Many systems exhibit a digit bias. For example, the first digit base 10 of the Fibonacci numbers, or of $2^n$, equals 1 not 10% or 11% of the time, as one would expect if all digits were equally likely, but about 30% of the time. This phenomenon, known as Benford's Law, has many applications, ranging from detecting tax fraud for the IRS to analyzing round-off errors in computer science. The central question is determining which data sets follow Benford's law. Inspired by natural processes such as particle decay, our work examines models for the decomposition of conserved quantities. We prove that in many instances the distribution of lengths of the resulting pieces converges to Benford behavior as the number of divisions grow. The main difficulty is that the resulting random variables are dependent, which we handle by a careful analysis of the dependencies and tools from Fourier analysis to obtain quantified convergence rates.

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.

We construct and analyze models for the fragmentation of a conserved quantity. Using a statistical model, we derive an approximation as well as bounds for the restricted partitioning problem, which we apply to the distribution of fragments. We also modify the canonical ensemble from statistical physics. Taken together, we set a threshold on the magnitude of the conserved quantity needed to result in power law behavior, as well as a threshold on the number of possible piece sizes in a special case. We also investigate variations on two specific fragmentation procedures, the directed and undirected fragmentations, for power law behavior. Calculations show that the undirected fragmentation exhibits power law behavior. We consider small perturbations to process rates and find that the multi-path fragmentation is affected less as the number of piece sizes grows. We confirm this numerical result using first-order perturbation theory.

We attempt to recover the results of Lemons for a probabilistic model of partitioning a conserved quantity. We derive an approximation for the number of partitions of integer X into parts given by a set H as well as the average number of partshn j i of a given part size x j . 1. What is Benford’s Law? Benford’s Law of Digit Bias concerns the distribution of the leading digits of the elements of a data set. A data set is Benford if the probability that the first digit of a set element is d is given by log 10 1 + 1 d .