Scaling Range and Cutoffs in Empirical Fractals

TL;DR

This paper investigates why experimental fractals typically exhibit a limited scaling range of only 0.5 to 2 decades, suggesting intrinsic system properties constrain the observed fractal behavior rather than experimental limitations.

Contribution

The study proposes a theoretical explanation for the limited scaling range in experimental fractals, linking it to intrinsic kinetic mechanisms in nonequilibrium aggregation systems.

Findings

Experimental fractal scaling ranges are mostly limited to less than two decades.

Scaling duration increases exponentially with the scaling range in aggregation systems.

Intrinsic properties of systems, not experimental setup, likely cause the limited scaling range.

Abstract

Fractal structures appear in a vast range of physical systems. A literature survey including all experimental papers on fractals which appeared in the six Physical Review journals (A-E and Letters) during the 1990's shows that experimental reports of fractal behavior are typically based on a scaling range $\Delta$ which spans only 0.5 - 2 decades. This range is limited by upper and lower cutoffs either because further data is not accessible or due to crossover bends. Focusing on spatial fractals, a classification is proposed into (a) aggregation; (b) porous media; (c) surfaces and fronts; (d) fracture and (e) critical phenomena. Most of these systems, [except for class (e)] involve processes far from thermal equilibrium. The fact that for self similar fractals [in contrast to the self affine fractals of class (c)] there are hardly any exceptions to the finding of $\Delta \le 2$ decades, raises the possibility that the cutoffs are due to intrinsic properties of the measured systems rather than the specific experimental conditions and apparatus. To examine the origin of the limited range we focus on a class of aggregation systems. In these systems a molecular beam is deposited on a surface, giving rise to nucleation and growth of diffusion-limited-aggregation-like clusters. Scaling arguments are used to show that the required duration of the deposition experiment increases exponentially with $\Delta$. Furthermore, using realistic parameters for surfaces such as Al(111) it is shown that these considerations limit the range of fractal behavior to less than two decades in agreement with the experimental findings. It is conjectured that related kinetic mechanisms that limit the scaling range are common in other nonequilibrium processes which generate spatial fractals.