Chem 120B
Problem Set 1
Due: January 28, 2011
1. (i) Show that the mean square fluctuation in any quantity
A
can be written in terms of its mean and
mean square values,
(
(
δA
)
2
)
=
(
A
2
) − (
A
)
2
.
(Here, as usual, the fluctuation in
A
is defined as
δA
=
A
− (
A
)
.)
(ii) Generalize this result to the case of two different fluctuation quantities, i.e., show that
(
δA δB
)
=
(
AB
) − (
A
)(
B
)
.
2. Consider two fluctuating variables
x
and
y
, whose statistics are described by a probability distribution
p
(
x, y
)
.
(i) If fluctuations in
x
and
y
are statistically independent, then their joint distribution
p
(
x, y
)
factorizes,
p
(
x, y
) =
f
(
x
)
g
(
y
)
, where
f
(
x
)
and
g
(
y
)
are the distributions of
x
and
y
, respectively. Explain why
this factorization is implied by independence of
x
and
y
. (It is not necessary to give a very detailed
account of your reasoning here, but it is important that you convince yourself of this fact.)
(ii) The average of a function
F
(
x, y
)
of these two variables can be written
(
F
)
=
integraldisplay
dx
integraldisplay
dy p
(
x, y
)
F
(
x, y
)
Consider a function
F
(
x, y
) =
A
(
x
)
B
(
y
)
that depends separably on
x
and
y
. Show that the average
of
F
also factorizes,
(
F
)
=
(
A
(
x
)
)(
B
(
y
)
)
.
(iii) Using your results from parts (i) and (ii), show that statistical independence of
x
and
y
implies
(
δx δy
)
= 0
,
where
δx
=
x
− (
x
)
and
δy
=
y
− (
y
)
are fluctuations of
x
and
y
about their mean values.
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 Spring '09
 JAMESAMES
 Physical chemistry, pH, Mean, Intensive and extensive properties, observation region

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