Rubber Physics (Part 4)

Instead of the distance vector $\vec r$ consider the number $n = \frac{\vec r \cdot \vec e_x}{l} = \frac{1}{l} \sum_{i = 1}^N \vec l_i \cdot \vec e_x = \sum_{i=1}^{N} s_i \in \mathbb{Z}$ with the sign $s_i \in \{+1,-1\}$ of the orientation vector $\vec l_i = s_i l \vec e_x$ . This number can be expressed by $n = N_+ - N_- = 2N_+ - N$ with the number of positive vectors, $N_+$ , and the number of negative vectors, $N_- = N - N_+$ . The probability that there are exactly $N_+$ positive orientated chain links correspond to a binomial distribution: $P_N (N_+) = \frac{N!}{N_+! (N - N_+)!} p_+^{N_+} p_-^{N - N_+} = \left(\begin{array}{c} N \\ N_+ \end{array}\right) p_+^{N_+} p_-^{N - N_+}$ with probabilites $p_+$ and $p_- =1 - p_+$ for the cases $s_i = +1$ and $s_i = -1$ , respectively. Here, both probabilities are the same, so that $p_+ = p_- = \frac{1}{2}$ . The mean value of the quatratic distance results $\langle \vec r^2 \rangle = \langle n^2 \rangle l^2 = (4 \langle N_+^2 \rangle - 4 \langle N_+ \rangle N + N^2) l^2$ These mean values $\langle N_+ \rangle$ and $\langle N_+^2\rangle$ can be estimated as follows $\begin{aligned} \langle N_+ \rangle &= \sum_{N_+ = 0}^N N_+ P_N(N_+) \\ &= \sum_{N_+ = 0}^N N_+ \frac{N!}{N_+! (N - N_+)!} p_+^{N_+} p_-^{N - N_+} \\ &= \sum_{N_+ = 1}^N \frac{N!}{(N_+-1)! (N - N_+)!} p_+^{N_+} p_-^{N - N_+} \\ &= \sum_{N_+ = 1}^N N p_+ \cdot \frac{(N- 1)!}{(N_+-1)! ((N -1) - (N_+-1))!} p_+^{N_+-1} p_-^{(N-1) - (N_+-1)}\\ &= N p_+ \sum_{N_+' = 0}^{N-1} \frac{(N- 1)!}{N_+'! ((N -1) - N_+')!} p_+^{N_+'} p_-^{(N-1) - N_+'}\\ &= N p_+ \sum_{N_+' = 0}^{N-1} P_{N-1}(N_+') \\ &= N p_+\\ \langle N_+^2 \rangle &= \sum_{N_+ = 0}^N N_+^2 P_N(N_+)\\ &= \sum_{N_+ = 0}^N N_+(N_+ - 1) P_N(N_+) + \sum_{N_+ = 0}^N N_+ P_N(N_+) \\ &= \sum_{N_+ = 0}^N N_+(N_+ -1)\frac{N!}{N_+! (N - N_+)!} p_+^{N_+} p_-^{N - N_+} + \langle N_+ \rangle \\ &= \sum_{N_+ = 2}^N \frac{N!}{(N_+ - 2)! (N - N_+)!} p_+^{N_+} p_-^{N - N_+} + N p_+ \\ &= \sum_{N_+ = 2}^N N(N-1) p_+^2 \frac{(N - 2)!}{(N_+ - 2)! ((N - 2) - (N_+ - 2))!} p_+^{N_+ -2} p_-^{(N-2) -(N_+ - 2)} + N p_+ \\ &= N(N-1) p_+^2 \sum_{N_+' = 0}^{N-2} \frac{(N - 2)!}{N_+'! ((N - 2) - N_+')!} p_+^{N_+'} p_-^{(N-2) -N_+'} + N p_+ \\ &= N(N-1) p_+^2 \sum_{N_+' = 0}^{N-2} P_{N-2}(N_+') + N p_+ \\ &= N(N-1) p_+^2 + N p_+ \end{aligned}$ Therefore, we get the final result $\begin{aligned} \langle \vec r^2 \rangle &= (4 (N(N-1) p_+^2 + N p_+) - 4 N^2 p_+ + N^2) l^2 \\ &= (4 N(N-1) (p_+^2 - p_+) + N^2) l^2\\ &= \left(4 N(N-1) \left(\frac{1}{4} - \frac{1}{2} \right) + N^2 \right) l^2 \\ &= \left(- N(N-1) + N^2 \right) l^2 = N l^2 \end{aligned}$ $\Rightarrow \quad \boxed{r_m = \sqrt{\langle \vec r^2 \rangle } = \sqrt{N} \cdot l = 100 \cdot l }$