Stochastic and deterministic processes

S. Ulam and J. von Neumann, On Combination of Stochastic and Deterministic Processes. Bulletin of the American Mathematical Society, 53 (1947) 1120,... 

Time series, power spectra, and phase portraits are shown in Fig. 1 for the BZ reaction for three different flow rates [21]. The power spectra for the periodic states in Figs. 1(a) and (c) contain an instrumentally sharp fundamental frequency and its harmonics, while the spectrum in (b) consists of broadband noise that is well above the instrumental noise level. This spectral noise could arise from either stochastic or deterministic processes. However, at least in principle it should be possible to distinguish stochastic and deterministic processes from the behavior of the power spectrum in the high frequency limit [34] for stochastic differential equations of order n, P(a)) 0) ", while for nonperiodic behavior given by deterministic differential equations, P( o)) exp(-ra)). To our knowledge this... 

Apparently the parameters of stochastic models are quite different from those of classic (deterministic) models where the permeability, the porosity, the pore radius, the tortuosity coefficient, the specific surface, and the coefficient of the effective diffusion of species represent the most used parameters for porous media characterization. Here, we will present the correspondence between the stochastic and deterministic parameters of a specified process, which has been modelled with a stochastic and deterministic model in some specific situations. 

Recently, Ugelstad et al. l969i proposed a semiempirtcal rate coefficient for radical desorption in vinyl chloride emulsion polymerization. On the other hand, Nomura et al. (1971, 1976) have derived a rate coefficient for radical desorption theoretically with both stochastic and deterministic approaches and have successfully applied it to vinyl acetate emulsion polymerization. They also pointed out that radical desorption from the particles and micelles played an important role in micellar particle formation, Fiiis et al. 1973 also derived the rate coefficient for radical desorption in a different way. Lift et al. (1981) discussed in more detail the chemical reactions incorporated in the physical process of radical desorption in the emulsion polymerization of vinyl acetate. 

The improved performance of the multiscale approach is due to the ability of orthonormal wavelets to approximately decorrelate most stochastic processes, and compress deterministic features in a small number of large wavelet coefficients. These properties permit representation of the prior probability distribution of the variables at each scale as a Gaussian or exponential function for stochastic and deterministic signals, respectively. Consequently, computationally expensive non-parametric methods need not be used for estimating the probability distribution of the coefficients at each scale. If the probability distribution of the contaminating errors and the prior can be represented as a Gaussian, the multiscale Bayesian approach provides... 

The disturbance variables are those over which the control engineer has no control. Disturbances may be stochastic (random) or deterministic. Stochastic disturbances arise from the natural variability of the process. Examples are short-term variations in flow rates caused by mechanical inaccuracies. Deterministic disturbances arise from known causes, and they usually occur at longer intervals. Examples are lot-to-lot variations in feedstock quality and changes in production rates mandated by the operation of some upstream or downstream process. Although the cause of such disturbances may be known, the disturbances themselves cannot be eliminated because of constraints external to the system. Some disturbances, stochastic and deterministic, may be measurable, but by definition they cannot be eliminated. However, the effect of such disturbances on the final product can be eliminated by compensating for them by adjusting the manipulated variables. This is the function of regulatory control. 

Stochastic V5. deterministic processing times When the processing times are not known with certainty, it is usually assumed that they follow a probability distribution with known mean and variance. 

Given the pre-eminence of the transfer function and frequency-domain-based approaches in process and chemical engineering, these two approaches will be discussed in greatest detail in this chapter. Nevertheless, information about the state-space-based approach will also be considered. This chapter will present the basic, univariate approach to time series analysis, which will be extended in Chap. 6 to consider the multivariate case containing both stochastic and deterministic components in order to model complex processes for process control, economic analysis, and simulation development. 

In an RHM the distribution of the possible structures could be uniform, where uniformity is a systematic parameter defined in the computational domain only. This observation allows both stochastic and deterministic conditions, as well as isotropic or anisotropic features to be considered. By reaching this point, the randomness of the final structure wRl be affected by initial conditions that introduce some deterministic touches to the final reconstructed heterogeneous material. Some approaches have proposed the use of electrodes with rather deterministic structures for electrochemical processes, for example, parallel and aligned CNTs, dots of controlled diameter catalysts supported on flat substrates, etc. 

For the second example, let us consider the random sphere model (RSM), which can be referred to as an intermediate deterministic-stochastic approach. This model and an appropriate mathematical apparatus were originally offered by Kolmogorov in 1937 for the description of metal crystallization [254], Later, this model became widely applicable for the description of phase transformations and other processes in PS, and usually without references to the pioneer work by Kolmogorov [134,149-152,228,255,256],... 

In contrast to liquid water, a detailed mechanistic understanding of the physical and chemical processes occurring in the evolution of the radiation chemical track in hydrocarbons is not available except on the most empirical level. Stochastic diffusion-kinetic calculations for low permittivity media have been limited to simple studies of cation-electron recombination in aliphatic hydrocarbons employing idealized track structures [56-58], and simplistic deterministic calculations have been used to model the radical and excited state chemistry [102]. While these calculations have been able to reproduce measured free ion yields and end product yields, respectively, the lack of a detailed mechanistic model makes it very difficult... 

Based on the mentioned analysis one can conclude that stochastic processes are phenomena that are neither completely random not strictly determined, i.e. random and deterministic phenomena are the left and right limits of stochastic phenomena. In order to find stochastic relationships the present-day engineering practice uses, apart from others, experiment and statistical calculation of obtained results. 

A rep < 1, Des < 1, the nucleation dynamics is stochastic in nature as a critical fluctuation in one, or more, order parameters is required for the development of a nucleus. For DeYep > 1, Des < 1 the chains become more uniformly oriented in the flow direction but the conformation remains unaffected. Hence a thermally activated fluctuation in the conformation can be sufficient for the development of a nucleus. For a number of polymers, for example PET and PEEK, the Kuhn length is larger than the distance between two entanglements. For this class of polymers, the nucleation dynamics is very similar to the phase transition observed in liquid crystalline polymers under quiescent [8], and flow conditions [21]. In fast flows, Derep > 1, Des > 1, A > A (T), one reaches the condition where the chains are fully oriented and the chain conformation becomes similar to that of the crystalline state. Critical fluctuations in the orientation and conformation of the chain are therefore no longer needed, as these requirements are fulfilled, in a more deterministic manner, by the applied flow field. Hence, an increase of the parameters Deiep, Des and A results into a shift of the nucleation dynamics from a stochastic to a more deterministic process, resulting into an increase of the nucleation rate. 

In all previous dissolution models described in Sections 5.1 and 5.2, the variability of the particles (or media) is not directly taken into account. In all cases, a unique constant (cf. Sections 5.1, 5.1.1, and 5.1.2) or a certain type of time dependency in the dissolution rate constant (cf. Sections 5.1.3, 5.2.1, and 5.2.2) is determined at the commencement of the process and fixed throughout the entire course of dissolution. Thus, in essence, all these models are deterministic. However, one can also assume that the above variation in time of the rate or the rate coefficient can take place randomly due to unspecified fluctuations in the heterogeneous properties of drug particles or the structure/function of the dissolution medium. Lansky and Weiss have proposed [130] such a model assuming that the rate of dissolution k (t) is stochastic and is described by the following equation ... 

So, we find that the mean behavior of the stochastic model is described by the deterministic model we have already developed. The fundamental difference between the stochastic and the deterministic model arises from the chance mechanism in the stochastic model that generates so-called process uncertainty, or stochastic error. 

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. 

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