Scale Invariance and Lack of Self-Averaging in Fragmentation

We derive exact statistical properties of a class of recursive fragmentation processes. We show that introducing a fragmentation probability 0<p<1 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.