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