Poisson differential equation

This represents the Poisson differential equation for the electrical potential U as a function of r and a charge density of a. 

In other words, solving the Poisson differential equation (26) for the mixing angle o(Q) does not suffer from the integration problem in Eq. (21) since the nonremovable part F has been eliminated from the right hand side of... 

Note that the mathematical symbol V stands for the second derivative of a function (in this case with respect to the Cartesian coordinates d fdx + d jdy + d jdz y, therefore the relationship stated in Eq. (41) is a second-order differential equation. Only for a constant dielectric Eq.(41) can be reduced to Coulomb s law. In the more interesting case where the dielectric is not constant within the volume considered, the Poisson equation is modified according to Eq. (42). 

The charge density is simply the distribution of charge throughout the system and has 1 units of Cm . The Poisson equation is thus a second-order differential equation (V the usual abbreviation for (d /dr ) + (f /dx/) + (d /dz )). For a set of point charges in constant dielectric the Poisson equation reduces to Coulomb s law. However, if the dielectr... 

The Poisson equation has been used for both molecular mechanics and quantum mechanical descriptions of solvation. It can be solved directly using numerical differential equation methods, such as the finite element or finite difference methods, but these calculations can be CPU-intensive. A more efficient quantum mechanical formulation is referred to as a self-consistent reaction field calculation (SCRF) as described below. 

To solve the Poisson equation, we must express as a p function of the coordinates of the system and solve the resulting second-order differential equation to obtain ip(x,y, z), from which trci and hence, AG, p - p°, and 7 can be calculated. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

The relation between the spatial distribution of the electrostatic potential /(jc) and the spatial distribution of charge density Qy(x) can be stated, generally, in terms of Poisson s differential equation. 

When inserting into (4.5), the term ZeR will be multiplied with the elements of the electric field gradient tensor V. Fortunately, the procedure can be restricted to diagonal elements Vu, because V is symmetric and, consequently, a principal axes system exists in which the nondiagonal elements vanish, = 0. The diagonal elements can be determined by using Poisson s differential equation for the electronic potential at point r = 0 with charge density (0), AV = Anp, which yields... 

Solve the differential equation (6.4) for the case when all neutron capture cross-sections are equal and show that it leads to the Poisson distribution for a single exposure r. Use this to derive an equivalent to Eq. (6.16) for this case when there is an exponential distribution of exposures as in Eq. (6.9). 

Assuming that a number of NMR data sets (e.g., 2-D or 3-D maps of displacement vectors resulting from an external periodic excitation) from an object are acquired, the remaining difficulty is their reconstruction into viscoelastic parameters. As written in Section 2 the basic physical equation is a partial differential equation (PDE, Eq. (3)) relating the displacement vector to the density, the attenuation, Young s modulus and Poisson s ratio of the medium. The reconstruction problem is indeed two-fold ... 

In the same way as described in Sec. 5.2 for a diifiise layer in aqueous solution, the differential electric capacity, Csc, of a space charge layer of semiconductors can be derived from the Poisson s equation and the Fermi distribution function (or approximated by the Boltzmann distribution) to obtain Eqn. 5-69 for intrinsic semiconductor electrodes [(Serischer, 1961 Myamlin-Pleskov, 1967 Memming, 1983] ... 

The third point implies that it is possible to develop a physical theory for ionic interactions that is relatively simple and still useful. The most frequently used is the Poisson-Boltzmann (P-B) equation, which combines the Poisson equation from classical electrostatics with the Boltzmann distribution from statistical mechanics. This is a second-order nonlinear differential equation and its solution depends on the geometry and the boundary conditions. 

The Poisson equation (see Equation (11.18)) gives the fundamental differential equation for potential as a function of charge density. The Debye-Hiickel approximation may be used to express the charge density as a function of potential as in Equation (11.28) if the potential is low. Combining Equations (11.24) and (11.32) gives... 

A. Friedman and K. Tintarev, Boundary asymptotics for solutions of the Poisson-Boltzmann equation, J. Differential Equations, 69 (1987), pp. 15-38. 

K. Tintarev, Fundamental solution of the Poisson-Boltzmann equation, in Differential Equations and Mathematical Physics, I. W. Knowles and Y. Saito, eds., Lecture Notes in Math. 1285, Springer-Verlag, Berlin, New York, 1987. 

Simultaneous measurements of d and osmotic pressure provide a relation between the separation of bilayers and their mutual repulsive pressure. Measurement of the electrostatic repulsion is, in fact, a determination of the electrostatic potential midway between bilayers relative to the zero of potential in the dextran reservoir. The full nonlinear Poisson-Boltzmann differential equation governing this potential has been integrated (I) from the midpoint to the bilayer surface to let us infer the surface potential. The slope of this potential at the surface gives a measure of the charge bound. 

Often this equation is referred to as the Poisson-Boltzmann equation. It is a partial differential equation of second order, which in most cases has to be solved numerically. Only for some simple geometries can it be solved analytically. One such geometry is a planar surface. 

For the simple case of a planar, infinitely extended planar surface, the potential cannot change in the y and z direction because of the symmetry and so the differential coefficients with respect to y and z must be zero. We are left with the Poisson-Boltzmann equation which contains only the coordinate normal to the plane x ... 

The first term in the evolution operator has the form of a Poisson bracket and evolution under this part of the operator can be expressed in terms of characteristics. The corresponding set of ordinary differential equations is... 

A non-linear mathematical model, which is a set of ordinary differential equations, for the process in the SPBER was developed.19 The model accounts for the heterogeneous electrochemical reaction and homogeneous reaction in the bulk solution. The lateral distributions of potential, current density and concentration in the reactor were studied. The potential distribution in the lateral dimension, x, of the packed bed was described by a one dimensional Poisson equation as ... 

The electrostatic field in the stationary state is described by the Poisson-Boltzmann equation. The PB model constitutes the fundamental equation of electrostatics and is based on the differential Poisson equation which describes the electrostatic potential 4>(r) in a medium with a charge density p(r) and a dielectric scalar field e(r) ... 

Moreira IS, Fernandes PA and Ramos MJ (2005) Accuracy of the numerical differentiation of the Poisson-Boltzmann equation, J Mol Struct (Theochem), 729 11-18... 

Poisson equation — In mathematics, the Poisson equation is a partial differential equation with broad utility in electrostatics, mechanical engineering, and theoretical physics. It is named after the French mathematician and physicist Simoon-Denis Poisson (1781-1840). In classical electrodynamics the Poisson equation describes the relationship between (electric) charge density and electrostatic potential, while in classical mechanics it describes the relationship between mass density and gravitational field. The Poisson equation in classical electrodynamics is not a basic equation, but follows directly from the Maxwell equations if all time derivatives are zero, i.e., for electrostatic conditions. The corresponding ( first ) Maxwell equation [i] for the electrical field strength E under these conditions is... 

Poisson-Boltzmann equation — The Poisson-Boltz-mann equation is a nonlinear, elliptic, second-order, partial differential equation which plays a central role, e.g., in the Gouy-Chapman (- Gouy, - Chapman) electrical -> double layer model and in the - Debye-Huckel theory of electrolyte solutions. It is derived from the classical -> Poisson equation for the electrostatic potential... 

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