Explaining irreversibility | American Journal of Physics

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A; citation_year=1987; citation_volume=35; citation_pages=3492-3498; " /><meta name="citation_reference" content="citation_title=Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen; citation_author=Boltzmann L.; citation_journal_title=Wien. Ber.; citation_year=1872; citation_volume=66; citation_pages=275-370; " /><meta name="citation_reference" content="citation_title=Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft; citation_author=Loschmidt J.; citation_journal_title=Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A; citation_year=1876; citation_volume=73; citation_pages=128-142; " /><meta name="citation_reference" content="citation_title=Über einen Satz der Dynamik und die mechanische Wärmelehre; citation_author=Zermelo E.; citation_journal_title=Ann. 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Phys.; citation_year=1901; citation_volume=309; citation_pages=553-563; " /><meta name="citation_reference" content="“Umkehr” means “reversal,” referring to the reversal of velocities in Loschmidt’s argument. “Einwand” means “objection.” " /><meta name="citation_reference" content="citation_title=Sur le probleme des trois corps et les équations de la dynamique; citation_author=Poincaré H.; citation_journal_title=Acta Math.; citation_year=1890; citation_volume=13; citation_pages=1-270; " /><meta name="citation_reference" content="“Wiederkehr” means “return” or “recur,” referring to the return to the original state in Zermelo’s argument. " /><meta name="citation_reference" content="If the period of each molecule divided into the longest period is an integer, the recurrence time will be equal to the longest period. It will not be astronomically long. Therefore, a probabilistic argument is necessary to establish that the recurrence time is extraordinarily long. 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Phys.; citation_year=2006; citation_volume=74; citation_pages=187-190; " /><meta name="citation_pdf_url" content="https://pubs.aip.org/aapt/ajp/article-pdf/76/7/643/13120824/643_1_online.pdf" /><meta name="dc.identifier" content="10.1119/1.2894523" /><meta name="description" content="I describe an approach that helps students understand why the behavior of macroscopic systems distinguishes between past and future, even though the underlying " /><meta property="og:updated_time" content="7/1/2008" /> <meta property="og:type" content="article" /> <meta property="og:title" content="Explaining irreversibility" /> <meta property="og:description" content="I describe an approach that helps students understand why the behavior of macroscopic systems distinguishes between past and future, even though the underlying " /> <meta property="og:url" content="/aapt/ajp/article/76/7/643/1038329/Explaining-irreversibility" /> <meta property="og:site_name" content="AIP Publishing" /> <meta property="og:image" content="https://aipp.silverchair-cdn.com/aipp/content_public/journal/ajp/76/7/10.1119_1.2894523/3/643_1_f1.jpeg?Expires=1735697734&Signature=3CfuOf1lVt1okUSE1puXzFeTpYiJ-6YctIJfRCkID5APENOvMDLL8XTbuHWhDWD19JZ~q0BdExNOS8ys5AdtPxktvolHuA1QTRazC9XXBwc9fUtSnYxYRZKxXCf-iqlBx57ojilS-NMKWeF80IlKmxIYm1uhyc6PPbZsytEdtYCObF~aEDG3ZoxRMHoguf5XvytNBNA-c4VhFnR3o7uuJDFGrU75KUYylm9f63Ina-tKoE4yuo519ELbBqxSCEPFz84h1V0YitnWOKW6Z7NFY01XNyRjn2IseAQA2gV91Ks33JXhp9YHk3H3O2CUxA~o4txR9hW3VaDQ4vmpvKRJrQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA" /> <meta property="og:image:url" content="https://aipp.silverchair-cdn.com/aipp/content_public/journal/ajp/76/7/10.1119_1.2894523/3/643_1_f1.jpeg?Expires=1735697734&Signature=3CfuOf1lVt1okUSE1puXzFeTpYiJ-6YctIJfRCkID5APENOvMDLL8XTbuHWhDWD19JZ~q0BdExNOS8ys5AdtPxktvolHuA1QTRazC9XXBwc9fUtSnYxYRZKxXCf-iqlBx57ojilS-NMKWeF80IlKmxIYm1uhyc6PPbZsytEdtYCObF~aEDG3ZoxRMHoguf5XvytNBNA-c4VhFnR3o7uuJDFGrU75KUYylm9f63Ina-tKoE4yuo519ELbBqxSCEPFz84h1V0YitnWOKW6Z7NFY01XNyRjn2IseAQA2gV91Ks33JXhp9YHk3H3O2CUxA~o4txR9hW3VaDQ4vmpvKRJrQ__&Key-Pair-Id=APKAIE5G5CRDK6RD3PGA" /> <meta property="og:image:alt" content="Fig. 1. 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Swendsen</a> <div class="al-author-info-wrap arrow-up stats-author-info-panel"> <div class="info-card-author authorInfo_ArticleTopInfo"> <div class="name-role-wrap"> <div class="info-card-name"> Robert H. Swendsen <span class="info-card-footnote"><a reveal-id="n1" data-open="n1" class="link link-ref link-reveal xref-author-notes"><sup>a)</sup></a></span> </div> </div> <div class="info-card-affilitation"> <div class="aff">Department of Physics, <div class="institution">Carnegie Mellon University</div>, Pittsburgh, Pennsylvania 15213</div> </div> <div class="info-card-search-label"> Search for other works by this author on: </div> <div class="info-card-search info-card-search-internal"> <a href="/search-results?f_AllAuthors=Robert+H.+Swendsen" rel="nofollow">This Site</a> </div> <div class="info-card-search info-card-search-pubmed"> <a href="http://www.ncbi.nlm.nih.gov/pubmed?cmd=search&term=Robert H. 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Phys.</em> 76, 643–648 (2008)</div> </div> <div class="pub-history-row citation-wrap-row clearfix"> <div class="ww-citation-wrap-doi"> <div class="citation-doi"> <a href="https://doi.org/10.1119/1.2894523">https://doi.org/10.1119/1.2894523</a> </div> </div> <div class="ww-citation-history-wrap js-history-dropdown-wrap"> <a href="javascript:;" class="history-label js-history-dropdown-trigger"> <span>Article history</span><i class="icon-history-small"></i> </a> <div class="ww-history js-history-entries-wrap"> <div class="history-entry"> <div class="wi-state">Received:</div> <div class="wi-date">September 25 2007</div> </div> <div class="history-entry"> <div class="wi-state">Accepted:</div> <div class="wi-date">February 16 2008</div> </div> </div> </div> </div> </div> </div> </div> <script> $(document).ready(function () { $('.article-top-widget').on('click', '.ati-toggle-trigger', function () { $(this).find('.icon-general-add, .icon-minus').toggleClass('icon-minus icon-general-add'); 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<div class="module-widget"> <div class="widget-items" data-widgetname="ArticleFulltext"> <div id="60602375" class="article-section-wrapper js-article-section js-content-section " data-section-parent-id="0"> <section class="abstract"><p>I describe an approach that helps students understand why the behavior of macroscopic systems distinguishes between past and future, even though the underlying classical or quantum microscopic equations are time-reversal invariant. This approach provides a qualitative understanding of irreversibility by an explicit calculation of irreversible behavior for a simple, time-reversal-invariant model system. The mathematics involved is accessible to upper-level undergraduate students.</p></section> </div> <div class="article-metadata-panel clearfix rs_skip"> <div class="widget-SolrResourceMetadata widget-instance-ContentMetadata_ArticleFulltext_Article"> <div class="content-metadata-taxonomies"> <div class="content-metadata"> <div class="content-metadata-taxonomies-title">Topics</div> <a href="/ajp/search-results?page=1&tax=C1763P29861">Molecular dynamics</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P2163P1720">Entropy</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P2163P47450">Irreversible process</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P3437P4840">Probability theory</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P4390P4396">Particle symmetry</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P4555P4555">Education</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P4555P4561">Teaching</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P5510P4553">Physicists</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P5789C1293">Maxwell-Boltzmann distribution</a><span>,</span> <a href="/ajp/search-results?page=1&tax=P5789P92261">Statistical mechanics theorems</a> </div> </div> </div> </div> <h2 scrollto-destination="60602475" id="60602475" data-section-title="REFERENCES" class="backreferences-title jumplink-heading">REFERENCES</h2> <div class=""> <div id="60602475" class="article-section-wrapper js-article-section js-content-section " data-section-parent-id="0"> <div class="ref-list js-splitview-ref-list"><div data-content-id="c1" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">1.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">P.</div> <div class="surname">Coveney</div></span> and <span class="name string-name"><div class="given-names">R.</div> <div class="surname">Highfield</div></span>, <div class="source">The Arrow of Time</div> (<div class="publisher-name">W. H. Allan</div>, London, <div class="year">1990</div>). This book contains a good discussion of the debate about irreversibility.<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=The%20Arrow%20of%20Time&author=P.%20Coveney&author=R.%20Highfield&publication_year=1990&book=The%20Arrow%20of%20Time" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c2" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">2.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">H. L.</div> <div class="surname">Frisch</div></span>, “<div class="article-title">An approach to equilibrium</div>,” <div class="source">Phys. Rev.</div> <div class="volume">109</div>, <div class="fpage">22</div>–<div class="lpage">29</div> (<div class="year">1958</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=An%20approach%20to%20equilibrium&author=H.%20L.%20Frisch&publication_year=1958&journal=Phys.%20Rev.&volume=109&pages=22-29" target="_blank">Google Scholar</a></div><div class="crossref-doi js-ref-link"><a class="openInAnotherWindow" href="http://dx.doi.org/10.1103/PhysRev.109.22" target="_blank">Crossref</a></div><div class="adsDoiReference hide"><a class="openInAnotherWindow" href="http://adsabs.harvard.edu/cgi-bin/basic_connect?qsearch=10.1103%2FPhysRev.109.22" target="_blank">Search ADS</a></div><div class="xslopenurl empty-target"><span class="inst-open-url-holders" data-targetId="10.1103%2FPhysRev.109.22"> </span></div> </div></div></div></div></div><div data-content-id="c3" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">3.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">A.</div> <div class="surname">Georgallas</div></span>, “<div class="article-title">Free expansion of an ideal gas into a box: An exactly solvable approach to equilibrium</div>,” <div class="source">Phys. Rev. A</div> <div class="volume">35</div>, <div class="fpage">3492</div>–<div class="lpage">3498</div> (<div class="year">1987</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Free%20expansion%20of%20an%20ideal%20gas%20into%20a%20box%3A%20An%20exactly%20solvable%20approach%20to%20equilibrium&author=A.%20Georgallas&publication_year=1987&journal=Phys.%20Rev.%20A&volume=35&pages=3492-3498" target="_blank">Google Scholar</a></div><div class="crossref-doi js-ref-link"><a class="openInAnotherWindow" href="http://dx.doi.org/10.1103/PhysRevA.35.3492" target="_blank">Crossref</a></div><div class="adsDoiReference hide"><a class="openInAnotherWindow" href="http://adsabs.harvard.edu/cgi-bin/basic_connect?qsearch=10.1103%2FPhysRevA.35.3492" target="_blank">Search ADS</a></div><div class="xslopenurl empty-target"><span class="inst-open-url-holders" data-targetId="10.1103%2FPhysRevA.35.3492"> </span></div> </div></div></div></div></div><div data-content-id="c4" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">4.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">L.</div> <div class="surname">Boltzmann</div></span>, “<div class="article-title">Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen</div>,” <div class="source">Wien. Ber.</div> <div class="volume">66</div>, <div class="fpage">275</div>–<div class="lpage">370</div> (<div class="year">1872</div>). The German word “Stoss” refers to a collision, “Zahl” means number, and “Ansatz” has become an English word for the starting point, or initial approximation.<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Weitere%20Studien%20%C3%BCber%20das%20W%C3%A4rmegleichgewicht%20unter%20Gasmolek%C3%BClen&author=L.%20Boltzmann&publication_year=1872&journal=Wien.%20Ber.&volume=66&pages=275-370" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c5" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">5.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">J.</div> <div class="surname">Loschmidt</div></span>, “<div class="article-title">Über den Zustand des Wärmegleichgewichtes eines Systems von Körpern mit Rücksicht auf die Schwerkraft</div>,” <div class="source">Sitzungsber. Akad. Wiss. Wien, Math.-Naturwiss. Kl., Abt. 2A</div> <div class="volume">73</div>, <div class="fpage">128</div>–<div class="lpage">142</div> (<div class="year">1876</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=%C3%9Cber%20den%20Zustand%20des%20W%C3%A4rmegleichgewichtes%20eines%20Systems%20von%20K%C3%B6rpern%20mit%20R%C3%BCcksicht%20auf%20die%20Schwerkraft&author=J.%20Loschmidt&publication_year=1876&journal=Sitzungsber.%20Akad.%20Wiss.%20Wien%2C%20Math.-Naturwiss.%20Kl.%2C%20Abt.%202A&volume=73&pages=128-142" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c6" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">6.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">E.</div> <div class="surname">Zermelo</div></span>, “<div class="article-title">Über einen Satz der Dynamik und die mechanische Wärmelehre</div>,” <div class="source">Ann. Phys.</div> <div class="volume">57</div>, <div class="fpage">485</div>–<div class="lpage">494</div> (<div class="year">1896</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=%C3%9Cber%20einen%20Satz%20der%20Dynamik%20und%20die%20mechanische%20W%C3%A4rmelehre&author=E.%20Zermelo&publication_year=1896&journal=Ann.%20Phys.&volume=57&pages=485-494" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c7" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">7.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">E.</div> <div class="surname">Zermelo</div></span>, “<div class="article-title">Über mechanische Erklärungen irreversibler Vorgänge. Eine Antwort auf Hrn. Boltzmann’s ‘Entgegnung’</div>,” <div class="source">Ann. Phys.</div> <div class="volume">59</div>, <div class="fpage">793</div>–<div class="lpage">801</div> (<div class="year">1896</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=%C3%9Cber%20mechanische%20Erkl%C3%A4rungen%20irreversibler%20Vorg%C3%A4nge.%20Eine%20Antwort%20auf%20Hrn.%20Boltzmann%E2%80%99s%20%E2%80%98Entgegnung%E2%80%99&author=E.%20Zermelo&publication_year=1896&journal=Ann.%20Phys.&volume=59&pages=793-801" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c8" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">8.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">M.</div> <div class="surname">Planck</div></span>, “<div class="article-title">Über das Gesetz der Energieverteilung im Normalspectrum</div>,” <div class="source">Ann. Phys.</div> <div class="volume">309</div>, <div class="fpage">553</div>–<div class="lpage">563</div> (<div class="year">1901</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=%C3%9Cber%20das%20Gesetz%20der%20Energieverteilung%20im%20Normalspectrum&author=M.%20Planck&publication_year=1901&journal=Ann.%20Phys.&volume=309&pages=553-563" target="_blank">Google Scholar</a></div><div class="crossref-doi js-ref-link"><a class="openInAnotherWindow" href="http://dx.doi.org/10.1002/(ISSN)1521-3889" target="_blank">Crossref</a></div><div class="adsDoiReference hide"><a class="openInAnotherWindow" href="http://adsabs.harvard.edu/cgi-bin/basic_connect?qsearch=10.1002%2F(ISSN)1521-3889" target="_blank">Search ADS</a></div><div class="xslopenurl empty-target"><span class="inst-open-url-holders" data-targetId="10.1002%2F(ISSN)1521-3889"> </span></div> </div></div></div></div></div><div data-content-id="c9" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">9.</span><div class="citation mixed-citation">“Umkehr” means “reversal,” referring to the reversal of velocities in Loschmidt’s argument. “Einwand” means “objection.”</div></div></div></div><div data-content-id="c10" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">10.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">H.</div> <div class="surname">Poincaré</div></span>, “<div class="article-title">Sur le probleme des trois corps et les équations de la dynamique</div>,” <div class="source">Acta Math.</div> <div class="volume">13</div>, <div class="fpage">1</div>–<div class="lpage">270</div> (<div class="year">1890</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Sur%20le%20probleme%20des%20trois%20corps%20et%20les%20%C3%A9quations%20de%20la%20dynamique&author=H.%20Poincar%C3%A9&publication_year=1890&journal=Acta%20Math.&volume=13&pages=1-270" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c11" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">11.</span><div class="citation mixed-citation">“Wiederkehr” means “return” or “recur,” referring to the return to the original state in Zermelo’s argument.</div></div></div></div><div data-content-id="c12" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">12.</span><div class="citation mixed-citation">If the period of each molecule divided into the longest period is an integer, the recurrence time will be equal to the longest period. It will not be astronomically long. Therefore, a probabilistic argument is necessary to establish that the recurrence time is extraordinarily long.</div></div></div></div><div data-content-id="c13" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">13.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">E. L.</div> <div class="surname">Hahn</div></span>, “<div class="article-title">Spin echos</div>,” <div class="source">Phys. Rev.</div><div class="pub-id-doi"><a href="https://doi.org/10.1103/PhysRev.80.580" class="link link-pub-id-doi openInAnotherWindow" target="_blank"> https://doi.org/10.1103/PhysRev.80.580</a></div> <div class="volume">80</div>, <div class="fpage">580</div>–<div class="lpage">594</div> (<div class="year">1950</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Spin%20echos&author=E.%20L.%20Hahn&publication_year=1950&journal=Phys.%20Rev.&volume=80&pages=580-594" target="_blank">Google Scholar</a></div><div class="crossref-doi js-ref-link"><a class="openInAnotherWindow" href="http://dx.doi.org/10.1103/PhysRev.80.580" target="_blank">Crossref</a></div><div class="adsDoiReference hide"><a class="openInAnotherWindow" href="http://adsabs.harvard.edu/cgi-bin/basic_connect?qsearch=10.1103%2FPhysRev.80.580" target="_blank">Search ADS</a></div><div class="xslopenurl empty-target"><span class="inst-open-url-holders" data-targetId="10.1103%2FPhysRev.80.580"> </span></div> </div></div></div></div></div><div data-content-id="c14" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">14.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">R. K.</div> <div class="surname">Pathria</div></span>, <div class="source">Statistical Mechanics</div> (<div class="publisher-name">Pergamon</div>, New York, <div class="year">1972</div>), pp. <div class="fpage">34</div>–<div class="lpage">36</div>.<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Statistical%20Mechanics&author=R.%20K.%20Pathria&publication_year=1972&book=Statistical%20Mechanics" target="_blank">Google Scholar</a></div> </div></div></div></div></div><div data-content-id="c15" xmlns:helper="urn:XsltStringHelper"><div class="ref false"><div class="ref-content "><span class="label title-label">15.</span><div class="citation mixed-citation"><span class="name string-name"><div class="given-names">R. H.</div> <div class="surname">Swendsen</div></span>, “<div class="article-title">Statistical mechanics of colloids and Boltzmann’s definition of the entropy</div>,” <div class="source">Am. J. Phys.</div><div class="pub-id-doi"><a href="https://doi.org/10.1119/1.2174962" class="link link-pub-id-doi openInAnotherWindow" target="_blank"> https://doi.org/10.1119/1.2174962</a></div> <div class="volume">74</div>, <div class="fpage">187</div>–<div class="lpage">190</div> (<div class="year">2006</div>).<div class="citation-links"><div class="google-scholar-ref-link js-ref-link"><a href="https://scholar.google.com/scholar_lookup?title=Statistical%20mechanics%20of%20colloids%20and%20Boltzmann%E2%80%99s%20definition%20of%20the%20entropy&author=R.%20H.%20Swendsen&publication_year=2006&journal=Am.%20J.%20Phys.&volume=74&pages=187-190" target="_blank">Google Scholar</a></div><div class="crossref-doi js-ref-link"><a class="openInAnotherWindow" href="http://dx.doi.org/10.1119/1.2174962" target="_blank">Crossref</a></div><div class="adsDoiReference hide"><a class="openInAnotherWindow" href="http://adsabs.harvard.edu/cgi-bin/basic_connect?qsearch=10.1119%2F1.2174962" target="_blank">Search ADS</a></div><div class="xslopenurl empty-target"><span class="inst-open-url-holders" data-targetId="10.1119%2F1.2174962"> </span></div> </div></div></div></div></div></div> </div>  </div> <div class="article-metadata-standalone-panel clearfix"></div> <div id="articlefulltext_figshare" class="figshare-wrapper" data-widgetname="filesViewerGeneric" data-showstats="True" data-showpageinfo="True" data-showsharebutton="True" data-showfiledetails="True" data-loaddoisbycollection="False"> </div> <div class="permissionstatement-section-wrapper "> <div class="copyright copyright-statement">© 2008 American Association of Physics Teachers.</div><div class="copyright copyright-year">2008</div><div class="copyright copyright-holder">American Association of Physics Teachers</div> </div>  <span id="UserHasAccess" data-userHasAccess="False"></span> </div> </div> </div> <div class="article-top-info-user-restricted-options">AAPT members receive access to the <em>American Journal of Physics</em> and <em>The Physics Teacher</em> as a member benefit. 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