236
DOC. 9
CRITICAL OPALESCENCE
general,
deviate
markedly
from
zero only
at
very
small values
of
A,
and
hence
also at
very
small values
of
X.
For
such small values
of
X,
the
terms
of
higher
than the
first
power
in
the
expression
for
A will
generally
make
a negligible
contribution
as
compared
with
that of the
second-power
terms.
If that
is
the
case, we can
substitute for
equation
(2a)
N
£a
[13] (2b)
dW
=
const.e
^
dX1...dXn,
which has
the
form
of the
Gaussian
error
law.
In
this
paper
we
shall confine ourselves to this most
important
special case.
It
follows
directly
from
(2b)
that the
mean
value
of
the
fluctuation
work
Av
allotted
to
the
parameter
Xv
is
A\
T
-
1~T2
-
RT°
(4)
A
=
-a
Xv
=
v
2
v
2N
Thus,
this
average
work
is
equal
to
one-third of
the
mean
kinetic
energy
of
a
monatomic
gas
molecule.
§ 3.
On Deviations in the
Spatial
Distribution of Fluids
and
Liquid
Mixtures
from a Uniform
Distribution
We denote
by
p0
the
mean
density
of
a
homogeneous
substance
or
the
mean
density
of
one component
of
a binary liquid
mixture. Because
of the
irregularity
of the thermal
motion,
the
density p
at
a
point
in
the
fluid will
generally
differ from
p0.
If the
liquid is
enclosed
in
a
cube
characterized,
with
respect
to
a
coordinate
system, by
0
x
L
0
y
L
and
0
z
L,
we can
put,
for the interior of
this
cube,
(5)
P
=
P0
+
A
A
=
Y*
y^ B
cos
2np-cos
2no
cos
2m
-
-
-
.
p" 2L 2L 2L
The
quantities p, o,
x
denote
positive
integers. However,
the
following
needs to
be
noted.
Strictly speaking, we
cannot
speak
of
the
density
of
a
fluid at
a spatial
point,
but
only
of the
mean
density
in
a
volume
whose
dimensions
are large
compared
with
the
mean
distance
between
neighboring
molecules.
For
this
reason,
the
terms of the series in which
one
of the
quantities p, a,
x
exceeds
certain
limits will have
no
physical meaning.
However,
we
will
see
from
the
following
that
this
circumstance
is
of
no
importance
to
us.
The
quantities
Bpaz
will
change over
time
such
that
on
the
average
they
will
be
zero.
Let
us now
seek the
statistical laws
that underlie the
quantities B.
The latter
play
the