Benford's law, Zipf's law and the pore properties in solids

Abstract

Abstract Benford's and Zipf's Laws have been investigated in large data sets of pore properties of 206 lab-made porous solids including spinels, aluminas, silicas, aluminophosphates, aluminometalates, MCM and SBA and materials. Such properties, like the mean pore diameters, the mean pore lengths and the mean pore anisotropies, were obtained by combining the specific surface areas and the specific pore volumes of the solids. All those parameters exhibit a distribution around a central value and their first digits obey, more or less, Benford's Law. Compliance with the Law depends on the spread of each distribution and improves exponentially with its standard deviation. The above data-sets do not follow Zipf’ Law because they refer to totally independent systems without any internal coherence property linked to their evolution. On the contrary Zipf's law holds over many orders of magnitude for the differential pore numbers and the differential pore radii of 324 experimental points estimated for lab made spinels and MCM silica, volcanic and magmatic porous rocks and a typical soil. The underlying reason is that those data sets exhibit an internal property of coherency which is the mechanism of pore development. This Zipfian behavior leads to impressive Benford distributions of the first digits of the above two differential quantities. It is assumed that extended Zipfian distributions lead to good Benford's Laws, but the reverse is not true since the phenomenon of Benfordness can also be observed from other distributions as far as they are sufficiently populated and extensively spread.