Scale Invariance and Lack of Self-Averaging in Fragmentation,

P.L. Krapivsky, I. Grosse, and E. Ben-Naim

We derive exact statistical properties of a class of recursive fragmentation processes. We show that introducing a fragmentation probability p leads to a purely algebraic size distribution in one dimension, P(x)~ x^{-2p}. In d dimensions, the volume distribution diverges algebraically in the small fragment limit, P(V)\sim V^{-\gamma} with \gamma=2p^{1/d}. Hence, the entire range of exponents allowed by mass conservation is realized. We demonstrate that this fragmentation process is non-self-averaging. Specifically, the moments Y_\alpha=\sum_i x_i^{\alpha} exhibit significant fluctuations even in the thermodynamic limit.

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