182
REVIEW
OF
BROWNIAN MOTION
N
F
-
Yf
Fj
(2) Ada
=
Ce
dp1...dpn,
da
where the integral
on
the
right-hand side is
extended
over
all combinations
of
those
state
variables
whose
value of
a
lies
between
a
and
a
+
da.
We
will confine ourselves
to
the
case
in which
the
nature
of the
problem
makes
it
immediately
evident
that
all (possible)
values of
a
have
the
same
[9]
probability
(frequency),
i.e., that the quantity
A
is
independent
of
a.
Imagine
now a
second
physical system
that differs
from
the
system
just
considered
by
the sole fact that it is acted
upon by a
force
of
potential
$(0), which
depends
only
on a.
If
E
is the
energy
of
the
system
considered
earlier, then
E
+
$
will
be
the
energy
of
the
system
considered
now, so
that
we
get
the
following
relation,
analogous to equation (1):
-
4f
(*+*)
dw1
=
C]
e dp1...dpn.
This, in
turn,
yields
a
relation
analogous
to equation
(2)
for the
probability
dW
that
at
an
arbitrarily
chosen
instant the value
of
a
will
lie
between
a
and
a +
da:
(I)
d¥
=
C'e
N
RT
(£+*)
dp
|...
dp
n
=
n V
6
11
e
N
RT
£
RT
$
Ada
§
da,
where
A'
is
independent
of
a.
This
relation,
which
corresponds exactly
to
the
exponential law
used
frequently
by
Boltzmann in
his
investigations
on
the
theory
of
gases,
is
[10]
characteristic for the molecular
theory
of
heat. It determines
how much
a
parameter
of
a
system
subjected to
a
constant
external
force
diverges
from
the
value
corresponding to
stable
equilibrium because of
random
molecular
motion.