Stochastic modeling on fragmentation process over lifetime and its dynamical scaling law of fragment distribution

TL;DR

This paper introduces a stochastic fragmentation model considering fragment lifetime as a size-dependent function, revealing conditions under which the fragment size distribution obeys a dynamical scaling law.

Contribution

It develops a new stochastic model incorporating size-dependent fragment lifetime and derives conditions for the emergence of a dynamical scaling law in fragment distributions.

Findings

Size distribution collapses into a single curve when lifetime is a power function of size.

Distribution does not obey scaling law if lifetime is logarithmic in size.

Scaling law holds under specific power-law lifetime conditions.

Abstract

We propose a stochastic model of a fragmentation process, developed by taking into account fragment lifetime as a function of their size based on the Gibrat process. If lifetime is determined by a power function of fragment size, numerical results indicate that size distributions at different times can be collapsed into a single time-invariant curve by scaling size by average fragment size (i.e., the distribution obeys the dynamical scaling law). If lifetime is determined by a logarithmic function of fragment size, the distribution does not obey the scaling law. The necessary and sufficient condition that the scaling law is obeyed is obtained by a scaling analysis of the master equation.