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Department of Physics
Prof. R. Bundschuh
The Ohio State University
Third Problem Set for Physics 847 (Statistical Physics II)
Winter quarter 2004
Important dates:
Feb 10 10:30am12:18pm midterm exam,
Mar 16 9:30am11:18am ±nal exam
Due date:
Tuesday, Jan 27
7. Ultrarelativistic gas
12 points
If the particles of a gas have velocities close to the speed of light
c
, their energy has to be
calculated relativistically. In the limit of massless particles (e.g., photons) which travel at the
speed of light, this relation between the momentum
~
p
i
of particle
i
and its energy
E
i
becomes
E
i
=
c

~
p
i

. Thus, the energy of a gas of
N
of these particles in a box of volume
V
is
H
(
~x
N
) =
c
N
∑
i
=1

~
p
i

all
~
r
i
inside volume
V
+
∞
otherwise
.
a) Calculate the partition function of such an ultrarelativistic gas. (Hint:
∞
R
0
x
2
exp(

x
)
dx
=
Γ(3) = 2.)
b) Calculate the free energy of the ultrarelativistic gas as a function of its natural variables
temperature
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 Winter '04
 BUNDSCHUH
 Physics

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