CINXE.COM

<!DOCTYPE html> <html class="client-nojs vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available" lang="en" dir="ltr"> <head> <meta charset="UTF-8"> <title>Entropy (order and disorder) - Wikipedia</title> <script>(function(){var className="client-js vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-sticky-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-feature-night-mode-enabled skin-theme-clientpref-day vector-toc-available";var cookie=document.cookie.match(/(?:^|; )enwikimwclientpreferences=([^;]+)/);if(cookie){cookie[1].split('%2C').forEach(function(pref){className=className.replace(new RegExp('(^| )'+pref.replace(/-clientpref-\w+$|[^\w-]+/g,'')+'-clientpref-\\w+( |$)'),'$1'+pref+'$2');});}document.documentElement.className=className;}());RLCONF={"wgBreakFrames":false,"wgSeparatorTransformTable":["",""],"wgDigitTransformTable":["",""],"wgDefaultDateFormat":"dmy", "wgMonthNames":["","January","February","March","April","May","June","July","August","September","October","November","December"],"wgRequestId":"d485b4b8-aa48-4550-b0e8-05dc9dcc41ac","wgCanonicalNamespace":"","wgCanonicalSpecialPageName":false,"wgNamespaceNumber":0,"wgPageName":"Entropy_(order_and_disorder)","wgTitle":"Entropy (order and disorder)","wgCurRevisionId":1213009411,"wgRevisionId":1213009411,"wgArticleId":7728392,"wgIsArticle":true,"wgIsRedirect":false,"wgAction":"view","wgUserName":null,"wgUserGroups":["*"],"wgCategories":["CS1 maint: multiple names: authors list","Articles with short description","Short description is different from Wikidata","Thermodynamic entropy","State functions"],"wgPageViewLanguage":"en","wgPageContentLanguage":"en","wgPageContentModel":"wikitext","wgRelevantPageName":"Entropy_(order_and_disorder)","wgRelevantArticleId":7728392,"wgIsProbablyEditable":true,"wgRelevantPageIsProbablyEditable":true,"wgRestrictionEdit":[],"wgRestrictionMove":[], "wgNoticeProject":"wikipedia","wgCiteReferencePreviewsActive":false,"wgFlaggedRevsParams":{"tags":{"status":{"levels":1}}},"wgMediaViewerOnClick":true,"wgMediaViewerEnabledByDefault":true,"wgPopupsFlags":0,"wgVisualEditor":{"pageLanguageCode":"en","pageLanguageDir":"ltr","pageVariantFallbacks":"en"},"wgMFDisplayWikibaseDescriptions":{"search":true,"watchlist":true,"tagline":false,"nearby":true},"wgWMESchemaEditAttemptStepOversample":false,"wgWMEPageLength":20000,"wgRelatedArticlesCompat":[],"wgCentralAuthMobileDomain":false,"wgEditSubmitButtonLabelPublish":true,"wgULSPosition":"interlanguage","wgULSisCompactLinksEnabled":false,"wgVector2022LanguageInHeader":true,"wgULSisLanguageSelectorEmpty":false,"wgWikibaseItemId":"Q5380796","wgCheckUserClientHintsHeadersJsApi":["brands","architecture","bitness","fullVersionList","mobile","model","platform","platformVersion"],"GEHomepageSuggestedEditsEnableTopics":true,"wgGETopicsMatchModeEnabled":false, "wgGEStructuredTaskRejectionReasonTextInputEnabled":false,"wgGELevelingUpEnabledForUser":false};RLSTATE={"ext.globalCssJs.user.styles":"ready","site.styles":"ready","user.styles":"ready","ext.globalCssJs.user":"ready","user":"ready","user.options":"loading","ext.cite.styles":"ready","ext.math.styles":"ready","skins.vector.search.codex.styles":"ready","skins.vector.styles":"ready","skins.vector.icons":"ready","ext.wikimediamessages.styles":"ready","ext.visualEditor.desktopArticleTarget.noscript":"ready","ext.uls.interlanguage":"ready","wikibase.client.init":"ready","ext.wikimediaBadges":"ready"};RLPAGEMODULES=["ext.cite.ux-enhancements","mediawiki.page.media","site","mediawiki.page.ready","mediawiki.toc","skins.vector.js","ext.centralNotice.geoIP","ext.centralNotice.startUp","ext.gadget.ReferenceTooltips","ext.gadget.switcher","ext.urlShortener.toolbar","ext.centralauth.centralautologin","mmv.bootstrap","ext.popups","ext.visualEditor.desktopArticleTarget.init", "ext.visualEditor.targetLoader","ext.echo.centralauth","ext.eventLogging","ext.wikimediaEvents","ext.navigationTiming","ext.uls.interface","ext.cx.eventlogging.campaigns","ext.cx.uls.quick.actions","wikibase.client.vector-2022","ext.checkUser.clientHints","ext.growthExperiments.SuggestedEditSession","wikibase.sidebar.tracking"];</script> <script>(RLQ=window.RLQ||[]).push(function(){mw.loader.impl(function(){return["user.options@12s5i",function($,jQuery,require,module){mw.user.tokens.set({"patrolToken":"+\\","watchToken":"+\\","csrfToken":"+\\"}); }];});});</script> <link rel="stylesheet" href="/w/load.php?lang=en&modules=ext.cite.styles%7Cext.math.styles%7Cext.uls.interlanguage%7Cext.visualEditor.desktopArticleTarget.noscript%7Cext.wikimediaBadges%7Cext.wikimediamessages.styles%7Cskins.vector.icons%2Cstyles%7Cskins.vector.search.codex.styles%7Cwikibase.client.init&only=styles&skin=vector-2022"> <script async="" src="/w/load.php?lang=en&modules=startup&only=scripts&raw=1&skin=vector-2022"></script> <meta name="ResourceLoaderDynamicStyles" content=""> <link rel="stylesheet" href="/w/load.php?lang=en&modules=site.styles&only=styles&skin=vector-2022"> <meta name="generator" content="MediaWiki 1.44.0-wmf.4"> <meta name="referrer" content="origin"> <meta name="referrer" content="origin-when-cross-origin"> <meta name="robots" content="max-image-preview:standard"> <meta name="format-detection" content="telephone=no"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/1200px-Boltzmann-molecules.svg.png"> <meta property="og:image:width" content="1200"> <meta property="og:image:height" content="1081"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/800px-Boltzmann-molecules.svg.png"> <meta property="og:image:width" content="800"> <meta property="og:image:height" content="720"> <meta property="og:image" content="https://upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/640px-Boltzmann-molecules.svg.png"> <meta property="og:image:width" content="640"> <meta property="og:image:height" content="576"> <meta name="viewport" content="width=1120"> <meta property="og:title" content="Entropy (order and disorder) - Wikipedia"> <meta property="og:type" content="website"> <link rel="preconnect" href="//upload.wikimedia.org"> <link rel="alternate" media="only screen and (max-width: 640px)" href="//en.m.wikipedia.org/wiki/Entropy_(order_and_disorder)"> <link rel="alternate" type="application/x-wiki" title="Edit this page" href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit"> <link rel="apple-touch-icon" href="/static/apple-touch/wikipedia.png"> <link rel="icon" href="/static/favicon/wikipedia.ico"> <link rel="search" type="application/opensearchdescription+xml" href="/w/rest.php/v1/search" title="Wikipedia (en)"> <link rel="EditURI" type="application/rsd+xml" href="//en.wikipedia.org/w/api.php?action=rsd"> <link rel="canonical" href="https://en.wikipedia.org/wiki/Entropy_(order_and_disorder)"> <link rel="license" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en"> <link rel="alternate" type="application/atom+xml" title="Wikipedia Atom feed" href="/w/index.php?title=Special:RecentChanges&feed=atom"> <link rel="dns-prefetch" href="//meta.wikimedia.org" /> <link rel="dns-prefetch" href="//login.wikimedia.org"> </head> <body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject mw-editable page-Entropy_order_and_disorder rootpage-Entropy_order_and_disorder skin-vector-2022 action-view"><a class="mw-jump-link" href="#bodyContent">Jump to content</a> <div class="vector-header-container"> <header class="vector-header mw-header"> <div class="vector-header-start"> <nav class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-dropdown" class="vector-dropdown vector-main-menu-dropdown vector-button-flush-left vector-button-flush-right" > <input type="checkbox" id="vector-main-menu-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-main-menu-dropdown" class="vector-dropdown-checkbox " aria-label="Main menu" > <label id="vector-main-menu-dropdown-label" for="vector-main-menu-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-menu mw-ui-icon-wikimedia-menu"></span> <span class="vector-dropdown-label-text">Main menu</span> </label> <div class="vector-dropdown-content"> <div id="vector-main-menu-unpinned-container" class="vector-unpinned-container"> <div id="vector-main-menu" class="vector-main-menu vector-pinnable-element"> <div class="vector-pinnable-header vector-main-menu-pinnable-header vector-pinnable-header-unpinned" data-feature-name="main-menu-pinned" data-pinnable-element-id="vector-main-menu" data-pinned-container-id="vector-main-menu-pinned-container" data-unpinned-container-id="vector-main-menu-unpinned-container" > <div class="vector-pinnable-header-label">Main menu</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-main-menu.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-main-menu.unpin">hide</button> </div> <div id="p-navigation" class="vector-menu mw-portlet mw-portlet-navigation" > <div class="vector-menu-heading"> Navigation </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-mainpage-description" class="mw-list-item"><a href="/wiki/Main_Page" title="Visit the main page [z]" accesskey="z"><span>Main page</span></a></li><li id="n-contents" class="mw-list-item"><a href="/wiki/Wikipedia:Contents" title="Guides to browsing Wikipedia"><span>Contents</span></a></li><li id="n-currentevents" class="mw-list-item"><a href="/wiki/Portal:Current_events" title="Articles related to current events"><span>Current events</span></a></li><li id="n-randompage" class="mw-list-item"><a href="/wiki/Special:Random" title="Visit a randomly selected article [x]" accesskey="x"><span>Random article</span></a></li><li id="n-aboutsite" class="mw-list-item"><a href="/wiki/Wikipedia:About" title="Learn about Wikipedia and how it works"><span>About Wikipedia</span></a></li><li id="n-contactpage" class="mw-list-item"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us" title="How to contact Wikipedia"><span>Contact us</span></a></li> </ul> </div> </div> <div id="p-interaction" class="vector-menu mw-portlet mw-portlet-interaction" > <div class="vector-menu-heading"> Contribute </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="n-help" class="mw-list-item"><a href="/wiki/Help:Contents" title="Guidance on how to use and edit Wikipedia"><span>Help</span></a></li><li id="n-introduction" class="mw-list-item"><a href="/wiki/Help:Introduction" title="Learn how to edit Wikipedia"><span>Learn to edit</span></a></li><li id="n-portal" class="mw-list-item"><a href="/wiki/Wikipedia:Community_portal" title="The hub for editors"><span>Community portal</span></a></li><li id="n-recentchanges" class="mw-list-item"><a href="/wiki/Special:RecentChanges" title="A list of recent changes to Wikipedia [r]" accesskey="r"><span>Recent changes</span></a></li><li id="n-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_upload_wizard" title="Add images or other media for use on Wikipedia"><span>Upload file</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> <a href="/wiki/Main_Page" class="mw-logo"> <img class="mw-logo-icon" src="/static/images/icons/wikipedia.png" alt="" aria-hidden="true" height="50" width="50"> <span class="mw-logo-container skin-invert"> <img class="mw-logo-wordmark" alt="Wikipedia" src="/static/images/mobile/copyright/wikipedia-wordmark-en.svg" style="width: 7.5em; height: 1.125em;"> <img class="mw-logo-tagline" alt="The Free Encyclopedia" src="/static/images/mobile/copyright/wikipedia-tagline-en.svg" width="117" height="13" style="width: 7.3125em; height: 0.8125em;"> </span> </a> </div> <div class="vector-header-end"> <div id="p-search" role="search" class="vector-search-box-vue vector-search-box-collapses vector-search-box-show-thumbnail vector-search-box-auto-expand-width vector-search-box"> <a href="/wiki/Special:Search" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only search-toggle" title="Search Wikipedia [f]" accesskey="f"><span class="vector-icon mw-ui-icon-search mw-ui-icon-wikimedia-search"></span> <span>Search</span> </a> <div class="vector-typeahead-search-container"> <div class="cdx-typeahead-search cdx-typeahead-search--show-thumbnail cdx-typeahead-search--auto-expand-width"> <form action="/w/index.php" id="searchform" class="cdx-search-input cdx-search-input--has-end-button"> <div id="simpleSearch" class="cdx-search-input__input-wrapper" data-search-loc="header-moved"> <div class="cdx-text-input cdx-text-input--has-start-icon"> <input class="cdx-text-input__input" type="search" name="search" placeholder="Search Wikipedia" aria-label="Search Wikipedia" autocapitalize="sentences" title="Search Wikipedia [f]" accesskey="f" id="searchInput" > <span class="cdx-text-input__icon cdx-text-input__start-icon"></span> </div> <input type="hidden" name="title" value="Special:Search"> </div> <button class="cdx-button cdx-search-input__end-button">Search</button> </form> </div> </div> </div> <nav class="vector-user-links vector-user-links-wide" aria-label="Personal tools"> <div class="vector-user-links-main"> <div id="p-vector-user-menu-preferences" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-userpage" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-dropdown" class="vector-dropdown " title="Change the appearance of the page's font size, width, and color" > <input type="checkbox" id="vector-appearance-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-appearance-dropdown" class="vector-dropdown-checkbox " aria-label="Appearance" > <label id="vector-appearance-dropdown-label" for="vector-appearance-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-appearance mw-ui-icon-wikimedia-appearance"></span> <span class="vector-dropdown-label-text">Appearance</span> </label> <div class="vector-dropdown-content"> <div id="vector-appearance-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <div id="p-vector-user-menu-notifications" class="vector-menu mw-portlet emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> <div id="p-vector-user-menu-overflow" class="vector-menu mw-portlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=Entropy+%28order+and+disorder%29" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=Entropy+%28order+and+disorder%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o" class=""><span>Log in</span></a> </li> </ul> </div> </div> </div> <div id="vector-user-links-dropdown" class="vector-dropdown vector-user-menu vector-button-flush-right vector-user-menu-logged-out" title="Log in and more options" > <input type="checkbox" id="vector-user-links-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-user-links-dropdown" class="vector-dropdown-checkbox " aria-label="Personal tools" > <label id="vector-user-links-dropdown-label" for="vector-user-links-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">Personal tools</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="User menu" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en"><span>Donate</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:CreateAccount&returnto=Entropy+%28order+and+disorder%29" title="You are encouraged to create an account and log in; however, it is not mandatory"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>Create account</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:UserLogin&returnto=Entropy+%28order+and+disorder%29" title="You're encouraged to log in; however, it's not mandatory. [o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>Log in</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> Pages for logged out editors <a href="/wiki/Help:Introduction" aria-label="Learn more about editing"><span>learn more</span></a> </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:MyContributions" title="A list of edits made from this IP address [y]" accesskey="y"><span>Contributions</span></a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:MyTalk" title="Discussion about edits from this IP address [n]" accesskey="n"><span>Talk</span></a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="Site"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="Contents" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">Contents</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Overview</span> </div> </a> <ul id="toc-Overview-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Phase_change" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Phase_change"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Phase change</span> </div> </a> <ul id="toc-Phase_change-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Entropy-driven_order" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Entropy-driven_order"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Entropy-driven order</span> </div> </a> <ul id="toc-Entropy-driven_order-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Adiabatic_demagnetization" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Adiabatic_demagnetization"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Adiabatic demagnetization</span> </div> </a> <ul id="toc-Adiabatic_demagnetization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Difficulties_with_the_term_"disorder"" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Difficulties_with_the_term_"disorder""> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Difficulties with the term "disorder"</span> </div> </a> <ul id="toc-Difficulties_with_the_term_"disorder"-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Entropy (order and disorder)</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="This article exist only in this language. Add the article for other languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-0" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">Add languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> <div class="after-portlet after-portlet-lang"><span class="uls-after-portlet-link"></span><span class="wb-langlinks-add wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5380796#sitelinks-wikipedia" title="Add interlanguage links" class="wbc-editpage">Add links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Entropy_(order_and_disorder)" title="View the content page [c]" accesskey="c"><span>Article</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a href="/wiki/Talk:Entropy_(order_and_disorder)" rel="discussion" title="Discuss improvements to the content page [t]" accesskey="t"><span>Talk</span></a></li> </ul> </div> </div> <div id="vector-variants-dropdown" class="vector-dropdown emptyPortlet" > <input type="checkbox" id="vector-variants-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-variants-dropdown" class="vector-dropdown-checkbox " aria-label="Change language variant" > <label id="vector-variants-dropdown-label" for="vector-variants-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">English</span> </label> <div class="vector-dropdown-content"> <div id="p-variants" class="vector-menu mw-portlet mw-portlet-variants emptyPortlet" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> </ul> </div> </div> </div> </div> </nav> </div> <div id="right-navigation" class="vector-collapsible"> <nav aria-label="Views"> <div id="p-views" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-views" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-view" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/Entropy_(order_and_disorder)"><span>Read</span></a></li><li id="ca-edit" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-history" class="vector-tab-noicon mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=history" title="Past revisions of this page [h]" accesskey="h"><span>View history</span></a></li> </ul> </div> </div> </nav> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-dropdown" class="vector-dropdown vector-page-tools-dropdown" > <input type="checkbox" id="vector-page-tools-dropdown-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-tools-dropdown" class="vector-dropdown-checkbox " aria-label="Tools" > <label id="vector-page-tools-dropdown-label" for="vector-page-tools-dropdown-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet" aria-hidden="true" ><span class="vector-dropdown-label-text">Tools</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-tools-unpinned-container" class="vector-unpinned-container"> <div id="vector-page-tools" class="vector-page-tools vector-pinnable-element"> <div class="vector-pinnable-header vector-page-tools-pinnable-header vector-pinnable-header-unpinned" data-feature-name="page-tools-pinned" data-pinnable-element-id="vector-page-tools" data-pinned-container-id="vector-page-tools-pinned-container" data-unpinned-container-id="vector-page-tools-unpinned-container" > <div class="vector-pinnable-header-label">Tools</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-page-tools.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-page-tools.unpin">hide</button> </div> <div id="p-cactions" class="vector-menu mw-portlet mw-portlet-cactions emptyPortlet vector-has-collapsible-items" title="More options" > <div class="vector-menu-heading"> Actions </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-more-view" class="selected vector-more-collapsible-item mw-list-item"><a href="/wiki/Entropy_(order_and_disorder)"><span>Read</span></a></li><li id="ca-more-edit" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit" title="Edit this page [e]" accesskey="e"><span>Edit</span></a></li><li id="ca-more-history" class="vector-more-collapsible-item mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=history"><span>View history</span></a></li> </ul> </div> </div> <div id="p-tb" class="vector-menu mw-portlet mw-portlet-tb" > <div class="vector-menu-heading"> General </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-whatlinkshere" class="mw-list-item"><a href="/wiki/Special:WhatLinksHere/Entropy_(order_and_disorder)" title="List of all English Wikipedia pages containing links to this page [j]" accesskey="j"><span>What links here</span></a></li><li id="t-recentchangeslinked" class="mw-list-item"><a href="/wiki/Special:RecentChangesLinked/Entropy_(order_and_disorder)" rel="nofollow" title="Recent changes in pages linked from this page [k]" accesskey="k"><span>Related changes</span></a></li><li id="t-upload" class="mw-list-item"><a href="/wiki/Wikipedia:File_Upload_Wizard" title="Upload files [u]" accesskey="u"><span>Upload file</span></a></li><li id="t-specialpages" class="mw-list-item"><a href="/wiki/Special:SpecialPages" title="A list of all special pages [q]" accesskey="q"><span>Special pages</span></a></li><li id="t-permalink" class="mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&oldid=1213009411" title="Permanent link to this revision of this page"><span>Permanent link</span></a></li><li id="t-info" class="mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=info" title="More information about this page"><span>Page information</span></a></li><li id="t-cite" class="mw-list-item"><a href="/w/index.php?title=Special:CiteThisPage&page=Entropy_%28order_and_disorder%29&id=1213009411&wpFormIdentifier=titleform" title="Information on how to cite this page"><span>Cite this page</span></a></li><li id="t-urlshortener" class="mw-list-item"><a href="/w/index.php?title=Special:UrlShortener&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEntropy_%28order_and_disorder%29"><span>Get shortened URL</span></a></li><li id="t-urlshortener-qrcode" class="mw-list-item"><a href="/w/index.php?title=Special:QrCode&url=https%3A%2F%2Fen.wikipedia.org%2Fwiki%2FEntropy_%28order_and_disorder%29"><span>Download QR code</span></a></li> </ul> </div> </div> <div id="p-coll-print_export" class="vector-menu mw-portlet mw-portlet-coll-print_export" > <div class="vector-menu-heading"> Print/export </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="coll-download-as-rl" class="mw-list-item"><a href="/w/index.php?title=Special:DownloadAsPdf&page=Entropy_%28order_and_disorder%29&action=show-download-screen" title="Download this page as a PDF file"><span>Download as PDF</span></a></li><li id="t-print" class="mw-list-item"><a href="/w/index.php?title=Entropy_(order_and_disorder)&printable=yes" title="Printable version of this page [p]" accesskey="p"><span>Printable version</span></a></li> </ul> </div> </div> <div id="p-wikibase-otherprojects" class="vector-menu mw-portlet mw-portlet-wikibase-otherprojects" > <div class="vector-menu-heading"> In other projects </div> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="t-wikibase" class="wb-otherproject-link wb-otherproject-wikibase-dataitem mw-list-item"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q5380796" title="Structured data on this page hosted by Wikidata [g]" accesskey="g"><span>Wikidata item</span></a></li> </ul> </div> </div> </div> </div> </div> </div> </nav> </div> </div> </div> <div class="vector-column-end"> <div class="vector-sticky-pinned-container"> <nav class="vector-page-tools-landmark" aria-label="Page tools"> <div id="vector-page-tools-pinned-container" class="vector-pinned-container"> </div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Interpretation of entropy as the change in arrangement of a system's particles</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Boltzmann-molecules.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/200px-Boltzmann-molecules.svg.png" decoding="async" width="200" height="180" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/300px-Boltzmann-molecules.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Boltzmann-molecules.svg/400px-Boltzmann-molecules.svg.png 2x" data-file-width="402" data-file-height="362" /></a><figcaption>Boltzmann's molecules (1896) shown at a "rest position" in a solid</figcaption></figure> <p>In <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a>, <a href="/wiki/Entropy" title="Entropy">entropy</a> is often associated with the amount of <a href="/wiki/Order_and_disorder" title="Order and disorder">order or disorder</a> in a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a>. This stems from <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a>' 1862 assertion that any <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">thermodynamic process</a> always "admits to being reduced [reduction] to the alteration in some way or another of the <i>arrangement</i> of the constituent parts of the <a href="/wiki/Working_body" class="mw-redirect" title="Working body">working body</a>" and that internal <a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">work</a> associated with these alterations is quantified energetically by a measure of "entropy" change, according to the following <a href="/wiki/Differential_equation" title="Differential equation">differential expression</a>:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int \!{\frac {\delta Q}{T}}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mspace width="negativethinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int \!{\frac {\delta Q}{T}}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd17f76f3bca1fe69adfb2a689f452a0981574cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.178ex; height:5.843ex;" alt="{\displaystyle \int \!{\frac {\delta Q}{T}}\geq 0}"></span></dd></dl> <p>where <span class="texhtml mvar" style="font-style:italic;">Q</span> = <a href="/wiki/Thermal_energy" title="Thermal energy">motional energy</a> ("heat") that is transferred <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversibly</a> to the system from the <a href="/wiki/Environment_(systems)" title="Environment (systems)">surroundings</a> and <span class="texhtml mvar" style="font-style:italic;">T</span> = the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a> at which the transfer occurs. </p><p>In the years to follow, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a> translated these 'alterations of arrangement' into a probabilistic view of order and disorder in <a href="/wiki/Gas-phase" class="mw-redirect" title="Gas-phase">gas-phase</a> molecular systems. In the context of entropy, "<i>perfect internal disorder</i>" has often been regarded as describing thermodynamic equilibrium, but since the thermodynamic concept is so far from everyday thinking, the use of the term in <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> has caused much confusion and misunderstanding. </p><p>In recent years, to interpret the concept of entropy, by further describing the 'alterations of arrangement', there has been a shift away from the words 'order' and 'disorder', to words such as 'spread' and <a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">'dispersal'</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=1" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>This "molecular ordering" entropy perspective traces its origins to molecular movement interpretations developed by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in the 1850s, particularly with his 1862 visual conception of molecular <a href="/wiki/Disgregation" title="Disgregation">disgregation</a>. Similarly, in 1859, after reading a paper on the diffusion of molecules by Clausius, Scottish physicist <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> formulated the <a href="/wiki/Maxwell_distribution" class="mw-redirect" title="Maxwell distribution">Maxwell distribution</a> of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>In 1864, <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>, a young student in Vienna, came across Maxwell's paper and was so inspired by it that he spent much of his long and distinguished life developing the subject further. Later, Boltzmann, in efforts to develop a <a href="/wiki/Kinetic_theory_of_gases" title="Kinetic theory of gases">kinetic theory</a> for the behavior of a gas, applied the laws of <a href="/wiki/Probability" title="Probability">probability</a> to Maxwell's and Clausius' molecular interpretation of entropy so as to begin to interpret entropy in terms of order and disorder. Similarly, in 1882 <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a> used the word "Unordnung" (disorder) to describe entropy.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=2" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To highlight the fact that order and disorder are commonly understood to be measured in terms of entropy, below are current science encyclopedia and science dictionary definitions of entropy: </p> <ul><li>A measure of the unavailability of a system's energy to do work; also a measure of disorder; the higher the entropy the greater the disorder.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li> <li>A measure of disorder; the higher the entropy the greater the disorder.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li> <li>In thermodynamics, a parameter representing the state of disorder of a system at the atomic, ionic, or molecular level; the greater the disorder the higher the entropy.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>A measure of disorder in the universe or of the unavailability of the energy in a system to do work.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <p>Entropy and disorder also have associations with <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">equilibrium</a>.<sup id="cite_ref-Landsberg-A_8-0" class="reference"><a href="#cite_note-Landsberg-A-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Technically, <i>entropy</i>, from this perspective, is defined as a thermodynamic property which serves as a measure of how close a system is to equilibrium—that is, to perfect internal <b>disorder</b>.<sup id="cite_ref-Encarta_9-0" class="reference"><a href="#cite_note-Encarta-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Likewise, the value of the entropy of a distribution of atoms and molecules in a <a href="/wiki/Thermodynamic_system" title="Thermodynamic system">thermodynamic system</a> is a measure of the disorder in the arrangements of its particles.<sup id="cite_ref-Greven_10-0" class="reference"><a href="#cite_note-Greven-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In a stretched out piece of rubber, for example, the arrangement of the molecules of its structure has an "ordered" distribution and has zero entropy, while the "disordered" kinky distribution of the atoms and molecules in the rubber in the non-stretched state has positive entropy. Similarly, in a gas, the <b>order</b> is perfect and the measure of entropy of the system has its lowest value when all the molecules are in one place, whereas when more points are occupied the gas is all the more disorderly and the measure of the entropy of the system has its largest value.<sup id="cite_ref-Greven_10-1" class="reference"><a href="#cite_note-Greven-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Systems_ecology" title="Systems ecology">systems ecology</a>, as another example, the entropy of a collection of items comprising a system is defined as a measure of their disorder or equivalently the relative likelihood of the instantaneous configuration of the items.<sup id="cite_ref-Ulanowicz_11-0" class="reference"><a href="#cite_note-Ulanowicz-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> Moreover, according to theoretical ecologist and chemical engineer <a href="/wiki/Robert_Ulanowicz" title="Robert Ulanowicz">Robert Ulanowicz</a>, "that entropy might provide a quantification of the heretofore subjective notion of disorder has spawned innumerable scientific and philosophical narratives."<sup id="cite_ref-Ulanowicz_11-1" class="reference"><a href="#cite_note-Ulanowicz-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> In particular, many biologists have taken to speaking in terms of the entropy of an organism, or about its antonym <a href="/wiki/Negentropy" title="Negentropy">negentropy</a>, as a measure of the structural order within an organism.<sup id="cite_ref-Ulanowicz_11-2" class="reference"><a href="#cite_note-Ulanowicz-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The mathematical basis with respect to the association entropy has with order and disorder began, essentially, with the famous <a href="/wiki/Boltzmann_formula" class="mw-redirect" title="Boltzmann formula">Boltzmann formula</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=k_{\mathrm {B} }\ln W\!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>ln</mi> <mo>⁡</mo> <mi>W</mi> <mspace width="negativethinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=k_{\mathrm {B} }\ln W\!}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6800bd7f3b41e6bab0266dc671151ef1afb8dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:12.354ex; height:2.509ex;" alt="{\displaystyle S=k_{\mathrm {B} }\ln W\!}"></span>, which relates entropy <i>S</i> to the number of possible states <i>W</i> in which a system can be found.<sup id="cite_ref-Jorgensen_13-0" class="reference"><a href="#cite_note-Jorgensen-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> As an example, consider a box that is divided into two sections. What is the probability that a certain number, or all of the particles, will be found in one section versus the other when the particles are randomly allocated to different places within the box? If you only have one particle, then that system of one particle can subsist in two states, one side of the box versus the other. If you have more than one particle, or define states as being further locational subdivisions of the box, the entropy is larger because the number of states is greater. The relationship between entropy, order, and disorder in the Boltzmann equation is so clear among physicists that according to the views of thermodynamic ecologists Sven Jorgensen and Yuri Svirezhev, "it is obvious that entropy is a measure of order or, most likely, disorder in the system."<sup id="cite_ref-Jorgensen_13-1" class="reference"><a href="#cite_note-Jorgensen-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> In this direction, the second law of thermodynamics, as famously enunciated by <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> in 1865, states that: </p> <div style="font-size:125%"> <style data-mw-deduplicate="TemplateStyles:r1244412712">.mw-parser-output .templatequote{overflow:hidden;margin:1em 0;padding:0 32px}.mw-parser-output .templatequotecite{line-height:1.5em;text-align:left;margin-top:0}@media(min-width:500px){.mw-parser-output .templatequotecite{padding-left:1.6em}}</style><blockquote class="templatequote"><p>The entropy of the universe tends to a maximum.</p></blockquote> </div> <p>Thus, if entropy is associated with disorder and if the entropy of the universe is headed towards maximal entropy, then many are often puzzled as to the nature of the "ordering" process and operation of <a href="/wiki/Evolution" title="Evolution">evolution</a> in relation to Clausius' most famous version of the second law, which states that the universe is headed towards maximal "disorder". In the recent 2003 book <i>SYNC – the Emerging Science of Spontaneous Order</i> by <a href="/wiki/Steven_Strogatz" title="Steven Strogatz">Steven Strogatz</a>, for example, we find "Scientists have often been baffled by the existence of spontaneous order in the universe. The <a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">laws of thermodynamics</a> seem to dictate the opposite, that nature should inexorably degenerate toward a state of greater disorder, greater entropy. Yet all around us we see magnificent structures—galaxies, cells, ecosystems, human beings—that have all somehow managed to assemble themselves."<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p><p>The common argument used to explain this is that, locally, entropy can be lowered by external action, e.g. solar heating action, and that this applies to machines, such as a refrigerator, where the entropy in the cold chamber is being reduced, to growing crystals, and to living organisms.<sup id="cite_ref-Encarta_9-1" class="reference"><a href="#cite_note-Encarta-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> This local increase in order is, however, only possible at the expense of an entropy increase in the surroundings; here more disorder must be created.<sup id="cite_ref-Encarta_9-2" class="reference"><a href="#cite_note-Encarta-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Brooks_15-0" class="reference"><a href="#cite_note-Brooks-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> The conditioner of this statement suffices that living systems are <a href="/wiki/Open_system_(systems_theory)" title="Open system (systems theory)">open systems</a> in which both <a href="/wiki/Heat" title="Heat">heat</a>, <a href="/wiki/Mass" title="Mass">mass</a>, and or <a href="/wiki/Mechanical_work" class="mw-redirect" title="Mechanical work">work</a> may transfer into or out of the system. Unlike temperature, the putative entropy of a living system would drastically change if the organism were thermodynamically isolated. If an organism was in this type of "isolated" situation, its entropy would increase markedly as the once-living components of the organism decayed to an unrecognizable mass.<sup id="cite_ref-Ulanowicz_11-3" class="reference"><a href="#cite_note-Ulanowicz-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Phase_change">Phase change</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=3" title="Edit section: Phase change"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Owing to these early developments, the typical example of entropy change Δ<i>S</i> is that associated with phase change. In solids, for example, which are typically ordered on the molecular scale, usually have smaller entropy than liquids, and liquids have smaller entropy than gases and colder gases have smaller entropy than hotter gases. Moreover, according to the <a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">third law of thermodynamics</a>, at <a href="/wiki/Absolute_zero" title="Absolute zero">absolute zero</a> temperature, crystalline structures are approximated to have perfect "order" and zero entropy. This correlation occurs because the numbers of different microscopic quantum energy states available to an ordered system are usually much smaller than the number of states available to a system that appears to be disordered. </p><p>From his famous 1896 <i>Lectures on Gas Theory</i>, Boltzmann diagrams the structure of a solid body, as shown above, by postulating that each <a href="/wiki/Molecule" title="Molecule">molecule</a> in the body has a "rest position". According to Boltzmann, if it approaches a neighbor molecule it is repelled by it, but if it moves farther away there is an attraction. This, of course was a revolutionary perspective in its time; many, during these years, did not believe in the existence of either atoms or molecules (see: <a href="/wiki/History_of_the_molecule" class="mw-redirect" title="History of the molecule">history of the molecule</a>).<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> According to these early views, and others such as those developed by <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">William Thomson</a>, if energy in the form of <a href="/wiki/Heat" title="Heat">heat</a> is added to a solid, so to make it into a liquid or a gas, a common depiction is that the ordering of the atoms and molecules becomes more random and chaotic with an increase in temperature: </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Solid-liquid-gas.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Solid-liquid-gas.svg/600px-Solid-liquid-gas.svg.png" decoding="async" width="600" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Solid-liquid-gas.svg/900px-Solid-liquid-gas.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9b/Solid-liquid-gas.svg/1200px-Solid-liquid-gas.svg.png 2x" data-file-width="1784" data-file-height="626" /></a><figcaption></figcaption></figure> <p>Thus, according to Boltzmann, owing to increases in thermal motion, whenever heat is added to a working substance, the rest position of molecules will be pushed apart, the body will expand, and this will create more <i>molar-disordered</i> distributions and arrangements of molecules. These disordered arrangements, subsequently, correlate, via probability arguments, to an increase in the measure of entropy.<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Entropy-driven_order">Entropy-driven order</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=4" title="Edit section: Entropy-driven order"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Entropy has been historically, e.g. by Clausius and Helmholtz, associated with disorder. However, in common speech, order is used to describe organization, structural regularity, or form, like that found in a crystal compared with a gas. This commonplace notion of order is described quantitatively by <a href="/wiki/Landau_theory" title="Landau theory">Landau theory</a>. In Landau theory, the development of order in the everyday sense coincides with the change in the value of a mathematical quantity, a so-called <a href="/wiki/Phase_transitions#Order_parameters" class="mw-redirect" title="Phase transitions">order parameter</a>. An example of an order parameter for crystallization is "bond orientational order" describing the development of preferred directions (the crystallographic axes) in space. For many systems, phases with more structural (e.g. crystalline) order exhibit less entropy than fluid phases under the same thermodynamic conditions. In these cases, labeling phases as ordered or disordered according to the relative amount of entropy (per the Clausius/Helmholtz notion of order/disorder) or via the existence of structural regularity (per the Landau notion of order/disorder) produces matching labels. </p><p>However, there is a broad class<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> of systems that manifest entropy-driven order, in which phases with organization or structural regularity, e.g. crystals, have higher entropy than structurally disordered (e.g. fluid) phases under the same thermodynamic conditions. In these systems phases that would be labeled as disordered by virtue of their higher entropy (in the sense of Clausius or Helmholtz) are ordered in both the everyday sense and in Landau theory. </p><p>Under suitable thermodynamic conditions, entropy has been predicted or discovered to induce systems to form ordered liquid-crystals, crystals, and quasicrystals.<sup id="cite_ref-19" class="reference"><a href="#cite_note-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-DamascenoScience2012_21-0" class="reference"><a href="#cite_note-DamascenoScience2012-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> In many systems, directional <a href="/wiki/Entropic_force#Colloids" title="Entropic force">entropic forces</a> drive this behavior. More recently, it has been shown it is possible to precisely engineer particles for target ordered structures.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Adiabatic_demagnetization">Adiabatic demagnetization</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=5" title="Edit section: Adiabatic demagnetization"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the quest for ultra-cold temperatures, a temperature lowering technique called <a href="/wiki/Adiabatic_demagnetization" class="mw-redirect" title="Adiabatic demagnetization">adiabatic demagnetization</a> is used, where atomic entropy considerations are utilized which can be described in order-disorder terms.<sup id="cite_ref-Halliday_23-0" class="reference"><a href="#cite_note-Halliday-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In this process, a sample of solid such as chrome-alum salt, whose molecules are equivalent to tiny magnets, is inside an insulated enclosure cooled to a low temperature, typically 2 or 4 kelvins, with a strong <a href="/wiki/Magnetic_field" title="Magnetic field">magnetic field</a> being applied to the container using a powerful external magnet, so that the tiny molecular magnets are aligned forming a well-ordered "initial" state at that low temperature. This magnetic alignment means that the magnetic energy of each molecule is minimal.<sup id="cite_ref-nasa_24-0" class="reference"><a href="#cite_note-nasa-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> The external magnetic field is then reduced, a removal that is considered to be closely <a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">reversible</a>. Following this reduction, the atomic magnets then assume random less-ordered orientations, owing to thermal agitations, in the "final" state: </p> <figure class="mw-halign-center" typeof="mw:File/Thumb"><a href="/wiki/File:Adiabatic-demagnitization.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Adiabatic-demagnitization.svg/275px-Adiabatic-demagnitization.svg.png" decoding="async" width="275" height="257" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/29/Adiabatic-demagnitization.svg/413px-Adiabatic-demagnitization.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/29/Adiabatic-demagnitization.svg/550px-Adiabatic-demagnitization.svg.png 2x" data-file-width="1541" data-file-height="1438" /></a><figcaption>Entropy "order"/"disorder" considerations in the process of <a href="/wiki/Adiabatic_demagnetization" class="mw-redirect" title="Adiabatic demagnetization">adiabatic demagnetization</a></figcaption></figure> <p>The "disorder" and hence the entropy associated with the change in the atomic alignments has clearly increased.<sup id="cite_ref-Halliday_23-1" class="reference"><a href="#cite_note-Halliday-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> In terms of energy flow, the movement from a magnetically aligned state requires energy from the thermal motion of the molecules, converting thermal energy into magnetic energy.<sup id="cite_ref-nasa_24-1" class="reference"><a href="#cite_note-nasa-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> Yet, according to the <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a>, because no <a href="/wiki/Heat" title="Heat">heat</a> can enter or leave the container, due to its adiabatic insulation, the system should exhibit no change in entropy, i.e. Δ<i>S</i> = 0. The increase in disorder, however, associated with the randomizing directions of the atomic magnets represents an entropy <i>increase</i>? To compensate for this, the disorder (entropy) associated with the <a href="/wiki/Temperature" title="Temperature">temperature</a> of the specimen must <i>decrease</i> by the same amount.<sup id="cite_ref-Halliday_23-2" class="reference"><a href="#cite_note-Halliday-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> The temperature thus falls as a result of this process of thermal energy being converted into magnetic energy. If the magnetic field is then increased, the temperature rises and the magnetic salt has to be cooled again using a cold material such as liquid helium.<sup id="cite_ref-nasa_24-2" class="reference"><a href="#cite_note-nasa-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Difficulties_with_the_term_"disorder""><span id="Difficulties_with_the_term_.22disorder.22"></span>Difficulties with the term "disorder"</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=6" title="Edit section: Difficulties with the term "disorder""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In recent years the long-standing use of term "disorder" to discuss entropy has met with some criticism.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-crutch_28-0" class="reference"><a href="#cite_note-crutch-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-29" class="reference"><a href="#cite_note-29"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Critics of the terminology state that entropy is not a measure of 'disorder' or 'chaos', but rather a measure of <a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">energy's diffusion</a> or dispersal to more microstates. Shannon's use of the term 'entropy' in information theory refers to the most compressed, or least dispersed, amount of code needed to encompass the content of a signal.<sup id="cite_ref-31" class="reference"><a href="#cite_note-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-33" class="reference"><a href="#cite_note-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Entropy" title="Entropy">Entropy</a></li> <li><a href="/wiki/Entropy_production" title="Entropy production">Entropy production</a></li> <li><a href="/wiki/Entropy_rate" title="Entropy rate">Entropy rate</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">History of entropy</a></li> <li><a href="/wiki/Entropy_of_mixing" title="Entropy of mixing">Entropy of mixing</a></li> <li><a href="/wiki/Entropy_(information_theory)" title="Entropy (information theory)">Entropy (information theory)</a></li> <li><a href="/wiki/Entropy_(computing)" title="Entropy (computing)">Entropy (computing)</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy (energy dispersal)</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second law of thermodynamics</a></li> <li><a href="/wiki/Entropy_(statistical_thermodynamics)" title="Entropy (statistical thermodynamics)">Entropy (statistical thermodynamics)</a></li> <li><a href="/wiki/Entropy_(classical_thermodynamics)" title="Entropy (classical thermodynamics)">Entropy (classical thermodynamics)</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><i>Mechanical Theory of Heat</i> – Nine Memoirs on the development of concept of "Entropy" by Rudolf Clausius [1850–1865]</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMahon,_Basil2003" class="citation book cs1">Mahon, Basil (2003). <i>The Man Who Changed Everything – the Life of James Clerk Maxwell</i>. Hoboken, NJ: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-470-86171-1" title="Special:BookSources/0-470-86171-1"><bdi>0-470-86171-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Man+Who+Changed+Everything+%E2%80%93+the+Life+of+James+Clerk+Maxwell&rft.place=Hoboken%2C+NJ&rft.pub=Wiley&rft.date=2003&rft.isbn=0-470-86171-1&rft.au=Mahon%2C+Basil&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAnderson2005" class="citation book cs1">Anderson, Greg (2005). <i>Thermodynamics of Natural Systems</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-84772-9" title="Special:BookSources/0-521-84772-9"><bdi>0-521-84772-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Thermodynamics+of+Natural+Systems&rft.pub=Cambridge+University+Press&rft.date=2005&rft.isbn=0-521-84772-9&rft.aulast=Anderson&rft.aufirst=Greg&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><i>Oxford Dictionary of Science</i>, 2005</span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><i>Oxford Dictionary of Chemistry</i>, 2004</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Barnes & Noble's <i>Essential Dictionary of Science</i>, 2004</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Gribbin's <i>Encyclopedia of Particle Physics</i>, 2000</span> </li> <li id="cite_note-Landsberg-A-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-Landsberg-A_8-0">^</a></b></span> <span class="reference-text">Landsberg, P.T. (1984). "Is Equilibrium always an Entropy Maximum?" J. Stat. Physics 35: 159–69.</span> </li> <li id="cite_note-Encarta-9"><span class="mw-cite-backlink">^ <a href="#cite_ref-Encarta_9-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Encarta_9-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Encarta_9-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text">Microsoft Encarta 2006. © 1993–2005 Microsoft Corporation. All rights reserved.</span> </li> <li id="cite_note-Greven-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-Greven_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Greven_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrevenKeller,_GerhardWarnercke,_Gerald2003" class="citation book cs1">Greven, Andreas; Keller, Gerhard; Warnercke, Gerald (2003). <i>Entropy – Princeton Series in Applied Mathematics</i>. Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-691-11338-6" title="Special:BookSources/0-691-11338-6"><bdi>0-691-11338-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+%E2%80%93+Princeton+Series+in+Applied+Mathematics&rft.pub=Princeton+University+Press&rft.date=2003&rft.isbn=0-691-11338-6&rft.aulast=Greven&rft.aufirst=Andreas&rft.au=Keller%2C+Gerhard&rft.au=Warnercke%2C+Gerald&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-Ulanowicz-11"><span class="mw-cite-backlink">^ <a href="#cite_ref-Ulanowicz_11-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Ulanowicz_11-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Ulanowicz_11-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Ulanowicz_11-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFUlanowicz2000" class="citation book cs1">Ulanowicz, Robert, E. (2000). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/growthdevelopmen0000ulan"><i>Growth and Development – Ecosystems Phenomenology</i></a></span>. toExcel Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-595-00145-9" title="Special:BookSources/0-595-00145-9"><bdi>0-595-00145-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Growth+and+Development+%E2%80%93+Ecosystems+Phenomenology&rft.pub=toExcel+Press&rft.date=2000&rft.isbn=0-595-00145-9&rft.aulast=Ulanowicz&rft.aufirst=Robert%2C+E.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fgrowthdevelopmen0000ulan&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKubatZeman,_J.1975" class="citation book cs1">Kubat, L.; Zeman, J. (1975). <i>Entropy and Information in Science and Philosophy</i>. Elsevier.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+and+Information+in+Science+and+Philosophy&rft.pub=Elsevier&rft.date=1975&rft.aulast=Kubat&rft.aufirst=L.&rft.au=Zeman%2C+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-Jorgensen-13"><span class="mw-cite-backlink">^ <a href="#cite_ref-Jorgensen_13-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Jorgensen_13-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJorgensenSvirezhev2004" class="citation book cs1">Jorgensen, Sven E; Svirezhev, Yuri M (2004). <i>Towards a Thermodynamic Theory for Ecological Systems</i>. Elsevier. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-08-044167-X" title="Special:BookSources/0-08-044167-X"><bdi>0-08-044167-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Towards+a+Thermodynamic+Theory+for+Ecological+Systems&rft.pub=Elsevier&rft.date=2004&rft.isbn=0-08-044167-X&rft.aulast=Jorgensen&rft.aufirst=Sven+E&rft.au=Svirezhev%2C+Yuri+M&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrogatz2003" class="citation book cs1">Strogatz, Steven (2003). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/syncemergingscie00stro"><i>the Emerging Science of Spontaneous Order</i></a></span>. Theia. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-7868-6844-9" title="Special:BookSources/0-7868-6844-9"><bdi>0-7868-6844-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=the+Emerging+Science+of+Spontaneous+Order&rft.pub=Theia&rft.date=2003&rft.isbn=0-7868-6844-9&rft.aulast=Strogatz&rft.aufirst=Steven&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsyncemergingscie00stro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-Brooks-15"><span class="mw-cite-backlink"><b><a href="#cite_ref-Brooks_15-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrooksWiley,_E.O.1988" class="citation book cs1">Brooks, Daniel, R.; Wiley, E.O. (1988). <i>Entropy as Evolution – Towards a Unified Theory of Biology</i>. University of Chicago Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-226-07574-5" title="Special:BookSources/0-226-07574-5"><bdi>0-226-07574-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+as+Evolution+%E2%80%93+Towards+a+Unified+Theory+of+Biology&rft.pub=University+of+Chicago+Press&rft.date=1988&rft.isbn=0-226-07574-5&rft.aulast=Brooks&rft.aufirst=Daniel%2C+R.&rft.au=Wiley%2C+E.O.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span><span class="cs1-maint citation-comment"><code class="cs1-code">{{<a href="/wiki/Template:Cite_book" title="Template:Cite book">cite book</a>}}</code>: CS1 maint: multiple names: authors list (<a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">link</a>)</span></span> </li> <li id="cite_note-16"><span class="mw-cite-backlink"><b><a href="#cite_ref-16">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCercignani1998" class="citation book cs1"><a href="/wiki/Carlo_Cercignani" title="Carlo Cercignani">Cercignani, Carlo</a> (1998). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/ludwigboltzmannm0000cerc"><i>Ludwig Boltzmann: The Man Who Trusted Atoms</i></a></span>. Oxford University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-850154-1" title="Special:BookSources/978-0-19-850154-1"><bdi>978-0-19-850154-1</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ludwig+Boltzmann%3A+The+Man+Who+Trusted+Atoms&rft.pub=Oxford+University+Press&rft.date=1998&rft.isbn=978-0-19-850154-1&rft.aulast=Cercignani&rft.aufirst=Carlo&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fludwigboltzmannm0000cerc&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-17"><span class="mw-cite-backlink"><b><a href="#cite_ref-17">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoltzmann1896" class="citation book cs1">Boltzmann, Ludwig (1896). <i>Lectures on Gas Theory</i>. Dover (reprint). <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68455-5" title="Special:BookSources/0-486-68455-5"><bdi>0-486-68455-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Lectures+on+Gas+Theory&rft.pub=Dover+%28reprint%29&rft.date=1896&rft.isbn=0-486-68455-5&rft.aulast=Boltzmann&rft.aufirst=Ludwig&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-18"><span class="mw-cite-backlink"><b><a href="#cite_ref-18">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFvan_AndersKlotsaAhmedEngel2014" class="citation journal cs1">van Anders, Greg; Klotsa, Daphne; Ahmed, N. Khalid; Engel, Michael; Glotzer, Sharon C. (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4234574">"Understanding shape entropy through local dense packing"</a>. <i>Proc Natl Acad Sci USA</i>. <b>111</b> (45): E4812–E4821. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1309.1187">1309.1187</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2014PNAS..111E4812V">2014PNAS..111E4812V</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.1073%2Fpnas.1418159111">10.1073/pnas.1418159111</a></span>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4234574">4234574</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/25344532">25344532</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proc+Natl+Acad+Sci+USA&rft.atitle=Understanding+shape+entropy+through+local+dense+packing&rft.volume=111&rft.issue=45&rft.pages=E4812-E4821&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4234574%23id-name%3DPMC&rft_id=info%3Abibcode%2F2014PNAS..111E4812V&rft_id=info%3Aarxiv%2F1309.1187&rft_id=info%3Apmid%2F25344532&rft_id=info%3Adoi%2F10.1073%2Fpnas.1418159111&rft.aulast=van+Anders&rft.aufirst=Greg&rft.au=Klotsa%2C+Daphne&rft.au=Ahmed%2C+N.+Khalid&rft.au=Engel%2C+Michael&rft.au=Glotzer%2C+Sharon+C.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4234574&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-19"><span class="mw-cite-backlink"><b><a href="#cite_ref-19">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOnsager1949" class="citation journal cs1">Onsager, Lars (1949). "The effects of shape on the interaction of colloidal particles". <i>Annals of the New York Academy of Sciences</i>. <b>51</b> (4): 627. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1949NYASA..51..627O">1949NYASA..51..627O</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fj.1749-6632.1949.tb27296.x">10.1111/j.1749-6632.1949.tb27296.x</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:84562683">84562683</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+the+New+York+Academy+of+Sciences&rft.atitle=The+effects+of+shape+on+the+interaction+of+colloidal+particles&rft.volume=51&rft.issue=4&rft.pages=627&rft.date=1949&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A84562683%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fj.1749-6632.1949.tb27296.x&rft_id=info%3Abibcode%2F1949NYASA..51..627O&rft.aulast=Onsager&rft.aufirst=Lars&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-20"><span class="mw-cite-backlink"><b><a href="#cite_ref-20">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHaji-AkbariEngelKeysZheng2009" class="citation journal cs1">Haji-Akbari, Amir; Engel, Michael; Keys, Aaron S.; Zheng, Xiaoyu; Petschek, Rolfe G.; Palffy-Muhoray, Peter; Glotzer, Sharon C. (2009). "Disordered, quasicrystalline and crystalline phases of densely packed tetrahedra". <i>Nature</i>. <b>462</b> (7274): 773–777. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1012.5138">1012.5138</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009Natur.462..773H">2009Natur.462..773H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fnature08641">10.1038/nature08641</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/20010683">20010683</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:4412674">4412674</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Disordered%2C+quasicrystalline+and+crystalline+phases+of+densely+packed+tetrahedra&rft.volume=462&rft.issue=7274&rft.pages=773-777&rft.date=2009&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A4412674%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2009Natur.462..773H&rft_id=info%3Aarxiv%2F1012.5138&rft_id=info%3Apmid%2F20010683&rft_id=info%3Adoi%2F10.1038%2Fnature08641&rft.aulast=Haji-Akbari&rft.aufirst=Amir&rft.au=Engel%2C+Michael&rft.au=Keys%2C+Aaron+S.&rft.au=Zheng%2C+Xiaoyu&rft.au=Petschek%2C+Rolfe+G.&rft.au=Palffy-Muhoray%2C+Peter&rft.au=Glotzer%2C+Sharon+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-DamascenoScience2012-21"><span class="mw-cite-backlink"><b><a href="#cite_ref-DamascenoScience2012_21-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDamascenoEngelGlotzer2012" class="citation journal cs1">Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "Predictive Self-Assembly of Polyhedra into Complex Structures". <i>Science</i>. <b>337</b> (6093): 453–457. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1202.2177">1202.2177</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2012Sci...337..453D">2012Sci...337..453D</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fscience.1220869">10.1126/science.1220869</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/22837525">22837525</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7177740">7177740</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science&rft.atitle=Predictive+Self-Assembly+of+Polyhedra+into+Complex+Structures&rft.volume=337&rft.issue=6093&rft.pages=453-457&rft.date=2012&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7177740%23id-name%3DS2CID&rft_id=info%3Abibcode%2F2012Sci...337..453D&rft_id=info%3Aarxiv%2F1202.2177&rft_id=info%3Apmid%2F22837525&rft_id=info%3Adoi%2F10.1126%2Fscience.1220869&rft.aulast=Damasceno&rft.aufirst=Pablo+F.&rft.au=Engel%2C+Michael&rft.au=Glotzer%2C+Sharon+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-22"><span class="mw-cite-backlink"><b><a href="#cite_ref-22">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGengvan_AndersDoddDshemuchadse2019" class="citation journal cs1">Geng, Yina; van Anders, Greg; Dodd, Paul M.; Dshemuchadse, Julia; Glotzer, Sharon C. (2019). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6611692">"Engineering Entropy for the Inverse Design of Colloidal Crystals from Hard Shapes"</a>. <i>Science Advances</i>. <b>5</b> (7): eeaw0514. <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://arxiv.org/abs/1712.02471">1712.02471</a></span>. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2019SciA....5..514G">2019SciA....5..514G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1126%2Fsciadv.aaw0514">10.1126/sciadv.aaw0514</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6611692">6611692</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/31281885">31281885</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Science+Advances&rft.atitle=Engineering+Entropy+for+the+Inverse+Design+of+Colloidal+Crystals+from+Hard+Shapes&rft.volume=5&rft.issue=7&rft.pages=eeaw0514&rft.date=2019&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6611692%23id-name%3DPMC&rft_id=info%3Abibcode%2F2019SciA....5..514G&rft_id=info%3Aarxiv%2F1712.02471&rft_id=info%3Apmid%2F31281885&rft_id=info%3Adoi%2F10.1126%2Fsciadv.aaw0514&rft.aulast=Geng&rft.aufirst=Yina&rft.au=van+Anders%2C+Greg&rft.au=Dodd%2C+Paul+M.&rft.au=Dshemuchadse%2C+Julia&rft.au=Glotzer%2C+Sharon+C.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC6611692&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-Halliday-23"><span class="mw-cite-backlink">^ <a href="#cite_ref-Halliday_23-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Halliday_23-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Halliday_23-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHallidayResnick,_Robert1988" class="citation book cs1">Halliday, David; Resnick, Robert (1988). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/fundamentalsofph3edhall_d7b7"><i>Fundamentals of Physics, Extended 3rd ed</i></a></span>. Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-81995-6" title="Special:BookSources/0-471-81995-6"><bdi>0-471-81995-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Physics%2C+Extended+3rd+ed.&rft.pub=Wiley&rft.date=1988&rft.isbn=0-471-81995-6&rft.aulast=Halliday&rft.aufirst=David&rft.au=Resnick%2C+Robert&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Ffundamentalsofph3edhall_d7b7&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-nasa-24"><span class="mw-cite-backlink">^ <a href="#cite_ref-nasa_24-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-nasa_24-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-nasa_24-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><a rel="nofollow" class="external text" href="https://imagine.gsfc.nasa.gov/docs/teachers/lessons/xray_spectra/background-adr.html">NASA – How does an Adiabatic Demagnetization Refrigerator Work ?</a></span> </li> <li id="cite_note-25"><span class="mw-cite-backlink"><b><a href="#cite_ref-25">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDenbigh_K.1981" class="citation book cs1">Denbigh K. (1981). <i>The Principles of Chemical Equilibrium: With Applications in Chemistry and Chemical Engineering</i>. London: Cambridge University Press. pp. 55–56.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Principles+of+Chemical+Equilibrium%3A+With+Applications+in+Chemistry+and+Chemical+Engineering.&rft.place=London&rft.pages=55-56&rft.pub=Cambridge+University+Press&rft.date=1981&rft.au=Denbigh+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-26"><span class="mw-cite-backlink"><b><a href="#cite_ref-26">^</a></b></span> <span class="reference-text"><a href="/wiki/Edwin_Thompson_Jaynes" title="Edwin Thompson Jaynes">Jaynes, E.T.</a> (1989). Clearing up mysteries—the original goal, in <i>Maximum Entropy and Bayesian Methods</i>, J. Skilling, Editor, Kluwer Academic Publishers, Dordrecht, pp. 1–27, page 24.</span> </li> <li id="cite_note-27"><span class="mw-cite-backlink"><b><a href="#cite_ref-27">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrandy2008" class="citation book cs1">Grandy, Walter T. Jr. (2008). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=SnMF37J50DgC"><i>Entropy and the Time Evolution of Macroscopic Systems</i></a>. Oxford University Press. pp. 55–58. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-954617-6" title="Special:BookSources/978-0-19-954617-6"><bdi>978-0-19-954617-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Entropy+and+the+Time+Evolution+of+Macroscopic+Systems&rft.pages=55-58&rft.pub=Oxford+University+Press&rft.date=2008&rft.isbn=978-0-19-954617-6&rft.aulast=Grandy&rft.aufirst=Walter+T.+Jr.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DSnMF37J50DgC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> <li id="cite_note-crutch-28"><span class="mw-cite-backlink"><b><a href="#cite_ref-crutch_28-0">^</a></b></span> <span class="reference-text">Frank L. Lambert, 2002, "<a rel="nofollow" class="external text" href="http://franklambert.net/entropysite.com/cracked_crutch.html">Disorder—A Cracked Crutch for Supporting Entropy Discussions</a>," <i>Journal of Chemical Education</i> 79: 187.</span> </li> <li id="cite_note-29"><span class="mw-cite-backlink"><b><a href="#cite_ref-29">^</a></b></span> <span class="reference-text">Carson, E. M., and Watson, J. R., (Department of Educational and Professional Studies, King's College, London), 2002, "<a rel="nofollow" class="external text" href="http://www.rsc.org/pdf/uchemed/papers/2002/p2_carson.pdf">Undergraduate students' understandings of entropy and Gibbs Free energy</a>," University Chemistry Education – 2002 Papers, Royal Society of Chemistry.</span> </li> <li id="cite_note-30"><span class="mw-cite-backlink"><b><a href="#cite_ref-30">^</a></b></span> <span class="reference-text">Sozbilir, Mustafa, PhD studies: Turkey, <i>A Study of Undergraduates' Understandings of Key Chemical Ideas in Thermodynamics</i>, Ph.D. Thesis, Department of Educational Studies, The University of York, 2001.</span> </li> <li id="cite_note-31"><span class="mw-cite-backlink"><b><a href="#cite_ref-31">^</a></b></span> <span class="reference-text">Shannon, C.E. (1945). <i>A mathematical theory of cryptography</i>, Memorandum for file, MM-45-110-98, 135 pages, page 20; found in File 24 at page 203 in <i>Claude Elwood Shannon: Miscellaneous Writings</i> edited by N.J.A. Sloane, and Aaron D. Wyner (revision of 2013), Mathematical Sciences Research Center, AT&T Bell Laboratories, Murray Hill, NJ; previously partly published by IEEE Press.</span> </li> <li id="cite_note-32"><span class="mw-cite-backlink"><b><a href="#cite_ref-32">^</a></b></span> <span class="reference-text">Gray, R.M. (2010). <i>Entropy and Information Theory</i>, Springer, New York NY, 2nd edition, p. 296.</span> </li> <li id="cite_note-33"><span class="mw-cite-backlink"><b><a href="#cite_ref-33">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMark_Nelson2006" class="citation web cs1">Mark Nelson (24 August 2006). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20180301161215/http://marknelson.us/2006/08/24/the-hutter-prize/">"The Hutter Prize"</a>. Archived from <a rel="nofollow" class="external text" href="http://marknelson.us/2006/08/24/the-hutter-prize/">the original</a> on 2018-03-01<span class="reference-accessdate">. Retrieved <span class="nowrap">2008-11-27</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Hutter+Prize&rft.date=2006-08-24&rft.au=Mark+Nelson&rft_id=http%3A%2F%2Fmarknelson.us%2F2006%2F08%2F24%2Fthe-hutter-prize%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEntropy+%28order+and+disorder%29" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Entropy_(order_and_disorder)&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Lambert, F. L. <a rel="nofollow" class="external text" href="http://franklambert.net/entropysite.com/">Entropy Sites — A Guide</a></li> <li>Lambert, F. L. <a rel="nofollow" class="external text" href="http://franklambert.net/entropysite.com/shuffled_cards.html"><i>Shuffled Cards, Messy Desks, and Disorderly Dorm Rooms – Examples of Entropy Increase? Nonsense!</i></a> Journal of Chemical Education</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Entropy_(order_and_disorder)&oldid=1213009411">https://en.wikipedia.org/w/index.php?title=Entropy_(order_and_disorder)&oldid=1213009411</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Thermodynamic_entropy" title="Category:Thermodynamic entropy">Thermodynamic entropy</a></li><li><a href="/wiki/Category:State_functions" title="Category:State functions">State functions</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_maint:_multiple_names:_authors_list" title="Category:CS1 maint: multiple names: authors list">CS1 maint: multiple names: authors list</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_is_different_from_Wikidata" title="Category:Short description is different from Wikidata">Short description is different from Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 10 March 2024, at 17:12<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. By using this site, you agree to the <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Terms_of_Use" class="extiw" title="foundation:Special:MyLanguage/Policy:Terms of Use">Terms of Use</a> and <a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy" class="extiw" title="foundation:Special:MyLanguage/Policy:Privacy policy">Privacy Policy</a>. Wikipedia® is a registered trademark of the <a rel="nofollow" class="external text" href="https://wikimediafoundation.org/">Wikimedia Foundation, Inc.</a>, a non-profit organization.</li> </ul> <ul id="footer-places"> <li id="footer-places-privacy"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Privacy_policy">Privacy policy</a></li> <li id="footer-places-about"><a href="/wiki/Wikipedia:About">About Wikipedia</a></li> <li id="footer-places-disclaimers"><a href="/wiki/Wikipedia:General_disclaimer">Disclaimers</a></li> <li id="footer-places-contact"><a href="//en.wikipedia.org/wiki/Wikipedia:Contact_us">Contact Wikipedia</a></li> <li id="footer-places-wm-codeofconduct"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Universal_Code_of_Conduct">Code of Conduct</a></li> <li id="footer-places-developers"><a href="https://developer.wikimedia.org">Developers</a></li> <li id="footer-places-statslink"><a href="https://stats.wikimedia.org/#/en.wikipedia.org">Statistics</a></li> <li id="footer-places-cookiestatement"><a href="https://foundation.wikimedia.org/wiki/Special:MyLanguage/Policy:Cookie_statement">Cookie statement</a></li> <li id="footer-places-mobileview"><a href="//en.m.wikipedia.org/w/index.php?title=Entropy_(order_and_disorder)&mobileaction=toggle_view_mobile" class="noprint stopMobileRedirectToggle">Mobile view</a></li> </ul> <ul id="footer-icons" class="noprint"> <li id="footer-copyrightico"><a href="https://wikimediafoundation.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/static/images/footer/wikimedia-button.svg" width="84" height="29" alt="Wikimedia Foundation" loading="lazy"></a></li> <li id="footer-poweredbyico"><a href="https://www.mediawiki.org/" class="cdx-button cdx-button--fake-button cdx-button--size-large cdx-button--fake-button--enabled"><img src="/w/resources/assets/poweredby_mediawiki.svg" alt="Powered by MediaWiki" width="88" height="31" loading="lazy"></a></li> </ul> </footer> </div> </div> </div> <div class="vector-settings" id="p-dock-bottom"> <ul></ul> </div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-8p8br","wgBackendResponseTime":146,"wgPageParseReport":{"limitreport":{"cputime":"0.313","walltime":"0.413","ppvisitednodes":{"value":1588,"limit":1000000},"postexpandincludesize":{"value":40537,"limit":2097152},"templateargumentsize":{"value":1214,"limit":2097152},"expansiondepth":{"value":12,"limit":100},"expensivefunctioncount":{"value":1,"limit":500},"unstrip-depth":{"value":1,"limit":20},"unstrip-size":{"value":79954,"limit":5000000},"entityaccesscount":{"value":0,"limit":400},"timingprofile":["100.00% 307.242 1 -total"," 69.53% 213.635 1 Template:Reflist"," 42.25% 129.800 13 Template:Cite_book"," 20.16% 61.946 1 Template:Short_description"," 14.14% 43.445 5 Template:Cite_journal"," 11.15% 34.271 2 Template:Pagetype"," 8.70% 26.726 5 Template:Main_other"," 4.97% 15.265 1 Template:SDcat"," 3.57% 10.953 1 Template:Cquote"," 2.92% 8.970 1 Template:Cite_web"]},"scribunto":{"limitreport-timeusage":{"value":"0.173","limit":"10.000"},"limitreport-memusage":{"value":5138082,"limit":52428800}},"cachereport":{"origin":"mw-web.codfw.main-f69cdc8f6-fhqv7","timestamp":"20241122143337","ttl":2592000,"transientcontent":false}}});});</script> <script type="application/ld+json">{"@context":"https:\/\/schema.org","@type":"Article","name":"Entropy (order and disorder)","url":"https:\/\/en.wikipedia.org\/wiki\/Entropy_(order_and_disorder)","sameAs":"http:\/\/www.wikidata.org\/entity\/Q5380796","mainEntity":"http:\/\/www.wikidata.org\/entity\/Q5380796","author":{"@type":"Organization","name":"Contributors to Wikimedia projects"},"publisher":{"@type":"Organization","name":"Wikimedia Foundation, Inc.","logo":{"@type":"ImageObject","url":"https:\/\/www.wikimedia.org\/static\/images\/wmf-hor-googpub.png"}},"datePublished":"2006-11-01T18:56:34Z","dateModified":"2024-03-10T17:12:33Z","image":"https:\/\/upload.wikimedia.org\/wikipedia\/commons\/0\/05\/Boltzmann-molecules.svg","headline":"interpretation of entropy as a measure of disorder"}</script> </body> </html>