Fractals and fragmentation

If the number-size distribution of objects satisfies the condition N ∼ r−D, then a fractal is defined with a fractal dimension D. In many cases, fragmentation results in a fractal distribution. This is taken as evidence that the fragmentation mechanism is scale invariant. Fragments produced by weathering, explosions, and impacts often satisfy a fractal distribution condition over a wide range of scales. Most correlations for number versus size for meteorites, asteroids, and interstellar grains also satisfy the fractal condition. Fractal behavior implies scale invariance; the renormalization group approach is often applicable to scale invariant processes. Two models are considered for a renormalization group approach to fragmentation; the models yield a fractal behavior but give different values for the fractal dimension. These results indicate that the fractal dimension is a measure of the fragility of the fragmented material.

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