Functional Integration Flexible Polymer Chains

Although the Wiener integral formulation for the distribution functions of flexible polymer chains rests upon general considerations of random walks and Brownian motion, it is easily introduced, heuristically, through the concept of an equivalent chain. In this section, only those flexible polymer chains are considered which are composed of equivalent gaussian links. Here L is the maximum contour length of the real chain at full extension, and (R ) for the equivalent chain is taken to be that for the real chain. Thus we have... 

The representation of a flexible polymer chain by an equivalent random flight model chain which is subsequently taken to be a continuous space curve leads directly to the introduction of a functional integral representation of polymer chains. 

Therefore, in an attempt to obtain simple analytic expressions for the distribution functions of stiff polymer chains, condition (5.2a) is relaxed. The relaxation of this condition is in the original spirit of the use of Wiener integrals. If this condition were imposed for flexible polymer chains, the Wiener measure would be 2[t s)] exp (—3L/2/) and would give equal weight (measure) to all continuous configurations of the polymer. Thus the use of (5.2a) would not yield the correct gaussian distribution for flexible chains. 

The techniques of functional integration are naturally introduced into the study of the statistical mechanics of polymer systems because they are ideally suited to the description of simple generic models of flexible and stiff ideal polymer chains. Thus we discuss ideal polymer chains to introduce the simple generic models which then naturally lead to the use of functional integrals. 

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