a) A polymer can be constructed as a three dimensional random walk where the position of the n+1-th monomer is given by

$\vec{r}_{n+1}=\vec{r}_{n}+a\hat{u}_{n+1}$

where $a$ is the monomer spacing and $\hat{u}$ is a random unit vector. The length of the polymer made by $N+1$ monomers can be estimated by

$\ell= \sqrt{\langle |\vec{r}_{N}-\vec{r}_{0}|^2 \rangle}}$

where the average is taken over the random orientation of the unit vectors $\hat{u}_n$.

b) Define the isobaric ensemble that associated with the natural variables $P$,$T$,$N$.

c) Consider the the Hamiltonian of exercise #6 a) in the grandcanonical ensemble.