Scaling Theory of Fragmentation

Abstract
We develop a scaling theory for linear fragmentation processes, for general breakup kernels characterized by a homogeneity index λ>0. We discuss the existence of scaling, and show that the scaled cluster-size distribution φ(x) generally decays with the scaled mass x as x2exp(axλ) as x. For small x, φ(x) approaches the log-normal form, exp(aln2x), if the kernel has a small-size cutoff, and a power-law form in the absence of a cutoff. We also show that λ<0 leads to a shattering transition. Finally, we outline the essential features of a nonlinear fragmentation process.

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