First-Digit Law in Nonextensive Statistics

Nonextensive statistics, characterized by a nonextensive parameter $q$, is a promising and practically useful generalization of the Boltzmann statistics to describe power-law behaviors from physical and social observations. We here explore the unevenness of the first digit distribution of nonextensive statistics analytically and numerically. We find that the first-digit distribution follows Benford’s law and fluctuates slightly in a periodical manner with respect to the logarithm of the temperature. The fluctuation decreases when $q$ increases, and the result converges to Benford’s law exactly as $q$ approaches 2. The relevant regularities between nonextensive statistics and Benford’s law are also presented and discussed.

💡 Research Summary

This paper investigates the relationship between nonextensive statistical mechanics, characterized by the entropic index (q), and Benford’s first‑digit law, which describes the logarithmic distribution of leading digits in many naturally occurring data sets. Nonextensive statistics, introduced by Tsallis, generalizes the Boltzmann–Gibbs formalism through the (q)-exponential function
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