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Entropy02:39

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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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Entropy and the Second Law of Thermodynamics01:20

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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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Entropy Change in Reversible Processes01:10

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In the Carnot engine, which achieves the maximum efficiency between two reservoirs of fixed temperatures, the total change in entropy is zero. The observation can be generalized by considering any reversible cyclic process consisting of many Carnot cycles. Thus, it can be stated that the total entropy change of any ideal reversible cycle is zero.
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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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Fractional Deng Entropy and Extropy and Some Applications.

Mohammad Reza Kazemi1, Saeid Tahmasebi2, Francesco Buono3

  • 1Department of Statistics, Faculty of Science, Fasa University, Fasa 746-168-6688, Iran.

Entropy (Basel, Switzerland)
|June 2, 2021
PubMed
Summary
This summary is machine-generated.

This study introduces fractional Deng entropy and extropy, novel measures for quantifying uncertainty within Dempster-Shafer theory (DST). These new fractional measures are analyzed and applied to pattern recognition classification problems.

Keywords:
Deng entropy and extropyclassification and discriminationfractional entropymeasures of uncertainty

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Area of Science:

  • Information Theory
  • Artificial Intelligence
  • Mathematical Physics

Background:

  • Dempster-Shafer theory (DST) utilizes Deng entropy and extropy to quantify uncertainty.
  • Extropy is recognized as the dual concept of entropy.
  • Existing measures may not fully capture complex uncertainty dynamics.

Purpose of the Study:

  • Introduce and define fractional Deng entropy and fractional Deng extropy.
  • Compare these new fractional measures with existing uncertainty quantification methods in DST.
  • Demonstrate the utility of fractional Deng entropy and extropy in pattern recognition tasks.

Main Methods:

  • Development of fractional calculus-based definitions for Deng entropy and extropy.
  • Theoretical analysis of the properties of fractional Deng entropy and extropy, including their maxima.
  • Application and evaluation of the proposed measures in a pattern recognition classification problem.

Main Results:

  • Successfully defined and presented fractional Deng entropy and fractional Deng extropy.
  • Demonstrated that fractional Deng entropy and extropy exhibit distinct behaviors and maxima.
  • Showcased the effectiveness of the new fractional measures in a classification task, highlighting their importance.

Conclusions:

  • Fractional Deng entropy and extropy offer a more nuanced approach to uncertainty quantification in DST.
  • These novel measures provide valuable tools for analyzing complex systems, particularly in pattern recognition.
  • The study underscores the significance of fractional extensions in information theory and evidence theory.