Benford's Law in Statistical Physics

Benford's law is the observation that for many datasets, the distribution of their first significant digit follows a non-uniform law; the probability that is the leading digit is .
The Boltzmann–Gibbs (BG) and the Fermi–Dirac (FD) distributions are two frequently occurring statistical laws. This Demonstration shows that BG and FD distributions both fluctuate slightly in a periodic manner around the Benford distribution with the temperature of the system. The red points in the figure present Benford's law, while the blue ones present the first digit distributions of BG (or FD) statistics.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


In this Demonstration, the first control is , where is the Boltzmann constant and is the system's temperature. This Demonstration shows clearly that for both distributions, the distribution of the first digit conforms approximately to Benford’s law with only slight fluctuations. By changing , you will find that the blue points oscillate periodically around the stationary red points. Another statistical law, the Bose–Einstein (BE) distribution, conforms to Benford's law exactly at all temperatures and therefore is not included here.
[1] S. Newcomb, "Note on the Frequency of the Use of Digits in Natural Numbers," American Journal of Mathematics, 4, 1881 pp. 39–40.
[2] F. Benford, "The Law of Anomalous Numbers," Proceedings of the American Philosophical Society 78(4), 1938 pp. 551–572.
[3] L. Shao and B-Q. Ma, "The Significant Digit Law in Statistical Physics," Physica A 389, 2010 pp. 3109–3116.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+