Poisson

The mathematics is completed by one additional theorem relating the divergence of the gradient of the electrical potential at a given point to the charge density at that point through Poisson s equation... 

Marmur [12] has presented a guide to the appropriate choice of approximate solution to the Poisson-Boltzmann equation (Eq. V-5) for planar surfaces in an asymmetrical electrolyte. The solution to the Poisson-Boltzmann equation around a spherical charged particle is very important to colloid science. Explicit solutions cannot be obtained but there are extensive tabulations, known as the LOW tables [13]. For small values of o, an approximate equation is [9, 14]... 

For the quantum mechanical case, p and Ware operators (or matrices in appropriate representation) and the Poisson bracket is replaced by the connnutator [W, p] If the distribution is stationary, as for the systems in equilibrium, then Bp/dt = 0, which implies... 

A stationary ensemble density distribution is constrained to be a functional of the constants of motion (globally conserved quantities). In particular, a simple choice is pip, q) = p (W (p, q)), where p (W) is some fiinctional (fiinction of a fiinction) of W. Any such fiinctional has a vanishing Poisson bracket (or a connnutator) with Wand is thus a stationary distribution. Its dependence on (p, q) through Hip, q) = E is expected to be reasonably smooth. Quanttun mechanically, p (W) is die density operator which has some fiinctional dependence on the Hamiltonian Wdepending on the ensemble. It is also nonnalized Trp = 1. The density matrix is the matrix representation of the density operator in some chosen representation of a complete orthononnal set of states. If the complete orthononnal set of eigenstates of die Hamiltonian is known ... 

The model used is the RPM. The average electrostatic potential ifr) at a distance r away from an ion / is related to tire charge density p.(r) by Poisson s equation... 

Vlachy V 1999 Ionic effects beyond Poisson-Boltzmann theory Ann. Rev. Phys. Chem. 50 145... 

The linear response of a system is detemiined by the lowest order effect of a perturbation on a dynamical system. Fomially, this effect can be computed either classically or quantum mechanically in essentially the same way. The connection is made by converting quantum mechanical conmuitators into classical Poisson brackets, or vice versa. Suppose tliat the system is described by Hamiltonian where denotes an... 

An identical expression holds classically [32] if- //) times the conmuitator is replaced by the classical Poisson bracket. 

Figure Bl.10.1. Poisson distribution for fi(left vertical axis). Cumulative Poisson distribution for fi= l(right vertical axis). The cumulative distribution is the sum of the values of the distribution from 0 to n, where n =, 2, 3, 4, 5 on this graph.

When the rate measurement is statistically distributed about the mean, the distribution of events can be described by the Poisson distribution, Prrf O, given by... 

Weinert M 1981 Soiution of Poisson s equation beyond Ewaid-type methods J. Math. Phys. 22 2433... 

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

More sophisticated approaches to describe double layer interactions have been developed more recently. Using cell models, the full Poisson-Boltzmann equation can be solved for ordered stmctures. The approach by Alexander et al shows how the effective colloidal particle charge saturates when the bare particle charge is increased [4o]. Using integral equation methods, the behaviour of the primitive model has been studied, in which all the interactions between the colloidal macro-ions and the small ions are addressed (see, for instance, [44, 45]). 

The set of coupled Poisson equations (50) can, in principle, be solved with any appropriate choice of boundary conditions for Pi j(qx)- There is one choice. 

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. 

It is possible to introduce a generalized Poisson bracket by considering two general differentiable functions/(z,z ) andg(z,z ) and write... 

Figure 10. Level spaeitig distributions P s/ s)) for the cone states of the first-excited electronic doublet state of Li3 with consideration of GP effects [12] (a) Ai symmetry (b) A2 symmetry (c) E symmetry (d) full spectrum. Also shown by the solid lines are the corresponding fits to a Poisson distribution.

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