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Advanced Topics in Equilibrium Statistical
Mechanics
Glenn Fredrickson
Lecture 5
Reading: Chandler 7.2, 7.3, 7.4, 7.5
Recap:
•
dilute to moderately dilute gases treated by virial expansions about
ρ
=
0(
z
= 0) state.
•
easily extended to molecular ﬂuids and dilute mixtures.
•
Also applicable to dilute solutions of solute in solvents, including polymers
and colloids, but not electrolyte solutions as we shall see.
4. Liquid State Theory
A. Distribution functions and radiation scattering
It is clear that virial expansions, i.e.
expansions about the
ρ
= 0 ideal gas
reference state, are not very useful in condensed liquid phases.
Instead, we
make note of the following central idea in liquid state theory known as the “van
der Waals picture of liquids” and exploited heavily by Chandler, Andersen, and
Weeks in the 1970’s:
* Due to the harsh repulsions
between molecules at close range, the
structure and thermodynamics of liquids are determined primarily
by packing eFects that depend on molecular shape.
We shall see that in liquids of nearly spherical molecules like A or CH
4
, this
suggests new expansion about a dense
hard sphere reference system
, where the
expansion is in the strength of the relatively weak attractive
interactions.
1
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View Full DocumentTo discuss “liquid structure”, we need to introduce the concept of reduced
distribution functions. For simplicity, we work in the canonical ensemble. Re
call that at equilibrium, particle positions and momenta are independently dis
tributed, the former described by:
P
(
r
N
)=
e
−
βU
(
r
N
)
R
d
r
N
e
−
(
r
N
)
=
1
Q
c
e
−
(
r
N
)
This is the joint probability distribution of observing particle 1 at
r
1
, ..., and
particle
N
at
r
N
. Suppose we now integrate over
r
3
,
r
4
, ...,
r
N
to create a
“reduced distribution function”
P
(2)
(
r
1
,
r
2
Z
d
r
3
...
Z
d
r
N
P
(
r
N
)
This is the joint probability distribution of observing particle 1 at
r
1
and par
ticle 2 at
r
2
. However, if we were observing a ﬂuid, we would not be able to
distinguish 1 and 2 from any other particles. Thus,
ρ
(2)
(
r
1
,
r
2
N
(
N
−
1)
P
(2)
(
r
1
,
r
2
)
gives the joint probability distribution of observing any
particle at
r
1
and any
other at
r
2
.
We can now generalize this to de±ne the
n

particle reduced distribution func
tion
by:
ρ
(
n
)
(
r
1
,...,
r
n
N
!
(
N
−
n
)!
Z
d
r
n
+1
Z
d
r
n
+2
Z
d
r
N
P
(
r
N
)
The functions are normalized such that
Z
d
r
n
ρ
(
n
)
=
N
!
(
N
−
n
)!
n
=
1
:
ρ
(1)
(
r
1
) should clearly be independent of position
r
1
in a ﬂuid that is ho
mogeneous (uniform) on average. This is true in the gas and liquid states, but
clearly not in a crystal phase. To be normalized properly,
ρ
(1)
must correspond
to the average density
ρ
(1)
(
r
1
ρ
≡
N
V
n
=
2
:
ρ
(2)
(
r
1
,
r
2
) should depend only on

r
1
−
r
2
≡
r
12
in a homogeneous, isotropic
ﬂuid (liquid or gas state). Hence
ρ
(2)
(
r
1
,
r
2
ρ
(2)
(
r
12
).
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 Spring '09

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