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PHY 418: Statistical Mechanics I
Prof. S. Teitel stte@pas.rochester.edu  Spring 2013
Problem Set 7
Due Monday, April 29, in lecture
 Problem 1 [15 points]
Consider a degenerate Fermi gas of noninteracting, nonrelativisitic, particles in two dimensions (this might be a model for electrons in a thin metallic film).
a) Find the density of states g(ε).
b) Find the Fermi energy and the T=0 energy density.
c) Using the the fact that the particle density n is given by
find the chemical potential as a function of temperature, µ(T), for fixed density n, by doing this integral exactly.
You may have to look up an integral in an integral handbook! Using the exact expression for µ(T), find a simpler approximation that holds at low T<<T_{F}. Does µ(T) have a power series expansion in T at low T, like we found in three dimensions with the Sommerfeld expansion?
 Problem 2 [15 points]
a) Find the chemical potential µ for an ideal, nonrelativistic, three dimensional 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? You may wish to review Lecture 23 to do this problem.)
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 noninteracting gas of spin zero bosons, whose energymomentum relationship is given by ε(p) = Ap^{s}, for some fixed positive numbers A and s. The dimensionality of the gas is the number d, i.e. the volume of the gas is V = L^{d}, for a system of length L. In the following parts, we are considering the behavior in the thermodynamic limit of V → ∞.
a) For what values of s and d will there exist BoseEinstein condensation at sufficiently low temperatures?
b) For the case that there is BoseEinstein condensation, write an expression that gives how the condensate density n_{o} depends on the temperature T, the condensation temperature T_{c}, and the total density n.
c) Show that the pressure is related to the energy by, p = (s/d)(E/V)
d) Can a gas of photons in d = 3 undergo BoseEinstein condensation?
 Problem 4 [15 points]
N Fermions A of spin 1/2 (i.e. g_{s}=2) are introduced into a large volume V at
temperature T. Two Fermions may combine to create a Boson with
spin 0 (i.e. g_{s}=1) via the interaction,
A + A ↔ A_{2}
Creation of the molecule A_{2} costs energy ε_{o} > 0.
At equilibrium, the system will contain N_{F} Fermions and
N_{B} Bosons. Provide expressions from which the ratio
N_{B}/N_{F} can be calculated, and perform the
calculation explicitly for T=0. What would this (T=0) ratio be,
if the particles were classical (i.e. quantum statistics could be neglected).
Explain the difference.

