Random phase model

A dilferent approach to the calculation of transport properties above a mobility edge was used by Hindley (1970) and Friedman (1971). Both authors discuss the random phase model (RPM) proposed by Mott (1967, 1972) and Cohen (1970). In the RPM a delocalized state 0 (r) can be written as a linear combination of atomic orbitals ij/ centered at R ... 

Davis and Mott s hypothesis simplifies considerably the following discussions and we shall adopt it. However, its validity is subject to justification in any particular theoretical model of the electronic structure of an amorphous solid. Hindley (1970) has shown that it follows from his random phase model for the wavefunctions in amorphous semiconductors. 

Friedman (1971) found a solution to this problem. He calculated the Hall mobility jjl using the random phase model which served also as the basis for the mobility expression Eq. (5.7). As in the case of hopping conduction of a small polaron (Holstein and Friedman (1968)) Friedman assumed that the applied magnetic field modifies the phase of the transfer integral between sites. A minimum of three sites, which are mutual nearest... 

The random phase model is based on assumed wave functions of the sort shown in Fig. 2.9(b) in which the phase fluctuates randomly from site to site corresponding to the condition A a. 

Various mathematical concepts and techniques have been used to derive the functions that describe the different types of dispersion and to simplify further development of the rate theory two of these procedures will be discussed in some detail. The two processes are, firstly, the Random Walk Concept [1] which was introduced to the rate theory by Giddings [2] and, secondly, the mathematics of diffusion which is both critical in the study of dispersion due to longitudinal diffusion and that due to solute mass transfer between the two phases. The random walk model allows the relatively simple derivation of the variance contributions from two of the dispersion processes that occur in the column and, so, this model will be the first to be discussed. 

Motivated by a puzzling shape of the coexistence line, Kierlik et al. [27] have investigated the model with Lennard-Jones attractive forces between fluid particles as well as matrix particles and have shown that the mean spherical approximation (MSA) for the ROZ equations provides a qualitatively similar behavior to the MFA for adsorption isotherms. It has been shown, however, that the optimized random phase (ORPA) approximation (the MSA represents a particular case of this theory), if supplemented by the contribution of the second and third virial coefficients, yields a peculiar coexistence curve. It exhibits much more similarity to trends observed in... 

Note that large density fluctuations are suppressed by construction in a random lattice model. In order to include them, one could simply simulate a mixture of hard disks with internal conformational degrees of freedom. Very simple models of this kind, where the conformational degrees of freedom affect only the size or the shape of the disks, have been studied by Fraser et al. [206]. They are found to exhibit a broad spectrum of possible phase transitions. 

Dunn et al. (D7) measured axial dispersion in the gas phase in the system referred to in Section V,A,4, using helium as tracer. The data were correlated reasonably well by the random-walk model, and reproducibility was good, characterized by a mean deviation of 10%. The degree of axial mixing increases with both gas flow rate (from 300 to 1100 lb/ft2-hr) and liquid flow rate (from 0 to 11,000 lb/ft2-hr), the following empirical correlations being proposed ... 

Wood and Hill consider that the role of fluoride in these glasses is uncertain. Phase-separation studies suggest that the structure of the glass might relate to the crystalline species formed, in which case a microcrystallite glass model is appropriate. But other evidence cited above on the structure-breaking role of fluoride is compatible with a random network model. 

For the second example, let us consider the random sphere model (RSM), which can be referred to as an intermediate deterministic-stochastic approach. This model and an appropriate mathematical apparatus were originally offered by Kolmogorov in 1937 for the description of metal crystallization [254], Later, this model became widely applicable for the description of phase transformations and other processes in PS, and usually without references to the pioneer work by Kolmogorov [134,149-152,228,255,256],... 

An adsorbed phase can exhibit monolayer adsorption as well as multilayer adsorption. Surface flow in the presence of multilayer adsorption can be accounted for in the models described in the previous section. For example Okazaki and Tamon (1981) describe multilayer diffusion in their random hopping model. 

As discussed in the previous sections, LSD works well, and GGA does better, by modelling the system-averaged hole in a way which respects the sum rules, amongst other things. A much better way to construct a hybrid functional [54,68] would be to construct a hybrid hole, in which the small-separation contributions would be modelled in LSD (or GGA), which works well here, and the large-separation contributions would be modelled by some approximation designed to work at large distances, e.g., the random phase approximation [35], or the correct asymptotic limit [19],... 

Random substitutional models are used for phases such as the gas phase or simple metallic liquid and solid solutions where components can mix on any atial position which is available to the phase. For example, in a simple body-centred cubic phase any of the components could occupy any of the atomic sites which define the cubic structure as shown below (Fig. 5.1). 

In the 1950s, many basic nuclear properties and phenomena were qualitatively understood in terms of single-particle and/or collective degrees of freedom. A hot topic was the study of collective excitations of nuclei such as giant dipole resonance or shape vibrations, and the state-of-the-art method was the nuclear shell model plus random phase approximation (RPA). With improved experimental precision and theoretical ambitions in the 1960s, the nuclear many-body problem was born. The importance of the ground-state correlations for the transition amplitudes to excited states was recognized. 

The study of the intra-phase mass transfer in SCR reactors has been addressed by combining the equations for the external field with the differential equations for diffusion and reaction of NO and N H 3 in the intra-porous region and by adopting the Wakao-Smith random pore model to describe the diffusion of NO and NH3 inside the pores [30, 44]. The solution of the model equations confirmed that steep reactant concentration gradients are present near the external catalyst surface under typical industrial conditions so that the internal catalyst effectiveness factor is low [27]. 

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