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In-Depth Information
webs of interest satisfy the conditions laid out above, but let us see where the present
line of argument takes us.
The manner in which problems are commonly formulated in diffusion theory is to
specify the initial distribution of the quantity of interest; here we continue to refer to
that quantity as particles, and use the diffusion equation to determine the evolution of
the web away from that initial state. An efficient way to solve the partial differential
equation is to use Fourier transforms. The Fourier transform of the number density in
one dimension is
n
(
k
,
t
)
,sofrom(
3.107
) we obtain
D
∞
−∞
2
n
∂
n
(
k
,
t
)
e
ikq
∂
(
q
,
t
)
=
dq
.
(3.111)
q
2
∂
t
∂
Assuming that the particle concentration and its gradient vanish asymptotically at all
times, that is,
∂
(
,
)/∂
|
q
=±∞
=
(
=±∞
,
)
=
,
n
q
t
q
n
q
t
0
we can integrate the equation
by parts twice to obtain
∂
n
(
k
,
t
)
Dk
2
=−
n
(
k
,
t
).
(3.112)
∂
t
The solution to this equation in terms of the initial value of the particle concentration is
e
−
Dk
2
t
(
,
)
=
(
,
)
.
n
k
t
n
k
0
(3.113)
Hence by inverting the Fourier transform we obtain
∞
e
−
ikq
e
−
Dk
2
t
dk
∞
−∞
1
2
e
iky
n
n
(
q
,
t
)
=
(
y
,
0
)
dy
.
(3.114)
π
−∞
A particularly important initial distribution has all the mass locally concentrated and
is given by a delta-function weight at a point
q
=
q
0
,
n
(
q
,
t
=
0
)
=
δ(
q
−
q
0
),
(3.115)
which, when substituted into (
3.114
), yields
∞
1
2
e
−
ik
(
q
−
q
0
)
e
−
Dk
2
t
dk
n
(
q
,
t
)
=
π
−∞
1
2
Dt
e
−
(
q
−
q
0
)
=
√
4
4
Dt
π
≡
P
(
q
−
q
0
;
t
).
(3.116)
The function
P
is the probability that a particle initially at
q
0
diffuses to
q
at
time
t
and is a Gaussian propagator with a variance that grows linearly in time. Here we
have assumed that the mass distribution is normalized to unity, thereby allowing us to
adopt a probabilistic interpretation. Consequently, we can write for the general solution
to the one-dimensional diffusion equation
(
q
−
q
0
;
t
)
∞
n
(
q
,
t
)
=
P
(
q
−
q
0
;
t
)
n
(
q
0
,
0
)
dq
0
,
(3.117)
−∞
which can be obtained formally by relabeling terms in (
3.114
).
