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leading to the experimental determination of Avogadro's number in 1926. Perrin,
giving a physicist's view of mathematics in 1913, stated that curves without tangents
(derivatives) are more common than those special, but interesting ones, like the cir-
cle, that have tangents. In fact he was quite adamant in his arguments emphasizing
the importance of non-analytic functions for describing complex physical phenomena,
such as Brownian motion. Thus, there are valid physical reasons for looking for these
types of functions, but the scientific reasons became evident to the general scientific
community only long after the mathematical discoveries made by Weierstrass and other
mathematicians.
A phenomenological realization of the effect of these discrete processes is given by
random-walk models in which the moving entity, the random walker, is assumed to have
a prescribed probability of taking certain kinds of steps. We show that after
N
steps,
where
N
is a large number, the probability of being a given distance from the starting
point can be calculated. This strategy for modeling random phenomenon has a long
lineage and is very useful for understanding the properties of certain simple stochastic
processes such as the diffusion of ink in water, the spreading of rumors, the flow of heat
and certain kinds of genetic transfer, to name just a few. A simple random walk is then
generalized to capture the inverse power-law behavior that we have observed in so many
complex phenomena.
Finally, the influence of memory on the dynamics of linear complex webs is
discussed using the fractional calculus. This type of complexity is a manifestation
of the non-locality of the web's dynamics, so that the complete history of the web's
behavior determines its present dynamics. This history-dependence became even more
significant when the web was again embedded in an environment that introduces
a random force. The resulting fractional random walk is a relatively new area of
investigation in science [
44
].
A number of the analyses pursued in foregoing chapters had to be cut short because
some of the technical concepts associated with random variables, such as the definition
of a probability density, had not been properly introduced. As we know, it is difficult
to proceed beyond superficial pleasantries without a proper introduction. To proceed
beyond the superficial, we now present the notion of probabilities, probability densities,
characteristic functions and other such things and replace the dynamics of web
observables with the dynamics of web distribution functions. Here again we strive
more for practicality than for mathematical elegance. We begin with the simplest
of probability arguments having to do with random walks, which form the intuitive
basis of probability theory in the physical sciences. From there we proceed to the
more mathematically sophisticated concept of chaos as another way of introducing
uncertainty into the modeling.
The concept of chaos, as mentioned, was developed as a response to scientists' inabil-
ity to obtain closed-form solutions in any but a handful of nonlinear ordinary differential
equations. This inability to obtain analytic solutions required the invention of a totally
different way of examining how the dynamics of a complex web unfolds. The methods
for analyzing general nonlinear dynamic webs are ultimately based on mapping tech-
niques to provide visualization of the underlying trajectories. Consequently, we begin
