Publication Numbers: 26,28.
Complex structures are common in nature. Their
analysis and
characterization is a difficult task, particularly
in the
case of structures which cannot be characterized
by one or
a few length scales.
The fractal concepts appear ideal for describing
such multi-scale
phenomena and the dynamics that generates them.
For example, in surface growth phenomena one
commonly observes
rough surfaces as well as cluster structures
which can be characterized
by their fractal dimensions.
Due to the reasons described above,
there has been much experimental effort
in the past fifteen years or so
to explore fractal structures in empirical systems.
Concurrently, models that give rise to a variety
of fractal structures
were introduced and studied extensively.
While these models typically exhibit scale-invariant
fractal structures,
empirical fractals tend to be bounded by upper
and lower cutoffs.
To examine the experimental status of the cutoffs
we performed a survey of experimental studies
and found that
experimental reports of fractal behavior are
typically based on a scaling range
$\Delta$
which spans less than two decades with very few
exceptions.
To obtain insight about this we have analyzed
the growth of diffusion-limited-aggregation-like
clusters
on surfaces in molecular beam epitaxy [26].
Using scaling relations for island growth
we obtained the dependence of the upper cutoff
on the duration
of the experiment. We obtained indications that
in these systems
the limited range may result from the fact that
the scaling range
(in decades) increases slowly (logarithmically)
with the duration
of the experiment, namely the experimental time
increases exponentially
as a function of
$\Delta$.
Interestingly, this resembles the situation in
the theory of
algorithmic complexity, where for intractable
problems the
computation time depends exponentially on the
input size