The Local Formulation for the Modified Green's Function Method

The green's function method in quantum statistics

Simulations that are meant to determine the steady-state distribution of a diffusible solute such as oxygen in tissues have typically used finite difference methods to solve the diffusion equation. Finite difference methods require a tissue mesh with enough points to resolve oxygen gradients near and between discrete blood vessels. The large number of points that are typically required can make these calculations very slow. In this paper, we investigate a numerical method known as the Green's function method which is not bound by the same constraint. The Green's function method is expected to yield an accurate oxygen distribution more quickly by requiring fewer mesh points. Both methods were applied to calculate the steady state oxygen distribution in a model simulation region. When the Green's function calculation used meshes with 1/2, 1/4 and, 1/8 of the resolution required for the finite-difference mesh, there was good agreement with the finite difference calculation in all cases. When the volume of the domain was increased 8-fold the Green's function method was able to calculate the O2 field in 22 minutes, whereas the finite difference calculation is expected to take approximately 1 week. The number of steps required for the Green's function calculation increases quadratically with the number of points in the tissue mesh. As a result, small meshes are calculated very quickly using Green's functions, while for larger mesh sizes this method experiences a significant decrease in efficiency.

In this paper, the free vibration of an annular membrane consisting of three con- centric segments is considered. The frequency equation and mode shapes are obtained by the use of the Green's function method. A numerical example to vibration problem of non- homogeneous annular membrane is presented. solution of the problem and vibration analysis of membranes with discontinuously varying thickness is presented. In reference (2) the finite element method in the analysis was also used. The free vibration problem of annular membrane with many discontinuous variation of the density is the subject of paper (3). Although the formulation of the problem deals with the vibration of membrane which consist of m segments (each of constant density), the numerical examples concern the antisymmetric modes of composite membrane consisting of two segments. In this case the frequency equation is obtained by setting the determinant of a 4x4 matrix of coefficients to a derived system of equation, which equals zero. In many papers various methods are applied to solution of the eigenvalue problems. The authors of paper (4) in solving the eigenproblem for annular membrane propose the method of fundamental solution. In this method the free space Green's function are ap- plied. In this paper the free vibration problem of a composite annular membrane con- sisting of three segments of constant densities is presented. The solution of the problem (frequency equation, mode shapes) is derived by using the properties of Green's functions corresponding to the Helmholtz operator in an annular domain. An example of numerical frequency analysis is given.

A new coarse-mesh computational method for the numerical solution of heat conduction and fluid flow problems is formally developed and applied to sample problems. The method is based upon formal use of Green's functions, which are defined locally over subdomains of the original system under consideration. The formal development of the local Green's function method for the solution of heat conduction problems is presented and discussed. Numerical solutions of sample problems for one-dimensional heat conduction with constant thermal conductivity, one-dimensional heat conduction with temperature-dependent thermal conductivity, and two-dimensional heat conduction with constant thermal conductivity are given, and these results are compared with results obtained using the finite difference and finite element methods. The formal development of the local Green's function method for the solution of fluid flow problems is then also presented and discussed; the numerical solution of a sample problem for simple one-dimensional incompressible fluid flow with viscous heating is also given.

The elastic response of a vacancy in a semi-infinite fcc copper lattice containing a free surface is calculated by using a new multiscale Green's function method. The method treats the lattice distortion near the vacancy at the atomistic level and the free surface at the macroscopic continuum level in the same formalism. The lattice is modeled using the lattice statics Green's function that fully accounts for the discrete atomistic structure of the lattice and can model a large crystallite containing a million atoms without excessive CPU effort. The method is especially useful for modeling the elastic response of nanocrystals containing point defects in which surfaces and interfaces play important roles. The method bridges the length scales seamlessly by relating the microscopic lattice distortion near a point defect to measurable macroscopic parameters of the solid such as the strain and the displacement field at a free surface. Using the interatomic potential derived by Cleri and Rosato, the lattice distortion, relaxation energy, and relaxation volume due to a vacancy are calculated in an otherwise perfect copper lattice for a million-atom model containing a free (100) surface. The calculated value of the relaxation volume is in excellent agreement with the observed value. Numerical results are also presented for the strain and the displacement fields at the free surface due to a vacancy and the interaction energy between a vacancy and the free surface in anisotropic semi-infinite copper.

Resonance plays a key role in the formation of many physical phenomena. The complex-scaled Green's function method provides a powerful tool for exploring resonance. In this paper, we combine this method with the theory describing deformed nuclei with the formalism presented. Taking $^{45}\text{S}$ as an example, we elaborate numerical details and demonstrate how to determine the resonance parameters. The results are compared with those obtained by the complex scaling method and the coupled-channel method and satisfactory agreement is obtained. In particular, the present scheme focuses on the advantages of the complex scaling method and the Green's function method and is more suitable for the exploration of resonance.

The formulation of the Green's-function method, recently suggested by Ziman, for solving the energy-band problem in a so-called "muffin-tin" potential (zero outside spheres representing the various atoms, and spherically symmetrical within each sphere) is transformed to a form very much like that of the Augmented-plane-wave (APW) method, but cannot be transformed exactly into the APW formulation. The wave function set up by the Green's-function method is a superposition of plane waves satisfying the wave equation inside as well as outside the spheres representing the atoms, having a discontinuity of slope at the surfaces of the atomic spheres, enough to produce the correct solution of the Schr\"odinger problem outside the spheres. To get correct wave functions inside the atoms, one would have to construct augmented plane waves from these plane waves.

Ab initio green's function calculations on highly conducting polymers: Effects of electron correlation and aperiodicity

We have calculated the Debye-Waller factor (DWF) of Cu from a model that was used successfully in earlier calculations of anharmonicity by Cowley and Shukla. The present calculation has been carried out using quasiharmonic theory, the lowest-order (λ2) anharmonic perturbation theory, and a Green's function (GF) method which sums an infinite series of the λ2−type anharmonic terms. The static approximation ω → 0 in the cubic contribution to the self-energy of the GF, introduced in the earlier work on the DWF by Shukla and Hubschle is further justified by showing that in the high-temperature limit the exact results for the λ2 anharmonic contributions (cubic and quartic) to the Helmholtz free energy are given in this approximation. Results for the DWF are also obtained for a modified version of the Morse potential with λ2 perturbation theory (PT) and the GF method. The GF results are in excellent agreement with the experimental Mossbauer and X-ray data in the entire temperature range, 300 K  T  120...

<sec><p indent="0mm">Exotic nuclei far from the β stability line have become important scientific goals in the studies of experimental physics at large scientific facilities and theoretical research due to the rich new physics. In these exotic nuclei, the neutron or proton Fermi level is close to the continuum threshold. The pairing correlation could scatter the valence neutrons or protons into the continuum. This leads to the extended neutron or proton density distributions in these exotic nuclei. Therefore, properly describing the pairing correlation and continuum is crucial for studying the structures and properties of these exotic nuclei. The Hartree-Fock-Bogoliubov (HFB) theory is one of the promising tools for describing exotic nuclei. In this theory, people usually solve the HFB equation directly in the coordinate space or the Woods-Saxon basis. In the coordinate space, when employing box boundary conditions to discretize the continuum, the behavior of the wave functions at the boundaries will be affected by the size of the box. Meanwhile, one could not obtain the energy and width of the resonant states directly from the discretized continuum states. The Green’s function (GF) is a simple and effective tool for handling the continuum, which has been widely used in nuclear structure research. The GF can be established by using the wave functions that satisfy the equation of motion and the proper bound boundary conditions for the bound and continuum states. Then one can use the loop integral of the GF on the complex energy plane to construct the particle density and level density. In this way, the continuum states can be included in the density with proper boundary conditions, and thus the extended density distribution can be properly described. Furthermore, the GF on the complex energy plane and the level density can be used to identify the resonant states directly. </sec><sec> This paper briefly reviews the development of the GF method in the continuum density functional theory, focusing on its application in describing halo phenomena and single-particle resonance states in exotic nuclei. For the description of the halo phenomena, this review takes the neutron-rich Zr as examples to show the results given by Skyrme HFB calculations with the GF method. The extended neutron particle and pair density distributions can be more properly described by the GF method, compared to the box-discretized method. The self-consistency to deal with the pairing correlation and the continuum is important to determine the asymptotic neutron particle and pair density distributions. Furthermore, with the level density obtained by the loop integral of the GF on the complex energy plane, one can describe both the bound and resonant states on the same footing. Recently, in the relativistic mean field theory, a new method to obtain precise information for the resonant energy and width is proposed by using directly the poles of the GF on the complex energy plane. This helps explain the conservation and breaking of pseudospin symmetry in the nucleon single-particle levels. Additionally, Green’s function method is easily compatible with various theoretical models, thus in the future it can be further applied to describe the exotic nuclear collective resonances and nuclear reaction processes. </sec>

The fracture behaviors of nanoscale sp-bonded materials have been studied using the molecular dynamics and lattice Green's function methods. The initial atomic structures of the crack are determined both from the elastic solutions as well as from those by lattice Green's function method for the infinite systems. Firstly, we calculate the Green function for the defective lattice, with dislocation and crack, by solving the Dyson equation, appropriate for absolute zero temperature, After the lattice Green functions of the absolute zero temperature have been determined, the lattice parameters and interatomic force constants are adjusted to fit to materials at temperature T. In general, we have found that the lattice trapping and stress intensity factors for dislocation emission K Ile , The fracture and strength properties are also investigated for the nanocrystailine materials like semiconductor quantum wire and nanotubes. The O(N) tight-binding molecular dynamics (TBMD) method is used to analyze the reconstruction of atomic bonding near the crack tip as well as the cleaved surface. We compare the fracture behavior of nanoscale materials with those of corresponding bulk-size materials.

Efficient calculational techniques for reflection high energy electron diffraction (RHEED) are reported for surfaces with large periodic supercells. A fast Fourier transform approach enables the computer time scaling of a conventional RHEED calculation to be reduced to n2 log (n), where n is the number of diffracted beams used in the calculation. The special technique needed to implement this for arbitrary incident azimuths with symmetry optimization is detailed. A Green's function method is also introduced which is particularly suitable for calculations for imperfect surfaces. This combines the conventional approach to RHEED for dealing with substrate diffraction with a Green's function treatment for an imperfect surface of supercells and has n log (n) time scaling. Techniques for matching the results of the conventional and Green's function treatments at the substrate–surface interface are given. In addition, numerical procedures for solving the resulting equations are described and a selection of illustrative results is presented.

This paper, on one hand, is an introduction to thermo‐field dynamics (TFD), on the other hand a derivation of some interesting features of Green's functions in the frame of thermo‐field dynamics. It is proved that in the language of TFD, Green's functions at finite temperature take a very simple form. Through the applications of TFD to many‐body systems it is shown that TFD is a straightforward method in dealing with finite temperature many‐body systems. As an application of Green's function method in TFD, we derive a Kogan‐type formula for the non‐equilibrium dynamic conductivity or laser‐pulse heated condensed matters, where electrons and ions are held at different temperatures.

An analytic and numerical solution with spectral Green's function method for transport equation in spherical geometry

Investigation of the complete valence shell of formic acid by electron momentum spectroscopy and Green's function methods