Probability generating functions

To generate the necessary distribution functions, the ratio of is used to approximate the true molecular weight distribution by a Schulz-Zimm distribution. It is also assumed that the reactive functional groups are distributed randomly on the polymer chain. The Schulz-Zimm parameters used to calculate distribution functions and probability generating functions (see below) are defined as follows ... 

The function r defines a probability generating function. Similar expressions can be written for P(Fg° ") and PsCPa )-... 

Equations 22 and 23 can be solved numerically using the method described in Ref. 5. For oligomers, the probability generating functions are calculated by the appropriate sums. For random copolymers analytical expressions for and t can be written for a polymer or crosslinker using the appropriate Schulz-Zimm parameters (5) ... 

For a random copolymer, the probability generating functions, and rp are given by the following (5) ... 

Writing the sums in terms of probability generating functions leads to the following expression ... 

If the mixture has gelled, the program proceeds to calculate P(Fa° ) and P(Fg° ) using a binary search method (lines 2510-2770). This method is more convenient that the earlier approach of Bauer and Budde (10) who used Newton s method, since derivatives of the functions are not required. The program also calculates the probability generating functions used to calculate sol fractions and the two crosslink densities (lines 2800-3150). Finally, the sol fraction and crosslink densities are calculated and printed out (lines 3160-3340). The program then asks for a new percents of reaction for the A and B groups. To quit enter a percent reaction for A of >100. 

The basis of model calculations for copolymerization, branching and cross-linking processes is the stochastic theory of Flory and Stockmayer (1-3). This classical method was generalized by Gordon and coworkers with the more powerful method of probability generating functions with cascade substitution for describing branching processes (4-6). With this method it is possible to treat much more complicated reactions and systems (7-9). 

In this study computational results are presented for a six-component, three-stage process of copolymerization and network formation, based on the stochastic theory of branching processes using probability generating functions and cascade substitutions (11,12). 

In the ideal case, all reactive groups have the same reactivity irrespective of the shape and size of the hyperbranched molecules and no rings are formed. Then, the distribution of units in different reaction states is expressed by the following probability generating function (pgf), F0n(Z, z) ... 

Probability Generating Function. For a discrete random variable, x, the function... 

Exercise. Let Xj be an infinite set of independent stochastic variables with identical distributions P(x) and characteristic function G(k). Let r be a random positive integer with distribution pr and probability generating function /(z). Then the sum 7 = Xl+X2 + +Xr is a random variable show that its characteristic function is f G k)). [This distribution of 7 is called a compound distribution in feller i, ch. XII.]... 

Thus the variance is always greater than that of the pure Poisson distribution with the same average. Also express the probability generating function of pn in the characteristic function of 0(a) and conclude that the moments of a are equal to the factorial moments of n compare (1.2.15). 

This is an equation for F(z, t 1,0) alone, which determines the probability generating function and hence the distribution, once y(i) is known. The treatment of the branching process is thereby reduced to solving a nonlinear integral equation. Unfortunately this can only be done explicitly for very few choices of y(r). 

There are many ways of solving the differential-difference equation (2.1) we choose a method that can also be used in more general cases. The essential tool is the probability generating function F(z, t) defined in 1.2 ... 

Exercise. Establish the relation between the probability generating function and the characteristic function. Derive the identities (2.4) and (2.5) from the known properties of the latter. 

The theory of cascade processes uses the formalism of probability generating functions (p.g.f.)... 

The classical approach to crosslinking statistics, as especially developed by Flory and Stockmayer, is capable of treating the simpler type of network formation processes, but becomes mathematically intractable in the more complicated cases (unequal reactivities, cyclization). Of the various recently proposed alternatives, the link probability generating function approach of Gordon et al., which is based upon the theory of... 

Since probability-generating functions are not often used in polymer science, we first give a brief outline of the properties before applying them to the problems of branched polymers. 

Since there are three different functional groups, the corresponding probability-generating functions for the reaction of these groups will also be different. These are easily found with the reactivity matrix A given by Eq. (C.34)... 

The factor Ci = C2 and C3 introduce the influence of the neighboured functionalities but we wish to make sure that a reaction of only two functional groups is not inhibited. One verifies from a substitution F0 (Ft (s)) that this is indeed achieved by the special choice of the factors and c3 in Eq. (E.50). The three constants Cq, c2 and c3 have to be determined such that they fulfil the normalization condition for the probability-generating functions. This means a solution of the three coupled equations... 

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