1
Homework 5 (due November 15)
Problem 51 (3 pts.)
Let the dispersion relationship for an ideal Bose gas particles be given by the following scaling law:
ε
(
p
)
∼
p
α
What should be the dimensionality of the physical space
d
for this system to be capable of Bose condensation?
Solution:
Bose condensation is possible if
µ
= 0
can be reached at a
fi
nite density, i.e. when the following integral converges:
ρ
c
=
4
π
(2
π
~
)
3
∞
Z
0
∙
exp
μ
±
(
p
)
T
¶
−
1
¸
−
1
p
2
dp
Divergency is only possible at small
p.
in this limit,
[exp (
±
(
p
)
/T
)
−
1]
∼
p
α
.
Hence, the following integral should
converge:
P
Z
0
p
d
−
α
−
1
dp
We conclude that
d > α
Problem 52 (4 pts.)
Find the energy density
U
of a photonic gas
in
d
dimensions, as a function of temperature
T
(assume all the
fundamental constants to be the same as in our world). By using the standard thermodynamic relationships,
fi
nd the
corresponding entropy density
s
, free energy density
f
, and radiation pressure
P
.
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 Winter '04
 ANON
 mechanics, Work, Statistical Mechanics, Magnetic Field, Fundamental physics concepts, Energy density

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