# On the Kinetics of Multi-dimensional Fragmentation

Source: [On the Kinetics of Multi-dimensional Fragmentation](<https://www.sota2.com/research/paper/68fa24422c2a3760193a1afd>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-24T09:00:16.554Z

Authors: P. Singh, M. K. Hassan

Published: 1996-07-04T11:10:25.000Z

## 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.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/cond-mat/9607027>)

[PDF](<https://arxiv.org/pdf/cond-mat/9607027.pdf>)