According to Benford's first digit law, the frequency of appearance of the first digit d taken from various sets of data is log(1+(1/d)) where d = 1, 2,…9. I present a thermodynamic derivation of this law that follows from the unbiased partitioning of a conserved quantity whose pieces may fragment and combine.

The frequency of the first digits of numbers drawn from an exponential probability density oscillate around the Benford frequencies. Analysis, simulations and empirical evidence show that datasets must have at least 10,000 entries for these oscillations to emerge from finite-sample noise. Anecdotal evidence from population data is provided.

We demonstrate that the various equilibria that can be realized by two, identical, inflated, party balloons, each attached to one end of a rigid plastic tube, are stable—a demonstration based on A N Gent’s equation of state, which relates the pressure of the air contained with a thin elastic sphere to the sphere radius. Measurements on 12 inch party balloons illustrate these stable equilibria.

A Student's Guide to Entropy - August 2013