Coefficient of variation for the substrate particles

4) and it is characteristic of a nonlinear system exhibiting chaotic behavior. Thus, heterogeneity may be at the origin of fluctuations, and fluctuations are the prelude of instability and chaotic behavior. 

Consequently, the observed process uncertainty may actually be an important part of the system and the expression of a structural heterogeneity. When the fluctuations in the system are small, it is possible to use the traditional deterministic approach. But when fluctuations are not negligibly small, the obtained differential equations will give results that are at best misleading, and possibly very wrong if the fluctuations can give rise to important effects. With these concerns in mind, it seems only natural to investigate an approach that incorporates the small volumes and small number of particle populations and may actually play an important part. 

Aside from the continuity assumption and the discrete reality discussed above, deterministic models have been used to describe only those processes whose operation is fully understood. This implies a perfect understanding of all direct variables in the process and also, since every process is part of a larger universe, a complete comprehension of how all the other variables of the universe interact with the operation of the particular subprocess under study. Even if one were to find a real-world deterministic process, the number of interrelated variables and the number of unknown parameters are likely to be so large that the complete mathematical analysis would probably be so intractable that one might prefer to use a simpler stochastic representation. A small, simple stochastic model can often be substituted for a large, complex deterministic model since the need for the detailed causal mechanism of the latter is supplanted by the probabilistic variation of the former. In other words, one may deliberately introduce simplifications or errors in the equations to yield an analytically tractable stochastic model from which valid statistical inferences can be made, in principle, on the operation of the complex deterministic process. 

Stochastic models have much to offer at the present time in strengthening the theoretical foundation and in extending the practical utility of the widespread deterministic models. After all, in a mathematical sense, the deterministic model is a special limiting case of a stochastic model. 

The master of the oracle at Delphi does not say anything and does not conceal anything, only hints. 

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