Ofer Malcai

Is the Geometry of Nature Fractal

The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.

The Limited Scaling Range of Empirical Fractals

The notion of the abundance of fractals is critically re-examined in light of surprising data regarding the scaling range in empirical reports on fractality.

SCALING RANGE AND CUTOFFS IN EMPIRICAL FRACTALS

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\char21{}E and Letters) during the 1990s shows that experimental reports of fractal behavior are typically based on a scaling range

Theoretical analysis and simulations of the generalized Lotka-Volterra model

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Long-Time Fluctuations in a Dynamical Model of Stock Market Indices

that spans only 0.5\char21{}2 decades. This range is limited by upper and lower cutoffs either because further data are 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

Pattern formation and a clustering transition in power-law sequential adsorption.

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Co-Evolutionist Stochastic Dynamics: Emergence of Power Laws

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

Emergence of Fractal Clusters in Sequential Adsorption Processes

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The Forbes 400 and the Pareto wealth distribution

. 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 that generate spatial fractals.

POWER-LAW DISTRIBUTIONS AND LEVY-STABLE INTERMITTENT FLUCTUATIONS IN STOCHASTIC SYSTEMS OF MANY AUTOCATALYTIC ELEMENTS

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 individuals in a society, as well as of the market values of firms in the stock market. The individual wealths or market values are given by a set of time dependent variables w(i), i=1,...,N. The equations include a stochastic autocatalytic term (representing investments), a drift term (representing social security payments), and a time dependent saturation term (due to the finite size of the economy). The w(i)s turn out to exhibit a power-law distribution of the form P(w) approximately w(-1-alpha). It is shown analytically that the exponent alpha can be expressed as a function of one parameter, which is the ratio between the constant drift component (social security) and the fluctuating component (investments). This result provides a link between the lower and upper cutoffs of this distribution, namely, between the resources available to the poorest and those available to the richest in a given society. The value of alpha is found to be insensitive to variations in the saturation term, which 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