33-341 — Thermal Physics I
Department of Physics, Carnegie Mellon University, Fall Term 2018, Deserno
Problem sheet #15
45. Electric dipole in a homogeneous electric field (5 points, due on Monday)
A three-dimensional electric dipole, characterized by a unit vector d, has the energy H(d) = −d·Ewhen placed in a
homogeneous electric field E. It will also have some translational and rotation al kinetic energy, but we will ignore this for now.
1. Choosing the direction of the electric field as the ˆz-axis, express the dipole’s energy in spherical polar coordinates.
2. Write down the canonical partition function Zfor this system; ignore the momentum integrals and factors of h.
Hint: don’t forget that this choice of coordinates implies a
Jacobian
(functional determinant) in your integral.
3. Define the polarization d|| =d·(E/E), which is the projection of the dipole moment along the direction of the external
field. Without explicitly evaluating Z, prove that hd||i=−∂ F/∂E , where F=−kBTln(Z)is the free energy.
Hint: Explicitly write down the canonical average and use parameter differentiation to rewrite your expression. (You
know anyways what you want to find—work backwards if you need to.)
4. Now evaluate the partition function and derive an explicit formula for hd|| i. Express your answer using the function
L(x) = coth(x)−x−1, which is often called the “Langevin function”. Plot the result using suitably normalized axes!
46. Electric dipole in a homogeneous electric field—continued (5 points, due on Tuesday)
1. Expand the average polarization hd|| ifrom the previous problem to lowest nontrivial order for small values of βdE .
2. The (isothermal) susceptibility of the polarization can be defined as χT=∂hd|| i
∂E T=−∂2F
∂E2. Show that in the limit
E→0or T→ ∞ this converges towards a constant that does not depend on the applied field.
3. The susceptibility is a response function. Derive the associated fluctuation-response-theorem kBT χT=hd2
||i − hd|| i2.
47. Heat capacity of stick-like molecules (5 points, due on Wednesday)
At not too low and not too high temperature, linear molecules (such as O2, N2, or CO2) can be viewed as linear sticks. Their
motion can be decomposed into some overall translation and some rotation. For the latter, the kinetic energy can be written
as Erot =1
2I(L2
1+L2
2), where L1and L2are the angular momentum of rotation about the two axes perpendicular to the
molecule’s axis (rotation around its axis makes no sense, says quantum mechanics). Given that the kinetic degrees of freedom
of each individual molec ule are hence (px, py, pz, L1, L2), what is the average kinetic energy of such a molecule in the canonical
state? And what is therefore the isochoric specific heat of diatomic gases such as O2?
Hint: Remember problem 44!
48. A microscopic model of rubber elasticity (5 points, due on Friday)
A simple model of a polymer is a chain of Nlinks, each of
length a, which are connected via perfectly flexible “hinges”.
Let’s say we pull on this chain with some tension τ. Each link
ican be characterized by two angles ϑiand ϕithat describe its
orientation relative to the pulling direction.
1. If a link is not perfectly aligned with the pulling direction,
it has to do work against the external pulling force, and
this suggests an obvious potential energy for each link
and, consequently, for the entire chain. Write it down!
N−1
a
L
τ
1
2
3
4
N
τ
2. Calculate the canonical partition function Zand the free energy Fof this system; ignore the momentum part.
3. What is the expected length hLiof the chain, i. e., the distance between the two end points of the chain (which must be
aligned with the pulling direction)? Explicitly calculate hLias a function of τ,T,a, and N.
4. Calculate ∂hLi
∂T τin the limit of small forces. Marvel at the sign. You’ve begun to unravel the mystery of problem 38!