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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 statesdepend on
a parameter x ≥ 0, with two of the energies
degenerate.
1 = 2 = bx2 −
1
2cx
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
thatminimizes the free energy. Find x as
function of T . Show that there is an abruptchange
(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
expandthe 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 characterizedby the parameter x. The phase
transition is known as the ‘cooperative Jahn-Teller’
phasetransition.
2. One-dimensional polymer: Consider a polymer formed by
connecting N disc-shaped moleculesinto a one-dimensional chain.
Each molecule can allign along either its long axis (of length2a)
or short axis (length a). The energy of the monomer aligned
along its shorter axis ishigher by , that is, the total energy
is H = m, where m is the number of
monomers standingup.
(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, L(T, N ), of the
polymer.
3. The electrical resistivity ρ of a metal at room
temperature is proportional to the probabilitythat an electron is
scattered by the vibrating atoms in the lattice, and this
probability isin turn proportional to the mean square amplitude of
vibration of these atoms. Assumingclassical statistics to be valid
in this temperature range, what is the dependence of theelectrical
resistivity ρ 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 ω . The
centripetal accelerationω2r acting on a particle of mass
m may then be replaced by an equivalent centrifugal
forcemω2r in the rotating frame of reference.
(a) Find how the relative density ρ(r) of molecules varies
with their radial distance r fromthe axis of
rotation.
(b) Show quantitatively how the molecular weight of the
macromolecules can be determinedif the density ratio ρ1/ρ2
at the radii r1 and r2 is measured by
optical means.
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− − P.T.O.
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The following need not be submitted but may be attempted for
practice. Part (a) of the first ques-tion 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 inwhich 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) arealready
open.(a) Show that the partition function can be summed in the
form
QN = 1 − exp[−β (N +
1)
1 − exp(−β)
(b) In the (low temperature) limit >> kBT ,
find the average number of open links.[The model is a very
simplified model of the unwinding of two-stranded DNA molecules
seeC. Kittel, American Journal of Physics 37, 917 (1969).]
2. Work out the thermodynamics of the simple one-dimensional
model for rubber elasticity thatI had sketched in class.
[Bowley-Sanchez pg 78]. If you are interested you may also look
at
a slightly more sophisticated model in Callen’s 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 toB/2, what becomes the
temperature of the system? If we slowly reduce the magneticfield to
zero, what becomes the temperature of the system? (Hint: the
entropy remainsunchanged in the above adiabatic process.)
(b) The spin system is now in thermal contact with an ideal gas
of N particles in a volumeV . Initially, the two
systems have a temperature T . Assume gµBB >>
kBT . If weslowly reduce the magnetic field to zero, what
becomes the temperature of the gas?
February 22, 2016
