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." '' '''Constantino Tsallis''' <ref>http://www.mlahanas.de/Greeks/new/Tsallis.htm</ref>
:'' "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>
'''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. ?).
'''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>.
==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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*[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://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://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==

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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S} ,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathrm d} S = \frac{\delta Q_{\mathrm {reversible}}} {T} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q} is the heat and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature.

Statistical mechanics[edit]

In statistical mechanics entropy is defined by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. S \right. := -k_B \sum_{i=1}^W p_i \ln p_i}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} is the index for the microstates, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_i} is the probability that microstate i is occupied. In the microcanonical ensemble this gives:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left.S\right. = k_B \ln W}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W} (sometimes written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Omega} ) 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_q:= k_B \frac{1-\sum_{i=1}^W p_i^q}{q-1}}

where is the Tsallis index [5]. As Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q \rightarrow 1 } 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]

  1. ↑ http://www.mlahanas.de/Greeks/new/Tsallis.htm
  2. ↑ R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie 125 pp. 353-400 (1865)
  3. ↑ 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
  4. ↑ Constantino Tsallis "Possible generalization of Boltzmann-Gibbs statistics", Journal of Statistical Physics 52 pp. 479-487 (1988)
  5. ↑ Filippo Caruso and Constantino Tsallis "Nonadditive entropy reconciles the area law in quantum systems with classical thermodynamics", Physical Review E 78 021102 (2008)

Related reading

External links[edit]

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