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, which we handle by using tools from Fourier analysis and irrationality exponents to obtain quantified convergence rates. Our method can be applied to many other systems; as an example, we show that the n! entries in the determinant expansions of n × n matrices with entries independently drawn from nice random variables converges to Benford's Law.
Optical pump-probe techniques are used to generate and measure electron spin polarization in a gallium arsenide epilayer in which the electron spin coherence time exceeds the mode-locked laser repetition period. Resonant spin amplification occurs at magnetic fields at which the electron spin polarization excited by successive laser pulses add constructively. Measurements of Kerr rotation as a function of applied magnetic field reveal nuclear spin polarization that aligns either with or against the external magnetic field depending on whether the applied magnetic field is being decreased or increased. Furthermore, the nuclear spin polarization magnitude varies in proportion to the perpendicular net electron spin polarization as the latter changes due to resonant spin amplification and other causes. We also report an experimental technique that reveals a minutes-long memory of precise field history in the electron-nuclear spin system.
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.
Applying a voltage to a semiconductor sample generates a current-induced electron spin polarization (CISP). Using an ultrafast mode-locked laser and lock-in detection scheme, we measure CISP on an indium gallium arsenide epilayer via Faraday rotation and extract the spin generation rate. While the measured spin polarization initially increases linearly with electric field as observed in previous work, larger applied voltages lead to a decreasing spin generation rate. We show that we can recover the linear dependence of spin generation rate with electric field even at larger applied voltages by modifying the applied voltage waveform to reduce heating and multiplying by an appropriate correction factor. Future CISP studies can utilize this technique to investigate CISP under larger applied electric fields.
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