Some Stochastic Processes in Polymer Systems

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Abstract
Certain distributional problems involving polymer configurations can be treated as special classes of stochastic processes known as regenerative processes. These processes have the property that the interval (of time or of length) between two consecutive events is a random variable. The method of regeneration point is applied to the problem of random crystallization of polymers and to the problem of force—length relationships in a one‐dimensional simulation of a polymer network. By assuming that the events of randomly placing crystalline units on a polymer chain are statistically independent, and that the probability of the first event occurring in a given interval is a simple step function, the direct application of the method of regeneration point leads to a well‐known equation from the theory of molecular distribution in one‐dimensional hard‐core fluids. In the second problem it is assumed that a single polymer chain consists of mesh points connected by flexible chains. A restriction is imposed that these mesh points cannot pass through each other. The molecular distribution functions for these mesh points are derived with the help of the regeneration‐point process. By applying this method, the relationships of the network extension to the fixed force are derived. It is also found that the affine transformation rule for the force‐biased distribution of chain lengths holds strictly only if the array represents a Poisson distribution of mesh points.

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