Abstract | 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(≥2) elements with probability pk. From the continuous time evolution equation, the size distribution function P(E;t) can be derived exactly in terms of the variable z=−logE, 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: pk∝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 t1/4exp[−αtlogE‾‾‾‾‾‾‾‾‾√][−logE]−3/4 with α being a positive constant, whereas for η<1 it is proportional to Eη−1t1/4exp[−αtlogE‾‾‾‾‾‾‾‾‾√][−logE]−3/4 with additional time-dependent corrections that are evaluated accurately with the saddle-point method. |