Dynamics of interval fragmentation and asymptotic distributions

TL;DR

This paper analyzes the evolution and asymptotic behavior of fragmentation processes starting from a single element, deriving exact size distribution functions and exploring different power-law fragmentation probabilities.

Contribution

The study provides exact solutions for the size distribution in fragmentation processes and characterizes their asymptotic behavior based on the power-law exponent.

Findings

Distribution for large times and small sizes derived explicitly.

Asymptotic forms depend on the power-law exponent ta.

Different regimes (ta>1 and ta<1) exhibit distinct distribution behaviors.

Abstract

We study the general fragmentation process starting from one element of size unity (E=1). At each elementary step, each existing element of size $E$ can be fragmented into $k\,(\ge 2)$ elements with probability $p_k$. From the continuous time evolution equation, the size distribution function $P(E;t)$ can be derived exactly in terms of the variable $z= -\log E$, with or without a source term that produces with rate $r$ additional elements of unit size. Different cases are probed, in particular when the probability of breaking an element into $k$ elements follows a power law: $p_k\propto k^{-1-\eta}$. The asymptotic behavior of $P(E;t)$ for small $E$ (or large $z$) is determined according to the value of $\eta$. When $\eta>1$, the distribution is asymptotically proportional to $t^{1/4}\exp[\sqrt{-\alpha t\log E}][-\log E]^{-3/4}$ with $\alpha$ being a positive constant, whereas for $\eta<1$ it is proportional to $E^{\eta-1}t^{1/4}\exp[\sqrt{-\alpha t\log E}][-\log E]^{-3/4}$ with additional time-dependent corrections that are evaluated accurately with the saddle-point method.