Information Technology Reference
In-Depth Information
differential equations that control the dynamics of individual particle trajectories. It is
equally clear that the connection between the fundamental laws of motion controlling
the individual particle dynamics and the observed large-scale dynamics cannot be made
in any straightforward way.
In a previous section we investigated the scaling properties of processes described
by certain stochastic differential equations. The scaling in the web response was a con-
sequence of the inverse power-law correlation. We now consider a fractional Langevin
equation in which the fractional derivatives model the long-time memory in the web
dynamics. It is determined that the solutions to such equations describe multifractal
statistics and we subsequently apply this model to a number of complex life phenomena,
including cerebral blood flow and migraines.
5.1
Transition to fractional dynamics
The differentiable nature of macroscopic physical phenomena, in a sense, is a natural
consequence of the microscopic randomness and of the related non-differentiability as
well, due to the key role of the central limit theorem (CLT). Recall that in the CLT the
quantities being added together are statistically independent, or at most weakly depen-
dent on one another, in order for the theorem to be applicable and normal statistics to
emerge. Once a condition of time-scale separation is established, in the long-time limit
the memory of the non-differentiable character of the microscopic dynamics is lost and
Gaussian statistics result. This also means that ordinary differential equations can again
be used on the macroscopic scale, even if the microscopic dynamics are incompatible
with the adoption of ordinary calculus methods.
On the other hand, if there is no time-scale separation between macroscopic and
microscopic levels of description, the non-differentiable nature of the microscopic
dynamics is transmitted to the macroscopic level. This can, of course, result from a
non-integrable Hamiltonian at the microscopic level so that the microscopic dynamics
are chaotic and the time scale associated with their dynamics diverges. Here we express
the generalized Langevin equation (
3.46
)as
2
t
dV
(
t
)
t
)
t
)
dt
+
ξ(
=−
0
ξ
(
t
−
V
(
t
),
(5.1)
dt
where
is the coupling coefficient between the Brownian particle velocity and the
ξ
(
ξ
(
booster,
is the autocorrelation function for the
doorway variable. We choose the autocorrelation function to be a hyperbolic distribution
of the form
t
)
is the doorway variable and
t
)
T
T
μ
−
1
ξ
(
t
)
=
(5.2)
+
t
and the microscopic time scale is given by
T
T
μ
−
1
∞
0
ξ
(
∞
T
μ
−
τ
=
t
)
dt
=
dt
=
(5.3)
+
t
2
0
