Independent variables theory approach

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

Extended nonequilibrium thermodynamics theory is often applied to flowing polymer solutions. This theory includes relevant fluxes and additional independent variables in describing the flowing polymer solutions. Other contemporary thermodynamic approaches for this problem are GENERIC formalism, matrix method, and internal variables (Jou and Casas-Vazquez, 2001), which are summarized in the following sections. 

However, this approach is too simple, for the following reason. If the experimental quantities x and y are measured independently, in the sense that the corresponding uncertainties do not depend on one another and are not correlated with one another in any sense, then the proposed upper limit for the probable range for W, i.e., (x y) (SEx + SEy), assumes that simultaneous observation of the highest probable values for x and y will occur with a probability equal to that of cases of partial cancellation of positive deviations from the mean value in one and negative deviations in the other (similarly for the lower limit). This assumption is simply not vahd and, when the possibility of mutual cancellation of errors is taken into account within the theory of the normal distribution (Section 8.2.3), the appropriate formula for combining uncertainties in simple sums and/or differences of several independently variable measured quantities, can be shown to be ... 

Changing the distance between the critical points requires a new variable (in addition to the three independent fractional concentrations of the four-component system). As illustrated by Figure 5, the addition of a fourth thermodynamic dimension makes it possible for the two critical end points to approach each other, until they occur at the same point. As the distance between the critical end points decreases and the height of the stack of tietriangles becomes smaller and smaller, the tietriangles also shrink. The distance between the critical end points (see Fig. 5) and the size of the tietriangles depend on the distance from the tricritical point. These dependencies also are described scaling theory equations, as are physical properties such as iuterfacial... 

Rates of addition to carbonyls (or expulsion to regenerate a carbonyl) can be estimated by appropriate forms of Marcus Theory. " These reactions are often subject to general acid/base catalysis, so that it is commonly necessary to use Multidimensional Marcus Theory (MMT) - to allow for the variable importance of different proton transfer modes. This approach treats a concerted reaction as the result of several orthogonal processes, each of which has its own reaction coordinate and its own intrinsic barrier independent of the other coordinates. If an intrinsic barrier for the simple addition process is available then this is a satisfactory procedure. Intrinsic barriers are generally insensitive to the reactivity of the species, although for very reactive carbonyl compounds one finds that the intrinsic barrier becomes variable. ... 

In this section we show how the general form of Renner-Teller interaction matrices can be obtained at any order in the phonon variables and with electron orbital functions of different symmetry (p-like, < like, /-like, etc.). For this purpose, we use an intuitive approach [18] based on the Slater-Koster [19] technique and its generalization [20] to express crystal field or two-center integrals in terms of independent parameters in the tight-binding band theory [21] then we apply standard series developments in terms of normal coordinates. 

Once the mathematical formalism of theoretical matrix mechanics had been established, all players who contributed to its development, continued their collaboration, under the leadership of Niels Bohr in Copenhagen, to unravel the physical implications of the mathematical theory. This endeavour gained urgent impetus when an independent solution to the mechanics of quantum systems, based on a wave model, was published soon after by Erwin Schrodinger. A real dilemma was created when Schrodinger demonstrated the equivalence of the two approaches when defined as eigenvalue problems, despite the different philosophies which guided the development of the respective theories. The treasured assumption of matrix mechanics that only experimentally measurable observables should feature as variables of the theory clearly disqualified the unobservable wave function, which appears at the heart of wave mechanics. 

Recently, an entropy-based approach has been introduced to compare the intrinsic and extrinsic variability of different descriptors, independent of their units and value ranges. The method was originally introduced in communication theory and is based on Shannon entropy, which calculates descriptor-entropy values using histogram representations. Shannon entropy is defined as ... 

Phenomenological approaches have been very successful in some areas, e.g. Miedema theory for predicting many quantities in metallurgy. The essential task in phenomenological theories is to identify a suitable set of physically meaningful variables, which are linearly independent , to characterize the materials. Experimental data are then correlated against this set of variables and functional relations are fitted. 

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