Discrete orientation models

Another most important question in anomalous dielectric relaxation is the physical interpretation of the parameters a and v in the various relaxation formulas and what are the physical conditions that give rise to these parameters. Here we shall give a reasonably convincing derivation of the fractional Smoluckowski equation from the discrete orientation model of dielectric relaxation. In the continuum limit of the orientation sites, such an approach provides a justification for the fractional diffusion equation used in the explanation of the Cole-Cole equation. Moreover, the fundamental solution of that equation for the free rotator will, by appealing to self-similarity, provide some justification for the neglect of spatial derivatives of higher order than the second in the Kramers-Moyal expansion. In order to accomplish this, it is first necessary to explain the concept of the continuous-time random walk (CTRW). 

We shall now almost exclusively concentrate on the fractal time random walk excluding inertial effects and the discrete orientation model of dielectric relaxation. We shall demonstrate how in the diffusion limit this walk will yield a fractional generalization of the Debye-Frohlich model. Just as in the conventional Debye relaxation, a fractional generalization of the Debye-Frohlich model may be derived from a number of very different models of the relaxation process (compare the approach of Refs. 22, 23, 28 and 34—36). The advantage of using an approach based on a kinetic equation such as the fractional Fokker-Planck equation (FFPE) however is that such a method may easily be extended to include the effects of the inertia of the dipoles, external potentials, and so on. Moreover, the FFPE (by use of a theorem of operational calculus generalized to fractional exponents and continued fraction methods) clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to include fractional dynamics. 

We shall now demonstrate how the CTRW in the diffusion limit may be used to justify the fractional diffusion equation. We consider an assembly of permanent dipoles constrained to rotate about a fixed axis (the dipole is specified by the angular coordinate unit circle with fixed angular spacing A. We note that A may not necessarily be fixed for example, if we have a Gaussian distribution of jumps, the standard deviation of A serves as a fixed quantity. A typical dipole may remain in a fixed orientation at a given site for an arbitrary long waiting time. It may then reorient to another discrete orientation site. This is the discrete orientation model. 

Another difficulty concerns the choice of the models. On the one hand, the discrete orientation model, which leads to an exponential decay of the measured parameter with time [see Eq. (D.3)], does not exactly... 

B. BASIC DISCRETE ORIENTATION MODELS. The simplest model is the symmetrical two-level model in which the vector hops between two opposite directions. The lineshape has a simple analytic expression, which may be expressed by... 

The fast component is clearly related to electronic polarization, Pfast = Pd, while the slow component, connected to nuclear motions of the solvent molecules, is often called the orientational polarization (Pslow = Pot), or inertial component (PsioW = Pin)- This simplified model has been developed and applied by many authors we shall recall here Marcus (see the papers already quoted), who first had the idea of using Psiow as a dynamical coordinate. For description of solvent dynamical coordinates in discrete solvent models see Warshel (1982) and other papers quoted in Section 9. 

Figure 2.16. The discrete orientations of rods in Warner and Wang s model (1992c).

Orientational distributions near the interface are rather broad consequently, simple discrete-state models of interfacial structure are unrealistic. 

A Fade analysis of Zwanzig s model by Runnels and Colvin< has shown that the character of the transition is stable against increase in the order of the approximation. However, the model does not converge to the Onsager limit with increasing number of discrete orientations and Straley has concluded that it does not adequately represent real liquid crystal systems. 

Table 2.1 aims to give a flavour of typical models created during the past 20 years. The selection of the discussed models is restricted by full 3D FE models of unit cells, where yam paths are continuous and smooth. The list does not include (semi-) analytical, orientation averaging, inclusion, discrete mosaic models as far as they are founded on completely different modelling principles. 

Another two-dimensional, discrete element model was applied by Cartaxo and Rocha [43]. In this work, only the dynamic phenomena were investigated, that is, heat and mass transfer between the phases were not considered. Thns, the inflnence of the momentum coupling between the discrete particles and the conveying air on the air radial velocity and the mass concentration profiles was presented. An object-oriented numerical model was developed to simulate the conveying of large spherical particles (3 mm) through 9.14 m vertical tube with 7.62 cm bore size. 

A transition to an ordered state can occur in a fluid of small rigid rods of sufficiently high concentration due to steric repulsion alone, as was shown by Onsager. In the lattice model of FloryS, strong steric interaction is imposed by the requirement that the monomers occupy only lattice sites. Repulsive hard rod interaction between spherpcylinders has also been investigated in continuum models which are not restricted to small sets of discrete orientations and improve the quantitative agreement with experiment. ... 

Zeigler, B. R (1987). Hierarchical, Modular Discrete-Event Modelling in an Object-Oriented Envirionment. Simulation 49(5) 219-230. 

On the Extension of Goal-Oriented Error Estimation and Hierarchical Modeling to Discrete Lattice Models. 

We use the modeling language SMV and the model-checker NuSMV2 3]. SMV enables the declaration of integer variables and constraints on their behavior. NuSMV builds transparently the Cartesian product of the ranges of all variables. When no constraint is declared, all the combinations of variable values (i.e., states) are possible and all transitions between each pair of states are implicitly declared. Constraints are then added to delete undesired states and transitions. As for variables, time is discrete. It is modeled by the operator next (). NuSMV is well-adapted to our variable-oriented modeling approach. Moreover, the implicit transition declaration is convenient for modeling the whole physically possible behavior. 

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