238
DOC.
9 CRITICAL OPALESCENCE
p(p)
=
i|»(v),
we
obtain
the
even
simpler
expression
L3
.
.3
D2
(6)
a
x
where
one
has to
substitute the
values
of the
quantities
v
and
dfy/Sv2
for the state
of
ideal
thermodynamic
equilibrium.
We
note
that the
coefficients B
appear
only as
squares
in
the
expression
for
A,
and
not
as
double
products. Thus,
the
quantities
B
are
parameters
of the
system
of the
same
kind
as
those
seen
in
equations
(2b)
and
(4)
in
the
preceding
section.
The
quantities
B
obey
therefore
(independently
of
each
other)
the Gaussian
error law,
and
equation
(4) yields directly
LVa2+iF~
-
RT°
f"
"
TT
(7)
_v3_ZB
The
statistical
properties
of
our
system
are
thus
completely
determined,
or
reduced
to
the
thermodynamically
ascertainable function
i|j.
It should be noted that the
omission
of
the terms with
A3
etc.
is permissible
only
if
c2ty/ldv2
for the
ideal
thermodynamic equilibrium
is
not too small
or even
vanishes.
The
latter
occurs
in
the
case
of
fluids
or
liquid
mixtures
that
are exactly
in
the
critical state.
Within
a
certain
(very small)
region
about the critical
state,
the formulas
(6)
and
(7)
become
invalid.
However,
there
is
no difficulty,
in
principle,
in
completing
the
theory
by
[15]
taking
into consideration the
terms
of
higher
order
in
the
coefficients.5
[16]
§
4.
A Calculation of
the
Deflection
of
Light from an
Infinitely
Slightly
Inhomogeneous, Absorption-free
Medium
Now
that
we
have
obtained
from
Boltzmann's
principle
the
statistical law
according
to
which
the
density
of
a
uniform substance
or
the
mixing
ratio of
a
mixture varies with
position,
we
will
investigate
the influence of the medium
on a
ray
of
light traveling
through
it.
Again
let
p
=
p0
+
A
denote the
density
at
some point
of the
medium,
or,
if
we are
dealing
with
a mixture,
the
spatial density
of
one
of the
components.
Let the
light ray
under consideration
be
monochromatic.
As
regards
this
ray,
the medium
can
be
characterized
by
the
refractive index
g
or by
the
apparent
dielectric constant
e
that
corresponds
to
the
frequency involved,
and
which
is
connected
with
the
refractive index
by
the relation
g
=
Jz
.
We
put
(8)
e
=
+
I I
A
= +
I~3p1J0
where both
i
and
A
should
be treated
as infinitesimally
small
quantities.
[14]
5
Cf. M.
v. Smoluchowski, l.c.,
p.
215.