Dynamics of interval fragmentation and asymptotic distributions

摘要: We study the general fragmentation process starting from one element of size unity (E = 1). At each elementary step, existing E can be fragmented into k ( ⩾ 2) elements with probability pk. From continuous time evolution equation, distribution function P(E; t) derived exactly in terms variable z −log E, or without a source term that produces rate r additional unit size. Different cases are probed, particular when breaking an k follows power law: pk∝k−1 − η. The asymptotic behavior for small (or large z) is determined according to value When η > 1, asymptotically proportional α being positive constant, whereas < 1 it time-dependent corrections evaluated accurately saddle-point method.