Deterministic, derivative formulation

In summary, models can be classified in general into deterministic, which describe the system as cause/effect relationships and stochastic, which incorporate the concept of risk, probability or other measures of uncertainty. Deterministic and stochastic models may be developed from observation, semi-empirical approaches, and theoretical approaches. In developing a model, scientists attempt to reach an optimal compromise among the above approaches, given the level of detail justified by both the data availability and the study objectives. Deterministic model formulations can be further classified into simulation models which employ a well accepted empirical equation, that is forced via calibration coefficients, to describe a system and analytic models in which the derived equation describes the physics/chemistry of a system. 

Model formulation. After the objective of modelling has been defined, a preliminary model is derived. At first, independent variables influencing the process performance (temperature, pressure, catalyst physical properties and activity, concentrations, impurities, type of solvent, etc.) must be identified based on the chemists knowledge about reactions involved and theories concerning organic and physical chemistry, mainly kinetics. Dependent variables (yields, selectivities, product properties) are defined. Although statistical models might be better from a physical point of view, in practice, deterministic models describe the vast majority of chemical processes sufficiently well. In principle model equations are derived based on the conservation law ... 

Since a number of particles involved in any reaction event are small, a change in concentration is of the order of 1 /V. Therefore, we can use for the system with complete particle mixing the asymptotic expansion in this small parameter 1 /V. The corresponding van Kampen [73, 74] procedure (see also [27, 75]) permits us to formulate simple rules for deriving the Fokker-Planck or stochastic differential equations, asymptotically equivalent to the initial master equation (2.2.37). It allows us also to obtain coefficients Gij in the stochastic differential equation (2.2.2) thus liquidating their uncertainty and strengthening the relation between the deterministic description of motion and density fluctuations. 

What are the advantages of the formulation, (18.8) The term t x) is the net flux of the deterministic kinetics, (18.6), and the derivative of the state function (j> is the species specific affinity, (/tx — for the linear case, or (/Xx — /x x) for the nonlinear case, the driving force for the reaction toward a stationary state. Thus we have a flux-driving force relation. Second, the formulation is symmetric with respect to /+(x) and t (x), which is not the case with other formulations. Third, the state function 4> determines the probability distribution of fluctuations in x from its value at the stationary state, see (2.34). Further, as we shall show shortly, the term D x) is a measure of the strength... 

One of the major barriers to the effective and general use of 3D database searching methods is the formulation of a useful and valid 3D query. This section will cite a number of methods for the derivation and construction of 3D queries the continuing theme is that these 3D templates can be computed deterministically from a variety of data sources. Figure 2 provides a graphical summary of the following subsections each icon is keyed to the relevant section number. 

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