On the Kinetics of Multi-dimensional Fragmentation

TL;DR

This paper derives exact solutions for the kinetics of fragmentation of multi-dimensional hypercuboids, analyzing their scaling behavior and shattering transitions for different fragmentation rates and daughter distributions.

Contribution

It introduces two classes of exact solutions for multi-dimensional fragmentation models, expanding understanding of their scaling and shattering phenomena.

Findings

Identification of scaling solutions in both classes

Analysis of shattering transition conditions

Explicit forms of solutions for specific fragmentation rates

Abstract

We present two classes of exact solutions to a geometric model which describes the kinetics of fragmentation of $d$-dimensional hypercuboid-shaped objects. The first class of exact solutions is described by a fragmentation rate $a({x_1},...,{x_d}) = 1$ and daughter distribution function $b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}})= {{(\a_1 + 2)x_1^{\a_1}}\over{x_1^{\p(\a_1+1)}}}...{{(\a_d+2)x_d^{\a_d}}\over {x_d^{\p(\a_d+1)}}}$. The second class of exact solutions is described by a fragmentation rate $ a({x_1},...,{x_d}) = {{{x_1}^{\a_1}}...{{x_d}^{\a_d}}/{2^d}}$ and a daughter distribution function $b({x_1},..,{x_d} | {{x_{1}^{\p}}},...,{{x_{d}^{\p}}}) = {2^d}{\d(x_1 - {{x_{1}^\p}}/2)...\d(x_d - {{x_{d}^\p}}/2)}$. Each class of exact solutions is analyzed in detail for the presence of scaling solutions and the occurrence of shattering transitions; the results of these analyses are also presented.