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Related Concept Videos

Entropy Change in Reversible Processes01:10

Entropy Change in Reversible Processes

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.
Entropy02:39

Entropy

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...
Entropy01:18

Entropy

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...
Entropy and the Second Law of Thermodynamics01:20

Entropy and the Second Law of Thermodynamics

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...
Entropy and the Second Law of Thermodynamics01:26

Entropy and the Second Law of Thermodynamics

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...
The Entropy as a State Function01:14

The Entropy as a State Function

Consider an arbitrary process that moves between two specific states (A and B) in a cyclic manner. This process is reversible and broken down into smaller parts that each follow a Carnot cycle. A Carnot cycle has two isothermal (constant temperature) processes. During these processes, the ratio of the amount of heat transferred to their respective temperature remains constant. The other two processes in the Carnot cycle are also reversible but adiabatic, which means they occur without any heat...

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Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Published on: July 19, 2016

On entropy rate for the complex domain.

Wei Xiong1, Tülay Adalı, Yi-Ou Li

  • 1University of Maryland Baltimore County, Baltimore, MD 21250, USA.

IEEE Transactions on Signal Processing : a Publication of the IEEE Signal Processing Society
|July 17, 2010
PubMed
Summary
This summary is machine-generated.

We developed a general entropy rate formula for complex Gaussian random processes using a widely linear model. This method improves order selection for correlated data, including functional magnetic resonance imaging (fMRI) signals.

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Area of Science:

  • Signal processing
  • Information theory
  • Statistical modeling

Background:

  • Complex Gaussian random processes are prevalent in various scientific fields.
  • Existing entropy rate formulas often lack generality for noncircular processes.
  • Order selection in complex-valued time series analysis remains challenging.

Purpose of the Study:

  • To derive a general entropy rate formula for complex Gaussian random processes.
  • To extend independent and identically distributed (i.i.d.) sampling schemes to the complex domain.
  • To apply the derived formula and sampling scheme to improve order selection for complex data, such as fMRI.

Main Methods:

  • Utilizing a widely linear model to derive the entropy rate formula.
  • Extending an i.i.d. sampling scheme to the complex domain for correlated samples.
  • Applying information-theoretic criteria for order selection.

Main Results:

  • A general entropy rate formula applicable to both circular and noncircular complex Gaussian processes was derived.
  • An extended sampling scheme effectively improves estimation performance for correlated complex data.
  • The approach demonstrated effectiveness in order selection for simulated and real fMRI data.

Conclusions:

  • The derived entropy rate formula provides a versatile tool for analyzing complex Gaussian processes.
  • The extended sampling scheme enhances the reliability of information-theoretic criteria in complex domains.
  • This work offers significant advancements for analyzing complex-valued time series data, particularly in neuroimaging applications.