Beyond Zipf's law: Pore ranking in solids by Beta distributions

Abstract Zipfian laws of the form Frequency = f (Size) have been applied in the past for ranking collections of random pores in solids. Nevertheless, the results convey little information about the development of the compared quantities. In the present work we have probed the ranking of sizes R and frequencies N of pores according to their cardinality using Beta functions. This mathematical expression has strong physical correspondence with a Multiplication-Mutation model, proposed previously for the development of biological systems. The ranking of R's and N's provides two almost mirror sigmoid curves whose combination yields a Zipfian relation between N and R. In addition, the Beta ranking functions can be analyzed into two components, an exponential and a power one. The combination of those functions provides two different power laws corresponding to the statistical development of internal and external pores. Similar methodology can be applied in other systems described by Beta relations.