CHEM 350 Lectures 11 and 12, January 27 and 29, 2010
Ensemble Averages.
•
variable
v
: having
M
possible values
v
1
, v
2
,
· · ·
, v
M
: corresponding probabilities
p
1
, p
2
,
· · ·
, p
M
–
N
systems in ensemble with variable
v
having value
v
r
in
N
r
=
N
p
r
of them
– with so many possibilities in principle, we should be willing to settle for a little less
detail. For example, we might be interested in first instance only in the
average value
of
the variable (other names:
mean value
,
ENSEMBLE AVERAGE
) defined by
h
v
i ≡
1
N
(
N
1
v
1
+
N
2
v
2
+
· · ·
+
N
M
v
M
) =
M
X
r
=1
p
r
v
r
.
•
average of a function of
v
:
h
f
(
v
)
i
=
M
X
r
=1
p
r
f
(
v
r
)
.
This represents a linear operation, since
h
cf
(
v
)
i
=
c
h
f
(
v
)
i
h
af
(
v
) +
bg
(
v
)
i
=
a
h
f
(
v
)
i
+
b
h
g
(
v
)
i
.
•
CAUTION: this does not necessarily mean that evaluation of products is simple!
h
f
(
u
)
g
(
v
)
i ≡
M
X
r
=1
M
X
s
=1
p
rs
f
(
u
r
)
g
(
v
s
)
.
Now, if we
assume
that the variables are statistically independent, then
p
rs
=
p
r
p
s
, and
h
f
(
u
)
g
(
v
)
i
=
M
X
r
=1
M
X
s
=1
p
r
p
s
f
(
u
r
)
g
(
v
s
)
=
h
f
(
u
)
i h
g
(
v
)
i
1
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•
All this is fine if we know all values of
p
r
. BUT, how do we evaluate such averages without
knowing all values of
p
r
?
Let’s consider an example that is of considerable interest to us: an ideal gas with
N
atoms
enclosed in a container of volume
V
T
. Let’s also divide the box into two parts, of volumes
V
and
V
0
, such that
V
+
V
0
=
V
T
. Treat the atoms as statistically independent, and construct
an ensemble of such containers of the ideal gas.
If
p
is the probability that an atom is found in volume
V
, then
q
= 1

p
is the probability
that an atom is found in volume
V
0
.
At equilibrium, we can write for
p
and
q
the expressions
p
=
V
V
T
,
q
=
V
0
V
T
Question
:
What is the probability p(N,n) that n out of the N atoms will be found in volume V (with
Nn hence in V’)?
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 Spring '10
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