Homework Set # 6 - Due October 6

1. Problem 7.14

2. Problem 7.19

3. Problem 7.20

4. Problem 7.21

5. 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.

6. 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.