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On the other hand, we know that the Gaussian distribution can be written in
normalized form as
q
2
1
e
−
.
q
2
P
(
q
)
=
2
2
(3.118)
q
2
π
However, multiplying (
3.107
)by
q
2
and integrating the rhs by parts yields
q
2
d
=
2
D
,
(3.119)
dt
which integrates to
q
2
t
;
=
2
Dt
,
(3.120)
and therefore the Gaussian can be written
q
2
1
e
−
,
q
2
P
(
q
,
t
)
=
2
2
;
t
(3.121)
q
2
π
;
t
with the second moment given by (
3.120
).
It is also interesting to note that (
3.121
) can be derived from the scaled form of the
probability density
F
q
t
δ
1
t
δ
(
,
)
=
P
q
t
(3.122)
/
√
t
,
by setting
δ
=
1
/
2, in which case, with
y
=
q
1
√
4
y
2
4
D
e
−
F
(
y
)
=
.
(3.123)
π
D
This is the property of scaling in the context of the probability density; in fact, when
a stochastic variable is said to scale, this is what is meant. It is not literally true that
Q
(
H
Q
, but rather the scaling occurs from the probability density of the form
(
3.122
) and the relation between the two scaling parameters
λ
t
)
=
λ
(
t
)
δ
and
H
depends on the
functional form of
F
.
Thus, when the physical conditions for simple diffusion are satisfied we obtain the
distribution of Gauss. From this we can infer that the complex web phenomena that
do not have Gauss statistics do not satisfy these physical conditions and we must look
elsewhere for unifying principles. So let us generalize our considerations beyond the
simple diffusion equation and investigate the behavior of the probability densities for
more general dynamical networks.
(
·
)
3.3.1
The Fokker-Planck equation (FPE)
The stochastic differential equations distinguish between the deterministic forces acting
on a particle through a potential and the random forces produced by interactions with
the environment. In the present discussion of a single-variable equation of motion the
dynamical variable is denoted by
Q
(
t
)
, the Langevin equation is
