Exact Green Function Research Articles

Diffusion‐Influenced Excited‐State Reversible Geminate ABCD Reaction in the Presence of an External Field

We obtained the exact Green functions, in the Laplace domain, for a diffusion-influenced excited-state reversible geminate ABCD reaction with two different lifetimes and quenching processes under a constant external field in one dimension. Analytic expressions for the survival probabilities of the initial and final states are obtained in the time domain at short and long times, respectively. The short-time approximations obtained in this work are valid for t</K/(-1), where K depends on several parameters of the system. The analysis of the long-time asymptotic behaviors reveals rather complex kinetic transitions dependent upon the field and lifetimes. We also find a destructive interplay leading to the reduction in the number of kinetic transitions similar to that found for the excited-state geminate ABC reaction with an external field in one dimension.

Point source excitation of a layered sphere: direct and far-field inverse scattering problems

A layered sphere is excited by a time-harmonic spherical acoustic wave, generated by a point source located either in the interior or in the exterior of the sphere. The sphere's core may be acoustically soft, hard, resistive or penetrable. Significant applications such as radiation from the neuron currents lying inside the human brain and localization and shape reconstruction of buried objects in layered media motivate the investigation of direct and inverse scattering problems involving such types of scatterers and excitations. The exact Green's function is determined by solving the corresponding boundary-value problem, by applying a combination of Sommerfeld's and T-matrix methods. We then introduce the low-frequency assumption and extract from the determined exact Green's function the low-frequency far-field results for a small layered sphere. The spherical wave low-frequency far-fields obtained reduce to those due to plane wave incidence on a layered sphere and also recover as special cases several classic results of the literature concerning the exterior spherical wave excitation of homogeneous spheres. The derived low-frequency far-field expansions are then utilized in order to establish inverse scattering algorithms for the determinations of (i) the sphere's center and the layers' radii, (ii) the layers' physical parameters and (iii) the location of the point source. The distance of the point source from the sphere's centre plays a significant role in the development of these algorithms. Several numerical results are included concerning the far-field interactions between the point source and the layered sphere.

The Green function of neutral gluons in color magnetic background field at finite temperature

In SU(2) gluodynamics, the tensor structure of the exact Green function of gluons neutral with respect to a homogeneous chromomagnetic field at finite temperature is derived. It is expressed through 10 tensors and corresponding form factors. These tensors constitute an algebra with respect to anticommutation. The structure constants of the algebra are calculated. The spectrum of gluons is derived from the location of poles of this Green function for the case of form factors computed in one-loop order. The high temperature asymptotics for the form factors and the spectra of different gluon states in this limit are calculated.

Coulomb corrections to the Delbrück scattering amplitude at low energies

In this paper we study the Coulomb corrections to the Delbr\uck scattering amplitude. We consider the limit when the energy of the photon is much less than the electron mass. The calculations are carried out in the coordinate representation using the exact relativistic Green function of an electron in a Coulomb field. The resulting relative corrections are of the order of a few percent for a large charge of the nucleus. We compare the corrections with the corresponding ones calculated through the dispersion integral of the pair production cross section and also with the magnetic loop contribution to the $g$ factor of a bound electron. The last one is in a good agreement with our results but the corrections calculated through the dispersion relation are not.

Stochastic quantization method for spinning particles in a gravitational plane wave field

Abstract The exact and analytic Green functions for spinning relativistic particles in interaction with a gravitational plane wave field are obtained within the Stochastic Quantization Method of Parisi and Wu. We have separated the classical calculations from those related to the quantum fluctuations. The problem has been solved by using a perturbative treatment via the Langevin equation relying on phase and configuration spaces formulation.

Exact Green's Functions and Bosonization of a Luttinger Liquid Coupled to Impedances

The exact Green’s functions of a Luttinger Liquid (LL) connected to impedances are computed at zero and finite temperature. It is also shown that if the resistances are equal to the characteristic impedance of the Luttinger liquid, then the finite Luttinger liquid connected to resistors is equivalent to an infinite Luttinger liquid. Impedance boundary conditions (IBC) include also as a special limit the case of open boundary conditions, which are explicitly recovered. Finally bosonization for a LL with IBC is proven to hold.

Explicit algebraic influence coefficients: a one-dimensional transient aquifer model

This work makes explicit an algebraic expression giving the matrix of transient influence coefficients associated with a one-dimensional semi-confined aquifer model. The domain studied is divided into a series of connected and completely mixed compartments over which the governing equation is discretized. The discrete equations obtained are solved for the compartmental hydraulic head and used to derive the algebraic expression in question. The basic properties of the so-called algebraic influence coefficients are investigated. In particular, their consistency with the exact Green function is highlighted. Finally, the newly derived influence coefficients are applied to a simplified aquifer system in order to formulate and solve the problem of identifying illegal groundwater pumping.

Diffuse fields in dynamic elasticity

In this article the problem of Green function retrieval from correlations is approached from a theoretical point of view and for this purpose an integral identity is considered: a representation theorem of the correlation type for an inhomogeneous region embedded in a homogeneous space. The full homogeneous case is studied with the theorem and it is concluded that, in the resulting field, the energy is equipartitioned. In infinite space this means that the ratio of P and S energy densities stabilizes to a constant value. That equipartition is reached in the classical sense is also demonstrated. Thus, in infinite space the energy densities associated with the possible degrees of freedom tend to share in equal parts the available energy. The representation theorem permits the verification of the well known result that by averaging correlations of motions from diffuse, equipartitioned fields within an inhomogeneous, anisotropic, elastic medium it is possible to retrieve its Green function. As a result of this it is shown that the average autocorrelation of the diffuse displacement field at a point is proportional to the imaginary part of the Green function at the source precisely at this point. As a consequence, the energy density of the diffuse field is proportional to the trace of the imaginary part of the Green tensor at the source. Thus, the analytical form of the Green function permits the establishment, in and around an inhomogeneous region, of the theoretical energy density of a diffuse field. In both homogeneous and inhomogeneous cases ( i.e. localized elastic inclusions or cavities) the equipartition of the background illumination (the so called incident field in scattering theory) is a necessary and sufficient condition to retrieve the exact Green function from correlations. Local effects lead to energy ratios that fluctuate in space and frequency. The boundary of a half-space produces in its interior fluctuations of energy densities that are local effects of the diffuse field. These results may be useful to assess the diffuse nature of seismic ground motion from a limited set of observation points and to detect the presence of a target by its signature in the distribution of diffuse energy.

Green function theory versus quantum Monte Carlo calculations for thin magnetic films

In this work we compare numerically exact quantum Monte Carlo (QMC) calculations and Green function theory (GFT) calculations of thin ferromagnetic films including second-order anisotropies. Thereby, we concentrate on easy-plane systems, i.e., systems for which the anisotropy favors a magnetization parallel to the film plane. We discuss these systems in perpendicular external field, i.e., $B$ parallel to the film normal. GFT results are in good agreement with QMC for high enough fields and temperatures. Below a critical field or a critical temperature, no collinear stable magnetization exists in GFT. On the other hand, QMC gives finite magnetization even below those critical values. This indicates that there occurs a transition from noncollinear to collinear configurations with increasing field or temperature. For slightly tilted external fields, a rotation of magnetization from out-of-plane to in-plane orientation is found with decreasing temperature.

Effective poroelastic properties of transversely isotropic rock-like composites with arbitrarily oriented ellipsoidal inclusions

The present work is devoted to the determination of the macroscopic poroelastic properties of transversely isotropic geomaterials or rock-like composites with arbitrarily oriented ellipsoïdal inhomogeneities. The key problem to solve is to separate respective effects of matrix anisotropy and of inhomogeneities’ orientation distribution and shape. Based on a numerical integration of the exact Green function provided by Pan and Chou [Pan, Y.C., Chou, T.W., 1976. Point force solution for an infinite transversely isotropic solid. Journal of Applied Mechanics 43, 608–612] for the transversely isotropic media, the Hill tensor P is obtained for an arbitrarily oriented oblate spheroidal inclusion. The integrate of the Hill tensor is performed in an intermediate system coordinate attached to the inhomogeneity and with one axis equal to the symmetry axis of the transversely isotropic matrix. This choice of system coordinate allows to simplify numerical calculations and to gain accuracy because partial integrate can be performed analytically. The obtained Hill tensor is next used to study the effect of matrix anisotropy, pores systems and microstructure-related parameters on the overall effective poroelastic properties in transversely isotropic rocks. As an example, a two step-homogenization scheme is presented to test the ability of the proposed model in the evaluation of effective Biot tensor and Biot modulus and stiffness tensor. With the help of an orientation distribution function (ODF) the anisotropy due to the pore systems is also accounted. Numerical applications are finally carried out for anisotropic porous rocks both composed of a matrix with solid minerals constituents and a pore space.

Force on a neutral atom near conducting microstructures

We derive the non-retarded energy shift of a neutral atom for two different geometries. For an atom close to a cylindrical wire we find an integral representation for the energy shift, give asymptotic expressions, and interpolate numerically. For an atom close to a semi-infinite halfplane we determine the exact Green's function of the Laplace equation and use it derive the exact energy shift for an arbitrary position of the atom. These results can be used to estimate the energy shift of an atom close to etched microstructures that protrude from substrates.

Exact analytical solution to the electronic transport in anN-coupled quantum dot array

Based on the exact Green's function calculation for an $N$-coupled quantum dot array, the rigorous expressions for the dc current, differential conductance, and density of states, which show analytically algebraic structures, are obtained. Our new findings indicate that for the quantum transport properties there exists an optimal tunneled coupling for which the density of states of the electrons in the quantum dot is smoothened to the fullest extent and consequently the dc current and differential conductance reach their extrema.

Green's function retrieval by cross‐correlation in case of one‐sided illumination

The cross‐correlation of acoustic wave fields at two receivers yields the exact Green's function between these receivers, provided the receivers are surrounded by sources on a closed surface. In most practical situations the sources are located on an open surface and as a consequence the illumination of the receivers is one‐sided. In this Letter we discuss the conditions for accurate Green's function retrieval for the situation of one‐sided illumination. It appears that the Green's function retrieval method benefits from the fact that the earth is inhomogeneous, without relying on assumptions about disorder.

Diffusion‐Influenced Reversible Trapping Problem in the Presence of an External Field

We investigate the field effect on the diffusion-influenced reversible trapping problem in one dimension. The exact Green function for a particle undergoing diffusive motion between two static reversible traps with a constant external field is obtained. From the Green function, we derive the various survival probabilities. Two types of trap distribution for the many-body problem are considered, the periodic and random distributions. The mean survival probability is obtained for the crossing-forbidden case for the two types of trap distribution. For the periodic distribution it decays exponentially. For the random trap distribution, similar to the irreversible case, there exists a critical field strength at which the long time asymptotic behavior undergoes a kinetic transition from the power law to exponential behaviors. The difference between equilibrium concentrations for the two types of trap distribution due to the fluctuation effect of trap concentration vanishes as the field strength increases.

Green's function for radially symmetric waves in inhomogeneous anisotropic solids

AbstractAn exact Green's function is presented for radially symmetric waves in cylindrically inhomogeneous anisotropic solids. It is found that the wave which passed through the center (coordinate origin) looses the logarithmic singularity at its front, when a material parameter is a half of odd integer. Further, the time‐harmonic Green's function is derived. It is also found that the wave nature of the Green's function vanishes when the wave velocity is a linear function of the radial distance.

Liouville field theory with heavy charges, I. The pseudosphere

We work out the perturbative expansion of quantum Liouville theory on the pseudosphere starting from the semiclassical limit of a background generated by heavy charges. By solving perturbatively the Riemann-Hilbert problem for the Poincare' accessory parameters, we give in closed form the exact Green function on the background generated by one finite charge. Such a Green function is used to compute the quantum determinants i.e. the one loop corrections to known semiclassical limits thus providing the resummation of infinite classes of standard perturbative graphs. The results obtained for the one point function are compared with the bootstrap formula while those for the two point function are compared with the existing double perturbative expansion and with a degenerate case, finding complete agreement.

A generalized CAD model for the full‐wave modelling of microstrip line and its regular and irregular discontinuities

AbstractIn this paper we present a new theoretical model for the modelling of the microstrip line as well as two types of discontinuities: regular (open end, step, bend and T‐ and cross‐junctions) and irregular (stub and bent‐stub). The two‐dimensional exact dyadic Green function of a grounded dielectric slab has been used with the Galerkin's technique. The subdivision of the discontinuity in a network of juxtaposed unit cells, characterized by their own longitudinal and transversal current distributions, allowed the treatment of a large class of irregular discontinuities in addition to the regular discontinuities. The obtained results have been commented and compared with those of different approaches and with experimental results where a good concordance has been observed. Copyright © 2006 John Wiley &amp; Sons, Ltd.

Revamped braneworld gravity

Gravity in five-dimensional braneworld backgrounds often exhibits problematic features, including kinetic ghosts, strong coupling, and the van Dam-Veltman-Zakharov (vDVZ) discontinuity. These problems are an obstacle to producing and analyzing braneworld models with interesting and potentially observable modifications of 4d gravity. We examine these problems in a general ${\mathrm{AdS}}_{5}/{\mathrm{AdS}}_{4}$ setup with two branes and localized curvature from arbitrary brane kinetic terms. We use the interval approach and an explicit straight gauge-fixing. We compute the complete quadratic gauge-fixed effective 4d action, as well as the leading cubic order corrections. We compute the exact Green's function for gravity as seen on the brane. In the full parameter space, we exhibit the regions which avoid kinetic ghosts and tachyons. We give a general formula for the strong coupling scale, i.e., the energy scale at which the linearized treatment of gravity breaks down, for relevant regions of the parameter space. We show how the vDVZ discontinuity can be naturally but nontrivially avoided by ultralight graviton modes. We present a direct comparison of warping versus localized curvature in terms of their effects on graviton mode couplings. We exhibit the first example of Dvali-Gabadadze-Porrati (DGP)-like crossover behavior in a general warped setup.

Semiclassical and quantum Liouville theory

We develop a functional integral approach to quantum Liouville field theory completely independent of the hamiltonian approach. To this end on the sphere topology we solve the Riemann-Hilbert problem for three singularities of finite strength and a fourth one infinitesimal, by determining perturbatively the Poincaré accessory parameters. This provides the semiclassical four point vertex function with three finite charges and a fourth infinitesimal. Some of the results are extended to the case of n finite charges and m infinitesimal. With the same technique we compute the exact Green function on the sphere on the background of three finite singularities. Turning to the full quantum problem we address the calculation of the quantum determinant on the background of three finite charges and of the further perturbative corrections. The zeta function regularization provides a theory which is not invariant under local conformal transformations. Instead by employing a regularization suggested in the case of the pseudosphere by Zamolodchikov and Zamolodchikov we obtain the correct quantum conformal dimensions from the one loop calculation and we show explicitly that the two loop corrections do not change such dimensions. We then apply the method to the case of the pseudosphere with one finite singularity and compute the exact value for the quantum determinant. Such results are compared to those of the conformal bootstrap approach finding complete agreement.

A neural-network method for the analysis of multilayered shielded microwave circuits

In this paper, a neural-network-based method for the analysis of practical multilayered shielded microwave circuits is presented. Using this idea, a radial basis function neural network (RBFNN) is trained to approximate the space-domain multilayered media boxed Green's functions used in the integral-equation (IE) method. Once the RBFNN has been trained, the outputs of the neural network (NN) replace the exact Green's functions, during the numerical solution of the IE. The computation of the RBFNN output values is very fast in comparison with the numerical methods used to calculate the exact Green's functions. This paper describes two novel strategies for efficiently training the RBFNN. In the first strategy, the input space of the RBFNN is divided into several spatial and frequency regions. The spatial subdivision is extended for the first time to both observation and source regions. In addition, the subdivision of the observation points regions is applied in a novel manner to the whole cross section of the metallic box. The second strategy combines the above region subdivision with an adaptive selection of the neurons variances in each region. The accuracy and the computational gain achieved with the NN method proposed makes possible the implementation of computer-aided-design tools that can be used for the analysis and design of integrated shielded microwave circuits (e.g., monolithic microwave integrated circuit devices) on a real-time basis.