Statistical distribution of the contour length

1 Statistical distribution of the contour length In the previous sections we regarded the primitive chain as an inexten-sible string of contour length L. In reality, the contour length of the primitive chain fluctuates with time, and the fluctuation sometimes plays an important role in various dynamical processes. 

First we consider the statistical distribution of the contour length. Since the primitive chain represents a set of conformations of the Rouse chain, the probability that a certain conformation of the primitive chain is realized is proportional to o), the number of the coiflormations of the Rouse chain which are represented by that primitive chain. The simplest hypothesis is to take the polymer as a random walk confined within a tube. Then a is calculated by the method described in Section 2.3.2 (see Appendix 6.1). The result is 

At first sight eqn (6.68) may seem to imply a paradoxical result since the entropy increases with decreasing L, the chain will contract to the state of L = 0 if its ends are not fixed. This argument is wrong. The collapse happens if the chain is confined in an infinitely long tube of given conformation, but does not happen in the case of a network, where the Rouse chain explores many tubes. To calculate the statistical distribution of L in the network, one has to take into account the multiplicity of the state specified by L. Let Q(L) be the number of primitive paths which have length L, then the probability that a primitive chain has a contour 

Q(L) can be estimated by the number of random walks consisting of Lloo 

This shows explicitly that the step length a of the primitive chain is of the same order as the tube diameter Oq. The average of the fluctuation is calculated from eqn (6.71) as 

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