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Entropy02:39

Entropy

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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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Entropy01:18

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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.
When an ideal gas expands isothermally, the disorder in the gas increases. From the molecular perspective, the gas molecules have more volume to move around in.
Consider an infinitesimal step in the expansion, which...
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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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The Second Law of Thermodynamics01:14

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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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Third Law of Thermodynamics02:38

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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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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.
The statement can be further generalized to prove that entropy is a state function. Take a cyclic process between any two points on a p-V diagram.
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Statistical analysis and first-passage-time applications of a lognormal diffusion process with multi-sigmoidal logistic mean.

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Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
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Generalized Entropies, Variance and Applications.

Abdolsaeed Toomaj1, Antonio Di Crescenzo2

  • 1Department of Mathematics and Statistics, Faculty of Basic Sciences and Engineering, Gonbad Kavous University, Basirat Blvd., Shahid Fallahi Street, Gonbad Kavous 4971799151, Golestan Province, Iran.

Entropy (Basel, Switzerland)
|December 8, 2020
PubMed
Summary

This study explores generalized cumulative residual entropy, revealing its connection to Poisson processes and reliability theory. New findings offer insights into random lifetimes, system reliability, and distribution characterizations.

Keywords:
generalized cumulative entropygeneralized cumulative residual entropymean residual lifestochastic ordersvariance

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Area of Science:

  • Reliability Theory
  • Information Theory
  • Probability and Statistics

Background:

  • The generalized cumulative residual entropy (GCRE) is an emerging measure of dispersion for random lifetimes.
  • Existing research lacks comprehensive analysis of GCRE's properties and its relationships with other statistical measures.

Purpose of the Study:

  • To derive new mathematical expressions and bounds for the generalized cumulative residual entropy.
  • To investigate the dynamic properties of GCRE and its dual measure, generalized cumulative entropy.
  • To establish connections between GCRE, non-homogeneous Poisson processes, and reliability concepts.

Main Methods:

  • Stochastic comparisons of random lifetimes using GCRE.
  • Analysis of GCRE in the context of residual lifetimes and aging notions.
  • Characterization of distributions and analysis of k-out-of-n systems.

Main Results:

  • Established an intimate connection between GCRE and non-homogeneous Poisson processes.
  • Derived new expressions, bounds, and stochastic comparisons for GCRE.
  • Investigated dynamic versions of GCRE and generalized cumulative entropy, yielding results for reliability theory.
  • Provided a characterization for the exponential distribution and analyzed k-out-of-n systems.

Conclusions:

  • The study deepens the understanding of generalized cumulative residual entropy and its dual measure.
  • Findings offer significant contributions to reliability theory, particularly concerning system analysis and distribution characterization.
  • The research highlights the utility of GCRE in analyzing random lifetimes and complex systems.