Physics 418: Statistical Mechanics I
Prof. S. Teitel email@example.com ----- Spring 2005
Problem Set 6
Due Thursday, April 7, in lecture
- Problem 1 [20 points]
Consider a degenerate Fermi gas of non-interacting, non-relativisitic, particles in two dimensions (this might be a model for electrons in a thin metallic film).
a) Find the density of states g().
b) Find the Fermi energy and the T=0 energy density.
c) Consider the Sommerfeld expansion for the density n=N/V, as we did in lecture to compute the chemical potential µ(T) at finite temperature in three dimensions. What does it seem to imply about the temperature dependence of the chemical potention in two dimensions?
d) Show that in fact the relative change in µ(T) from µ(0)=F is of order exp(-F/kBT). This explains why the Sommerfeld expansion does not work: because
exp(-F/kBT) is not an analytic function of T at T=0! [Hint 1: follow the steps we took in deriving the Sommerfeld expansion up until the point where we made the Taylor series expansion of (x). At this step, remember that there was one other subtle approximation that was made beside doing the Taylor expansion (it has to do with the limits of integration). Think about this other approximation and how to do it better. Or, Hint 2: you can try to do the integral exactly! This will give you the exact answer, but it is less fun.]
- Problem 2 [15 points]
a) Find the chemical potential µ for an ideal (non-relativistic) Fermi
gas at low temperature, to second order in T, at fixed pressure p.
Note, this is different from what we did in lecture - there we computed µ at fixed density
N/V. (Hint: what is E/V at fixed p?)
b) The Gibbs free energy is related to the chemical potential
by G(T,p,N) = Nµ(T,p). The entropy can be derived from the
Gibbs free energy, and hence from the chemical potential, by using,
S = - (dG/dT)p,N = - N(dµ/dT)p
Using your result from part (a), find the entropy of the Fermi gas
to lowest order in T. Does your result agree with that given in
Pathria §8.1 Eq.(41), as derived there from the Helmholtz free energy?
- Problem 3 [15 points]
Consider a non-interacting ideal gas of non-relativistic bosons (i.e. (p) = p2/2m) in d dimensions, where d is any number.
Show that there is no Bose Einstein condensation for any d less than or equal to two.