  Many mathematical, man-made and natural systems exhibit a leading-digit bias,
where a first digit (base 10) of 1 occurs not 11\% of the time, as one would
expect if all digits were equally likely, but rather 30\%. This phenomenon is
known as Benford's Law. Analyzing which datasets adhere to Benford's Law and
how quickly Benford behavior sets in are the two most important problems in the
field. Most previous work studied systems of independent random variables, and
relied on the independence in their analyses.
  Inspired by natural processes such as particle decay, we study the dependent
random variables that emerge from models of 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,
and give several conjectures for other fragmentation processes. The main
difficulty is that the resulting random variables are dependent. We handle this
by using tools from Fourier analysis and irrationality exponents to obtain
quantified convergence rates as well as introducing and developing techniques
to measure and control the dependencies. The construction of these tools is one
of the major motivations of this work, as our approach can be applied to many
other dependent systems. As an example, we show that the $n!$ entries in the
determinant expansions of $n\times n$ matrices with entries independently drawn
from nice random variables converges to Benford's Law.
