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Homework #4
— PHYS 603 — Spring 2007
Deadline:
Tuesday, April 17, 2007, in class
Professor Victor Yakovenko
Oﬃce: 2115 Physics
Web page: http://www2.physics.umd.edu/˜yakovenk/teaching/
Textbook: Gregory H. Wannier,
Statistical Physics
Dover 1987 reprint of the 1966 edition, ISBN 048665401X
Do not forget to write your name and the homework number!
Each problem is worth 10 points.
Ch. 4 The GibbsBoltzmann Distribution
1. Problem 4.2, Derive the Maxwell distribution
p
(
v
)
for the speed
v
.
2. Problem 4.3, Compare
h
v
i
with
q
h
v
2
i
3. Problem 4.6, Calculate the most probable speed
v
*
.
4. Energy distribution
p
(
E
)
for the 3D Maxwell gas.
(a)
Derive the probability density
p
(
E
) for the distribution of energy
E
of an atom in
a threedimensional (3D) Maxwell gas. The deﬁnition of the probability density
is
dP
=
p
(
E
)
dE
, where
dP
is the probability to have energy in the interval
(
E,E
+
dE
).
Hint:
Follow the discussion around Eq. (4.35) for one particle
n
= 1.
(b)
From the probability density
p
(
E
) derived above, calculate the most probable
energy
E
*
which maximizes
p
(
E
). Compare
E
*
with
mv
2
*
/
2, where
v
*
is the most
probable speed maximizing
p
(
v
), as derived in Problem 3. Are
E
*
and
mv
2
*
/
2
equal? If not, can you explain why?
5. Problem 4.7, Mean kinetic energy in a leak.
Hint:
To calculate the mean kinetic energy for a leak in the
x
direction, calculate
the energy ﬂux
h
Ev
x
ρ
(
E
)
i
and divide it by the particle ﬂux
h
v
x
ρ
(
E
)
i
. Here
E
=
m
(
v
2
x
+
v
2
y
+
v
2
x
)
/
2 is the energy of an atom,
ρ
[
E
(
v
)] is the GibbsBoltzmann distribution
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 Spring '08
 V.Yakovenco
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