234
DOC.
9
CRITICAL OPALESCENCE
(entropy
S0)
W
is
of the order of
magnitude one, so
that
we
then
have,
with order-of–
magnitude
accuracy,
N
W
=
e*
(s
-
s0)
From this
we can
conclude
that the
probability
dW that the
quantities
A1 ...
Xn
lie
between and
A1
+
dX1,
...,
Xn
and
Xn
+
dXn
is
given,
in
order of
magnitude, by
the
equation2
"(S
- 50)
dW-e*
.dX.l
...dX
n
in
the
case
when
the
system
is
determined
only incompletely (in
the
phenomenological
sense)
by
X1
...
Xn3.
To
be
exact,
dW
still differs from
the
given
expression by a
factor
f,
so
that
we
must set
N
dW
=
e*
iß
-
sn)
•
f. dX-..
•
dX
J
1
n7,
where
f
will
be
a
function of
X1
...
Xn
and
its
order of
magnitude
will
be
such
that
it
does not affect
the order of
magnitude
of the factor
on
the
right-hand side.4
We
now
form dW
for
the immediate
vicinity
of
an entropy
maximum.
If
the
Taylor
expansion converges
in
the
region
considered,
we
may
put
5
= S0
-
1
0
2
s
XX
+..
fl\
p
V
/
=/o +
JL
dX
• • •
y
if,
for the
state
of
maximum
entropy,
X1
=
X2
=
...
Xn
=
0.
Since
we are dealing
with
an
entropy maximum,
the double
sum
in
the
expression
for
S
is
essentially positive.
One
can
therefore introduce
new
variables in
place
of
the
A's, so
that the
above
double
sum
is
transformed
into
a
simple
sum
in
which
only
squares
of the
new variables,
which
are
again
denoted
by
X,
will
appear.
We
get
AT
dW
=
const.
e
™
E
/o
+
E
Kxv
dX
dX-.
* •
dX
1 n
2 We will
assume
that
regions
with extensions
of
observable size have
a
finite extension in
X.
3
The
manifold
of
possible
states would otherwise be
only (n
-
1)-dimensional on
account
of the
energy principle.
4
We do not know
anything
about the order of
magnitude
of the
derivatives
of
the
function
f
with
respect
to
X.
But
we
will
assume
in
what
follows
that the
derivatives
of
f
are
of the
same
order
of
magnitude
as
the function
f
itself.