Scale invariance and lack of self-averaging in fragmentation

We derive exact statistical properties of a recursive fragmentation process. We show that introducing a fragmentation probability 0<p<1 leads to a purely algebraic size distribution, P(x) approximately x(-2p), in one dimension. In d dimensions, the volume distribution diverges algebraically in the small fragment limit, P(V) approximately 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 as the moments Y(alpha)= summation operator(i)x(alpha)(i) exhibit significant sample to sample fluctuations.