Fragmentation energy

Motivated by a problem arising in mining industry, we estimate the energy ${\cal E}(\eta)$ which is needed to reduce a unit mass to fragments of size at most $\eta$ in a fragmentation process, when $\eta\to0$. We assume that the energy used by the instantaneous dislocation of a block of size $s$ into a set of fragments $(s_1,s_2,...)$, is $s^\beta \varphi(s_1/s,s_2/s,..)$, where $\varphi$ is some cost-function and $\beta$ a positive parameter. Roughly, our main result shows that if $\alpha>0$ is the Malthusian parameter of an underlying CMJ branching process (in fact $\alpha=1$ when the fragmentation is mass-conservative), then ${\cal E}(\eta)\sim c \eta^{\beta-\alpha}$ whenever $\beta < \alpha$. We also obtain a limit theorem for the empirical distribution of fragments with size less than $\eta$ which result from the process. In the discrete setting, the approach relies on results of Nerman for general branching processes; the continuous setting follows by considering discrete skeletons. We also provide a direct approach to the continuous setting which circumvents restrictions induced by the discretization