http://debianws.lexgopc.com/wiki143/index.php?action=history&feed=atom&title=Gibbs_algorithmGibbs algorithm - Revision history2026-05-04T21:47:53ZRevision history for this page on the wikiMediaWiki 1.43.1http://debianws.lexgopc.com/wiki143/index.php?title=Gibbs_algorithm&diff=2129605&oldid=previmported>Jlwoodwa: tag as one source2024-03-13T03:50:34Z<p>tag as one source</p>
<p><b>New page</b></p><div>{{short description|Criterion for choosing a probability distribution in statistical mechanics}}<br />
{{Distinguish |Gibbs sampler}}<br />
{{one source |date=March 2024}}<br />
[[FILE:Josiah Willard Gibbs -from MMS-.jpg|thumb|200px|Josiah Willard Gibbs]]<br />
In [[statistical mechanics]], the '''Gibbs algorithm''', introduced by [[J. Willard Gibbs]] in 1902, is a criterion for choosing a [[probability distribution]] for the [[statistical ensemble]] of [[microstate (statistical mechanics)|microstate]]s of a [[thermodynamic system]] by minimizing the average log probability<br />
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:<math> \langle\ln p_i\rangle = \sum_i p_i \ln p_i \, </math><br />
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subject to the probability distribution {{math|''p<sub>i</sub>''}} satisfying a set of constraints (usually expectation values) corresponding to the known [[macroscopic]] quantities.<ref name=Dewar>{{cite book|first=Roderick C. |last=Dewar|chapter=4. Maximum Entropy Production and Non-equilibrium Statistical Mechanics|editor-last1=Kleidon|editor-first1=A.|title=Non-equilibrium thermodynamics and the production of entropy : life, earth, and beyond|series=Understanding Complex Systems|url=https://archive.org/details/nonequilibriumth00klei |url-access=limited |date=2005|publisher=Springer|location=Berlin|isbn=9783540224952|pages=41–55|doi=10.1007/11672906_4}}</ref> in 1948, [[Claude E. Shannon|Claude Shannon]] interpreted the negative of this quantity, which he called [[entropy_(Information_theory)|information entropy]], as a measure of the uncertainty in a probability distribution.<ref name=Dewar/> In 1957, [[E.T. Jaynes]] realized that this quantity could be interpreted as missing information about anything, and generalized the Gibbs algorithm to non-equilibrium systems with the [[principle of maximum entropy]] and [[maximum entropy thermodynamics]].<ref name=Dewar/><br />
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Physicists call the result of applying the Gibbs algorithm the [[Gibbs distribution]] for the given constraints, most notably Gibbs's [[grand canonical ensemble]] for open systems when the average energy and the average number of particles are given. (See also ''[[Partition function (mathematics)|partition function]]'').<br />
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This general result of the Gibbs algorithm is then a [[maximum entropy probability distribution]]. Statisticians identify such distributions as belonging to [[exponential family|exponential families]].<br />
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==References==<br />
{{Reflist}}<br />
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{{DEFAULTSORT:Gibbs Algorithm}}<br />
[[Category:Statistical mechanics]]<br />
[[Category:Particle statistics]]<br />
[[Category:Entropy and information]]<br />
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