Stochastic events

For certain types of stochastic or random-variable problems, the sequence of events may be of particular importance. Statistical information about expected values or moments obtained from plant experimental data alone may not be sufficient to describe the process completely. In these cases, computet simulations with known statistical iaputs may be the only satisfactory way of providing the necessary information. These problems ate more likely to arise with discrete manufactuting systems or solids-handling systems rather than the continuous fluid-flow systems usually encountered ia chemical engineering studies. However, there ate numerous situations for such stochastic events or data ia process iadustries (7—10). 

Particle-Bubble Attachment. In the above, principles leading to creation of desired hydrophobicity/hydrophihcity of the particles has been discussed. The next step is to create conditions for particle-bubble contact, attachment, and their removal, which is simply described as a combination of three stochastic events with which are associated the probability of particle-bubble colhsion, probabihty of attachment, and probability of retention of attachment. The first term is controlled by the hydrodynamic conditions prevaihng in the flotation unit. The second is determined by the surface forces. The third is dependent on the s irvival of the laden bubble by liq ud t irbulence and impacts by the other suspended particles. A detailed description of the hydrodynamic and other physical aspects of flotation is found in the monograph by Schulze (19 ). 

Nasmyth There are many examples where this is a stochastic event. Stochastic can refer to genes flicking on and off, or it can be choosing a position on the cell and marking that point — budding in yeast is a good example of this. Also, in the lateral inhibition, it is which one of the neuroblasts will win out. Once you have established this, you have created a focus for generating asymmetry. 

Goodwin That is similar to what happens in Fucus. There is a symmetrical cell, and even in the absence of any polarization due to light, it will break symmetry and produce an axis. There is probably a similar sort of stochastic event that triggers some kind of polymerization or pattern. 

Kotlikoff It also requires a sufficient number of those stochastic events to occur in time, such that you can observe it. I can imagine a situation where there are stochastic events that are below the threshold of resolution. 

The solar system abundances of the elements are the result of the Big Bang, which produced hydrogen and helium, 7.5 billion years of stellar nucleosynthesis, which produced most of the rest of the elements, and the physical processes that mixed the materials together to form the Sun s parent molecular cloud. The unique features of the solar system composition may also reflect the stochastic events that occurred in the region where the Sun formed just prior to solar system formation. 

It must be emphasized that the complex systems at all levels in our hierarchy are dynamical (interactive). There is a constant motion of the whole and of the parts. Therefore relationships, interactions or transactions that take place do so as stochastic events. This is precisely what Poincare described, who was perhaps the major founding father of post-Newtonian science. The fact that purposive encounters do occur as a part of biologically significant processes may be viewed either as a result of purely random events, or it may be characteristic of the behaviour of complex systems poised at the edge of chaos. 

Furthermore, other types of toxic effect may also be stochastic events, if a reactive metabolite interacts with a critical protein or affects a gene involved with development of the embryo, for example. 

Segregated or corpuscular models regard biomass as a population of individual cells. Consequently, the corresponding mathematical model is based on statistical equations. Such models are valuable for describing the variations in a given populations such as the age distribution amongst the cells. This approach is also useful for describing stochastic events, in which case probability and statistics are applied. 

In a record obtained by the patch clamp technique, the channel is closed for much of the time (i.e. no current flows across the patch of membrane that contains it), but at irregular intervals the channel opens for a short time, producing a pulse of current. Successive current pulses are always of much the same size in any one experiment, suggesting that the channel is either open or closed, and not half open (there are exceptions to this rule). The durations of the pulses, however, and the intervals between them, vary in an apparently random fashion from one pulse to the next. Hence the openings and closings of channels are stochastic events. This means that, as with many other molecular processes, we can predict when they will occur only in terms of statistical probabilities. But one of the most useful features of the patch clamp method is that it allows observation of these stochastic changes in single ion channels as they actually happen individual protein molecules can be observed in action. 

Populations of endangered species often consist of small numbers that may be trapped in the so-called extinction vortex. This means that the combination of inbreeding, demographic stochasticity, and genetic drift leads to feedback loops that make small populations even smaller (Caughley 1994). According to Klok (2000), the minimum viable population size is species specific because the life history of a species can have a large impact on the outcome of the aforementioned stochastic events. Moreover, the MVP size will also depend on the quality of the habitat and, consequently, on the temporal and spatial distribution and availability of pollutants. 

Bedaux and Kooijman 1994 Kooijman 1996 Newman and McCloskey 1996, 2000 Zhao and Newman 2007). This is not just an academic discussion the 2 theories lead to different time courses of mortality at constant exposure (Kooijman 1996) (see Figure 2.10) and have very different consequences for sequential exposure (Newman and McCloskey 2000 Zhao and Newman 2007). In reality, both sensitivity difference and stochasticity are likely to play a role in mortality. Individuals also differ in sensitivity, especially in field populations, but there is clearly a substantial stochastic component involved in mortality that cannot be ignored. The method to deal with stochastic events in time is survival analysis or time-to-event analysis (see Bedaux and Kooijman 1994 Newman and McCloskey 1996). For industrial practices, this method has a long history as failure time analysis (see, e.g., Muenchow 1986). Bedaux and Kooijman (1994) link survival analysis to a TK model to describe survival as a function of time (i.e., the hazard rate is taken proportional to the concentration above a threshold value). Newman and McCloskey (1996) take an empirical relationship between external concentration and hazard rate. 

As another example of hybrid simulation touched upon above, Haseltine and Rawlings (2002) treated fast reactions either deterministically or with Langevin equations and slow reactions as stochastic events. Vasudeva and Bhalla (2004) presented an adaptive, hybrid, deterministic-stochastic simulation scheme of fixed time step. This scheme automatically switches reactions from one type to the other based on population size and magnitude of transition probability. 

How do moons and rings form The solid bodies around the giant planets formed as a consequence of the assembly of the giant planets, but stochastic events such as large collisions may have played crucial roles. For example, we do not know whether the massive Saturnian rings are as old as Saturn itself. 

The current burst model is potentially powerful in providing explanations for many mechanistic and morphological aspects involved in the formation of PS. However, as recognized by Foil et al. themselves, it would be extremely difficult for such a unified model to be expressed in mathematical form because it has to include all of the conditional parameters and account for all of the observed phenomena. Fundamentally, all electrochemical behavior is in nature the statistical averages of the numerous stochastic events at a microscopic scale and could in theory be described by the oscillation of the reactions on some microscopic reaction units which are temporally and spatially distributed. Ideally, a single surface atom would be the smallest dimension of such a unit and the integration of the contribution of all of the atoms in time and space would then determine a specific phenomenon. In reality, it is not possible because one does not know with any certainty the reactivity functions of each individual atoms. The difficulty for the current burst model would be the establishment of the reactivity functions of the individual reaction units. Also, some of the assumptions used in this model are questionable. For example, there is no physical and chemical foundation for the assumption that the oxide covering the reaction unit is... 

From these considerations it appears that freezing is a stochastic event considering a single sample and only a probability of freezing can be defined. Therefore, the freezing femperatures of identical samples are expected to be scattered around a mean value given by T, which becomes lower as the samples get smaller. For example for identical water samples of a few cm, this temperature is found to be around — 15°C, for a few mm they are around — 20°C and for a few /rm around — 40°C. 

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