From the discussion of fluctuations in the number of particles on the left side of the box,
or coin flips, we found in lecture:
"
N
L
(
)
2
=
N
L
2
#
N
L
2
N
L
=
h
i
i
=
1
N
$
"
N
L
(
)
2
=
h
i
h
j
j
=
1
N
$
i
=
1
N
$
#
h
i
i
=
1
N
$
h
j
j
=
1
N
$
At this point, we used the statistical independence of the particles to eliminate all terms
where i
≠
j:
h
i
h
j
=
h
i
h
j
for
i
"
j
At this point, we are left with only terms where i = j, thus we eliminate one of the
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 Spring '09
 JAMESAMES
 Physical chemistry, pH, Probability theory, #, Dice, i=1, Summations

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