DOC.
9
CRITICAL OPALESCENCE
233
deduced,
as we
know,
from
the theorem that the
entropy
of
a
total
system
that
is
composed
of
subsystems equals
the
sum
of
the
entropies
of
the
subsystems.
Equation
(1)
can
thus
be
proved
for
all
of the
states
Z
that
belong
to
the
same
value
of
energy.
The
following
objection
can
be raised
against
this
interpretation
of Boltzmann's
principle:
one
cannot
speak
of the
statistical
probability
of
a state,
but
only
of that of
a
state
region.
The latter
is
defined
by a
portion
g
of the
"energy
surface"
E(X1...kn)
=
[7]
0.
Obviously,
W
tends
toward
zero along
with
the
size
of the chosen
portion
of the
energy
surface.
For
this
reason, equation (1)
would
be
totally
meaningless
if the
relation
between
S and W
were
not
of
a
quite
special
kind.
That
is
to
say,
lg
W appears
in
the
equation
(1)
multiplied
by
the
very
small
factor
R/N.
If
one
imagines
that
W
has
been obtained for
a region
Gw
just
large
enough
that
its dimensions lie
on
the border
of
the
perceptible,
then
lg
W will have
a
certain
value.
If
the
region
is
reduced
perhaps
e10
times,
then the
right-hand
side will
only
be diminished
by
the
vanishingly
small
quantity
10(R/N) on
account
of the reduction
in
the
size
of the
region. Thus,
if
the
dimensions of
the
region are
indeed chosen
small
compared
with
perceptible dimensions,
but nevertheless
large
enough
for
R/N
lg
Gw/G
to
be
a numerically negligible quantity,
then
equation (1)
will have
a sufficiently
exact
meaning.
We
have
assumed
so
far that
X1...ln
determines
completely,
in
the
phenomenological
sense,
the
state
of
the
system
in
question.
However,
equation
(1)
also
retains
its
meaning
undiminished
if
we
seek the
probability
of
a
state
that
is
incompletely
determined
in
the
phenomenological sense.
For let
us
seek the
probability
of
a
state
that
is
defined
by
specific
values
of
X1
...
Xv
(where
v
n),
while
the
values
of
Av
...
Xn
are
left
[8]
indeterminate.
Among
all
the
states with
the
values
A1 ...
Av,
those
values
of
Av
...
Xn
will
be
far and
away
the
most
frequent
which
make the
entropy
of
the
system
at constant
[9]
A1
...
Xn
a
maximum.
In that
case,
equation
(1)
will
hold between
this maximum value
of the
energy
and the
probability
of
this state.
§
2.
On the Deviations
from a
State of
Thermodynamic Equilibrium
[10]
Let
us now
draw conclusions
from
equation
(1)
regarding
the relation between the
thermodynamic
and
statistical
properties
of
a system.
Equation
(1) yields
immediately
the
probability
of
a
state if
its entropy
is
given.
We
have
seen, though,
that
this
relation
is
not
exact;
instead,
only
the order of
magnitude
of the
probability
W
of the
state in
question can
be determined
from
a
known
S. Nevertheless,
it
is
possible
to
derive
exact
relationships concerning
the
statistical
behavior of
a system
from
equation
(1)
in
cases
where the
range
of
the
state variables
for
which W has values
for the
kind
under
consideration
can
be
regarded as
infinitely
small.
It
follows from
equation
(1)
that
W
=
const.
eR
.
This
equation
is
valid to
an
order
of
magnitude
if
each state
Z
is
assigned a
small
region
of the order of
magnitude
of
perceptible
regions.
The order of
magnitude
of the
constant
is
determined
by taking
into
account
that for
the state
of
maximum
entropy