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Canonical partition function for an ultrarelativistic gas in a power-law trap
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Physics
Branch of physics
Statistical physics
Canonical ensemble
Canonical partition function
Classical ideal-gas partition function
Created
2026-09-24
Updated
2026-09-24
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For
energy
p
c
+
U
(
x
)
,
Z
1
=
π
2
(
ℏ
c
)
3
(
k
B
T
)
3
∫
e
−
U
(
x
)
/
(
k
B
T
)
d
3
x
.
(1)
If
U
(
r
)
=
r
2
n
/
V
2
n
/3
, then
Z
1
=
π
(
ℏ
c
)
3
4
V
(
k
B
T
)
3
+
3/
(
2
n
)
I
n
,
I
n
=
∫
0
∞
u
2
e
−
u
2
n
d
u
.
(2)
Consequently
p
V
=
N
k
B
T
, while the
mean
energy
and
variance
are
N
A
k
B
T
and
N
A
(
k
B
T
)
2
with
A
=
3
+
3/
(
2
n
)
.
Table of contents
Most likely radius in an isotropic power-law trap
Canonical partition function for an ultrarelativistic gas in a power-law trap
Most likely radius in an isotropic power-law trap
0
0
0
Canonical partition function for an ultrarelativistic gas in a power-law trap
The radial
density
is proportional to
r
2
e
−
r
2
n
/
(
V
2
n
/3
k
B
T
)
, whose mode is
r
=
V
1/3
(
n
k
B
T
)
1/
(
2
n
)
.
(1)
Ancestors
(7)
Classical ideal-gas partition function
Canonical partition function
Canonical ensemble
Statistical physics
Branch of physics
Physics
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(1)
Past exam of the mathematics course of the University of Cambridge
/
2024
/
ii
/
Paper 1
/
36B
/
c
/
i
/
Solution
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