Investigation of the kinetic equation of cascade fragmentation theory at not self-similar subdivision

TL;DR

This paper studies a non-self-similar cascade fragmentation process, deriving an evolution equation for the probability distribution and analyzing its long-term behavior, which differs from classical Kolmogorov law.

Contribution

It introduces a new kinetic equation for cascade fragmentation with a specific probability density form and analyzes its asymptotic solution.

Findings

Derived the evolution equation for the probability distribution density.

Found the limit solution at large time for a specific probability density.

Demonstrated the solution differs from the Kolmogorov law.

Abstract

The cascade kinetic fragmentation process of solids is investigated when the condition probability density of splinter formation do not depends on time and has the property $P(\rho, r, t) = P(\rho/r)$. It is obtained the evolution equation for the probability distribution density in terms of its Mellin transformation. In the particular case $P(\rho/r) = C (\rho/r)^\alpha$, the limit solution of this equation at $t \to \infty$ is found. It differs essentially from the Kolmogorov law.