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: $p_k\propto k^{-1-η}$. The asymptotic behavior of $P(E;t)$ for small $E$ (or large $z$) is determined according to the value of $η$. When $η>1$, the distribution is asymptotically proportional to $t^{1/4}\exp[\sqrt{-αt\log E}][-\log E]^{-3/4}$ with $α$ being a positive constant, whereas for $η<1$ it is proportional to $E^{η-1}t^{1/4}\exp[\sqrt{-αt\log E}][-\log E]^{-3/4}$ with additional time-dependent corrections that are evaluated accurately with the saddle-point method.

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