428
DOC.
27
DISCUSSION OF
DOC. 26
incorrect. Boltzmann's
equation
is
usually applied
in
the
following
way:
one
starts out
from
a specific
elementary
theory (e.g.,
molecular
mechanics),
determines the
probability
of
a
state
theoretically,
and
calculates
the
entropy
from
this
by means
of Boltzmann's
equation
in
order
to
learn,
finally,
the
thermodynamic properties
of
the
system.
But
one
can
also
proceed
in
the
reverse
direction: from the
empirically
ascertained thermal
behavior of the
system,
one
determines the
entropy
values
of the
individual
states,
and
from these
one
calculates
the
probabilities
of the
states
by
means
of Boltzmann's
equation.
To illustrate
this
way
of
applying
Boltzmann's
principle,
let
me use
the
following
[4]
example:
Suppose
a cylindrical
vessel
contains
a
liquid,
in which
there
shall
be
suspended
a
particle
whose
weight
exceeds
by
P the
liquid displaced by
it.
According
to
thermodynamics,
the
particle
should
sink
to
the bottom
and
remain there. From the
point
of
view
of the kinetic
theory
of
heat,
in
an
incessant
fluctuation the
particle
will
change
its
height
above the bottom
in
irregular succession,
without
ever coming
to rest.
To
lift
the
particle
to
the
height
z
above
the
bottom,
one
has to
perform
the work
Pz.
In order for the
energy
of the
system
not to
change
in this
process, one
must simulta-
neously
withdraw
from the
system
an
amount
of heat that
is
equivalent
to this
work, so
that the
entropy
of the
system
as a
function of the
height
z
of the
particle
is
expressed
by
Pz
S
=
const.
-
-. T
Using
Boltzmann's
equation, one
obtains
from this
the
probability
W
that the
particle
will
be found
at
the
height
z
at
an
arbitrary
instant of
time:
_Pz
W
=
Cew
[5]
This
is
the
law
that Perrin
actually
obtained
from his observations.
It
is
clear that
this
relation
expresses
the
state
of
affairs
established
by
Perrin
only
if
the
probability
W
has
been defined
in
the
manner
indicated
above.
This
simple example
also
provides
a
beautiful illustration of Boltzmann's
conception
Pz
of
an
irreversible
process.
Namely,
if
P
is
not
far
too
small,
then the
exponent
-
will
kT
be
of
considerable
magnitude
for
fairly large z
because of the
smallness
of
the constant
k
(=
R_N);
thus,
W will
be
small
and
will
decrease
very
rapidly
with
increasing
z.
If
one
raises
the
particle
to
a
certain
height
above
the bottom and then
leaves it
alone,
in
the
overwhelming
majority
of
cases
the
particle
will sink to
the bottom
in
an
almost
perpendicular
line
and
with almost constant
velocity
(an
irreversible
process
in
the
sense