The Significant Digit Law in Statistical Physics

TL;DR

This paper explores how Benford's law applies to key physical distributions, revealing that Bose-Einstein distribution always follows it exactly, while Boltzmann-Gibbs and Fermi-Dirac distributions fluctuate around it.

Contribution

It demonstrates the universal applicability of Benford's law to fundamental physical statistics and uncovers their specific behaviors relative to the law.

Findings

Bose-Einstein distribution conforms exactly to Benford's law at all temperatures.

Boltzmann-Gibbs and Fermi-Dirac distributions fluctuate around Benford's law with temperature.

Benford's law may be a fundamental pattern in physical statistics.

Abstract

The occurrence of the nonzero leftmost digit, i.e., 1, 2, ..., 9, of numbers from many real world sources is not uniformly distributed as one might naively expect, but instead, the nature favors smaller ones according to a logarithmic distribution, named Benford's law. We investigate three kinds of widely used physical statistics, i.e., the Boltzmann-Gibbs (BG) distribution, the Fermi-Dirac (FD) distribution, and the Bose-Einstein (BE) distribution, and find that the BG and FD distributions both fluctuate slightly in a periodic manner around the Benford distribution with respect to the temperature of the system, while the BE distribution conforms to it exactly whatever the temperature is. Thus the Benford's law seems to present a general pattern for physical statistics and might be even more fundamental and profound in nature. Furthermore, various elegant properties of Benford's law, especially the mantissa distribution of data sets, are discussed.