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Related Concept Videos

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

Entropy

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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.
The relation  between entropy and disorder can be illustrated with the example of the phase change of ice to water. In ice, the molecules are located at specific sites giving a solid state, whereas, in a liquid form, these molecules are much freer to move. The molecular arrangement has therefore become more randomized. Although the change in average...
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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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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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Standard Entropy Change for a Reaction03:00

Standard Entropy Change for a Reaction

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Entropy is a state function, so the standard entropy change for a chemical reaction (ΔS°rxn) can be calculated from the difference in standard entropy between the products and the reactants.
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Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
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A generalized permutation entropy for noisy dynamics and random processes.

José M Amigó1, Roberto Dale1, Piergiulio Tempesta2

  • 1Centro de Investigación Operativa, Universidad Miguel Hernández, 03202 Elche, Spain.

Chaos (Woodbury, N.Y.)
|March 23, 2021
PubMed
Summary
This summary is machine-generated.

We introduce a generalized permutation entropy to address the divergence issue of standard permutation entropy in time series analysis, particularly for noisy or random data. This new method provides finite complexity measures for practical applications.

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Area of Science:

  • Complexity Science
  • Time Series Analysis
  • Information Theory

Background:

  • Permutation entropy is a popular complexity measure for deterministic time series, based on ordinal patterns.
  • Its popularity stems from convergence to Kolmogorov-Sinai entropy and computational simplicity.
  • Standard permutation entropy diverges for noisy or random time series, limiting its practical use.

Purpose of the Study:

  • To propose a generalized permutation entropy that remains finite for noisy and random time series.
  • To extend the applicability of permutation entropy to real-world data exhibiting imperfections.

Main Methods:

  • Developed a generalized permutation entropy within the framework of group entropies.
  • The method involves a data symbolic quantization using rank vectors (ordinal patterns).
  • Theoretical analysis and numerical illustrations were performed on random processes and noisy deterministic signals.

Main Results:

  • The proposed generalized permutation entropy yields finite values even when standard permutation entropy diverges.
  • Demonstrated the method's effectiveness on random processes with varying dependencies and noisy deterministic signals.
  • The generalized approach provides a more robust measure of complexity for practical time series data.

Conclusions:

  • The generalized permutation entropy offers a finite and practical complexity measure for time series analysis.
  • This advancement overcomes limitations of standard permutation entropy in the presence of noise and randomness.
  • The findings are relevant for analyzing complex systems in various scientific domains.