A notion which has been advanced intensively in the past two decades, is that fractal geometry describes well the irregular face of Nature. We were prompted by Marder's recent article in Science [1], to comment here on the applicability of this wide-spread notion.Marder summarizes a simulation study of fractured silicon nitride by Kalia et. al [2] which successfully mimics experimental data, and emphasizes the role of fractal geometry in describing complex-geometry physical structures, in general. Specifically, the results of Kalia's et. al were interpreted as "showing that this mechanism... leads to fractal fracture surfaces". However, upon examining Kalia's results (Fig. 4 in Ref. [2]) one finds that Marder's statement is based on four exponents, all of which hold over less than one order of magnitude.We recall that a fractal object, in the purely mathematical sense, requires infinitely many orders of magnitude of the power law scaling, and that a consequent interpretation of experimental results as indicating fractality requires, "many" orders of magnitude. We also recall that, for instance, in the celebrated fractal Koch flake, one order of magnitude means about two iterations in the construction and that such two-iterations Koch curve is not a fractal object. It is our feeling that Marder, like many others in the scientific community, may have been swayed by the wide spread image and belief that many-orders fractality abounds in experimental documentation.In a recent detailed statistical data analysis we have shown that this is not the case, at 1 least in the original sense of the concept [3]: We found that reported experimental fractality in a wide range of physical systems is typically based on a scaling range which spans over only 0.5 -2.0 decades. The survey was based on all experimental papers reporting fractal analysis of data which appeared over a period of seven years in all Physical Review journals (Phys. Rev. A to E and Phys. Rev. Lett., 1990Lett., -1996. In these papers an empirical fractal dimension, D, was calculated from various relations between a property, P , and the resolution, r, of the general form( 1) where k is the prefactor for the power law and the exponent is a simple function of D.In most cases, fitting the data to Eq.(1) was done through its linear log-log presentation.Typically, the range of the linear behavior terminated on both sides either because further data is not accessible or due to crossover bends. A histogram of the number of orders of magnitude used to declare fractality, covering all of the 96 relevant reports, was prepared and is reproduced in Fig. 1. A clear picture emerges from it: the scaling range of experimentally declared fractality is extremely limited, centered around 1.3 orders of magnitude, spanning mainly, as mentioned above, between 0.5 and 2.0 [4]. This stands in stark contradistinction to the public image of the status of experimental fractals.It seems that the most acute questions to be asked in view of this data are: Is the limited range inherent?; ...

No 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 ∆ 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 ∆ ≤ 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 ∆. 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.

The dynamics of generalized Lotka-Volterra systems is studied by theoretical techniques and computer simulations. These systems describe the time evolution of the wealth distribution of namely between the resources available to the poorest and those available to the richest in a given society. The value of α is found to be insensitive to variations in the saturation term, that represent the expansion or contraction of the economy. The results are of much relevance to empirical studies that show that the distribution of the individual wealth in different countries during different periods in the 20th century has followed a power-law distribution with 1 < α < 2.

A generic model of stochastic autocatalytic dynamics with many degrees of freedom wi, i = 1, . . . , N is studied using computer simulations. The time evolution of the wi's combines a random multiplicative dynamics wi(t + 1) = λwi(t) at the individual level with a global coupling through a constraint which does not allow the wi's to fall below a lower cutoff given by c ·w, wherew is their momentary average and 0 < c < 1 is a constant. The dynamic variables wi are found to exhibit a power-law distribution of the form p(w) ∼ w −1−α . The exponent α(c, N ) is quite insensitive to the distribution Π(λ) of the random factor λ, but it is non-universal, and increases monotonically as a function of c. The "thermodynamic" limit N → ∞ and the limit of decoupled free multiplicative random walks c → 0 do not commute: α(0, N ) = 0 for any finite N while α(c, ∞) ≥ 1 (which is the common range in empirical systems) for any positive c. The time evolution ofw(t) exhibits intermittent fluctuations parametrized by a (truncated) Lévy-stable distribution Lα(r) with the same index α. This non-trivial relation between the distribution of the wi's at a given time and the temporal fluctuations of their average is examined and its relevance to empirical systems is discussed.