G25.2651: Statistical Mechanics

Problem set #3

Due: March 3, 2003

1.

Calculate the volume fluctuations given by

in the isothermal-isobaric ensemble. Express your answer in terms of the isothermal compressibility given by

Show that and hence vanish in the thermodynamic limit.

2.

onsider an ideal gas of N particles at temperature T inside a cylinder with radius a and a vertical extent L. The cylinder rotates along its axis (taken as the z-axis) with angular velocity . In addition, a uniform gravitational field g acts vertically in the negative z direction. In this case, the Hamiltonian for a single particle is given by

where is the z component of the cross product.

a.

Calculate the canonical partition function.

b.

Suppose that the cylinder is equipped with a movable piston that allows only the length, L, to change in such a way as to maintain a constant pressure. What is the partition function in this case?

Hint: It may help you to know that the binomial expansion is

c.

Imagine, once again, the case of fixed cylinder volume. Using cylindrical coordinates , imagine dividing the cylinder into rings of radius r with radial thickness and height . Within a ring, we can assume that r and z are constant. Also within each ring, we can work in the grand canonical ensemble. Show generally (i.e., even in the presence of interactions between the particles), that the grand canonical partition function satisfies:

where is the grand canonical partition function for and g=0. is the volume of the ring. What is ?

3.

A simple model for polymers is the so called Gaussian random coil model, which is illustrated in the figure below:

 

In this model, the endpoint particles at positions and are fixed in space, while the remaining N particles are free to move. Assume all particles have the same mass m. The N particles interact with each other via a nearest-neighbor harmonic potential of the form:

where is the frequency of the harmonic coupling between neighboring particles. In the above expression, we adopt the convention that and .

We now want to calculate the canonical partition function of this polymer at temperature T.

a.

Write down the expression for . Why does Q have these arguments?

b.

Consider the following change of integration variables in the expression for :

By performing this change of variables, calculate the canonical partition function.

Hint: Note that the transformation is defined recursively. How should you start the recursion?

c.

Suppose the endpoints at and are no longer fixed. A quantity of central interest in the study of polymers is the so called mean-square end-to-end distance , which can be measured by light scattering experiments. Determine this ensemble average when the endpoints are free to move.

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