Green’s function, and is useful for finding concentration profiles for more
complicated initial conditions.
In particular, consider that the initial con
centration distribution is given by
c
(
x,
0) =
δ
(
x
), where
δ
(
x
) is the Dirac
delta function and basically means that there is a spike at the origin.
In
particular, you will show that
G
(
x, t
) =
1
√
4
πDt
e

x
2
4
Dt
,
(2)
where we introduce the notation
G
(
x, t
) to signify that this is the concentra
tion profile for the special case in which the initial concentration is the spike
at the origin as represented by the delta function.
To obtain the solution, we will Fourier transform the diffusion equation
in the spatial variable
x
according to the Fourier transform convention
˜
f
(
k
) =
1
2
π
Z
∞
∞
f
(
x
)
e

ikx
dx,
(3)
and
f
(
x
) =
Z
∞
∞
˜
f
(
k
)
e
ikx
dk.
(4)
Using these definitions, Fourier transform both sides of the diffusion equation
to arrive at the ordinary differential equation
d
˜
c
(
k, t
)
dt
=

Dk
2
˜
c
(
k, t
)
.
(5)
Solve this differential equation to obtain ˜
c
(
k, t
) and make sure to use the
initial condition
c
(
x,
0) =
δ
(
x
) to find ˜
c
(
k,
0). Then invert the Fourier trans
form on ˜
c
(
k, t
) to find
c
(
x, t
).
NOTE: You will need to use completion of
2
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the square to carry out the inversion.
Make sure you explain all of your
steps. We are big on having you not only do the analysis correctly, but also
to explain what you are doing and why you are doing it. Also, explain why
I said this is the solution for “free space”. Why would this solution fail to
describe diffusion in a finite box?
(b) Using the solution we obtained above, find
h
x
i
and
h
x
2
i
. In general, we
have that
h
x
n
i
=
R
∞
∞
x
n
c
(
x, t
)
dx
R
∞
∞
c
(
x, t
)
dx
.
(6)
Explain what you find for both the first and second moments of the distribu
tion as a function of time and explain how it relates to the estimated diffusion
time
t
=
L
2
/D
which we use to find the time scale for diffusion over a length
L
. Using the EinsteinStokes relation given by
D
=
k
B
T
6
πηa
,
(7)
where
η
is the viscosity which for water is
η
water
= 10

3
Pa s
and
a
is the
radius of the diffusing particle, estimate the diffusion constant for a protein
in water and make a loglog plot of diffusion time vs distance (with distances
ranging from 1 nm to 1 m) and comment on its biological significance. Also,
make a plot of the solution for the point source as a function of time by
showing
c
(
x, t
) at various times
t
using the same diffusion constant.
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 Winter '09
 Fourier Series, Dirac delta function, PBOC, pnr pl −nr

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