Equipartitions and a distribution for numbers: A statistical model for Benford's law

A statistical model for the fragmentation of a conserved quantity is analyzed, using the principle of maximum entropy and the theory of partitions. Upper and lower bounds for the restricted partitioning problem are derived and applied to the distribution of fragments. The resulting power law directly leads to Benford's law for the first digits of the parts.

Figure 1

Average number of parts $\langle n_j\rangle$ when partitioning a number $X$ into the set $\left\{x_j\right\}=\{1,2,3,4,5,6,7,8,9,10\}$ for various values of $X=\{25,50,100,200,400,800\}$ (increasing $X$ from bottom to top). The dots are exact calculations using all partitions and the solid lines the maximum entropy result using Eq. (5). Also shown is the equipartition result of Eq. (7), for $X=1600$ (dashed).

Figure 2

Example fragment distributions using the generalized power law Eq. (21), with $x_1=1, x_2=1000$ (see text) and $a=1$ (black) and $a=2$ (gray). Also shown are the corresponding power laws (dashed).