Single Green Function Research Articles

Weakly nonlinear thermoacoustics for stacks with slowly varying pore cross-sections

A general theory of thermoacoustics is derived for arbitrary stack pores. Previous theoretical treatments of porous media are extended by considering arbitrarily shaped pores with the only restriction that the pore cross-sections vary slowly in the longitudinal direction. No boundary-layer approximation is necessary. Furthermore, the model allows temperature variations in the pore wall. By means of a systematic approach based on dimensional analysis and small parameter asymptotics, we derive a set of ordinary differential equations for the mean temperature and the acoustic pressure and velocity, where the equation for the mean temperature follows as a consistency condition of the assumed asymptotic expansion. The problem of determining the transverse variation is reduced to finding a Green's function for a modified Helmholtz equation and solving two additional integral equations. Similarly the derivation of streaming is reduced to finding a single Green's function for the Poisson equation on the desired geometry.

Thermodynamics of phonon-modulated tunneling centers

In recent years tunneling centers have frequently been used to explain the unusual thermodynamic properties of disordered materials; in these approaches, however, the effect of the tunneling-phonon interaction is neglected. The present study considers the archetype model of phonon-assisted tunneling, which is well known from other areas of tunneling physics (quantum diffusion, etc.). It is shown that the full thermodynamic information can be rigorously extracted from a single Green function. An extended factorization procedure beyond Hartree-Fock is introduced, which is checked by sum rules as well as by exact Goldberger-Adams expansions. The phonon-modulated internal energy and specific heat are calculated for different power-law coupling setups.

The effect on the quality of high-energy beams of nonlinearities in the field of magnetic quadrupoles

Nonlinearities in the field of magnetic quadrupoles will affect the quality of any beam in which they are used. Good field-regions or image-quality determinations can be made by exploring the phase space transmitted by the beam line. The error field of each quadrupole disturbs the trajectory of any particle passing through it. We present an efficient method for approximating the effect of this disturbance. The integral equation of motion is solved approximately by a single Green's function iteration, giving an expression linear in the non-quadrupole terms in the magnetic field. This expression can then be evaluated by a linear approximation to the particle orbits in the quadrupole. The field is decomposed into multipoles, the effects of different orders are generated recursively, and the results are summed to obtain the net disturbance.

Multiple scattering of waves in a medium of randomly distributed particles and derivation of the transport equation

In a previous paper, the characteristic functional of the wave function has been found to be the solution of a basic equation which has the same form as the original wave equation and is given in terms of the characteristic functional of the refractive index. In this paper, the theory is applied to the case of scalar wave propagation in a medium of randomly distributed particles, and the transport equation is derived by use of an analytical procedure. The renormalization of the medium and of the one‐particle scattering matrix is explicitly introduced, and is found to play an essential role for energy conservation. The Fourier transform of the resulting single Green's function (in the infinite medium) has a set of poles of infinite number. The transport equation is expressed by a series of residue values at the poles of the original and complex conjugate Green's functions, and has a wider range of applicability than that of the conventional transport equation. As a method of solving the transport equation, a set of eigenfunctions is introduced, and the solution is obtained in terms of the eigenfunction series. The diffusion function associated with each eigenfunction is also obtained.

The unitarity equations for three-particle scattering.

Bound-state amplitudes for three-particle scattering are defined in terms of the residues of the poles of a single Green's function. Unitarity equations are derived from the Hilbert relation for this Green's function.

Equations for the Green functions in quantum electrodynamics

Using the Freese-Matthews-Salam equations for chronological products of field operators, equations are written for Green functions of many electrons and photons. It is shown that in order to find any single Green function an infinite recursive system of these equations has to be solved. After adding terms containing the external electric current and external electromagnetic potential to the Lagrangian, this system is reduced to one equation containing functional derivatives of higher orders. It is shown that all relations and equations become much simpler when the definition of the Green function is appropriately changed.