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Homework 2 (due October 6)
Problem 21 (3 pts.)
The classical expression for entropy,
S
=
−
h
log
p
k
i
=
−
X
states
p
k
log
p
k
is derived under assumption that all microstates of the system have the same statistical weight. Generalize this result
for the case of states with arbitrary individual statistical weights,
w
1
,w
2
, ..., w
K
. Find the probability distribution
which corresponds to the maximum entropy. Note: that the only constrain is,
P
p
k
=1
.
Solution:
W
(
n
1
,n
2
, .....
, n
k
)=
N
!
Y
states
w
n
k
k
n
k
!
(1)
By using the Stirling formula,
h
S
i
=
log
W
N
=
1
N
"
log (
N
!) +
X
states
log
μ
w
n
k
k
n
k
!
¶
#
=
−
X
states
n
i
N
log
μ
n
k
Nw
k
¶
=
−
X
states
p
k
log
p
k
w
k
(2)
Apply the maximum entropy principle to
f
nd the probabilities:
δ
X
states
μ
p
k
log
p
k
w
k
+
αp
k
¶
=0
, therefore,
p
k
=
const
·
w
k
=
w
k
P
w
k
Problem 22 (3 pts.)
Find the free energy and heat capacity
C
V
of the system of
N
singleatom molecules of mass
m
each, in the
presence of uniform gravitational
f
eld
g
(in 3D).
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This note was uploaded on 09/28/2008 for the course PHYS 510 taught by Professor Anon during the Winter '04 term at Cornell University (Engineering School).
 Winter '04
 ANON
 mechanics, Work, Statistical Mechanics, Entropy

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