132
MOVEMENT OF SMALL
PARTICLES
and
only considering
the first
and
third
term
of the
right-hand
side:
[18] (1)
This is the familiar differential
equation
for
diffusion,
and
D
can
be
recognized
as
the diffusion coefficient.
Another
important
consideration
can
be
linked
to
this
development.
We
assumed
that all the individual particles
are
referred
to
the
same
coordinate
system.
However,
this is
not
necessary
since the
motions
of the individual
particles
are
mutually independent.
We
will
now
refer the
motion
of
each
particle
to
a
coordinate
system whose
origin
coincides
at
time
t
=
0
with
the
position of
the
center of gravity
of the
particle
in question, with the
difference that f(x,t)dx
now
denotes
the
number
of
particles
whose
X-coordinate
has
increased
between
the times
t
=
0
and
t
=
t
by a
quantity
lying between
x
and
x
+
dx. Thus,
the function
f
varies
according to
equation
(1)
in
this
case as
well.
Further, it
is
obvious
that for
x
0
and
t
=
0
we
must have
•+00
f(x,t)
=
0
and
f(x,t)dx
=
n
00
The
problem,
which
coincides with the
problem
of
diffusion
from
one
point
(neglecting
the interaction
between
the
diffusing
particles),
is
now
completely
determined
mathematically;
its solution is
x2
f(x,t)
=
n
^
4
iD
[t
The
frequency
distribution
of
the
changes
of
position
occurring
in the
arbitrary
time
t
is
thus the
same as
the distribution
of
random errors,
[19]
which
was
to
be
expected.
What
is of
importance, however,
is
how
the
constant
in the
exponent
is related
to
the coefficient of diffusion.
With
the
help
of
this
equation
we
now
calculate the
displacement
yX
in
the direction
of
the
X-axis that
a
particle
experiences
on
the
average,
or,
to
be
more
precise,
the