Poisson kernel

The Poisson kernel is the Lorentzian on the orientation sphere (cf. Sect. 9.8). 

Solution for Lorentz Distributions. For Lorentz distributions the solution is the Poisson kernel... 

In the standard setup W (y) is the profile of the primary beam in horizontal direction. In order to solve the smearing integral, the orientation distribution of the layer normals, g (0), is approximated by a Poisson kernel and W (y) is approximated by a shape function with the integral breadth 2ymax of the primary beam perpendicular to the plane of incidence. In the simplified result... 

In the earlier sections we have seen the GME of Eq. (59) in action with the memory kernel defined by Eq. (66). In Section X we have seen that non-Poisson renewal processes are characterized by aging. Thus, this is the right moment to establish the age of the GME of Eq. (59). In this section we show that this is the GME corresponding to a brand new condition. Thus, we have also to establish the form of the aged GME, if this ever exists. 

We apply to this equation the same remarks as those adopted for the comparison among Eqs. (316)—(318). We note, first of all, that the structure of Eq. (325) is very attractive, because it implies a time convolution with a Lindblad form, thereby yielding the condition of positivity that many quantum GME violate. However, if we identify the memory kernel with the correlation function of the 1/2-spin operator ux, assumed to be identical to the dichotomous fluctuation E, studied in Section XIV, we get a reliable result only if this correlation function is exponential. In the non-Poisson case, this equation has the same weakness as the generalized diffusion equation (133). This structure is... 

To propagate the solvent-structuring effect induced by the presence of the hydrophobic spheres, we replace the position-dependent dielectric by an integral kernel convoluted with the electric field at position r to represent the correlations with the field at neighboring positions r. This prompts us to replace the classical Poisson equation by the rigorous relation (see Chap. 14) ... 

Here d,e is the classic Poisson bracket, e is the total energy with the accoimt of self-consistent field, l(p) is the collision integral of Boltzmann-Landau type, the kernel r(p,pf) and the collision cross-section in l( ) are expressed through the amplitude of binary quasiparticle scattering. 

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