Process, continuous deterministic

A computational method was developed by Gillespie in the 1970s [381, 388] from premises that take explicit account of the fact that the time evolution of a spatially homogeneous process is a discrete, stochastic process instead of a continuous, deterministic process. This computational method, which is referred to as the stochastic simulation algorithm, offers an alternative to the Kolmogorov differential equations that is free of the difficulties mentioned above. The simulation algorithm is based on the reaction probability density function defined below. 

The particular form in which the continuous deterministic process will arise is precisely the new formalism of the calculus of variations that Bellman has propounded in his book (1957). We seek the maximum of the integral of some function of p(f) and q(f),... 

In this chapter we pass from the discrete to the continuous deterministic process, and the difference equations derived for the stages now become differential equations. It is scarcely surprising to find many features of the discrete process are retained by the continuous process in particular we know from the work of Denbigh (1944), and will prove here afresh, that the optimal temperature policy is disjoint in the case of a single reaction. However, this simplicity is lost when more than one reaction is taking place and we shall do well to examine the simple consecutive system A B C with some care, as it opens up the principal features of the general case. 

A better alternative for tissue growth modehng is the hybrid discrete-continuous approach. Hybrid discrete-continuous models employ a discrete algorithm to simulate the dynamics of a cell population, while processes such as diffusion and consumption of GFs or nutrients are described with a continuous, deterministic component usually based on partial differential equations. 

A brief coverage of stochastic processes in general, and of stochastic reaction kinetics in particular. Many dynamical systems of scientific and technological significance are not at the thermodynamic limit (systems with very large numbers of particles). Stochasticity then emerges as an important feature of their dynamic behavior. Traditional continuous-deterministic models, such as reaction rate... 

Although the reactions are discrete events, they are regarded as continuous-deterministic rate processes. 

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

Hydrate nucleation is the process during which small clusters of water and gas (hydrate nuclei) grow and disperse in an attempt to achieve critical size for continued growth. The nucleation step is a microscopic phenomenon involving tens to thousands of molecules (Mullin, 1993, p. 173) and is difficult to observe experimentally. Current hypotheses for hydrate nucleation are based upon the better-known phenomena of water freezing, the dissolution of hydrocarbons in water, and computer simulations of both phenomena. Evidence from experiments shows that nucleation is a statistically probable (not deterministically certain see Section 3.1.3) process. 

In polymer processing, the mathematical models are by and large deterministic (as are the processes), generally transport based, either steady (continuous process, except when dynamic models for control purposes are needed) or unsteady (cyclic process), linear generally only to a first approximation, and distributed parameter (although when the process is broken into small, finite elements, locally lumped-parameter models are used). 

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. 

As discussed in Section 1.5, the characterization of observables as random variables is ubiquitous in descriptions of physical phenomena. This is not immediately obvious in view of the fact that the physical equations of motion are deterministic and this issue was discussed in Section 1.5.1. Random functions, ordered sequences of random variable, were discussed in Section 1.5.3. The focus of this chapter is a particular class of random functions, stochastic processes, for which the ordering parameter is time. Time is a continuous ordering parameter, however in many practical situations observations of the random function z(Z) are made at discrete time 0 < Zi < t2, , < tn < T. In this case the sequence z(iz) is a discrete sample of the stochastic process z(i). 

In considering this question, elsewhere I have vigorously defended the view that life was virtually bound to arise under the conditions that prevailed at the site of its birth (1995, 2002, 2005). This opinion rests mainly on the fact that life must have arisen by chemical processes and that chemistry deals with highly deterministic, reproducible mechanisms. To be sure, contingency became added to chemical determinism when replicable molecules, most likely RNA, began to be made. This phenomenon introduced continuity into the course of events, but, also, the risk of its accidental loss by mutation. Many researchers have emphasized the chancy nature of such developments. 

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