Benford's law from Turing ensembles and integer partitions
Phys. Rev. E 112, 044149 – Published 28 October, 2025
DOI: https://doi.org/10.1103/xjlr-sg7r
Abstract
We develop two complementary generative mechanisms that explain when and why Benford's first-digit law arises. First, a probabilistic Turing machine (PTM) ensemble induces a geometric law for codelength. Maximizing its entropy under a constraint on halting length yields Benford statistics. This model shows a phase transition with respect to the halt probability. Second, a constrained partition model (Einstein-solid combinatorics) recovers the same logarithmic profile as the maximum entropy solution under a coarse-grained entropy-rate constraint, clarifying the role of nonergodicity (ensemble vs. trajectory averages). We also perform numerical experiments that corroborate our conclusions.
Physics Subject Headings (PhySH)
Corrections
26 December, 2025
Correction: Several proof corrections that were not implemented during the production process have now been made. A redundant paragraph at the end of the Introduction has been removed. Other minor changes have been made in the text below Eq. (1), in Eq. (31), and in the first sentence of the Acknowledgments.
