Geometric statistics and dynamic fragmentation

TLDR

Dynamic fragmentation processes produce particle size distributions that have been reviewed in prior work. The study seeks to model fragmentation as a random Poisson process, develop Mott’s theory and an alternative Poisson‑based model, extend the theory with statistical heterogeneity, and explore a maximum‑entropy application. The authors analyze one‑dimensional Poisson fragmentation, two‑ and three‑dimensional area and volume fragmentation, compare Mott’s theory and their Poisson model with analytic, Voronoi, Johnson–Mehl, and other computational studies, and benchmark the results against dynamic fragmentation data. Size distributions from random geometric fragmentation depend on the construction method, preventing a definitive choice between the two proposed distributions.

Abstract

The present study is focused on the distributions in particle size produced in dynamic fragmentation processes. Previous work on this subject is reviewed. We then examine the one-dimensional fragmentation problem as a random Poisson process and provide comparisons with expanding ring fragmentation data. Next we explore the two-dimensional (area) and, less extensively, the three-dimensional (volume) fragmentation problem. Mott’s theory of random area fragmentation is developed, and we propose an alternative application of Poisson statistics which leads to an exponential distribution in fragment size. Both theoretical distributions are compared with analytic and computer studies of random area geometric fragmentation problems, including those suggested by Mott, the Voronoi construction, a variation of the Johnson–Mehl construction, and several methods of our own. We find that size distributions from random geometric fragmentation are construction dependent, and that a conclusive choice between the two distributions cannot be made. A tentative application of the maximum entropy principle to fragmentation is discussed. The statistical theory is extended to include a concept of statistical heterogeneity in the fragmentation process. Finally, comparisons are made with various, dynamic fragmentation data.