Entropy: Difference between revisions

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:'' "Energy has to do with possibilities. Entropy has to do with the probabilities of those possibilities happening. It takes energy and performs a further epistemological step." ''
:'' "Energy has to do with possibilities. Entropy has to do with the probabilities of those possibilities happening. It takes energy and performs a further epistemological step." '' '''Constantino Tsallis''' <ref>http://www.mlahanas.de/Greeks/new/Tsallis.htm</ref>
::::: '''[[Constantino Tsallis]]''' <ref>http://www.mlahanas.de/Greeks/new/Tsallis.htm</ref>
'''Entropy''' was first described by [[Rudolf Julius Emanuel Clausius]] in 1865 <ref>[http://dx.doi.org/10.1002/andp.18652010702 R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie '''125''' pp. 353-400 (1865)]</ref>. The [[statistical mechanics | statistical mechanical]] desciption is due to [[Ludwig Eduard Boltzmann]] (Ref. ?). The word entropy originated from the Greek word meaning a turning or transformation "τροπή" <ref>[https://books.google.es/books?id=8LIEAAAAYAAJ&pg=PA357  Rudolf Clausius "The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies", London (1867) page 357]</ref>.
'''Entropy''' was first described by [[Rudolf Julius Emanuel Clausius]] in 1865 <ref>[http://dx.doi.org/10.1002/andp.18652010702 R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie '''125''' pp. 353-400 (1865)]</ref>. The [[statistical mechanics | statistical mechanical]] desciption is due to [[Ludwig Eduard Boltzmann]] (Ref. ?).
==Classical thermodynamics==
==Classical thermodynamics==
In [[classical thermodynamics]] one has the entropy, <math>S</math>,
In [[classical thermodynamics]] one has the entropy, <math>S</math>,
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:<math>S_q:= k_B \frac{1-\sum_{i=1}^W p_i^q}{q-1}</math>
:<math>S_q:= k_B \frac{1-\sum_{i=1}^W p_i^q}{q-1}</math>


where <math>q</math> is the ''Tsallis index''. As <math>q \rightarrow 1 </math> one recovers the standard expression for entropy.
where <math>q</math> is the ''Tsallis index'' <ref>[http://dx.doi.org/10.1103/PhysRevE.78.021102 Filippo Caruso and Constantino Tsallis "Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics", Physical Review E '''78''' 021102 (2008)]</ref>.
As <math>q \rightarrow 1 </math> one recovers the standard expression for entropy. This expression for the entropy is the cornerstone of [[non-extensive thermodynamics]].


==Arrow of time==
==Arrow of time==
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*[http://dx.doi.org/10.1063/1.881363 Joel L. Lebowitz "Boltzmann's Entropy and Time's Arrow", Physics Today '''46''' pp. 32-38 (1993)]
*[http://dx.doi.org/10.1063/1.881363 Joel L. Lebowitz "Boltzmann's Entropy and Time's Arrow", Physics Today '''46''' pp. 32-38 (1993)]
*[http://dx.doi.org/10.1023/A:1023715732166 Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics '''33''' pp. 467-490 (2003)]
*[http://dx.doi.org/10.1023/A:1023715732166 Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics '''33''' pp. 467-490 (2003)]
*[http://dx.doi.org/10.1103/PhysRevE.79.061103 Noah Linden, Sandu Popescu, Anthony J. Short, and Andreas Winter "Quantum mechanical evolution towards thermal equilibrium", Physical Review E '''79''' 061103 (2009)]
Books:
Books:
* Steven F. Savitt (Ed.) "Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time", Cambridge University Press (1997) ISBN 0521599458
* Steven F. Savitt (Ed.) "Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time", Cambridge University Press (1997) ISBN 0521599458
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*Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR '''124''' pp. 768-771 (1959)
*Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR '''124''' pp. 768-771 (1959)
*[http://dx.doi.org/10.1063/1.1670348 William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics '''49''' pp. 1981-1982 (1968)]
*[http://dx.doi.org/10.1063/1.1670348 William G. Hoover "Entropy for Small Classical Crystals", Journal of Chemical Physics '''49''' pp. 1981-1982 (1968)]
*Arieh Ben-Naim "Entropy Demystified: The Second Law Reduced to Plain Common Sense", World Scientific (2008) ISBN 978-9812832252
* Arieh Ben-Naim "Entropy Demystified: The Second Law Reduced to Plain Common Sense", World Scientific (2008) ISBN 978-9812832252
*Arieh Ben-Naim "Farewell to Entropy: Statistical Thermodynamics Based on Information",  World Scientific (2008) ISBN 978-981-270-707-9
* Arieh Ben-Naim "Farewell to Entropy: Statistical Thermodynamics Based on Information",  World Scientific (2008) ISBN 978-981-270-707-9
* Arieh Ben-Naim "Discover Entropy and the Second Law of Thermodynamics: A Playful Way of Discovering a Law of Nature" World Scientific Publishing (2010) ISBN: 978-981-4299-75-6
* Arieh Ben-Naim "Entropy and the Second Law Interpretation and Misss-Interpretations", World Scientific Publishing (2012) ISBN 978-981-4407-55-7
* Arieh Ben-Naim "Information, Entropy, Life and the Universe: What We Know and What We Do Not Know" World Scientific Publishing (2015) ISBN 978-981-4651-66-0
* Arieh Ben-Naim "Entropy The Truth, the Whole Truth, and Nothing But the Truth", World Scientific Publishing (2016) ISBN 978-981-3147-67-6
*[http://dx.doi.org/10.1063/1.4879553  Jose M. G. Vilar and J. Miguel Rubi "System-size scaling of Boltzmann and alternate Gibbs entropies", Journal of Chemical Physics '''140''' 201101 (2014)]
*[http://doi.org/10.1063/1.4972525 Misaki Ozawa and Ludovic Berthier "Does the configurational entropy of polydisperse particles exist?", Journal of Chemical Physics '''146''' 014502 (2017)]
*[http://dx.doi.org/10.1080/00268976.2016.1238523 Simin Yazdi Nezhad and Ulrich K. Deiters "Estimation of the entropy of fluids with Monte Carlo computer simulation", Molecular Physics '''115''' pp. 1074-1085 (2017)]
*[http://dx.doi.org/10.1063/1.4984965 Gérôme Faure, Rafael Delgado-Buscalioni, and Pep Español "The entropy of a complex molecule", Journal of Chemical Physics '''146''' 224106 (2017)]
 
==External links==
==External links==
*[http://www.mdpi.com/journal/entropy entropy] an international and interdisciplinary Open Access journal of entropy and information studies.
*[http://www.mdpi.com/journal/entropy entropy] an international and interdisciplinary Open Access journal of entropy and information studies.

Latest revision as of 14:26, 17 January 2018

"Energy has to do with possibilities. Entropy has to do with the probabilities of those possibilities happening. It takes energy and performs a further epistemological step." Constantino Tsallis [1]

Entropy was first described by Rudolf Julius Emanuel Clausius in 1865 [2]. The statistical mechanical desciption is due to Ludwig Eduard Boltzmann (Ref. ?). The word entropy originated from the Greek word meaning a turning or transformation "τροπή" [3].

Classical thermodynamics[edit]

In classical thermodynamics one has the entropy, ,

where is the heat and is the temperature.

Statistical mechanics[edit]

In statistical mechanics entropy is defined by

where is the Boltzmann constant, is the index for the microstates, and is the probability that microstate i is occupied. In the microcanonical ensemble this gives:

where (sometimes written as ) is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. This equation provides a link between classical thermodynamics and statistical mechanics

Tsallis entropy[edit]

Tsallis (or non-additive) entropy [4] is defined as (Eq. 1)

where is the Tsallis index [5]. As one recovers the standard expression for entropy. This expression for the entropy is the cornerstone of non-extensive thermodynamics.

Arrow of time[edit]

Articles:

Books:

  • Steven F. Savitt (Ed.) "Time's Arrows Today: Recent Physical and Philosophical Work on the Direction of Time", Cambridge University Press (1997) ISBN 0521599458
  • Michael C. Mackey "Time's Arrow: The Origins of Thermodynamic Behavior" (1992) ISBN 0486432432
  • Huw Price "Time's Arrow and Archimedes' Point New Directions for the Physics of Time" Oxford University Press (1997) ISBN 978-0-19-511798-1

See also:[edit]

References[edit]

Related reading

External links[edit]