Models of breakage and selection for particle size distributions

It is generally agreed that particle size distributions of sediments tend ideally to approximate the form of the lognormal probability law, but there is no single widely accepted explanation of how sedimentary processes generate the form of this law. Conceptually, and in its simplest form, sediment genesis involves the transformation of a parent rock mass into a particulate end product by processes that include size reduction and selection during weathering, transportation, and deposition. The many variables that operate simultaneously during this transformation can be shown to produce a distribution of particle sizes that approaches asymptotically the lognormal form when the effect of the variables is multiplicative. This was first shown by Kolmogorov (1941). Currently available models combine breakage and selection in differing degrees, but are similar in treating the processes as having multiplicative effects on particle sizes. The present paper, based on careful specification of the initial state, the nth breakage rule and the nth selection rule, leads to two stochastic models for particle breakage, and for both models the probability distributions of particle sizes are obtained. No attempt is made to apply these models to real world sedimentary processes, although this topic is touched upon in the closing remarks.