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Physics 5524
Statistical Mechanics
Problem Set 9
Due: Wednesday, Mar. 31
9.1 (From the Fall 2006 Qualifying Exam)
A system of
N
bosons in two dimensions has an energymomentum relationship
E
=
cp
3
/
2
and number density
n
=
N/A
(
A
is the area).
(a) Show that at low temperatures the system will Bose condense (unlike the 2D Bose gas
in Problem 8.2), and that the Bose condensation temperature
T
c
∼
n
α
. Find
α
.
(b) Show that the entropy below
T
C
goes like
S
∼
T
β
, and the “pressure” (which in two
dimensions has units of force per unit length) goes as
P
∼
T
γ
. Find
β
and
γ
.
9.2 Consider a
D
dimensional ideal Bose gas with single particle energy spectrum
E
(
~
k
) =
A

~
k

s
where
s
is some positive number. (
Note:
In doing this problem you should not have
to evaluate any integrals explicitly.)
(a) Determine the conditions on
D
and
s
for which this gas exhibits Bose condensation
with
T
c
>
0.
(b) For the cases when Bose condensation occurs, obtain an expression for the condensate
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 Spring '09
 Bonesteel
 mechanics, Energy, Momentum, Statistical Mechanics

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