Path weight generating functions

We have already mentioned that a generating function contains the same information as the corresponding probability distribution, and the distribution can be obtained by special expansion techniques from the p.g.f25,27-291 84,132-1345 The same technique of expansion can also be applied to the path-weight generating function and yields the properties of molecularly monodisperse fractions of the branched polymer. We will give an example later in Chap. D.I. (See also Appendix). 

Since in the present case s is a vector, the functions U (s) must also be vectors. Hence, the path-weight generating functions are... 

The cascade substitution for the construction of the path-weight generating function is achieved by... 

The path-weight generating function is obtained in the same way as in the case of the ABf polycondensates. 

For the average of this path-weight generating function, one arrives after differentiation at... 

The formalism of construction a path-weight generating function appears at first sight the same as for the random polycondensates, with the essential difference, however, that s and U in the equations for a polycondensate with functionality y for the monomers... 

The rest of the work consists in the cascade substitution for the construction of a path-weight generating function which is done as usual. Gordon and Parker have focussed their interest on the position of the gel point which became shifted from ac = 0.5000, in the random case, to ac = 0.5733, for the steric hindrance case. It will certainly be of interest to investigate whether... 

The Lagrange expansion technique can also be applied to the calculation of the particle-scattering factors Px (q) of branched or linear polymers of DP = x from the path-weight generating function of the polydisperse system. In Chap. C.I1I we have shown the equivalence... 

For Newtonian dynamics and a canonical distributions of initial conditions one can reject or accept the new path before even generating the trajectory. This can be done because Newtonian dynamics conserves the energy and the canonical phase-space distribution is a function of the energy only. Therefore, the ratio plz ]/p z at time 0 is equal to the ratio p[.tj,n ]/p z ° at the shooting time and the new trajectory needs to be calculated only if accepted. For a microcanonical distribution of initial conditions all phase-space points on the energy shell have the same weight and therefore all new pathways are accepted. The same is true for Langevin dynamics with a canonical distribution of initial conditions. 

In other words, each generation is weighted after differentiation with a function that depends on the path length. One easily verifies Eq. (C.84) for the random polycondensates when a - Gaussian statistics) is assumed. 

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