Representation of the Distribution Functions

Distribution functions can be classified as discontinuous or continuous. Discontinuous distribution functions are subdivided into frequency distributions and cumulative distributions. Continuous distribution functions are further classified as differential and integral distribution functions. 

The cumulative distributions give the summations over all statistical weights. They give the probability of finding the property E for values less than Ei. Typical cumulative distributions are then 

Discontinuous distribution functions can, of course, be transformed into continuous distribution functions when the difference between two neighboring properties is very small compared with the whole range of property values. Frequency distributions convert to the corresponding differential distributions and cumulative distributions convert to integral distributions, 

In the German scientific tradition, only molar distributions are described as frequency distributions. This terminology can lead to confusion since weight andz distributions can also be given in terms of frequency distributions. 

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Figure 2.14 gives a schematic representation of the distribution function of the free volume fraction F(f) vs. / for blends of various degrees of compatibility. When the components are mixed to a heterogeneity limit of 10-20 A, the curve of F f) will approach the curve for a molecularly mixed system. 

Figure 2.14. Schematic representation of the distribution function of free volume fraction F f) for various blend systems. Curves 1 and 2 show the shapes of the functions of components 1 and 2, respectively. Curves S., P.M., M.H., and P.F.M. indicate two separated phases, partially miscible, microheterogeneous, and perfectly miscible systems, respectively. The volume fraction of each component is 50% for all these blends. The broken line shows the F(f) of the components in the perfectly miscible system. (Manabe et al, 1971.)...

Note that both models yield satisfactory results on this point. However, it is important to apply the comparison to several types of results. For example. Figure 1.11 shows that, for the representation of the distribution function, Lennard-Jones and Devonshire s model, Eyring s model and the calculations performed by numerical simulation are very similar. Meanwhile, Figure 1.12, which gives the variation of the compressibility coefficient as a function of a reduced volume, illustrates the significant behavioral difference between the molecular dynamics simulation and Eyring s model, on the one hand, and Lennard-Jones/Devonshire s, Guggenheim s (see section 1.3.1)... 

Representation of the distribution function tis points on the surface of a sphere. 

We turn now to the orientational correlations which are of particular relevance for liquid crystals that is involving the orientations of the molecules with each other, with the vector joining them and with the director [17, 28]. In principal they can be characterised by a pair distribution function but in view of the large number of orientational coordinates the representation of the multi-dimensional distribution can be rather difficult. An alternative is to use distance dependent orientational correlation coefficients which are related to the coefficients in an expansion of the distribution function in an appropriate basis set [17, 28]. 

The chains of typical networks are of sufficient length and flexibility to justify representation of the distribution of their end-to-end lengths by the most tractable of all distribution functions, the Gaussian. This facet of the problem being so summarily dealt with, the burden of rubber elasticity theory centers on the connections between the end-to-end lengths of the chains comprising the network and the macroscopic strain. 

The Fourier representation provides us with a particular realization of this decomposition The vacuum Vp is realized by the Fourier component Po,o,...,o(7) of the distribution function. The abstract decomposition (4) can, however, also be realized in more general ways. Equation (7) expresses the fact that in a system without interactions, one cannot create correlations out of the vacuum. 

Additionally, from Fig. 29 one sees that, if, as proposed by Frost 42), a spherical gaussian function is a fair representation of the distribution of charge within an electride ion, there should he, as found by Slater 97>, a very good correlation, and in many cases practically an equality, between the atomic radii. . . and the calculated radius of maximum radial charge density in the outermost shell of the atom". 

In general case Eqs. (4.60) and (4.61) present infinite sets of the five-term (pentadiagonal) recurrence relations with respect to the index l. In certain special cases (t - 0 or a - 0), they reduce to three-term (tridiagonal) recurrence relations. In this section the sweep procedure for solving such relations is described. This method, also known as the Thomas algorithm, is widely used for recurrence relations entailed by the finite-difference approximation in the solution of differential equations (e.g., see Ref. 61). In our case, however, the recurrence relation follows from the exact expansion (4.60) of the distribution function in the basis of orthogonal spherical functions and free of any seal of proximity, inherent to finite-difference method. Moreover, in our case, as explained below, the sweep method provides the numerical representation of the exact solution of the recurrence relations. 

Note, that in given representations (4.114) and (4.116) of either of the distribution functions, the V ) parameter is a reference value and does not have a direct meaning of the mean volume. The latter is formed as a particular combination—[see Eqs. (4.115) and (4.116) of corresponding Vo and the distribution width. 

Numerical solution of Eqs. (4.263) and (4.264) yields a representation for the distribution function (4.260) accurate up to the terms of the order J 0. Using it to evaluate the reduced magnetization (x) and taking into account expansion (4.250), one can present the magnetic response as a sum of the frequency-dependent contributions as... 

The variational results are at odds with the phase diagram for the cuprates. Are they a good representation of the physics of the t-J model What does the variational parameter A0 really represent In a model with strong local interactions U t, there must be a transfer of the distribution function n(k)... 

The SCF wave functions we have used to calculate V (r) are written in terms of canonical one-electron orbitals Canonical form is not able to give a simple and evident visualization of a single bond or chemical group. A better representation of the wave function for this purpose is in terms of localized orbitals (LO s), which give a chemically more expressive picture of the electron distribution. It is well known that a one-determinant wave function, written in terms of canonical orbitals localized orbitals A. It is merely necessary to perform a suitable unitary transformation on the set

[Pg.143]

Here the lowest vibrational level is chosen as , = 0. The choice of —Sill as lower limit of integration in equation (1.70) corresponds to the representation of the step function for n,( i) in a quantized oscillator by a continuous function. This function distributes the increase of ni(E,) at mSf smoothly over the energy interval [we, + ejl, mSi — e,/2], see Fig. 1.20 m integer). 

Fig. 24. Schematic representation of radial distribution functions for crystalline (a) and amorphous materials (b). The broken line represents the radial distribution function Air r p for an ideal gas. The reduced radial distribution function G(r) = Air r p(r)-Pol for an amorphous solid is shown in part c.

Thus far we have the functional integral representation for the distribution functions involved in the configurational statistics of flexible polymers. By considering the polymer in the presence of an external field, we are able to relate the configurations of a polymer to the paths of a particle when this particle is undergoing Brownian or diffusive motion, or when it is evolving according to the laws of quantum mechanics. This establishes connections with other familiar concepts. 

In this section we consider functional Integral representations for the distribution functions for stiff polymer chains. Aside from providing a class of useful models of stiff polymer chains, these results illustrate the following ... 

To obtain quantitative representation of fractionation, a model for the thermodynamic properties of the copolymer + solvent + nonsolvent system and the original two-dimensional distribution function are required. Ratzsch et al. [46] presented the application of continuous thermodynamics to successive homopolymer fractionation procedures based on solubility differences. This method is now applied to copolymer fractionation. The liquid-liquid equilibria (LLE) of polymer solutions forms the thermodynamic background for these procedures. The introduction of the precipitation rate (23) permits calculation of the distribution functions in the sol and gel phases of every fractionation step, i, according to ... 

In the energy representation, the interaction energy between the solute and solvent is adopted for the one-dimensional coordinate of the distribution functions, and a functional for the solvation free energy is constructed from energy distribution functions in the solution and reference-solvent systems of interest. The introduction of the energy coordinate for distribution functions is a kind of coarse-graining procedure for reducing the information content of solute-solvent... 

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