Homework Set # 6 - Due October 4

1. Consider a system of N non-interacting particles. Each particle can be in one of the following 3 energy states: E_0=0, E_1=ε, and E_2=2ε. If the number of particles N in the system is such that 3-δ<N<3+δ.

a) How many states the system has and what are the energies of those states?

b) If we know that the energy E of the system is in the range 2ε-δ<E< 2ε+δ provide the probability P(ε) in the microcanonical ensemble and make a plot showing P(ε) vs ε.

c) Now provide the probability P(ε) in the canonical ensemble and make a plot showing P(ε) vs ε. Hint: Write the partition function Z and find β.

d) Now provide the probability ℙ(ε) in the grand-canonical ensemble and make a plot showing ℙ(ε,N=3) vs ε and ℙ(E=2ε,N) vs N. Hint: Write the grand-partition function and find μ; once ℙ(E,N_m)<0.001 you can assume that ℙ(E,N)=0.

2. Consider a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain. Each molecule can align along either its long axis (of length 2a) or short axis (length a). The energy of the monomer aligned along its shorter axis is higher by ε. That is, the total energy E=εU, where U is the number of monomers standing up.

a) Calculate the partition function, Z, of the polymer.

b) Find the relative probability for the monomer to be aligned along its short or long axis.

c) Calculate the average length <L(T,N)> of the polymer.

d) Obtained the average energy <E> of the polymer as a function of T and N.

e) Obtain U in terms of T and N.

3. Consider a monoatomic crystal with M atoms that can be placed in two different kind of positions: 1) normal, i.e., at the sites of a squared lattice and 2) interstitial, slightly displaced from the normal site. Assume that the crystal contains M normal sites and M interstitial sites. The atoms in the normal sites have energy e_0 while the atoms in the interstitial sites have energy e_1 with e_1>e_0. Assume that each site can be occupied by either 0 or 1 atoms.

a) Write the grand-canonical partition function for one single cell of this system, i.e., a cell composed by one normal and one interstitial site.

b) Now assume that the average number of atoms per cell is <N>=1 and obtain the chemical potential.

c) Now find the average population <n> of the interstitial site for one cell. Consider the limits e_1-e_0>>kT and e_1-e_0<<kT.

d) Assuming weak interactions find the average population <n> for the interstitial sites for the crystal formed by M cells in the limit e_1-e_0>>kT.

4. Problem 3.31

5. Problem 3.41 (Hint: Give a qualitative answer and set-up an equation for T_F in terms of T_S, T_G, N_S, and N_G, where the subscripts S, and G, indicate spins and gas and F stands for the final (equilibrium) temperature achieved by both systems.