Phenomenological procedure

The approach adopted, although the equations used are clearly empirical, is rather general and it surely represents a development with respect to phenomenological procedures describing relationships between structure and processing conditions. In the intention of the work, the kinetic parameters are the connections among such macroscopic observations. 

Fermi surface sheets which were used to calculate the dHvA frequencies in fig. 73 are shown in fig. 74. The Fermi surfaces proposed by Zwicknagl et al. (1990) and Yamagami and Hasegawa (1993) actually look similar to each other. In calculations of the former, adjustable parameters were used to fit the theoretical dHvA frequencies to the observed ones, while no such phenomenological procedure was employed in calculations of the latter. Therefore, we discuss the origins of dHvA branches on the basis of the Fermi surfaces calculated by Yamagami and Hasegawa. 

Since the susceptibilities can be extracted from the optical spectra of these active modes, a quantitative description based on dissipative tunneling techniques can be developed. Such a program should include the analysis of the motion of the reaction complex PES, with the dissipation of active modes taken into account. The advantage of this procedure is that it would allow one to confine the number of PES degrees of freedom to the relevant modes, and incorporate the environment phenomenologically. 

In reaction rate studies one s goal is a phenomenological description of a system in terms of a limited number of empirical constants. Such descriptions permit one to predict the time-dependent behavior of similar systems. In these studies the usual procedure is to try to isolate the effects of the different variables and to investigate each independently. For example, one encloses the reacting system in a thermostat in order to maintain it at a constant temperature. 

We recall that our wave equation represents a long wave approximation to the behavior of a structured media (atomic lattice, periodically layered composite, bar of finite thickness), and does not contain information about the processes at small scales which are effectively homogenized out. When the model at the microlevel is nonlinear, one expects essential interaction between different scales which in turn complicates any universal homogenization procedure. In this case, the macro model is often formulated on the basis of some phenomenological constitutive hypotheses nonlinear elasticity with nonconvex energy is a theory of this type. 

In a realistic simulation, one initiates trajectories from the reactant well, which are thermally distributed and follows the evolution in time of the population. If the phenomenological master equations are correct, then one may readily extract the rate constants from this time evolution. This procedure has been implemented successfully for example, in Refs. 93,94. Alternatively, one can compute the mean first passage time for all trajectories initiated at reactants and thus obtain the rate, cf. Ref 95. 

How is statistical thermodynamics used for deriving adsorption isotherms What are the similarities and differences between this procedure and the one based on phenomenological thermodynamics How is the kinetic theory of gases used for deriving adsorption isotherms ... 

Therefore, the simplest procedure to get the stochastic description of the reaction leads to the rather complicated set of equations containing phenomenological parameters / (equation (2.2.17)) with non-transparent physical meaning. Fluctuations are still considered as a result of the external perturbation. An advantage of this approach is a useful analogy of reaction kinetics and the physics of equilibrium critical phenomena. As is well known, because of their nonlinearity, equations (2.1.40) reveal non-equilibrium bifurcations [78, 113]. A description of diffusion-controlled reactions in terms of continuous Markov process - equation (2.2.15) - makes our problem very similar to the static and dynamic theory of critical phenomena [63, 87]. When approaching the bifurcation points, the systems with reactions become very sensitive to the environment fluctuations, which can even produce new nonequilibrium transitions [18, 67, 68, 90, 108]. The language developed in the physics of critical phenomena can be directly applied to the processes in spatially extended systems. 

A simpler procedure has been implemented in applications to electron transfer in collisions of ions with metal surfaces.[35] Returning to Eq.(31), the second line can be interpreted as describing the rate of change of coherence of p- and s-regions over time. This is not likely to change much from its initial value for short collision times, so that the equation can be replaced on the average by a phenomenological rate equation,... 

The phenomenological spin-orbit Hamiltonian ought not to be used for computing spin-orbit matrix elements, though. An example for a failure of such a procedure will be discussed in detail in the later subsection on a word of caution. 

The parameters A, B, D and TA are determined using a least squares fitting procedure. While A, B, and D are phenomenological parameters, TA equals the so-called time to arrival which is the time between contrast agent injection and appearance of the first contrast agent molecules within the ROI. 

The rates of decomposition [d/ml/dt = Gml(dNml/df)] obtained in FIK for molecular ions are generally normalized to give a normalized rate fej ( ) (phenomenological rate coefficient in some earlier papers) [43], This procedure assumes that the collection efficiency, G, is the same for all fragment ions and the stable molecular ions, viz. 

The phase-type distributions are designed to serve as retention-time distributions in semi-Markov models. To obtain the equations of the model for a phenomenological compartmental configuration, one has to follow the following procedure ... 

Phenomenology of the crystallization. The conversion versus time curves obtained at three different temperatures are shown in Figure 1. With the synthesis procedure used, the sigmoid curves were characterized by shorter induction periods than the traditional method (11,12). As expected, temperature had a strong effect on the rate of crystallization. The overall crystallization rates may be approximated by the reciprocal of the times of half conversion. From these values an apparent activation energy of 22 1 kcal/mol was obtained. With respect to literature data, this value exceeds that reported, for instance, for zeolite Na-X (1,4) but compares well with the 19.8 kcal/mol found for ZSM-11 (13). 

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