Most-probable distribution

In so doing, we obtain the condition of maximum probability (or, more properly, minimum probable prediction error) for the entire distribution of events, that is, the most probable distribution. The minimization condition [condition (3-4)] requires that the sum of squares of the differences between p and all of the values xi be simultaneously as small as possible. We cannot change the xi, which are experimental measurements, so the problem becomes one of selecting the value of p that best satisfies condition (3-4). It is reasonable to suppose that p, subject to the minimization condition, will be the arithmetic mean, x = )/ > provided that... 

Theoretical efforts a step beyond simply fitting standard statistical curves to fragment size distribution data have involved applications of geometric statistical concepts, i.e., the random partitioning of lines, areas, or volumes into the most probable distribution of sizes. The one-dimensional problem is reasonably straightforward and has been discussed by numerous authors... 

The most probable distribution of unbalance in the finally installed rotor, considering manufacturing tolerances, balancing residuals after low-speed balance, assembly tolerances, etc. 

This distribution is known as the Schultz-Flory or most probable distribution.2S The moments of the molecular weight distribution are ... 

A comment should be made on the dispersity of star polymers. If the arms each have a most probable distribution 2), dispersity of the star... 

There is thus assumed to be a one-to-one correspondence between the most probable distribution and the thermodynamic state. The equilibrium ensemble corresponding to any given thermodynamic state is then used to compute averages over the ensemble of other (not necessarily thermodynamic) properties of the systems represented in the ensemble. The first step in developing this theory is thus a suitable definition of the probability of a distribution in a collection of systems. In classical statistics we are familiar with the fact that the logarithm of the probability of a distribution w[n is — J(n) w n) In w n, and that the classical expression for entropy in the ensemble is20... 

To find the most probable distribution (largest W) we make an infinitesimal displacement in n, (with N held constant) and set the displacement equal to zero. That is... 

The mean of this most probable distribution is (obviously) / v, and the polydisper-sity (PD) is... 

The growing polymer chains have the most probable distribution defined by Equation (13.26). Typically, is large enough that PD 2 for the growing chains. It remains 2 when termination occurs by disproportionation. Example 13.5 shows that the polydispersity drops to 1.5 for termination by pure combination. The addition rules of Section 13.2.2 can be applied to determine 1.5 < PD < 2 for mixed-mode terminations, but disproportionation is the predominant form for commercial polymers. 

Various PIB architectures with aromatic finks are ideal model polymers for branching analysis, since they can be disassembled by selective link destmction (see Figure 7.7). For example, a monodisperse star would yield linear PIB arms of nearly equal MW, while polydisperse stars will yield linear arms with a polydispersity similar to the original star. Both a monodisperse and polydisperse randomly branched stmcture would yield linear PIB with the most-probable distribution of M jM = 2, provided the branches have the most-probable distribution. Indeed, this is what we found after selective link destruction of various DlBs with narrow and broad distribution. Recently we synthesized various PIB architectures for branching analysis. 

The viscosity average molecular weight depends on the nature of the intrinsic viscosity-molecular weight relationship in each particular case, as represented by the exponent a of the empirical relationship (52), or (55). However, it is not very sensitive to the value of a over the range of concern. For polymers having the most probable distribution to be discussed in the next chapter, it may be shown, for example, that... 

For a polymer possessing the most probable distribution, the mole fraction Na is given by Eq. (1) hence... 

The root-mean-square degree of polymerization for the most probable distribution is found to be... 

Distribution curves calculated for several values of / are shown in Fig. 56. Values of p have been adjusted to give the same number average (see Eq. 23), which also locates the maxima in the curves very nearly at the same abscissa value. The sharpening of the curves with increase in / is evident. The curve for /= 1, corresponding to the most probable distribution, is included for comparison. Even for /=2, which represents the linear polymer prepared by condensing... 

The portion of the polymer consisting of molecules terminated by transfer will conform to the most probable distribution, its average degree of polymerization being... 

Fig. 57.—Weight percent (100 w ) vs. number of units for polymers formed by successive addition of monomers to a fixed number of active centers, as calculated from Eq. (33) for the values of p indicated. The p = 500 curve is drawn to the scales along the upper and right-hand margins scales for the other curves are given along the lower and left-hand margins. The broken curve represents a most probable distribution, 5n = 101, shown for comparison.

At the gel point, (3 —l) = l/p, which with the foregoing expression gives Eq. (14), thus establishing equivalence of the two procedures. The primary molecules in a condensation polymer must almost invariably conform to a most probable distribution (see Chap. VIII). The random cross-linking of primary molecules otherwise distributed in size has no counterpart in polyfunctional condensation, therefore. 

RA4 molecules and random cross-linking of primary moleculeshaving the most probable distribution can be demonstrated also. In this case, however, it is desirable to define jy, in terms of the repeating unit... 

The velocity gradient leads to an altered distribution of configuration. This distortion is in opposition to the thermal motions of the segments, which cause the configuration of the coil to drift towards the most probable distribution, i.e. the equilibrium s configurational distribution. Rouse derivations confirm that the motions of the macromolecule can be divided into (N-l) different modes, each associated with a characteristic relaxation time, iR p. In this case, a generalised Maxwell model is obtained with a discrete relaxation time distribution. 

It might seem that the most probable distribution could be found by settihg the differential hW equal to zero. Or, as W and In W are maximum at the same point (if W 0), the differential of Eq. (27) can also be considered. The needed result is given by (see problem 3)... 

The most probable distribution of the four a electrons—the distribution that keeps them as far apart as possible—is at the vertices of a tetrahedron (Fig. 7a). The most probable arrangement of the four (3 electrons is also at the vertices of a tetrahedron (Fig. 7b). In a free atom these two tetrahedra are independent, so they can have any relative orientation giving, an overall spherical density. 

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