The first-digit frequencies in data of turbulent flows

Abstract

Considering the first significant digits (noted d) in data sets of dissipation for turbulent flows, the probability to find a given number (d=1 or 2 or... 9) would be 1/9 for an uniform distribution. Instead the probability closely follows Newcomb-Benford's law, namely P(d)=log(1+1/d). The discrepancies between Newcomb-Benford's law and first-digits frequencies in turbulent data are analysed through Shannon's entropy. The data sets are obtained with direct numerical simulations for two types of fluid flow: an isotropic case initialized with a Taylor-Green vortex and a channel flow. Results are in agreement with Newcomb-Benford's law in nearly homogeneous cases and the discrepancies are related to intermittent events. Thus the scale invariance for the first significant digits, which supports Newcomb-Benford's law, seems to be related to an equilibrium turbulent state, namely with a significant inertial range. A matlab/octave program is provided in appendix in such that part of the presented results can easily be replicated

First Digits’ Shannon Entropy

“…Many—though not all—distributions follow the NB Law more or less closely (the population of municipalities, microorganisms in a culture, spread of internet contents, capital growth, wages and prices, loss of value over time, street addresses [ 12 ], page numbers in literature citations [ 14 ]). Biau [ 15 ] analyses the discrepancies between the NB Law and first-digits frequencies in data of turbulent flows through Shannon’s entropy.…”

A New Stylometry Method Basing on the Numerals Statistic

“…Incomplete understanding [3] does not preclude the successful use of Benford's law to detecting fraud in accounting and auditing data [4] and election fraud [5]; the applications suggested extend from physics and astronomy [6,7] through seismology [8] to steganography [9] and scientometrics [10].…”