PH3201 StatMech Assgn3

Assignment 3
PH 3201 Basic Statistical Physics
The assignment is due on 1 March, 2016.

1. A particle can exist in only three states labelled by n = 1, 2, 3. The energies n of these states
depend on a parameter x 0, with two of the energies degenerate.
1
1 = 2 = bx2 cx
2
3 = bx2 + cx
where b and c are constants.
(a) Find the Helmoltz free energy per particle a(x, T ) = A(x, T )/N for a collection of N
such non-interacting particles.
(b) If x is allowed to freely vary at constant T , it will assume an equilibrium value x that
minimizes the free energy. Find x as function of T . Show that there is an abrupt
change (as function of temperature) in the value of x that minimizes the free energy.
Find the value of this transition temperature. [Hint: Assume x to be small and expand
the exponential to order x2 . Assume that x cannot be negative.]
This model can be used to describe ions in a crystal subject to uniform strain characterized
by the parameter x. The phase transition is known as the cooperative Jahn-Teller phase
transition.
2. One-dimensional polymer: Consider a polymer formed by connecting N disc-shaped molecules
into a one-dimensional chain. Each molecule can allign 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 , that is, the total energy is H = m, where m is the number of monomers standing
up.
(a) Calculate the partition function QN (T ) of the polymer.
(b) Find the relative probabilities for a monomer to be aligned along its short or long axis.
(c) Calculate the average length, hL(T, N )i, of the polymer.
3. The electrical resistivity of a metal at room temperature is proportional to the probability
that an electron is scattered by the vibrating atoms in the lattice, and this probability is
in turn proportional to the mean square amplitude of vibration of these atoms. Assuming
classical statistics to be valid in this temperature range, what is the dependence of the
electrical resistivity on the absolute temperature T?
4. A dilute solution of macromolecules (large molecules of biological interest) at temperature T
is placed in an ultracentrifuge rotating with angular velocity . The centripetal acceleration
2 r acting on a particle of mass m may then be replaced by an equivalent centrifugal force
m 2 r in the rotating frame of reference.
(a) Find how the relative density (r) of molecules varies with their radial distance r from
the axis of rotation.
(b) Show quantitatively how the molecular weight of the macromolecules can be determined
if the density ratio 1 /2 at the radii r1 and r2 is measured by optical means.
P.T.O.
The following need not be submitted but may be attempted for practice. Part (a) of the first question was in the midsem. The second problem and part (a) of the third problem were (hurriedly)
discussed in class.
1. A zipper has N links; each link has a state in which it is closed with energy 0 and state in
which it is open with energy . We require, however, that the zipper can only unzip from the
left end, and that the link number s can only open if all links to the left (1, 2, ..., s 1) are
already open.
(a) Show that the partition function can be summed in the form
QN =
1 exp[(N + 1)
1 exp()
(b) In the (low temperature) limit >> kB T , find the average number of open links.
[The model is a very simplified model of the unwinding of two-stranded DNA molecules see
C. Kittel, American Journal of Physics 37, 917 (1969).]
2. Work out the thermodynamics of the simple one-dimensional model for rubber elasticity that
I had sketched in class. [Bowley-Sanchez pg 78]. If you are interested you may also look at
a slightly more sophisticated model in Callens book.
3. Consider N spin-1/2 spins in a magnetic field B.
(a) Initially, the system has a temperature T . If we slowly reduce the magnetic field to
B/2, what becomes the temperature of the system? If we slowly reduce the magnetic
field to zero, what becomes the temperature of the system? (Hint: the entropy remains
unchanged in the above adiabatic process.)
(b) The spin system is now in thermal contact with an ideal gas of N particles in a volume
V . Initially, the two systems have a temperature T . Assume gB B >> kB T . If we
slowly reduce the magnetic field to zero, what becomes the temperature of the gas?
February 22, 2016

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Assignment 3

PH 3201 Basic Statistical Physics

The assignment is due on 1 March, 2016.

February 22, 2016

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