Phenomenological problems

Other complications include inhibition or enhancement by oxidation products COj and HjO (e.g., Foster and Masel). Rate expressions accounting for this have been developed (e.g., Golodets, p. 132). It is interesting that the mechanism of Equation (4) can explain many observations. However, this may be simply fortuitous because other mechanisms have been proposed that can explain more readily other phenomena (e.g., the selectivity to formaldehyde formation at low conversions) but that present other phenomenological problems. The conclusion to be drawn is that the formulation of a complex reaction model, however consistent it may be with experiment, may be unnecessary at best and misleading at worst. In a practical situation, the best model is the simplest one that is consistent with the available data and that allows reasonable extrapolations to conditions of interest to be made. 

Keywords cognitive science, collaborative problem solving, design experiments, evolution phase, falsification, formalization phase, instructional cycle, joystick, mental model, microworlds, motivation, Newtonian mechanics, phenomenological problems, physics, scientific inquiry, scientific method, ThinkerTools, transfer phase, transfer test... 

We hope the examples presented in this appendix will provide some pedagogical illustrations and applications of the qualitative theory developed in this book. The range of instances varies from phenomenological problems to applications. Since very few nonlinear systems can be analyzed without computers, we will perform numerical computations where necessary. At some points, our de facto presentation will bear a descriptive character, avoiding technical details of computations. The two packages which have been used in the preparation of this appendix are Content [182] and Dstool [164]. 

To improve the accuracy of the solution, the size of the time step may be decreased. The smaller is the time step, the smaller are the assumed errors in the trajectory. Hence, in contrast (for example) to the Langevin equation that includes the friction as a phenomenological parameter, we have here a systematic way of approaching a microscopic solution. Nevertheless, some problems remain. For a very large time step, it is not clear how relevant is the optimal trajectory to the reality, since the path variance also becomes large. Further-... 

In order to anticipate problems and to interpret observations under the extreme conditions of shock compression, it is necessary to consider structural and electronic characteristics of PVDF. Although the phenomenological piezoelectric properties of PVDF are similar to those of the piezoelectric crystals, the structure of the materials is far more complex due to its ferroelectric nature and a heterogeneous mixture of crystalline and amorphous phases which are strongly dependent on mechanical and electrical history. 

Models of a second type (Sec. IV) restrict themselves to a few very basic ingredients, e.g., the repulsion between oil and water and the orientation of the amphiphiles. They are less versatile than chain models and have to be specified in view of the particular problem one has in mind. On the other hand, they allow an efficient study of structures on intermediate length and time scales, while still establishing a connection with microscopic properties of the materials. Hence, they bridge between the microscopic approaches and the more phenomenological treatments which will be described below. Various microscopic models of this type have been constructed and used to study phase transitions in the bulk of amphiphihc systems, internal phase transitions in monolayers and bilayers, interfacial properties, and dynamical aspects such as the kinetics of phase separation between water and oil in the presence of amphiphiles. 

These apparent restrictions in size and length of simulation time of the fully quantum-mechanical methods or molecular-dynamics methods with continuous degrees of freedom in real space are the basic reason why the direct simulation of lattice models of the Ising type or of solid-on-solid type is still the most popular technique to simulate crystal growth processes. Consequently, a substantial part of this article will deal with scientific problems on those time and length scales which are simultaneously accessible by the experimental STM methods on one hand and by Monte Carlo lattice simulations on the other hand. Even these methods, however, are too microscopic to incorporate the boundary conditions from the laboratory set-up into the models in a reahstic way. Therefore one uses phenomenological models of the phase-field or sharp-interface type, and finally even finite-element methods, to treat the diffusion transport and hydrodynamic convections which control a reahstic crystal growth process from the melt on an industrial scale. 

There is a lively controversy concerning the interpretation of these and other properties, and cogent arguments have been advanced both for the presence of hydride ions H" and for the presence of protons H+ in the d-block and f-block hydride phases.These difficulties emphasize again the problems attending any classification based on presumed bond type, and a phenomenological approach which describes the observed properties is a sounder initial basis for discussion. Thus the predominantly ionic nature of a phase cannot safely be inferred either from crystal structure or from calculated lattice energies since many metallic alloys adopt the NaCl-type or CsCl-type structures (e.g. LaBi, )S-brass) and enthalpy calculations are notoriously insensitive to bond type. 

We have treated the problem from one point of view only, of course (for a phenomenological analysis of CA behavior as a function of lattice structure, see section 3.3). Many of the questions concerning the exact relationship between lattice topology and dynamics remain unanswered, awaiting a deeper study. We mention... 

Thermodynamic, statistical This discipline tries to compute macroscopic properties of materials from more basic structures of matter. These properties are not necessarily static properties as in conventional mechanics. The problems in statistical thermodynamics fall into two categories. First it involves the study of the structure of phenomenological frameworks and the interrelations among observable macroscopic quantities. The secondary category involves the calculations of the actual values of phenomenology parameters such as viscosity or phase transition temperatures from more microscopic parameters. With this technique, understanding general relations requires only a model specified by fairly broad and abstract conditions. Realistically detailed models are not needed to un-... 

The stochastic problem is to describe properly the time evolution of the Heisenberg operator d(t) averaged over all the realizations of collisional process in the interval (0,t). The averaging, performed in the impact theory, results in the phenomenological kinetic equation [170, 158]... 

We first consider the stmcture of the rate constant for low catalyst densities and, for simplicity, suppose the A particles are converted irreversibly to B upon collision with C (see Fig. 18a). The catalytic particles are assumed to be spherical with radius a. The chemical rate law takes the form dnA(t)/dt = —kf(t)ncnA(t), where kf(t) is the time-dependent rate coefficient. For long times, kf(t) reduces to the phenomenological forward rate constant, kf. If the dynamics of the A density field may be described by a diffusion equation, we have the well known partially absorbing sink problem considered by Smoluchowski [32]. To determine the rate constant we must solve the diffusion equation... 

The hydrodynamic interaction is introduced in the Zimm model as a pure intrachain effect. The molecular treatment of its screening owing to presence of other chains requires the solution of a complicated many-body problem [11, 160-164], In some cases, this problem can be overcome by a phenomenological approach [40,117], based on the Zimm model and on the additional assumption that the average hydrodynamic interaction in semi-dilute solutions is still of the same form as in the dilute case. 

In general, percolation is one of the principal tools to analyze disordered media. It has been used extensively to study, for example, random electrical networks, diffusion in disordered media, or phase transitions. Percolation models usually require approximate solution methods such as Monte Carlo simulations, series expansions, and phenomenological renormalization [16]. While some exact results are known (for the Bethe lattice, for instance), they are very rare because of the complexity of the problem. Monte Carlo simulations are very versatile but lack the accuracy of the other methods. The above solution methods were employed in determining the critical exponents given in the following section. 

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