Random phenomena, study

Data generally are classified as either deterministic or random. Deterministic means that the process under study can be described by an explicit mathematical relationship. Random means that the phenomenon under study cannot be described by an explicit mathematical function because each observation of the phenomenon is unique. A single representation of a random phenomenon is called a sample function. If, as in all experiments, the sample function is of finite length, then it is called a sample record. The set of all possible sample functions, y r), which the random process might produce, defines the random or stochastic process. The mean value and autocorrelation function for a random process are defined by... 

Worth specifying that after the presented detailed study centered on the matter of the propagation with the formation of Standing Waves -SW, we have reached the conclusion that the Borrmann effect is not a random phenomenon, but it is always the companion of the propagation in the crystal that suffered a dynamic scattering with X-ray on the thick crystal... 

In any particular situation, it is usually possible to give a variety of reasons why the observed quantity behaves in an erratic manner. The observed quantity may be critically dependent on certain parameters and the observed fluctuations attributed to slight variations of these parameters. The implication here is that the observed fluctuations appear erratic only because we have not taken the trouble to make a sufficiently precise analysis of the situation to disclose the pattern the observations are following. It is also possible, in some situations, to adopt the viewpoint that certain aspects of the phenomenon being studied are inherently unknowable and that the best physical laws we can devise to explain the phenomenon will have some form of randomness or unpredictability built into them. Such is the case, for example, with thermal noise voltages, which are believed to be governed by the probabilistic laws of quantum physics. 

In the last several decades, both experimental data and theoretical studies [5, 9, 13-15] have revealed the effect of similar defect aggregation in the course of the bimolecular A+B —> 0 reaction under permanent particle source (irradiation) - the phenomenon similar to that discussed in previous Chapters for the diffusion-controlled concentration decay. Radiation-induced aggregation of similar defects being observed experimentally at 4 K after prolonged X-ray irradiation [16] via both anomalously high for random distribution concentration of dimer F2 centres (two nearest F centres) and directly in the electronic microscope [17], permits to accumulate defect concentrations whose saturation value exceed by several times that of the Poisson distribution. 

The above studies support the notion that nucleation is a very stochastic phenomenon when the sample is held at constant temperature, compared to when the sample was cooled at a constant cooling rate. As suggested previously, the magnitude of the driving force can affect the degree of stochastic or random behavior of nucleation. For example, on the basis of extensive induction time measurements of gas hydrates, Natarajan (1993) reported that hydrate induction times are far more reproducible at high pressures (>3.5 MPa) than at lower pressures. Natarajan formulated empirical expressions showing that the induction time was a function of the supersaturation ratio. 

The onset of percolation and the conditions that produce this phase-transition phenomenon have been of considerable interest to many disciplines of science. The classical studies have focused on immobile ingredients in a system that increase in concentration by randomly adding to the collection of particles. It is possible to estimate the conditions leading to the onset of percolation under these circumstances. When the ingredients are in motion, this estimation is far more difficult. It is an obvious challenge that was tackled using cellular automata. 

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