The Phenomenological Approach

For modelling conformational transitions and nonlinear dynamics of NA a phenomenological approach is often used. This allows one not just to describe a phenomenon but also to understand the relationships between the basic physical properties of the system. There is a general algorithm for modelling in the frame of the phenomenological approach determine the dominant motions of the system in the time interval of the process treated and theti write... 

To start, we shall first flash on the phenomenological approach used to interpret the data. 

The only known alternatives for the analysis of macrosystems are by the phenomenological approach, known as thermodynamics, or by the use of... 

No attempt will be made here to extend our results beyond the simple lowest-order limiting laws the often ad hoc modifications of these laws to higher concentrations are discussed in many excellent books,8 11 14 but we shall not try to justify them here. As a matter of fact, for equilibrium as well as for nonequilibrium properties, the rigorous extension of the microscopic calculation beyond the first term seems outside the present power of statistical mechanics, because of the rather formidable mathematical difficulties which arise. The main interests of a microscopic theory lie both in the justification qf the assumptions which are involved in the phenomenological approach and in the possibility of extending the mathematical techniques to other problems where a microscopic approach seems necessary in the particular case of the limiting laws, obvious extensions are in the direction of other transport coefficients of electrolytes (viscosity, thermal conductivity, questions involving polyelectrolytes) and of plasma physics, as well as of quantum phenomena where similar effects may be expected (conductivity of metals and semi-... 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

The phenomenological approach developed mainly by Wyman, and used by Ackers et al., starts by defining all the binding constants as equilibrium constants for the binding process. Thus, is defined in terms of the standard free-energy change for the process, written symbolically as... 

In the phenomenological approach there is no way of interpreting (to which we have referred as the triplet correlation function). However, if one treats as if it were an "interaction energy f then the natural consequence would be to write, for a linear arrangement of three sites. 

In both cases the GPF formalism can, in principle, give the BI of a nonideal system, with respect to either the adsorbent or the ligand molecules. This is not possible if one uses the phenomenological approach as described in Section 2.3. 

Indeed, there are two approaches to the theory of binding phenomena. The first, the older, and the more common approach is the thermodynamic or the phenomenological approach. The central quantity of this approach is the binding polynomial (BP). This polynomial can easily be obtained for any binding system by viewing each step of the binding process as a chemical reaction. The mass action... 

The ab initio calculation of the transition rate between two electronic states with the emission of p phonons involves a very complicated sum over phonon modes and intermediate states. Due to this complexity, these sums are extremely difficult to compute however, it is just this complexity, which permits a very simple phenomenological theory to be used. There are an extremely large number of ways in which p phonons can be emitted and the sums over phonon modes and intermediate states are essentially a statistical average of matrix elements. In the phenomenological approach it is assumed that the ratio of thep-th and (p - l)-th processes will be given by a couphng constant characteristic of the matrix in which the rare earth is situated and not depending... 

Because of all these minor components (e.g., catalysts and inhibitors, added to major ones) the cure of vinyl ester resins is very complex, involving many competitive reactions. There are some new variables to account for, such as the inhibitor and initiator concentrations and induction time. Several papers [81,96,200,201] use the mechanistic approach, claiming that the phenomenological models do not explicitly include these facts, resulting in a new parameter characterization after each change in resin formulation [96]. Despite these arguments, the phenomenological approach is the most widely used and is based on an autocatalytic model which has been successfully applied to epoxy resins. Many authors [30,34,74,199,202,203] proposed the Equation 2.30 to describe the cure kinetic of unsaturated polyesters ... 

If we adopt the phenomenological approach, emotions are characterized by nine primary features (Frijda 1986 Ekman 1992) ... 

The Darken-type equation (D = [NB-DA + NA-DB,]-flh, see Section4.3.3) is obtained only if cross coefficients are zero. In order to evaluate these cross coefficients, kinetic theory beyond the phenomenological approach is needed [A. R. Allnatt, A. B. Lidiard (1993)]. 

The explanation provided for Mo03 is entirely phenomenological. Recently, it was proposed that the reactivity of various crystal surfaces can be estimated using the bond-strength model of active sites (72). A different ordering of the activity of the different crystal planes is obtained for propene oxidation than is obtained from the phenomenological approach. It will be interesting to apply this model to the data on methanol and ethanol oxidation. 

The difference this derivation has in comparison to the previous derivation of the nonlinear Schrodinger equation is that the nonlinearity is more fundamentally due to the non-Abelian wavefunction rather than from material coefficients. In effect these material coefficients and phenomenology behave as they do because the variable index of refraction is associated with non-Abelian electrodynamics. Ultimately these two views will merge, for the mechanisms on how photons interact with atoms and molecules will give a more complete picture on how non-Abelian electrodynamics participates in these processes. However, at this stage we can see that we obtain nonlinear terms from a non-Abelian electrodynamics that is fundamentally nonlinear. This is in contrast to the phenomenological approach that imposes these nonlinearities onto a fundamentally linear theory of electrodynamics. 

In the development of the second law and the definition of the entropy function, we use the phenomenological approach as we did for the first law. First, the concept of reversible and irreversible processes is developed. The Carnot cycle is used as an example of a reversible heat engine, and the results obtained from the study of the Carnot cycle are generalized and shown to be the same for all reversible heat engines. The relations obtained permit the definition of a thermodynamic temperature scale. Finally, the entropy function is defined and its properties are discussed. 

The theory of relaxation processes for a macromolecular coil is based, mainly, on the phenomenological approach to the Brownian motion of particles. Each bead of the chain is likened to a spherical Brownian particle, so that a set of the equation for motion of the macromolecule can be written as a set of coupled stochastic equations for coupled Brownian particles... 

One possible general approach to calculate the conductivity in proton conductors is the phenomenological approach applying the formalism of nonequilibrium thermodynamics [13] (Section 5.7.6) to calculate the FIGURE 8.10 Hydrogen hydrogen diffusion coefficient in oxides [36],... 

Entropy in an isolated system increases dS/dt> 0 until it reaches equilibrium dS/dt = 0, and displays a direction of change leading to the thermodynamic arrow of time. The phenomenological approach favoring the retarded potential over the solution to the Maxwell field equation is called the time arrow of radiation. These two arrows of time lead to the Einstein-Ritz controversy Einstein believed that irreversibility is based on probability considerations, while Ritz believed that an initial condition and thus causality is the basis of irreversibility. Causality and probability may be two aspects of the same principle since the arrow of time has a global nature. 

A scientific theory, like a mathematical system, never yields more than is built into it in the way of assumptions or postulates. The phenomenological approach presented in the preceding chapters could no more than characterize the kinetic behavior of systems in terms of the macroscopic variables used to describe them. We have obtained from this approach the kinetic quantities and rate constants or, in terms of the Arrhenius formulation, the frequency factors and the energies of activation. These quantities constitute our phenomenological category of kinetic language. If we are to relate them to the molecular properties of the reacting species, we must construct a new theory and a new nomenclature which starts with the molecule as the unit under consideration. 

For multisite reactions a site is only active when it is surrounded by the required number of vacant sites. The probability of such a configuration decreases as more sites become irreversibly covered Also, certain reaction steps are only possible when the sites are covered by a specific set of coinpounds. These aspects are not accounted for in the phenomenological approach applied in the preceding. 

Figure 3 illustrates how the dual site coking causes the selectivity for Q to decrease with increasing degree of site coverage at a rate which depends upon the number of sites in the cluster Clearly, the phenomenological approach cannot reveal the trends presented here ... 

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