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) Prove that the entropy $S$ is given by $S = -k_B tr \rho \log \rho$ where $\rho$ is the equilibrium density matrix in the canonical ensemble. First perform the calculation in the classical canonical ensemble, then generalize it to the quantum case.

c) Using the fugacity expansion construct the classical hard-spheres equation of state. Consider hard spheres of radius $a$ with interparticle potential

$V(r)=\infty$ if $r<a$

$V(r)=0$ if $r \ge a$