FIGURE 2. Problems 3 and 4
Problem 5. A model for rubber string is a one-dimensional chain of molecules, each with
length a. Molecules are joined at their ends in a way, that the next one can go left or right
with equal probability. If we denote $n_+$ the number of molecules going to the right and $n_-$
the number going to the left, they satisfy the relations
$n_+ + n_- = N$, $X = a(n_+ - n_-)$
where $N$ is their total number and $X$ is the length of the chain.
(a) show that the probability of having a given length $X$, denoted by $P(X)$, is given by
binomial distribution
(b) Using Sterling's formula, calculate the entropy of the chain $S(X) = k_B log(P(X))$
(c) Calculate the \"entropic force\"
$F = -k_B T \frac{\partial S(X)}{\partial X}$
for small chain $|x| \ll Na$ and near its maximal strenching $|x| \to Na$.
FIGURE 3. Sketch of the string for problem 5, with $N = 10$, $n_- = 2$, $n_+ =$
8, $X = 6a$.
