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

• Published in: Microporous and mesoporous materials 2021-04, Vol.317, p.110987, Article 110987
• Main Authors: Margellou, Antigoni G., Pomonis, Philippos J.
• Format: Article
• Language: English
• Subjects: Beta functions; Multiplication-mutation model; Pore ranking; Zipf law
• ISSN: 1387-1811; 1873-3093
• DOI: 10.1016/j.micromeso.2021.110987

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. [Display omitted] •The frequencies N and radii R of random pores have been ranked by Beta functions.•The Beta relations correspond to an Expansion–Mutation model used for biosystems.•The results provide two mirror sigmoid curves whose combination yields Zipf laws.•The Beta relations were split into their exponential and power law components.•These components yield power laws describing the internal and external porosity.