD), based on iterative use of Green functions on the sky plane to reconstruct the radial emission profiles of spherically symmetric sources. This approach works reasonably well in the case of optically thin emitting sources, which is not the case for SN 1993J since, as we find, the emission from the central part of SN 1993J further away from us is strongly or totally absorbed. We describe the GFD method and present our findings about the emission structure of the shell. We also present the expansion of SN 1993J based on a method complementary to GFD, which will be described elsewhere.

COUPLED BENDING–BENDING–TORSION VIBRATION ANALYSIS OF ROTATING PRETWISTED BLADES: AN INTEGRAL FORMULATION AND NUMERICAL EXAMPLES

A new approximate method is presented for the analysis of the modal characteristics of straight, pretwisted non-uniform blades corresponding to the coupled flapwise bending, chordwise bending and torsion of both rotating and non-rotating blades. An integral approach is described based on the use of Green functions (structural influence functions), which are used to develop the equations of motion. A clamped–free blade is analyzed and comparisons are made with numerical results for the literature. Several examples regarding specific aspects of the flapwise bending, coupled bending–bending, coupled bending–torsion and coupled bending–bending–torsion vibration analysis are presented. The method presented gives good results and can be used for modelling of turbomachine blades, aircraft propellers or helicopter rotor blades which may be considered as straight non-uniform beams with built-in pretwist.

A Comprehensive Energy Formulation for General Nonlinear Material Continua

By specialization to the continuum problem of a general formulation of the initial/boundary value problem for every nonpotential operator (Tonti, 1984) and by virtue of a suitable choice of the “integrating operator,” a comprehensive energy formulation is established. Referring to the small strain and displacement case in the presence of any inelastic generally nonlinear constitutive law, provided that it is differentiable, this formulation allows us to derive extensions of well-known principles of elasticity (Hu-Washizu, Hellinger-Reissner, total potential energy, and complementary energy). An illustrative example is given. Peculiar properties of the formulation are the energy characterization of the functional and the use of Green functions of the same problem in the elastic range for every inelastic, generally nonlinear material considered.

Crossed field excitation of an acoustic superlattice

This paper analyses theoretically the `crossed field' scheme for acoustic excitation of an acoustic superlattice. The electrical admittance is deduced by the use of Green functions. A configuration with an electromechanical coupling coefficient as high as 0.76 is found.

Surface scattering of x rays in thin films. Part I. Theoretical treatment

The diffuse scattering of x rays by thin films limited by rough interfaces is presented theoretically in terms of height–height correlation functions. The method employed, analogous to the use of Green functions, allows a rigourous treatment down to grazing incidences. An expression for the scattered intensity is obtained from the scattering cross section. The results are then discussed using a model correlation function, and the interpretation of x-ray reflectivity experiments is reexamined. We show that off-specular scans yield direct information about the correlations between interfaces which, in case of liquid thin films, can be used to determine elastic parameters. The generalized case of stratified media is examined in an appendix. This theory is compared to experimental results in the case of soap films in a companion article.

Fast potential theory. II. layer potentials and discrete sums

We present three new families of fast algorithms for classical potential theory, based on Ewald summation and fast transforms of Gaussians and Fourier series. Ewald summation separates the Green function for a cube into a high-frequency localized part and a rapidly-converging Fourier series. Each part can then be evaluated efficiently with appropriate fast transform algorithms. Our algorithms are naturally suited to the use of Green functions with boundary conditions imposed on the boundary of a cube, rather than free-space Green functions. Our first algorithm evaluates classical layer potentials on the boundary of a d-dimensional domain, with d equal to two or three. The quadrature error is O( h m ) +ε, where h is the mesh size on the boundary and m is the order of quadrature used. The algorithm evaluates the discretized potential using N elements at O( N) points in O( N log N) arithmetic operations. The constant in O( N log N) depends logarithmically on the desired error tolerance. Our second scheme evaluates a layer potential on the domain itself, with the same accuracy. It produces M d values using N boundary elements in O(( N + M d ) log M) arithmetic operations. Our third method evaluates a discrete sum of values of the Green function, of the type which occur in particle methods. It attains error ϵ at a cost O( N a log N), where a = 2 (1 + D d ) and D is the Hausdorff dimension of the set where the sources concentrate in the limit N √ ∞. Thus it is O( N log N) when the sources do not cluster too much and close to O( N log N) in the important practical case when the points are uniformly distributed over a hypersurface. We also sketch an O( N log N) algorithm based on special functions. Two-dimensional numerical results are presented for all three algorithms. Layer potentials are evaluated to second-order accuracy, in times which exhibit considerable speedups even over a reasonably sophisticated direct calculation. Discrete sum calculations are speeded up astronomically; our algorithm reduces the CPU time required for a calculation with 40,000 points from six months to one hour.

Direct energy transfer in the diffusion case, numerically treated by the use of Green functions

The energy decay function for an excited donor molecule surrounded by acceptors in diffusional motion is computed for a finite-size donor and arbitrary characteristics of the boundary. We evaluate the donor-acceptor---pair diffusion function via a Green-function approach for the Feynman-Kac equation in Laplace space. Quick and stable numerical solutions for the acceptor-averaged pair diffusion function, and thereby the energy decay functions, can be obtained, thus allowing model parametrization from time-resolved fluorescence measurements.

The nuclear density oscillations

Nuclear density oscillations coming from a semi-infinite nuclear matter calculation in the local density approximation are compared with the density oscillations of the Thorpe and Thouless Green function method. A new, simpler method without use of Green functions which easily leads to higher terms in the density expansion is proposed. The appearance of the first “bump” in the density on the rising density curve was justified.

Steady three-dimensional temperature field in cooled turbine blades

A method based on the use of Green functions is proposed for calculating the temperature field of cooled turbine blades. The method presumes the use of high-speed computers with large memories.

Introduction to Superconductivity

M Tinkham Maidenhead: McGraw Hill 1975 pp xiv + 296 price £11.50 Professor Tinkham writes with clarity, emphasizing the physical ideas underlying the theoretical approaches to superconductivity which form the basis of this excellent book. By avoiding certain techniques, such as the use of Green functions, and by analysing many practical situations, he has ensured that the text is accessible to graduate experimentalists and theoreticians alike.

Use of Green Functions in Atomic and Molecular Calculations. VI. The First-Order Perturbation to the Eigenfunction of the Helium-Atom Ground State

We present an analytical expression for the first-order perturbation correction to the eigenfunction of the He-atom ground state. The expression is derived from the previously obtained He-atom Green function and it has the form of a series expansion in terms of Legendre polynomials Pn(cosθ12) where θ12 is the angle between the position vectors of the two electrons. The coefficients of the expansion, which are functions of the electron coordinates (r1, r2), are obtained in the form of integral representations.

Use of Green Functions in Atomic and Molecular Calculations. II. The Green Function of the Helium Atom

We derive an analytical expression for the Green function of the helium atom, that is to say for the helium atom where the electron repulsion term (e2 / r12) has been neglected. It is outlined that this Green function may be used as a basis for a numerical iterative procedure to derive the exact eigenfunctions of the helium atom.

Use of Green Functions in Atomic and Molecular Calculations. IV. The Green Function of the Lithium Atom and of the Other Atoms

We derive an analytical expression for the Green function of the lithium atom, that is to say for the lithium atom where the electron repulsion terms (e2 / rij) have been neglected. It is shown that our derivation is easily generalized to all other atoms and ions.

Use of Green Functions in Atomic and Molecular Calculations. III. Second-Order Perturbation of the Hydrogen Molecular Ion

We have derived analytical expressions for the second-order perturbation energy of a hydrogen atom, perturbed by a point charge λ at a distance R. Here the charge λ is taken as the perturbation parameter. The calculations make use of Green-function techniques and of the hydrogen-atom Green function that we derived in a previous publication.

On the use of green functions in atomic and molecular calculations V. The modified green functions of the atoms

We use previously derived expressions of atomic Green functions for the derivation of the modified Green functions, which belong to the atomic ground states. These modified Green functions may be used for formal perturbation treatments.

Erratum: Use of Green Functions in Atomic and Molecular Calculations. I. The Green Function of the Hydrogen Atom

Erratum: Use of Green Functions in Atomic and Molecular Calculations. I. The Green Function of the Hydrogen Atom