A classical model of a rubber band

The rubber band is modeled as a one-dimensional polymer, that is, a chain of N monomers. Each monomer has a fixed length δ, and the monomers are connected end-to-end to form the polymer. Each monomer is completely free to rotate in two dimensions (there are no steric hinderances from the other monomers). A microscopic state of the rubber band is therefore described by the set of angles

describing the orientations of the monomers. [You can decide whether to define the angles with respect to the orientation of the previous monomer, or according to a fixed frame of reference. However, you must say which you've chosen!] The rubber band (polymer) is used to suspend a mass M above the floor. One end of the rubber band is attached to the ceiling, and the other end is attached to the mass, so that the mass is hanging above the floor by the rubber band. For simplicity, assume that the rubber band is weightless. You can ignore the mass of the monomers and their kinetic energy. The whole system is in thermal equilibrium at temperature T.

1. Calculate the energy of the system (polymer plus mass) for an arbitrary microscopic state θ of the monomers (ignoring the kinetic energy).

2. Find an expression for the canonical partition function in terms of a single, onedimensional integral.

3. Find an expression for the average energy U.

4. For very high temperatures, find the leading term in an expansion of the average energy U.