Entropy: Difference between revisions
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In [[statistical mechanics]] entropy is defined by | In [[statistical mechanics]] entropy is defined by | ||
:<math>\left. S \right. = -k_B \ | :<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]], | 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 '' | is the probability that microstate ''i'' 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 W</math> | ||
where <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. | 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]] | ||
==Tsallis entropy== | ==Tsallis 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> | Tsallis 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> | |||
==Arrow of time== | ==Arrow of time== | ||
Articles: | Articles: | ||
Revision as of 11:58, 30 May 2014
- "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."
Entropy was first described by Rudolf Julius Emanuel Clausius in 1865 [2]. The statistical mechanical desciption is due to Ludwig Eduard Boltzmann (Ref. ?).
Classical thermodynamics
In classical thermodynamics one has the entropy, ,
where is the heat and is the temperature.
Statistical mechanics
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
Tsallis entropy [3] is defined as (Eq. 1)
Arrow of time
Articles:
- 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)
- Milan M. Ćirković "The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann–Schuetz Argument", Foundations of Physics 33 pp. 467-490 (2003)
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:
References
- ↑ http://www.mlahanas.de/Greeks/new/Tsallis.htm
- ↑ R. Clausius "Ueber verschiedene für die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie", Annalen der Physik und Chemie 125 pp. 353-400 (1865)
- ↑ Constantino Tsallis "Possible generalization of Boltzmann-Gibbs statistics", Journal of Statistical Physics 52 pp. 479-487 (1988)
Related reading
- Karl K. Darrow "The Concept of Entropy", American Journal of Physics 12 pp. 183-196 (1944)
- E. T. Jaynes "Gibbs vs Boltzmann Entropies", American Journal of Physics 33 pp. 391-398 (1965)
- Daniel F. Styer "Insight into entropy", American Journal of Physics 86 pp. 1090-1096 (2000)
- S. F. Gull "Some Misconceptions about Entropy" in Brian Buck and Vincent A. MacAulay (Eds.) "Maximum Entropy in Action", Oxford Science Publications (1991)
- 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)
- 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
External links
- entropy an international and interdisciplinary Open Access journal of entropy and information studies.
- Joel L. Lebowitz "Time's arrow and Boltzmann's entropy", Scholarpedia, 3(4):3448 (2008)