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From these three equations one can derive two basic differential equations of motion
for the physical or appropriate biological network, one for the particle density and the
other for the total particle velocity. Traditional theory emphasizes the diffusion equation,
which is obtained by substituting (
3.102
)into(
3.104
) to obtain
∂
n
(
r
,
t
)
=∇·[
D
∇
n
(
r
,
t
)
]
.
(3.105)
∂
t
When the diffusion coefficient is independent of both space and time (
3.105
) simpli-
fies to
∂
(
,
)
n
r
t
2
n
=
D
∇
(
r
,
t
),
(3.106)
∂
t
which in one spatial dimension becomes
2
n
∂
n
(
q
,
t
)
D
∂
(
q
,
t
)
=
.
(3.107)
q
2
∂
t
∂
Once the particle density has been determined from the diffusion equation, the current
j
(
r
can be determined from (
3.102
).
An alternative way to describe diffusion is through the velocity field
v
,
t
)
Once
the velocity is known, the particle density can be obtained by combining (
3.102
) and
(
3.103
). In the case of a constant diffusion coefficient we can write
(
r
,
t
).
v
(
r
,
t
)
=−
D
∇
log
n
(
r
,
t
).
(3.108)
A differential equation for
v
(
r
,
t
)
is obtained by noting from (
3.103
) and (
3.104
) that
∂
v
(
r
,
t
)
∇
∂
∂
)
−
1
=−
D
log
n
(
r
,
t
)
=
D
[
n
(
r
,
t
∇·
n
(
r
,
t
)
v
(
r
,
t
)
]
∂
t
t
=
D
∇[∇ ·
v
(
r
,
t
)
+
v
(
r
,
t
)
·∇
log
n
(
r
,
t
)
]
,
(3.109)
so that inserting (
3.108
)into(
3.109
) yields
∂
v
(
r
,
t
)
=∇[
D
∇·
v
(
r
,
t
)
−
v
(
r
,
t
)
·
v
(
r
,
t
)
]
.
(3.110)
∂
t
Therefore, even elementary diffusion theory is nonlinear if one formulates the theory
in terms of the “wrong” variable. Here the equation for the velocity is quadratic in the
velocity. From this construction it is obvious that the solution to the quadratically non-
linear partial differential equation for the velocity field is proportional to the gradient of
the logarithm of the solution of the classical linear diffusion equation given by (
3.108
).
The relationships between the diffusion equation and a number of other nonlinear equa-
tions have been recognized over the years and were discussed in this context over thirty
years ago by Montroll and West [
34
].
Much of our previous attention focused on the distribution of a quantity of interest,
whether it is the number or magnitude of earthquakes, how wealth is distributed in a
population, how influence is concentrated, or the number of papers published by scien-
tists. We are now in a position to calculate the probability density associated with these
phenomena from first principles. The first principles are physically based and not all the
