Fragmentation energy

Motivated by a problem arising in the mining industry, we estimate the energy epsilon(eta) that is needed to reduce a unit mass to fragments of size at most eta in a fragmentation process, when eta -> 0. We assume that the energy used in the instantaneous dislocation of a block of size s into a set of fragments (s(1), s(2),...) is s(beta)phi(s(1)/s, s(2)/s,...), where phi 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 Crump-Mode-Jagers branching process (with alpha = 1 when the fragmentation is mass-conservative), then there exists a c is an element of (0, infinity) such that epsilon(eta) similar to c eta(beta-alpha) when beta < alpha. We also obtain a limit theorem for the empirical distribution of fragments of size less than eta 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.