Editing Entropy

:'' "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. ?).
==Classical thermodynamics==
In [[classical thermodynamics]] one has the entropy, <math>S</math>,
:<math>{\mathrm d} S = \frac{\delta Q_{\mathrm {reversible}}} {T} </math>

where <math>Q</math> is the [[heat]] and <math>T</math> is the [[temperature]]. 
==Statistical mechanics==
In [[statistical mechanics]] entropy is defined by

:<math>\left. S \right. := -k_B \sum_{i=1}^W p_i \ln p_i</math>

where <math>k_B</math> is the [[Boltzmann constant]], <math>i</math> is the index for the [[microstate |microstates]], and <math>p_i</math>
is the probability that microstate ''i'' is occupied. 
In the [[microcanonical ensemble]] this gives:

:<math>\left.S\right. = k_B \ln W</math>

where <math>W</math> (sometimes written as <math>\Omega</math>)
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 | classical thermodynamics]] and 
[[Statistical mechanics | statistical mechanics]]
==Tsallis entropy==
Tsallis (or ''non-additive'') entropy <ref>[http://dx.doi.org/10.1007/BF01016429 Constantino Tsallis "Possible generalization of Boltzmann-Gibbs statistics", Journal of Statistical Physics '''52''' pp. 479-487 (1988)]</ref> is defined as (Eq. 1)

:<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'' <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==
Articles:
*[http://dx.doi.org/10.1119/1.1942052 T. Gold "The Arrow of Time",  American Journal of Physics '''30''' pp. 403-410 (1962)]
*[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/10.1103/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:
* 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:==
*[[Entropy of a glass]]
*[[H-theorem]]
*[[Non-extensive thermodynamics]]
*[[Shannon entropy]]
==References==
<references/>
'''Related reading'''
*[http://dx.doi.org/10.1119/1.1990592 Karl K. Darrow "The Concept of Entropy",  American Journal of Physics '''12''' pp.  183-196 (1944)]
*[http://dx.doi.org/10.1119/1.1971557 E. T. Jaynes "Gibbs vs Boltzmann Entropies",  American Journal of Physics '''33''' pp. 391-398 (1965)]
*[http://dx.doi.org/10.1119/1.1287353 Daniel F. Styer "Insight into entropy",  American Journal of Physics '''86''' pp. 1090-1096 (2000)]
*[http://www.ucl.ac.uk/~ucesjph/reality/entropy/text.html S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)]
*[http://dx.doi.org/10.2174/1874396X00802010007 Efstathios E. Michaelides "Entropy, Order and Disorder", The Open Thermodynamics Journal '''2''' pp. (2008)]
*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)]
* 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 "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)]

==External links==
*[http://www.mdpi.com/journal/entropy entropy] an international and interdisciplinary Open Access journal of entropy and information studies.
*[http://dx.doi.org/10.4249/scholarpedia.3448 Joel L. Lebowitz "Time's arrow and Boltzmann's entropy", Scholarpedia, 3(4):3448 (2008)]
[[category: statistical mechanics]]
[[category: Classical thermodynamics]]