DOC. 20 THEORETICAL ATOMISM
243
there
is
no
upper
limit below which the
velocity
of the Brownian motion
must
be
confined;
on
the
contrary,
all
velocities
must occur,
no
matter
how
great they may
be. But the
greater
the
velocity
considered,
the
less
frequently
it
occurs,
and the
frequency
of
occurrence
of
a
velocity
decreases
very rapidly
with
its
magnitude.
We
call this
frequency
of
occurrence
of
a
velocity
its
"probability."
If
we impart a
considerable
velocity
c
to
the
particle by
external
means,
we bring
it
thereby
into
a
state
of
very
low
probability.
How will this
velocity change
in the
short time
x
if the
particle
is
left
to
itself?
According
to
the kinetic
theory,
this
experiment,
which
we imagine
to
have been carried
out
many
times,
will
not
always
give
the
same
result.
In
one
subset of the
experiments,
the
velocity
of the
particle
after the
lapse
of time
t
will be
greater
than the initial
velocity
c
(first case);
in the
rest
of the
experiments,
the
velocity
after the
lapse
of time
x
will be smaller than
c
(second case). However,
it is
evident that the second
case
will
occur
much,
much
more
frequently
than the first
case;
because
according
to
what
we
have said
before,
for
a
particle
left
to itself,
smaller velocities
occur
much
more
frequently
(more
probably)
than do
greater ones.
If the
particle
is
somewhat
large,
these
frequencies
are so greatly
different that
it is
practically impossible
to observe
the first
case
at all.
Thus,
Boltzmann solves the contradiction in
question by showing
that
even
though
it is
possible
in
principle, according
to
the kinetic
theory,
for there
to
be
a process
that is the
reverse
of
a
thermal
process
that
is
irreversible
according
to
thermodymanics,
the
probability
that such
a
process
will
actually
occur
is
vanishingly
small.
Thus, according
to
Boltzmann,
the
average
laws of
experience
lend the
appearance
of
irreversibility
to
thermal
processes.
Generalizing, we
can
assert
the
following proposition:
The
changes
of
state
of
an
isolated
system
occur
in such
a
way
that
(on
average) more probable
states
succeed
less
probable
ones.
One
sees
that the
probability
of
a
state must
have
a
fundamental
[16]
thermodynamic significance.
In
fact,
Boltzmann
was
able
to
show that the
thermody-
namically
defined
entropy S
of
a
state is
directly
connected with the
probability W
of the
same
state
according
to
the
equation
S
=
-lg
w,
N
where
R
and
N
are
the
previously
discussed
constants,
and
lg
W
denotes the natural
logarithm
of the
probability
of the
state
(cf.
Article
32). [17]
This
equation
connects
thermodynamics
with the molecular
theory.
It
even
yields
the statistical
probabilities
of the
states
of such
systems
for which
we are
unable
to
construct
a
molecular-theoretical model.
Thus,
Boltzmann's
magnificent
idea
is
of
significance
for
theoretical
physics
not
only
because it eliminated
an
apparent
contradiction in the
theory,
but
also,
most
importantly,
because
it
provided
a
heuristic