Non-uniform breakage-mechanism branching processes and degradation of long-chain polymers

Abstract

We consider the problem of number and weight distributions for breakage-mechanism branching processes whose break distributions are general. We derive a recursive relation between the expected empirical distribution after (n + 1) breaks and after n breaks, making use of length-biased sampling. Using this relation and the strong law of large numbers, we derive integro-differential equations for the asymptotic expected empirical distribution and its associated weight distribution. The mean of the asymptotic number distribution is derived using the integro-differential equation. We then provide approximate solutions to these equations and the moments of these approximations. Finally we apply these results to the case where the break distribution is a symmetric beta distribution.