Solutions of the Poisson Equation

By symmetry, the temperature gradient at x = 0, where the temperature has its maximum value is zero. Hence 

The total rate of heat generation in the element is 2q At, where A is the area of one of the large surfaces of the plate. The heat conduction rate out of each surface (q) is one half of this, so that 

In practice, the fuel region will be separated from the coolant by cladding of thickness c (Fig. 6.5). Since there is no heat generation in the cladding, equation (6.15) reduces to the Laplace form 

The temperature therefore decreases linearly through the cladding. 

The heat conducted per unit time through the cladding on one surface is given by the Fourier equation as k AiT — r )/c where is the thermal conductivity of the cladding. Equating this to the rate of heat conduction into the cladding, as given by equation (6.20), we obtain the relation 

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This justifies the use of the simpler language over the one. The solution of the Poisson equation and the boundary conditions used are explained in detail elsewhere [55]. Here, we will present some selected results. 

Davis, M. E., McCammon, J. A. Dielectric boundary smoothing in finite difference solutions of the poisson equation An approach to improve accuracy and convergence. J. Comp. Chem. 12 (1991) 909-912. 

Especially for the electrons, the fluid model has the advantage of a lower computational effort than the PIC/MC method. Their low mass (high values of the transport coefficients) and consequent high velocities give rise to small time steps in the numerical simulation (uAf < Aa) if a so-called explicit method is used. This restriction is easily eliminated within the fluid model by use of an implicit method. Also, the electron density is strongly coupled with the electric field, which results in numerical Instabilities. This requires a simultaneous implicit solution of the Poisson equation for the electric field and the transport equation for the electron density. This solution can be deployed within the fluid model and gives a considerable reduction of computational effort as compared to a nonsi-multaneous solution procedure [179]. Within the PIC method, only fully explicit methods can be applied. 

The description of the properties of this region is based on the solution of the Poisson equation (Eqs 4.3.2 and 4.3.3). For an intrinsic semiconductor where the only charge carriers are electrons and holes present in the conductivity or valence band, respectively, the result is given directly by Eq. (4.3.11) with the electrolyte concentration c replaced by the ratio n°/NA, where n is the concentration of electrons in 1 cm3 of the semiconductor in a region without an electric field (in solid-state physics, concentrations are expressed in terms of the number of particles per unit volume). 

In this framework, we have developed an analytical model based on a self-consistent solution of the Poisson equation using an adiabatic approximation for laser generated fast electrons [75], This model, briefly outlined in the following, allows the determination of the optimal target thickness to optimize the maximum proton (and ion) energies, as well as the particle number as a function of given UHC laser pulse parameters. 

As mentioned above, the PCM is based on representing the electric polarization of the dielectric medium surrounding the solute by a polarization charge density at the solute/solvent boundary. This solvent polarization charge polarizes the solute, and the solute and solvent polarizations are obtained self-consistently by numerical solution of the Poisson equation with boundary conditions on the solute-solvent interface. The free energy of solvation is obtained from the interaction between the polarized solute charge distribution and the self-... 

Values of GENp calculated from the GB approximation compare well to values obtained from numerical solution of the Poisson equation for similar collections of point charges [83,237,238], A very promising extension of the GB methods is provided by a new scaled pairwise approximation to the dielectric screening integrals [215],... 

The COSMO method is a solution of the Poisson equation designed primarily for the case of very high e [190], It takes advantage of an analytic solution for the case of a conductor (e = ). The difference between (l--)for the case of e = 80 and e = °°is only 1.3%, so this is a good approximation for water. Its use for the treatment of nonpolar solvents with e 2 depends on further approximations which have not yet been sufficiently tested to permit an evaluation of their efficacy. 

The Debye-Huckel theory was developed to extend the capacitor model and is based on a simplified solution of the Poisson equation. It assumes that the double layer is really a diffuse cloud in which the potential is not a discontinuous function. Again, the interest is in deriving an expression for the electrical potential function. This model states that there is an exponential relationship between the charge and the potential. The distribution of the potential is ... 

There are two other main directions for the calculation of the electrostatic interaction between the solute and a surrounding dielectric continuum for molecular-shaped cavities. Both require intensive numerical calculations and are thus slower than GB methods. The first direction is the direct numerical solution of the Poisson equation for the volume polarization P(r) at a position r of the dielectric medium ... 

Earlier calculation on many electron atomic systems under plasma was performed by Stewart and Pyatt [58], who estimated the variation of IP of several atoms using a finite temperature TF model. Applications of the density functional theory on these systems were reviewed by Gupta and Rajagopal [57], The calculations on many electron systems are mostly concerned with the hot and dense plasmas with the application of the IS model, or from general solutions of the Poisson equation for the potential function. The discussions using the average atom model in Section 3.3, Inferno model of Liberman in 3.4, STA model in 3.5, hydrodynamic model in... 

The authors demonstrated that the minimum of the functional in Equation (1.76) corresponds to the solution of the Poisson equation, Equation (1.73). However the value of the functional in the minimum correspond to minus the electrostatic energy. 

A connection to vector fields (1.119) is established by the notion that a is equal to the normal component of the polarization vector P(r) located on the external side of S. Polarization vanishes in the bulk of the medium provided the dielectric constant does not change there. The apparent charge cr(r) found in terms of numerical algorithms [12] is, in turn, a linear functional of p(r). Its computation is equivalent to a solution of the Poisson equation with proper matching conditions for [Pg.98]

Figure 3.31 As (due to orientational response of aqueous solvent) versus e, calculated for ET in a large binuclear transition metal complex (D (Ru2+/3+) and A (Co2+/3+) sites bridged by a tetraproline moiety) molecular-level results obtained from a nonlocal polarization response theory (NRFT, solid lines) continuum results are given by dashed lines, referring to numerical solution of the Poisson equation with vdW (cont./vdW) and SAS (cont./SAS) cavities, or as the limit of the NRFT results when the full k-dependent structure factor (5(k)) is replaced by 5(0) 5(k) for bulk water was obtained from a fluid model based on polarizable dipolar spheres (s = 1.8 refers to ambient water (square)). For an alternative model based on TIP3 water (where, nominally, 6 = ), ambient water corresponds to the diamond. (Reprinted from A. A. Milishuk and D. V. Matyushov, Chem Phys., 324, 172. Copyright (2006), with permission from Elsevier).

Consider solutions of the Poisson equation in one spatial dimension for a system with a uniform charge density /Cm the interval — /2 general solution for this domain, and discuss the possibilities for making this general solution periodic with period L. 

The Ewald potential is traditionally implemented as a lattice sum (Ziman, 1972 Leeuw et al, 1980). We just outlined a conceptualization of electrostatic interactions in periodic boundary conditions that involved adding a uniform neutralizing background for each charge, and the subsequent solution of the Poisson equation in periodic boundary conditions. Here we discuss the interconnections between that conceptualization and the traditional lattice sums, as presented in many sources, e.g. (Allen and Tildesley, 1987 Frenkel and Smit, 2002 Leeuw et al, 1980). 

We are discussing our manner to calculate the total energy for small molecules within the DV-Xa approximation by using only the monopol part of the potential in the solution of the Poisson equation. A discussion of the relativistic effects, including our results for heavy diatomic molecules, is followed by remarks on the choice of the exchange-correlation potential together with our results of calculations on molecules for the element 106 and their chemical interpretation. We conclude with results on very heavy correlation diagrams for collision systems with a united Z above 110. 

W. H. Orttung. Direct solution of the Poisson equation for biomolecules of arbitrary shape, polarizability density, and charge distribution. Ann. N. Y. Acad. Sci., 303 22-37 (1977). 

It is well known that the solution of the Poisson equation can be presented in the form of a volume integral (Zhdanov, 1988) ... 

If one assumes that the relation C = is valid for low concentrations [113,125-127], ( can be theoretically calculated from equation obtained from the solution of the Poisson equation for the Stern model for 1 1 electrolyte ... 

With (/) + known from the solution of the Poisson equation, the pressure and velocities at the new time level n+l are obtained from ... 

In what follows we take that the direction perpendicular to the surface and pointing into the solution as the positive x direction. At any point the potential S (x) = (x) — may be found as the solution of the Poisson equation (1.226), written in the form... 

The elliptic equation describes a steady-state or an equilibrium process within the region. Physically, steady state is not attained unless the net rate of generation is balanced with the net flux into the region. The physics is manifested in the existence of the solution. Thus, the elliptic equation does not admit a solution unless the condition for the existence of a steady state is satisfied. For p = 0, the condition for the existence of a solution of the Poisson equation is... 

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