Scale Invariance and Lack of Self-Averaging in Fragmentation

TL;DR

This paper derives exact statistical properties of recursive fragmentation processes, revealing algebraic size distributions, divergence in volume distributions, and demonstrating non-self-averaging behavior with significant fluctuations.

Contribution

It provides the first exact analytical characterization of size distributions and non-self-averaging properties in a class of fragmentation models.

Findings

Size distribution follows a power law P(x) ~ x^{-2p} in one dimension.

Volume distribution diverges algebraically as P(V) ~ V^{-eta} with eta=2p^{1/d}.

Fragmentation process exhibits non-self-averaging with large fluctuations in moments.

Abstract

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.