Dynamics of interval fragmentation and asymptotic distributions

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$\alpha$beingapositiveconstant,whereasfor$\eta<1$ it is proportional to $E^{\eta -1}{t}^{1\mathrm{/}4}\exp [\sqrt{-\alpha t\log E}][-\log E{]}^{-3\mathrm{/}4}$ with additional time-dependent corrections that are evaluated accurately with the saddle-point method.