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'''Entropy''' was first described by [[Rudolf Julius Emanuel Clausius]] in 1865 | {{Cleanup-rewrite}} | ||
'''Entropy''' was first described by [[Rudolf Julius Emanuel Clausius]] in 1865 (Ref. 1). The [[statistical mechanics | statistical mechanical]] desciption is due to [[Ludwig Eduard Boltzmann]] (Ref. ?). | |||
==Classical thermodynamics== | ==Classical thermodynamics== | ||
In [[classical thermodynamics]] one has the entropy, | In [[classical thermodynamics]] one has the '''entropy''', S, | ||
:<math>{\mathrm d} S = \frac{\delta Q_{\mathrm {reversible}}} {T} </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]]. | where <math>Q</math> is the [[heat]] and <math>T</math> is the [[temperature]]. | ||
==Statistical mechanics== | ==Statistical mechanics== | ||
In [[statistical mechanics]] entropy is defined by | In [[statistical mechanics]] the '''entropy''', S, is defined by | ||
:<math>\left. S \right. | :<math>\left. S \right. = -k_B \sum_m p_m \ln p_m</math> | ||
where <math>k_B</math> is the [[Boltzmann constant]], | where <math>k_B</math> is the [[Boltzmann constant]], ''m'' is the index for the [[microstate |microstates]], and <math>p_m</math> | ||
is the probability that microstate '' | is the probability that microstate ''m'' is occupied. | ||
In the [[microcanonical ensemble]] this gives: | In the [[microcanonical ensemble]] this gives: | ||
:<math>\left.S\right. = k_B \ln | :<math>\left.S\right. = k_B \ln \Omega</math> | ||
where <math> | where <math>\Omega</math> (sometimes written as <math>W</math>) | ||
is the number of microscopic configurations that result in the observed macroscopic description of the thermodynamic system. | 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 | This equation provides a link between [[Classical thermodynamics | classical thermodynamics]] and | ||
[[Statistical mechanics | statistical mechanics]] | [[Statistical mechanics | statistical mechanics]] | ||
==Arrow of time== | ==Arrow of time== | ||
Articles: | 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/10.1119/1.1942052 T. Gold "The Arrow of Time", American Journal of Physics '''30''' pp. 403-410 (1962)] | ||
* | * 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)] | ||
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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==See also:== | ==See also:== | ||
*[[Entropy of a glass]] | *[[Entropy of a glass]] | ||
*[[Shannon entropy]] | |||
*[[Tsallis entropy]] | |||
*[[H-theorem]] | *[[H-theorem]] | ||
==Interesting 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/10.1119/1.1990592 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/10.1119/1.1971557 E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics '''33''' pp. 391-398 (1965)] | ||
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*[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://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)] | *[http://dx.doi.org/10.2174/1874396X00802010007 Efstathios E. Michaelides "Entropy, Order and Disorder", The Open Thermodynamics Journal '''2''' pp. (2008)] | ||
== | ==References== | ||
#[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)] | |||
#Ya. G. Sinai, "On the Concept of Entropy of a Dynamical System," Doklady Akademii Nauk SSSR '''124''' pp. 768-771 (1959) | |||
[[category: statistical mechanics]] | #[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)] | ||
[[ | [[category:statistical mechanics]] | ||
[[Classical thermodynamics]] | |||