Linear superposition concept

Having represented the individual rich streams, we are now in a position to construct the rich composite stream. A rich composite stream represents the cumulative mass of the pollutant lost by all the rich streams. It can be readily obtained by using the diagonal nile for superposition to add up mass in the overlapped regions of streams. Hence, the rich composite stream is obtained by applying linear superposition to all the rich streams. Figure 3.4 illustrates this concept for two rich streams. 

Separability between electronic and nuclear states is fundamental to get a description in terms of a hierarchy of electronic and subsidiary nuclear quantum numbers. Physical quantum states, i.e. wavefiinctions 0(q,Q), are non-separable. On the contrary, there is a special base set of functions Pjt(q,Q) that can be separable in a well defined mode, and used to represent quantum states as linear superpositions over the base of separable molecular states. For the electronic part, the symmetric group offers a way to assign quantum numbers in terms of irreducible representations [17]. Space base functions can hence be either symmetric or anti-symmetric to odd label permutations. The spin part can be treated in a similar fashion [17]. The concept of molecular species can be introduced this is done at a later stage [10]. Molecular states and molecular species are not the same things. The latter belong to classical chemistry, the former are base function in molecular Hilbert space. 

No one of the equations introduced here are defined as in the standard Bom-Oppenheimer approach. The reason is that electronic base functions that depend parametrically on the geometry of the sources of external potential are not used. The concept of a quantum state with parametric dependence is different. This latter is a linear superposition the other are objects gathered in column vectors. 

In the early view, there are correspondence rules relating the primitive concept of state and observable to empirical reality. Observables are mapped on to the set of eigenvalues of a particular class of self-adjoint operators (e.g., Hamiltonians). The individual systems would occupy only one base state the amplitude appearing in the linear superposition in square modulus represents the probability to find one system occupying a base state when scanning the ensemble. 

Fortunately enough, there is a theory justifying this idea — quantum mechanics. Actually, in the full quantum-mechanical treatment, the one-dimensional particle K might be allowed the linear superpositions of the different positions, thus making the concept of the position (and consequently of trajectory ) physically meaningless. ... 

FIGURE 4.12 Schematic concept of the linear superposition of a series of thin-film point source solutions to model the transient diffusion of arbitrary concentration profiles. 

The Stress-Rang e Concept. The solution of the problem of the rigid system is based on the linear relationship between stress and strain. This relationship allows the superposition of the effects of many iadividual forces and moments. If the relationship between stress and strain is nonlinear, an elementary problem, such as a siagle-plane two-member system, can be solved but only with considerable difficulty. Most practical piping systems do, ia fact, have stresses that are initially ia the nonlinear range. Using linear analysis ia an apparendy nonlinear problem is justified by the stress-range concept... 

The concept of superposition finds important application in linear problems. However, rather than its casual use for some appreciation as done in this example, it requires elaboration beyond the scope of this text. 

First, we need a rule to predict the effect of time-varying loads on a viscoelastic model. When a combination of loads is applied to an elastic material, the stress (and strain) components caused by each load in turn can be added. This addition concept is extended to linear viscoelastic materials. The Boltzmann superposition principle states that if a creep stress ai is... 

If linear, a sine excitation input results in a sine response. However, the immittance concept can be extended to nonlinear networks, where a sine wave excitation leads to a nonsinusoidal response. Including a separate immittance value for each harmonic component of the response performs the necessary extension. In the linear region, the principle of superposition is valid. This means, for example, that the presence of strong harmonics in the applied current or voltage would not affect immittance determination at the fundamental frequency or a harmonic (Schwan, 1963). Some lock-in amplifiers can measure harmonic components, making it possible to analyze nonlinear phenomena and extend measurement to nonsinusoidal responses. 

The lowest frequency occurs when n = I and is called the fundamental. Doubling the frequency corresponds to raising the pitch by an octave. Those solutions having values of n > I are known as the overtones. As mentioned previously, one important property of waves is the concept of superposition. Mathematically, it can be shown that any periodic function that is subject to the same boundary conditions can be represented by some linear combination of the fundamental and its overtone frequencies, as shown in Figure 3.8. In fact, this type of mathematical analysis is known as a Fourier series. Thus, while the note middle-A on a clarinet, violin, and piano all have the same fundamental frequency of 440 Hz, the sound (or timbre) that the different instruments produce will be distinct, as shown in Figure 3.9. 

The Schapery Model. One of the earliest models of the nonlinear viscoelastic response of pol5nners to use the concept of a reduced time is due to Schapery (147-149). The model is based on thermodynamic considerations and has a form similar to the Boltzmann superposition principal described previously. The model time dependences, except for the shift factors, are the same as those obtained in the linear response regime. Hence, the model is relatively easy to implement and to determine the relevant material parameters. It results in a generalization of the generalized superposition principal developed by Leaderman (150). 

This concept could be extended to any other linear and nonlinear QSAR relationships, by calculating either n x n distance matrices D (especially suited for nonlinear relationships) or n X n covariance matrices C as similarity measures. For this purpose, all or only several relevant properties of the compounds are used to calculate the corresponding similarity matrices. No superposition of the molecules is necessary. If a distance matrix D is calculated from the X matrix of explanatory physicochemical properties n rows, m columns), then all Xij values must be normalized before, i.e., mean-value-centered and standardized, column by column. The great advantage of distance similarity index matrices is that no special models need to be defined in the case of nonlinear relationships on the other hand, problems may arise from significant intercorrelations between the different columns of the similarity matrices. 

Time-temperature superposition is of interest in two contexts. For the experimentalist it is the basis of a technique for substantially increasing the range of times or frequencies over which linear behavior can be determined. And for the polymer scientist, it may provide additional information about molecular structure. It was Ferry [1] who first provided a scientific basis for this procedure. The essence of the concept is that if all the relaxation phenomena involved in G t) have the same temperature dependency, then changing the temperature of a measurement will have the same effect on the data as shifting the data horizontally on the log(time) or log(frequency) axis. Let us say that a change in the temperature from a reference value Tg to a different temperature T has the following effect on all the relaxation times ... 

In Chapter 4, it was noted that linear viscoelastic behavior is observed only in deformations that are very small or very slow. The response of a polymer to large, rapid deformations is nonlinear, which means that the stress depends on the magnitude, the rate and the kinematics of the deformation. Thus, the Boltzmann superposition principle is no longer valid, and nonlinear viscoelastic behavior cannot be predicted from linear properties. There exists no general model, i.e., no universal constitutive equation or rheological equation of state that describes all nonlinear behavior. The constitutive equations that have been developed are of two basic types empirical continuum models, and those based on a molecular theory. We will briefly describe several examples of each type in this chapter, but since our primary objective is to relate rheological behavior to molecular structure, we will be most interested in models based on molecular phenomena. The most successful molecular models to date are those based on the concept of a molecule in a tube, which was introduced in Chapter 6. We therefore begin this chapter with a brief exposition of how nonlinear phenomena are represented in tube models. A much more complete discussion of these models will be provided in Chapter 11. 

The fundamental concept of the material clock or reduced time is similar to the principle described above in the discussion of time-temperature superposition. In the mechanical constitutive models, however, the change in the stress or deformation induces a shift in the material relaxation time. The fact that the time depends on the state of stress (or strain) or on its history leads to additional non-linearities in behavior from what is expected with, eg, the K-BKZ model. Physical explanations for the shifting material time are often based on free-volume ideas that are often invoked to explain time-temperature superposition. In addition, entropy changes have been invoked as have stress-activated processes. 

In order to predict the creep behavior and possibly the ensuing failure a number of approaches have been proposed. These are based respectively on the theory of viscoelasticity — including the concept of free volume — or on empirical representations of e(t) or of the creep modulus E(t) = ao/e(t). The framework of the linear theory of viscoelasticity permits the calculation of viscoelastic moduli from relaxation time spectra and their inter conversion. The reduction of stresses and time periods according to the time-temperature superposition principle frequently allows establishment of master-curves and thus the extrapolation to large values of t (cf. Chapter 2). The strain levels presently utilized in load bearing polymers, however, are generally in the non-linear range of viscoelasticity. This restricts the use of otherwise known relaxation time spectra or viscoelastic moduli in the derivation of e (t) or E (t). 

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