DOC.
16
131
Now
we
investigate
how
the coefficient
of
diffusion
depends on
Q,
restricting ourselves
again
to
the
case
that the
number
v
of particles
per
unit
volume
depends
only
on
x
and t.
Let
v
=
f(x,t)
be
the
number
of
particles
per
unit
volume;
we
then
calculate
the
distribution
of
the
particles
at
time
t+r from
their
distribution
at
time t.
From
the definition of the function
Q(A) we
can
easily
obtain the
number
of
particles
found at
time
t+r between
two
planes
perpendicular
to
the X-axis with abscissas
x
and
x+dx. We
obtain
f(x,t
+
r)dx
=
dx.
rA&-+00
A=-00
f(x
+
£±)(p(^dli
But
since
r
is
very
small,
we can
put
f{x,t
+
t)
=
f(x,t)
+ r
Further,
we
expand
f(x
+
A,t)
in
powers
of
A:
f(x
+
A.o
=
f(x,t)
+
A
g2£^'°...ad
inf.
We can
perform
this
expansion
under
the
integral
since
only
very
small values
of
A
contribute
anything
to
the latter.
We
obtain
r+oo
f
+
TFTT =
f
p(A)ZA
+
-oo
+00
oo
A^(A)rfA
+
|h(-
•+00
-oo
A2
d^
• «
On
the
right-hand
side,
the second, fourth,
etc., terms
vanish since
Q(x)
=
Q(-x),
while
among
the
first,
third, fifth,
etc., terms, each
subsequent term
is
very
small
compared
with the
one preceding
it.
From
this
equation
we
get,
by
taking
into
account
that
putting
r+oo
00
y(A)/A
=
1,
1
+0°
A 2
y^(A)/A =
D
-oo
[17]