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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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The first law of thermodynamics is quantitatively formulated via an equation relating the internal energy of a system, the heat exchanged by it, and the work done on it. A quantitative formulation of the second law of thermodynamics leads to defining a state function, the entropy.
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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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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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Ordinal Pattern Based Entropies and the Kolmogorov-Sinai Entropy: An Update.

Tim Gutjahr1, Karsten Keller1

  • 1Institute of Mathematics, University of Lübeck, D-23562 Lübeck, Germany.

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|December 8, 2020
PubMed
Summary

This study clarifies the relationship between permutation entropy and Kolmogorov-Sinai entropy. It establishes conditions under which permutation entropy serves as a lower bound for Kolmogorov-Sinai entropy, advancing complexity analysis.

Keywords:
Kolmogorov–Sinai entropyconditional entropyordinal patternspermutation entropy

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Area of Science:

  • Information Theory
  • Dynamical Systems Theory
  • Statistical Mechanics

Background:

  • Established relationships exist between ordinal pattern-based entropies and Kolmogorov-Sinai (KS) entropy.
  • The precise nature of these relationships, particularly for permutation entropy, remains incompletely understood.
  • Existing literature highlights potential equalities and bounds but lacks a comprehensive theoretical framework.

Purpose of the Study:

  • To update the understanding of the relationship between permutation entropy and KS entropy.
  • To theoretically investigate the combinatorial aspects linking ordinal pattern distributions to KS entropy.
  • To present new findings on conditional permutation entropy and its bounds relative to KS entropy.

Main Methods:

  • Theoretical analysis of ordinal pattern distributions.
  • Combinatorial investigations of entropy measures.
  • Development of new proofs for entropy bounds.
  • Examination of a previously proposed method for analyzing entropy relationships.

Main Results:

  • A new statement regarding conditional permutation entropy is provided.
  • A novel proof demonstrates that permutation entropy is an upper bound for KS entropy.
  • General conditions are established for permutation entropy to be a lower bound for KS entropy.

Conclusions:

  • The study significantly advances the theoretical understanding of permutation entropy and KS entropy.
  • New conditions are identified for permutation entropy as a lower bound, offering deeper insights into system complexity.
  • The findings contribute to a more complete picture of entropy measures in dynamical systems.