Equation for the Green Function

The conformational properties of a flexible polyelectrolyte chain of length L (L oo) and its spatial distribution of monomers follow from the probability density G(r, Ur, 0) (Green function), where r(0) = r and r(L) = r denote the positions of the polymer end points. The Green function itself follows from the equation [40, 41, 58-60, 150]  

Because the equation for G is linear, it can be solved by an eigenfunction expansion  

The term weak adsorption implies that the entropic free energy of a chain is comparable to its electrostatic attraction energy to the interface. The chain is assumed to be Gaussian and its conformations are only weakly perturbed by interactions with the surface. This is the most severe approximation of the current model. We also assume that the polyelectrolyte-density profile is built up near the adsorbing surface without disturbing the electrostatic potential and ionic distribution near the interface prescribed by the Poisson-Boltzmann theory. A more general 

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Stationary condition in view of (2) leads to an effective equation for the Green function... 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

Computing the renormalized electron and phonon Green functions and the corresponding self-energies involves the self-consistent solution of the following coupled equations for the Green functions ... 

The distinction between the dilute and semidilute regimes is in the magnitude of the ratio between the translational term and the rotational term in the equation for the Green function G (a, a i). Let r be defined by ... 

In the next section, we present the results of this approach for polyelectrolyte adsorption onto planar, cylindrical, and spherical surfaces. This is possible because the equation for the Green function reduces, in the corresponding separable coordinates, to a one-dimensional equation comparable to (32) in the ground-state approximation. We confirmed that the WKB applicability condition Q x)/Q x) < 1 is satisfied for aU three geometries. The approach applies particularly well above the adsorption-desorption transition, whereas it naturally fails in the proximity of the zero-potential point xq at which Q(xo) = 0. 

Here sin(Green function of the harmonic oscillator at t > 0. Inserting equation (24) into equation (16), one obtains... 

The equation for the vertex function can not be closed diagrammatically (Fig. 20). Nevertheless, it is possible to write close set of equations Redin s equations), which are exact equations for full Green functions written through a functional derivative. Hedin s equations are equations (373)-(376) and the equation for the vertex function... 

The nonequilibrium zeroth Green s functions are determined by the Dyson equations (62) and (63) on the Keldysh contour. The standard way to solve these equations is to perform a Fourier transform and then solve the algebraic matrix equations for the Green s functions. For the Keldysh functions, this procedure cannot be implemented in a straightforward way because of two time branches. Thus, we should find the Fourier transform for each Keldysh function after applying the Langreth s mapping procedure described in Section 2 [41, 45]. In particular for — t ), the Dyson... 

As is known from the differential equations theory, the Green functions are used for solution of the linear inhomogeneous differential equations ... 

As it does no longer depend on the variables qt,Pt, t, the above equation for the Green s function is also valid for the reduced density operator in the Wigner phase space. Therefore,... 

Analytical results for the Green function are obtained for a planar geometry only. For a planar surface, the linearized Poisson-Boltzmann equation yields the screened potential ... 

In some cases, one is not interested in the Green function but in the Hamiltonian. Grigore, Nenciu and Purice (1989) and Thaller (1992, p. 184) gave " ormula for the relativistic corrections to the non relativistic eigenstates of energy Eq. The following discussion is a bit abstract, but it will be illustrated by examples in the next two sections. Equation (2) is rewritten as... 

We conclude this section by giving the equation for the alloy Green s function matrix, which is relevant for electronic structure calculations. Elaborating Eq. (23) one finds... 

This differential equation can be replaced by an integral equation using the Green s function from which one can obtain E and tp(r). For this purpose, the Green s function G(r, r E) of the operator E — H, is defined as a solution of... 

In this section, we describe our model, and give a brief, self-contained account on the equations of the non-equilibrium Green function formalism. This is closely related to the electron and particle-hole propagators, which have been at the heart of Jens electronic structure research [7,8]. For more detailed and more general analysis, see some of the many excellent references [9-15]. We restrict ourselves to the study of stationary transport, and work in energy representation. We assume the existence of a well-defined self-energy. The aim is to solve the Dyson and the Keldysh equations for the electronic Green functions ... 

From this equation it follows that dg,A Pa is diagonal in the spin indices. We will therefore in the following put density variation 5p (r) determines the potential variation 5vs,(r) only up to a constant (see also [66] ). To find an explicit expression for the above functional derivative we must find an expression for the inverse density response function i A. In order to do this we make the following approximation to the Greens function (see Sharp and Horton [39], Krieger et al. [21]) ... 

Equations (46)-(48) lead to an expression of Dx/Dx0, in which Le = L and dc = d for rodlike polymers. Since Le/de = L/d = 50, f can be equated to unity in a good approximation. Estimating Djo/Dj from Eq. (58) with X = 0.025 and Fjo/Fxo from Eqs. (B4) and (B5) in Appendix B with Le/de = L/d = 50, we have calculated Dx/Dx0 as a function of the reduced concentration L3c The results are compared with Bitsanis et al. s simulation data in Fig. 16a. It can be seen that the theoretical solid curve for D /D 0 deviates downward slightly from the simulation data points, implying that the Green function method for Dx overestimates the entanglement effect compared to Bitsanis et al. s simulation. 

Translational diffusion of a particle can be described by the Green function. For simplicity, we consider here a one-dimensional diffusion process. Let x be the coordinate of the diffusing particle in its path. If no barriers are present in the path, the particle obeys the usual diffusion equation, and the unperturbed Green function associated with this diffusion equation is given by... 

Next we consider the case where more than one barrier (or perturbation element) appears in the diffusion path, and the particle receives multiple perturbations. Teraoka and Hayakawa [107] assumed that the Green function for this case satifies the following Dyson equation ... 

Quantum field theory provides an unambiguous way to find energy levels of any composite system. They are determined by the positions of the poles of the respective Green functions. This idea was first realized in the form of the Bethe-Salpeter (BS) equation for the two-particle Green function (see Fig. 1.2)... 

So far, the Lagrangian density for a homogenous problem (no sink or source term in the diffusion equation) has been considered, subject to the requirement that the approximate trial function, ip, can be forced to satisfy the boundary conditions. In this sub-section, these limitations are removed and the Lagrangian density for the Green s function developed. The Green s functions for the forward and backward time process satisfy the equations... 

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