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Related Concept Videos

Entropy02:39

Entropy

37.8K
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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Estimating Population Standard Deviation01:26

Estimating Population Standard Deviation

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When the population standard deviation is unknown and the sample size is large, the sample standard deviation s is commonly used as a point estimate of σ. However, it can sometimes under or overestimate the population standard deviation. To overcome this drawback, confidence intervals are determined to estimate population parameters and eliminate any calculation bias accurately. However, this only applies to random samples from normally distributed populations. Knowing the sample mean and...
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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 and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

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Consider an isolated system in which a hot object is placed in contact with a cold one. This is an irreversible process that eventually leads both objects to reach the same equilibrium temperature. It is crucial to note that the constituents of any substance exhibit increased disorder at higher temperatures. As a cold substance absorbs heat, its constituents become more disordered. The energy transfer from a hotter object to a cooler one increases the system's disorder or randomness. This...
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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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Related Experiment Video

Updated: Mar 29, 2026

Applications of EEG Neuroimaging Data: Event-related Potentials, Spectral Power, and Multiscale Entropy
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A Benchmark for Entropy Estimators.

Lucio M Calcagnile1, Angelo Di Garbo1,2, Stefano Galatolo3

  • 1Istituto di Biofisica, CNR, Via G. Moruzzi 1, 56124 Pisa, Italy.

Entropy (Basel, Switzerland)
|March 28, 2026
PubMed
Summary
This summary is machine-generated.

This study benchmarks entropy estimators for time series data. Approximate Entropy and symbolic methods accurately estimated Kolmogorov-Sinai entropy, unlike Sample and Permutation Entropy.

Keywords:
Kolmogorov–Sinai entropyLyapunov exponentcomplexitytime series

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Area of Science:

  • Dynamical Systems Theory
  • Information Theory
  • Time Series Analysis

Background:

  • Estimating Kolmogorov-Sinai entropy is crucial for characterizing complex dynamical systems.
  • Existing entropy estimators vary in accuracy and robustness across different data types and system dynamics.
  • Computer-assisted proofs offer certified entropy values with rigorous error bounds for benchmarking.

Purpose of the Study:

  • To quantitatively assess and compare the performance of widely used entropy estimators.
  • To evaluate estimators on diverse one-dimensional dynamical systems with known certified entropy.
  • To identify reliable entropy estimation techniques for numerical and symbolic time series data.

Main Methods:

  • Generated long time series orbits for four classes of one-dimensional interval maps.
  • Compared certified Kolmogorov-Sinai entropy values against estimates from Approximate Entropy, Sample Entropy, Permutation Entropy, symbolic Plug-In, and Non-Sequential Recursive Pair Substitution (NSRPS) methods.
  • Utilized Grassberger-type bias correction for symbolic Plug-In and NSRPS estimators.

Main Results:

  • Approximate Entropy and symbolic methods (Plug-In, NSRPS) demonstrated consistent accuracy within rigorous error bounds across all tested systems.
  • Sample Entropy systematically underestimated the true entropy.
  • Permutation Entropy exhibited significant biases, particularly for expanding maps lacking a Markov partition.

Conclusions:

  • The study provides a quantitative benchmark for evaluating entropy estimation techniques in deterministic dynamical systems.
  • Approximate Entropy and symbolic methods are recommended for reliable entropy estimation in similar systems.
  • Further research is needed to improve the accuracy of Sample and Permutation Entropy estimators for complex dynamics.