Homework 4
Problem 1. (see Pathria, 3.15)
Consider classical gas of
N
relativistic indistinguishable particles in volume
V
each
described by the Hamiltonian
H(p,q)=pc
. Using canonical ensemble theory
calculate
1.
Partition function of the gas
Q(N,V,T)
2.
Helmholtz free energy
A(N,V,T)
3.
Internal energy
U(N,V,T)
4.
Entropy of the gas
S(N,V,T)
5.
Density of states for the gas
g(E)
. Verify that indeed
S=k*ln[g]
Now you should really appreciate how easy is to work with canonical
distribution compared to microcanonical one as we have done in Problem
3 of Homework 2.
Solution
1. Partition function for a single particle is given by
/33
2
/
3
1
33
3
3
0
14
4
(,)
2
( )
pc kT
pc kT
VV
Q V T
e
d pd q
p dpe
kT
hh
h
c
π
∞
−−
===
∫∫
As a result
()
3
1
8
1
(,,)
[ (,)
]
!!
N
N
N
V
kT
QNVT
QVT
NN
h
c
⎡
⎤
==
⎢
⎥
⎣
⎦
2. Helmholtz free energy is
( , , )
ln ( , , )
ln8
3
ln
ln
ln(
)
3
ln(
)
(1 ln8 )
kT
ANVT
kT QNVT
N
kT
V
N
N
kT N N
hc
Vk
T
NkT
NkT
NkT
Nh
c
=−
−
+
−
−
−
+
3. Internal Energy is
,
/
3
1/
NV
AT
UNVT
kNT
T
∂
⎛⎞
⎜⎟
∂
⎝⎠
4. Entropy is
,
l
n
3 l
n
(
4l
n
8)
VN
AV
k
T
S N V T
Nk
Nk
Nk
TN
h
c
∂
=
+
+
+
∂
By the way, expressing it via the energy we obtain
l
n
n
(
n
3
VU
S N V E
Nk
Nk
Nk
h
c
=+
+
+
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View Full Documentwhich is identical to the solution of problem 3 in homework 2.
5. Density of states for the gas
''
33
3
31
1
/
3
'
3
3
'
11
(
8
)
()
(,,)
22
!
(
)
1(
8 )
(
(
8 )3
2
3
2
!(
)
!(
)
(3
1)!
!(
)
(3 )!
!
(3 )!
NE
ii
E
NN
N
x
N
N
N
N
i
N
N
i
Ve
gE
e QNV
d
d
N
h
c
VE
e
V N
E
V
N
E
dx
i
N hc
x
N hc
N
N hc
N
N
hc
N
β
ββ
π
ππ
+∞
−∞
−−
−
−
===
⎛⎞
==
=
⎜⎟
−
⎝⎠
∫∫
∫
which is identical to the result (divided by
h
3N
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 Spring '09
 pathria
 Work, Statistical Mechanics, Fundamental physics concepts, Helmholtz free energy, χ, dβ

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