Benford’s law and complex atomic spectra

Pain, Jean‐Christophe

doi:10.1103/physreve.77.012102

Cited by 38 publications

(31 citation statements)

References 23 publications

(25 reference statements)

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“…We found that Learner's distribution of electric-dipole lines possesses a fractal character, and that its dimension is close to 1/3. The rule remains a mystery, but can be related to other observed peculiarities of spectra such as Benford's law [37] for the distribution of strengths, which can be explained by scale invariance. Actually, the natural development of fractal analysis of sets with a complicated structure is their description as multifractals [56].…”

Section: Resultsmentioning

confidence: 99%

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Regularities and symmetries in atomic structure and spectra

Pain¹

2013

High Energy Density Physics

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The use of statistical methods for the description of complex quantum systems was primarily motivated by the failure of a line-by-line interpretation of atomic spectra. Such methods reveal regularities and trends in the distributions of levels and lines. In the past, much attention was paid to the distribution of energy levels (Wigner surmise, random-matrix model...). However, information about the distribution of the lines (energy and strength) is lacking. Thirty years ago, Learner found empirically an unexpected law: the logarithm of the number of lines whose intensities lie between 2 k I0 and 2 k+1 I0, I0 being a reference intensity and k an integer, is a decreasing linear function of k. In the present work, the fractal nature of such an intriguing regularity is outlined and a calculation of its fractal dimension is proposed. Other peculiarities are also presented, such as the fact that the distribution of line strengths follows Benford's law of anomalous numbers, the existence of additional selection rules (PH coupling), the symmetry with respect to a quarter of the subshell in the spinadapted space (LL coupling) and the odd-even staggering in the distribution of quantum numbers, pointed out by Bauche and Cossé.

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Section: Resultsmentioning

confidence: 99%

“…[37]) that the distribution of lines in a given transition array follows very well Benford's logarithmic law of significant digits -see Figs. 2 and 3.…”

Section: Presentationmentioning

confidence: 99%

Regularities and symmetries in atomic structure and spectra

Pain¹

2013

High Energy Density Physics

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“…Equation (8) follows immediately from Eqn. (6) in a natural way: each succeeding application of C regenerates the function h and thus reproduces the previous application, except for an overall constant.…”

Section: Iterating the Law Of Total Probability And The Invariant Funmentioning

confidence: 99%

“…We can extend and unify our discussion by examining the fixed-point functions, or more precisely the invariant functions, of the iteration procedure in Eqn. (8). This is done by replacing G n with h, and results in…”

Section: Iterating the Law Of Total Probability And The Invariant Funmentioning

confidence: 99%

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Ubiquity of Benford's law and emergence of the reciprocal distribution

2016

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We apply the Law of Total Probability to the construction of scale-invariant probability distribution functions (pdf's), and require that probability measures be dimensionless and unitless under a continuous change of scales. If the scale-change distribution function is scale invariant then the constructed distribution will also be scale invariant. Repeated application of this construction on an arbitrary set of (normalizable) pdf's results again in scale-invariant distributions. The invariant function of this procedure is given uniquely by the reciprocal distribution, suggesting a kind of universality. We separately demonstrate that the reciprocal distribution results uniquely from requiring maximum entropy for size-class distributions with uniform bin sizes.

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Benford’s law applied to aerobiological data and its potential as a quality control tool

et al. 2009

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Benford's phenomenological law gives the expected frequencies of the first significant digits of any given series of numbers. According to this law, the frequency of 1 is higher than that of 2; this in turn appears more often 3, and so on. Similarly, Benford's law can also be applied to the first two significant digits (i.e., from 10 to 99), and so on. Here, we show that gross data sets of daily pollen counts from three aerobiological stations (located in European cities with different features regarding vegetation and climatology) fit Benford's law for the first significant digits, but this is not always true for the data transformed by a correction factor used in aerobiological studies. That is to say, the biases introduced by rounding and lower and upper built-in limits in pollen counts are detected by Benford's law analysis. The analysis of the first two significant digits from transformed data is better explained by a Power law than Benford's law. We propose that Benford's law could be used as a quality control tool for numerical aerobiological data sets.

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Benford’s law and complex atomic spectra

Cited by 38 publications

References 23 publications

Regularities and symmetries in atomic structure and spectra

Regularities and symmetries in atomic structure and spectra

Ubiquity of Benford's law and emergence of the reciprocal distribution

Benford’s law applied to aerobiological data and its potential as a quality control tool

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