Complex system examples

All the phase diagrams reported above show a complete mutual solubility in the liquid state. The formation of a single phase in the liquid state corresponds to behaviour frequently observed in intermetallic (binary and complex) systems. Examples, however, of a degree of immiscibility in the liquid state are also found in selected intermetallic systems. Fig. 2.16 shows a few binary systems in which such immiscibility can be observed (existence of miscibility gaps in the liquid state). All the three... 

Diatomic molecules have only one vibrational mode, but VER mechanisms are paradoxically quite complex (see examples C3.5.6.1 and C3.5.6.2). Consequently there is an enonnous variability in VER lifetimes, which may range from 56 s (liquid N2 [18]) to 1 ps (e.g. XeF in Ar [25]), and a high level of sensitivity to environment. A remarkable feature of simpler systems is spontaneous concentration and localization of vibrational energy due to anhannonicity. Collisional up-pumping processes such as... 

In this section we examine some examples of cross-linked step-growth polymers. The systems we shall describe are thermosetting polymers of considerable industrial importance. The chemistry of these polymerization reactions is more complex than the hypothetical AB reactions of our models. We choose to describe these commercial polymers rather than model systems which might conform better to the theoretical developments of the last section both because of the importance of these materials and because the theoretical concepts provide a framework for understanding more complex systems, even if they are not quantitatively successful. 

The holistic thermodynamic approach based on material (charge, concentration and electron) balances is a firm and valuable tool for a choice of the best a priori conditions of chemical analyses performed in electrolytic systems. Such an approach has been already presented in a series of papers issued in recent years, see [1-4] and references cited therein. In this communication, the approach will be exemplified with electrolytic systems, with special emphasis put on the complex systems where all particular types (acid-base, redox, complexation and precipitation) of chemical equilibria occur in parallel and/or sequentially. All attainable physicochemical knowledge can be involved in calculations and none simplifying assumptions are needed. All analytical prescriptions can be followed. The approach enables all possible (from thermodynamic viewpoint) reactions to be included and all effects resulting from activation barrier(s) and incomplete set of equilibrium data presumed can be tested. The problems involved are presented on some examples of analytical systems considered lately, concerning potentiometric titrations in complex titrand + titrant systems. All calculations were done with use of iterative computer programs MATLAB and DELPHI. 

Another common approach is to use an information-processing model to classify human errors. The classification models the information processing which occurs when a person operates and controls complex systems such as processing plants. One such classification (Rouse and Rouse, 1983) identifies six steps in information processing. Exhibit 6.1 lists the six steps, and provides some examples of errors that can occur at each of these steps. 

Thermodynamic data show that the stabilities of the caesium chloride-metal chloride complexes are greater than the conesponding sodium and potassium compounds, and tire fluorides form complexes more readily tlrair the chlorides, in the solid state. It would seem that tire stabilities of these compounds would transfer into tire liquid state. In fact, it has been possible to account for the heats of formation of molten salt mixtures by the assumption that molten complex salts contain complex as well as simple anions, so tlrat tire heat of formation of the liquid mixtures is tire mole fraction weighted product of the pure components and the complex. For example, in the CsCl-ZrCU system the heat of formation is given on each side of tire complex compound composition, the mole fraction of the compound... 

There are a variety of ways to express absolute QRA results. Absolute frequency results are estimates of the statistical likelihood of an accident occurring. Table 3 contains examples of typical statements of absolute frequency estimates. These estimates for complex system failures are usually synthesized using basic equipment failure and operator error data. Depending upon the availability, specificity, and quality of failure data, the estimates may have considerable statistical uncertainty (e.g., factors of 10 or more because of uncertainties in the input data alone). When reporting single-point estimates or best estimates of the expected frequency of rare events (i.e., events not expected to occur within the operating life of a plant), analysts sometimes provide a measure of the sensitivity of the results arising from data uncertainties. 

Time reversibility. Newton s equation is reversible in time. Eor a numerical simulation to retain this property it should be able to retrace its path back to the initial configuration (when the sign of the time step At is changed to —At). However, because of chaos (which is part of most complex systems), even modest numerical errors make this backtracking possible only for short periods of time. Any two classical trajectories that are initially very close will eventually exponentially diverge from one another. In the same way, any small perturbation, even the tiny error associated with finite precision on the computer, will cause the computer trajectories to diverge from each other and from the exact classical trajectory (for examples, see pp. 76-77 in Ref. 6). Nonetheless, for short periods of time a stable integration should exliibit temporal reversibility. 

This procedure constitutes an application of the steady-state approximation [also called the quasi-steady-state approximation, the Bodenstein approximation, or the stationary-state hypothesis]. It is a powerful method for the simplification of complicated rate equations, but because it is an approximation, it is not always valid. Sometimes the inapplicability of the steady-state approximation is easily detected for example, Eq. (3-143) predicts simple first-order behavior, and significant deviation from this behavior is evidence that the approximation cannot be applied. In more complex systems the validity of the steady-state approximation may be difficult to assess. Because it is an approximation in wide use, much critical attention has been directed to the steady-state hypothesis. 

A house is a complex system of interacting parts that contribute to overall performance including comfort, energy use, health, maintenance, and longevity. For example, a common air distribution system utilizes supply ducts running through the attic and return ducts tied directly to the air handler inside the home. If the ductwork is not properly sealed and there are combustion appliances in the home, this configuration can lead to health and fire hazards because the... 

Perhaps the quintessential example of a complex system is the human brain, which, consisting of something on the order of 10 neurons with lO -lO coniiec-... 

Exact computability in this sense, however, is achieved only at the cost of being able to obtain approximate solutions. Perturbation analysis, for example, is rendered virt ially meaningless in this context. It is not s irprising that traditional investigatory methodologies are not very well suited to studies of complex systems. Since the behavior of such models can generally be obtained only through explicit simulation, the computer becomes the one absolutely indispensable research tool. 

These patterns are an example of what are sometimes called dissipative structures, which arise in many complex systems. Dissipative structures are dynamical patterns that retain their organized state by persistently dissipating matter and energy into an otherwise thermodynamically open environment. 

Gases, fluids, crystals, and lasers are all examples of complex systems that are familiar to ns from physics. Chemical reactions, in which a large number of molecules conspire to produce new molecules, are also good examples. From biology, we have DNA molecules built up from amino acids, cells built from molecules, and organisms built from colls. 

It is one thing to describe as we have done informally above, even qualitatively, what a complex system is, and to conjure up myriad examples of complex systems. It is quite another to quantify the notion of complexity itself, to describe the relationship between complexity and information, and/or to understand the role that complexity plays in various physical and/or computational contexts. Each of these difficult problems in fact remains very much open. While we may find it easy enough to distinguish a complex object from a less complex object, it is far from trivial to furnish anything that goes beyond a vague characterization as to how we have done so. Some recent attempts at quantifying the notion of complexity are sketched below. 

The final comment is that algorithmic complexity is really a better measure of the degree of randomness in a system rather than its complexity. For example,... 

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