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Related Concept Videos

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Salt particles that have dissolved in water never spontaneously come back together in solution to reform solid particles. Moreover, a gas that has expanded in a vacuum remains dispersed and never spontaneously reassembles. The unidirectional nature of these phenomena is the result of a thermodynamic state function called entropy (S). Entropy is the measure of the extent to which the energy is dispersed throughout a system, or in other words, it is proportional to the degree of disorder of a...
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A pure, perfectly crystalline solid possessing no kinetic energy (that is, at a temperature of absolute zero, 0 K) may be described by a single microstate, as its purity, perfect crystallinity,and complete lack of motion means there is but one possible location for each identical atom or molecule comprising the crystal (W = 1). According to the Boltzmann equation, the entropy of this system is zero.
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The second law of thermodynamics can be stated quantitatively using the concept of entropy. Entropy is the measure of disorder of the system.
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In the quest to identify a property that may reliably predict the spontaneity of a process, a promising candidate has been identified: entropy. Scientists refer to the measure of randomness or disorder within a system as entropy. High entropy means high disorder and low energy. To better understand entropy, think of a student’s bedroom. If no energy or work were put into it, the room would quickly become messy. It would exist in a very disordered state, one of high entropy. Energy must be...
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The process of surrounding a solute with solvent is called solvation. It involves evenly distributing the solute within the solvent. The rule of thumb for determining a solvent for a given compound is that like dissolves like. A good solvent has molecular characteristics similar to those of the compound to be dissolved. For example, polar solutions dissolve polar solutes, and apolar solvents dissolve apolar solutes. A polar solvent is a solvent that has a high dielectric constant (ϵ...
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In any measurement, the precision of the measuring tool is an essential factor. An ordinary ruler, for example, can measure length to the closest millimeter; a caliper, on the other hand, can measure length to the nearest 0.01 mm. As a result, the caliper is a more precise measurement tool because it can measure extremely minute changes in length. The measurements will be more accurate if the measuring tool is more precise.
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First Digits' Shannon Entropy.

Welf Alfred Kreiner1

  • 1Faculty of Natural Sciences, University of Ulm, Einsteinallee 11, D-89069 Ulm, Germany.

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|July 8, 2023
PubMed
Summary

The study explores Shannon entropy in numerical data, finding that first-digit distributions deviate from the Newcomb-Benford Law. This deviation impacts entropy calculations, revealing lower entropy values for non-standard distributions.

Keywords:
BoltzmannClausiusNewcomb–Benford lawShannon entropyVenusalphabetcalcitecratersdensity functionentropyexoplanetsfirst-digit phenomenonfragmentgraniteinformationmarblemineralpopulationsprobabilityscale invariancespace probesstatistical thermodynamicsstock pricesstreet addresseswages

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Area of Science:

  • Information Theory
  • Statistical Analysis
  • Data Science

Background:

  • Entropy quantifies information, measuring average binary digits for character transmission.
  • The Newcomb-Benford Law (NB Law) describes the typical frequency distribution of leading digits in numerical datasets.
  • Deviations from the NB Law can occur, with leading digit '1' appearing significantly more often than predicted.

Purpose of the Study:

  • To calculate and compare Shannon entropy (H) for various first-digit distributions.
  • To investigate how deviations from the Newcomb-Benford Law affect entropy values.
  • To analyze entropy in diverse datasets, including those not following the NB Law.

Main Methods:

  • Calculating Shannon entropy (H) based on the probabilities of first-digit occurrences.
  • Analyzing datasets exhibiting standard NB Law distributions.
  • Analyzing datasets with non-standard first-digit distributions, including those derived from power functions.
  • Comparing entropy values across different data distributions.

Main Results:

  • The Shannon entropy for first digits following the NB distribution is H = 2.88 bits per digit.
  • Datasets deviating from the NB Law showed lower entropy values, such as 2.76 bits per digit (Venus crater diameters) and 2.04 bits per digit (mineral fragment weights).
  • Specific distributions, where '1' is over 40 times more frequent than '9', can be modeled by a power function with a negative exponent (p > 1).

Conclusions:

  • Shannon entropy provides a measure of information content in first-digit distributions.
  • Deviations from the Newcomb-Benford Law lead to reduced entropy.
  • Entropy analysis reveals distinct information characteristics in datasets beyond those typically described by the NB Law.