Homework #4
— PHYS 603 — Spring 2008
Deadline:
Thursday, March 6, 2008, in class
Professor Victor Yakovenko
Oﬃce: 2115 Physics
Web page: http://www2.physics.umd.edu/˜yakovenk/teaching/
Textbook: Silvio Salinas,
Introduction to Statistical Physics
Springer, 2001, ISBN 0387951199
Do not forget to write your name and the homework number!
Each problem is worth 10 points.
Ch. 4 Microcanonical Ensemble
1. Problem 4.1, Spin chain with energy
S
2
j
.
2. Problem 4.4, Entropy of the Boltzmann gas.
Derive the general equations shown in this problem. In the last sentence, I believe
the answer

k
B
T
ln
T
is wrong. To derive the entropy of the Boltzmann gas, use the
following procedure.
In the spirit of Eq. (4.51), in the continuous limit the occupation numbers become
N
j
N
=
P
j
→
P
(
p
) =
e

p
2
/
2
mT
Z
1
,
(1)
where the normalizing factor
Z
1
is given by a formula similar to Eq. (4.49), but written
more accurately
Z
1
=
Z
d
3
pd
3
r
(2
π
¯
h
)
2
e

p
2
/
2
mT
=
V
ˆ
√
2
πmT
2
π
¯
h
!
3
(2)
The entropy per particle
s
=

∑
j
P
j
ln
P
j
becomes
s
=

Z
d
3
pd
3
r
(2
π
¯
h
)
2
P
(
p
) ln
P
(
p
) =

Z
d
3
pd
3
r
(2
π
¯
h
)
2
P
(
p
)
ˆ

p
2
2
mT

ln
Z
1
!
= ln
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 Fall '08
 V.Yakovenco
 Physics, Energy, Work, Statistical Mechanics, Entropy, pj ln pj, UMD qualiﬁer problem

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