Perturbation Poisson equation

An expression for e(k) in the case of a Fermi gas of free electrons can be obtained by considering the effect of an introduced point charge potential, small enough so the arguments of perturbation theory are valid. In the absence of this potential, the electronic wave functions are plane waves V 1/2exp(ik r), where V is the volume of the system, and the electron density is uniform. The point charge potential is screened by the electrons, so that the potential felt by an electron, O, is due to the point charge and to the other electrons, whose wave functions are distorted from plane waves. The electron density and the potential are related by the Poisson equation,... 

As pointed out in the Introduction, it is customary in the treatment of such systems to assume local electro-neutrality (LEN), that is, to omit the singularly perturbed higher-order term in the Poisson equation (1.9c). Such an omission is not always admissible. We shall address the appropriate situations at length in Chapter 5 and partly in Chapter 4. We defer therefore a detailed discussion of the contents of the local electro-neutrality assumption to these chapters and content ourselves here with stating only that this assumption is well suited for a treatment of the phenomena to be considered in this chapter. 

If we neglect the expansion term, we see that the first equation reduces to the usual Poisson equation. The last equation insures that the two Bardeen potentials are similar since in general the anisotropic stresses are small (they are negligible for non relativistic matter as well as for a scalar field). Note that in the absence of any form of matter all the scalar metric perturbations are 0. In addition to the scalar perturbations, there exists one equation for the tensor modes ... 

Jeans, who at first demonstrated the nature of this so called Jeans instability, considered perturbations in a uniform infinite gas. Considering the continuity equation, the equation of motion and the Poisson equation ... 

The typical space-size characterizing a plasma is the Debye radius, which is a linear measure of electroneutrality and shielding of external electric fields. The typical plasma time scale and typical time of plasma response to the external fields is determined by the plasma frequency illustrated in Fig. 3-19. Assume in a one-dimensional approach that all electrons at X > 0 are initially shifted to the right on the distance xq, whereas heavy ions are not perturbed and remain at rest. This results in an electric field, which pushes the electrons back. If = 0 at X < 0, this electric field acting to restore the plasma quasi-neutrality can be found at x > xq from the one-dimensional Poisson equation as... 

The potential in the double layer satisfies the Poisson equation therefore the perturbation of electric field i// is given by... 

MacGillivray [98] was the first to provide a mathematical justification for the LEN assumption on the basis of the perturbation theory. The idea behind such a justification is that an order of magnitude analysis of the Poisson equation, Eq. (8), shows that... 

The electrostatic contribution can be obtained by solving the Poisson equation in the continuum dielectric approximation. Although this approximation has not been systematically tested for interfacial systems, its recent applications to bulk solutions proved to be highly successful [45,46]. The conventional continuum, dielectric model can be considered as an implementation of second-order perturbation theory [47]. The first-order term is assumed to vanish and all terms beyond the second order are neglected. It is not clear, however, how well these approximations hold near an interface. In particular, interfacial solvent molecules have preferred orientations due to the interfacial excess electric field. They will, therefore, not be randomly oriented around the cavity volume of the solute - a requisite for the first-order term to vanish. Furthermore, it has... 

Notice that the perturbation method has replaced the Helmholtz equation with a system of Laplace and Poisson equations. By observing that polar coordinates can be used and that there is no 9 dependence, that is m = u(f), we get... 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

Those using analytical methods (often via perturbation approaches, which necessitate some level of approximation such as linearization of the Poisson-Boltzmann equation) to incorporate increasing complexity into the model of interfacial structure. 

The linear response of a system is determined by the lowest order effect of a perturbation on a dynamical system. Formally, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical commutators into classical Poisson brackets, or vice versa. Suppose that the system is described by Hamiltonian H + where denotes an external perturbation that may depend on time and generally does not commute with H. The density matrix equation for this situation is given by the Bloch equation [32]... 

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