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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> <button aria-controls="toc-Overview-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Overview subsection</span> </button> <ul id="toc-Overview-sublist" class="vector-toc-list"> <li id="toc-Quasi-radiationless_non-equilibrium_thermodynamics_of_matter_in_laboratory_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Quasi-radiationless_non-equilibrium_thermodynamics_of_matter_in_laboratory_conditions"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions</span> </div> </a> <ul id="toc-Quasi-radiationless_non-equilibrium_thermodynamics_of_matter_in_laboratory_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_equilibrium_thermodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Local_equilibrium_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Local equilibrium thermodynamics</span> </div> </a> <ul id="toc-Local_equilibrium_thermodynamics-sublist" class="vector-toc-list"> <li id="toc-Local_equilibrium_thermodynamics_with_materials_with_"memory"" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Local_equilibrium_thermodynamics_with_materials_with_"memory""> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Local equilibrium thermodynamics with materials with "memory"</span> </div> </a> <ul id="toc-Local_equilibrium_thermodynamics_with_materials_with_"memory"-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Extended_irreversible_thermodynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Extended_irreversible_thermodynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Extended irreversible thermodynamics</span> </div> </a> <ul id="toc-Extended_irreversible_thermodynamics-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Basic_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Basic concepts</span> </div> </a> <ul id="toc-Basic_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stationary_states,_fluctuations,_and_stability" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stationary_states,_fluctuations,_and_stability"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Stationary states, fluctuations, and stability</span> </div> </a> <ul id="toc-Stationary_states,_fluctuations,_and_stability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Local_thermodynamic_equilibrium" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Local_thermodynamic_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Local thermodynamic equilibrium</span> </div> </a> <button aria-controls="toc-Local_thermodynamic_equilibrium-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Local thermodynamic equilibrium subsection</span> </button> <ul id="toc-Local_thermodynamic_equilibrium-sublist" class="vector-toc-list"> <li id="toc-Ponderable_matter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ponderable_matter"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Ponderable matter</span> </div> </a> <ul id="toc-Ponderable_matter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Milne's_definition_in_terms_of_radiative_equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Milne's_definition_in_terms_of_radiative_equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Milne's definition in terms of radiative equilibrium</span> </div> </a> <ul id="toc-Milne's_definition_in_terms_of_radiative_equilibrium-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Entropy_in_evolving_systems" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Entropy_in_evolving_systems"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Entropy in evolving systems</span> </div> </a> <button aria-controls="toc-Entropy_in_evolving_systems-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Entropy in evolving systems subsection</span> </button> <ul id="toc-Entropy_in_evolving_systems-sublist" class="vector-toc-list"> <li id="toc-Entropy_in_non-equilibrium" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Entropy_in_non-equilibrium"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Entropy in non-equilibrium</span> </div> </a> <ul id="toc-Entropy_in_non-equilibrium-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Flows_and_forces" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Flows_and_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Flows and forces</span> </div> </a> <button aria-controls="toc-Flows_and_forces-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Flows and forces subsection</span> </button> <ul id="toc-Flows_and_forces-sublist" class="vector-toc-list"> <li id="toc-Onsager_reciprocal_relations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Onsager_reciprocal_relations"> <div class="vector-toc-text"> <span class="vector-toc-numb">7.1</span> <span>Onsager reciprocal relations</span> </div> </a> <ul id="toc-Onsager_reciprocal_relations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Speculated_extremal_principles_for_non-equilibrium_processes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Speculated_extremal_principles_for_non-equilibrium_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Speculated extremal principles for non-equilibrium processes</span> </div> </a> <ul id="toc-Speculated_extremal_principles_for_non-equilibrium_processes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-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">10</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">11</span> <span>References</span> </div> </a> <button aria-controls="toc-References-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle References subsection</span> </button> <ul id="toc-References-sublist" class="vector-toc-list"> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> <span class="vector-toc-numb">11.1</span> <span>Sources</span> </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">12</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-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">13</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" title="Table of Contents" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" 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Available in 24 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-24" 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">24 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%AA%D8%B1%D9%85%D9%88%D8%AF%D9%8A%D9%86%D8%A7%D9%85%D9%8A%D9%83_%D8%A7%D9%84%D9%84%D8%A7%D8%AA%D9%88%D8%A7%D8%B2%D9%86" title="ترموديناميك اللاتوازن – Arabic" lang="ar" hreflang="ar" data-title="ترموديناميك اللاتوازن" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D1%9E%D0%BD%D0%B0%D0%B2%D0%B0%D0%B6%D0%BD%D0%B0%D1%8F_%D1%82%D1%8D%D1%80%D0%BC%D0%B0%D0%B4%D1%8B%D0%BD%D0%B0%D0%BC%D1%96%D0%BA%D0%B0" title="Нераўнаважная тэрмадынаміка – Belarusian" lang="be" hreflang="be" data-title="Нераўнаважная тэрмадынаміка" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Termodin%C3%A0mica_del_no_equilibri" title="Termodinàmica del no equilibri – Catalan" lang="ca" hreflang="ca" data-title="Termodinàmica del no equilibri" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%A8%D0%B0%D0%B9%D0%BB%D0%B0%D1%88%D1%83%D1%81%C4%83%D1%80%D0%BB%C4%83%D1%85%D0%BB%D0%B0_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0" title="Шайлашусăрлăхла термодинамика – Chuvash" lang="cv" hreflang="cv" data-title="Шайлашусăрлăхла термодинамика" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Nerovnov%C3%A1%C5%BEn%C3%A1_termodynamika" title="Nerovnovážná termodynamika – Czech" lang="cs" hreflang="cs" data-title="Nerovnovážná termodynamika" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Nichtgleichgewichtssystem" title="Nichtgleichgewichtssystem – German" lang="de" hreflang="de" data-title="Nichtgleichgewichtssystem" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Termodin%C3%A1mica_del_no_equilibrio" title="Termodinámica del no equilibrio – Spanish" lang="es" hreflang="es" data-title="Termodinámica del no equilibrio" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Termodinamika_itzulezin" title="Termodinamika itzulezin – Basque" lang="eu" hreflang="eu" data-title="Termodinamika itzulezin" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D8%AA%D8%B1%D9%85%D9%88%D8%AF%DB%8C%D9%86%D8%A7%D9%85%DB%8C%DA%A9_%D8%BA%DB%8C%D8%B1%D8%AA%D8%B9%D8%A7%D8%AF%D9%84%DB%8C" title="ترمودینامیک غیرتعادلی – Persian" lang="fa" hreflang="fa" data-title="ترمودینامیک غیرتعادلی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Thermodynamique_hors_%C3%A9quilibre" title="Thermodynamique hors équilibre – French" lang="fr" hreflang="fr" data-title="Thermodynamique hors équilibre" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B9%84%ED%8F%89%ED%98%95%EC%97%B4%EC%97%AD%ED%95%99" title="비평형열역학 – Korean" lang="ko" hreflang="ko" data-title="비평형열역학" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Termodinamica_del_non_equilibrio" title="Termodinamica del non equilibrio – Italian" lang="it" hreflang="it" data-title="Termodinamica del non equilibrio" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%AA%D7%A8%D7%9E%D7%95%D7%93%D7%99%D7%A0%D7%9E%D7%99%D7%A7%D7%94_%D7%9C%D7%90_%D7%91%D7%A9%D7%99%D7%95%D7%95%D7%99_%D7%9E%D7%A9%D7%A7%D7%9C" title="תרמודינמיקה לא בשיווי משקל – Hebrew" lang="he" hreflang="he" data-title="תרמודינמיקה לא בשיווי משקל" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Nepusiausviroji_termodinamika" title="Nepusiausviroji termodinamika – Lithuanian" lang="lt" hreflang="lt" data-title="Nepusiausviroji termodinamika" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%BC%D0%BD%D0%BE%D1%82%D0%B5%D0%B6%D0%BD%D0%B0_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0" title="Нерамнотежна термодинамика – Macedonian" lang="mk" hreflang="mk" data-title="Нерамнотежна термодинамика" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%97%E0%A9%88%E0%A8%B0-%E0%A8%B8%E0%A9%B0%E0%A8%A4%E0%A9%81%E0%A8%B2%E0%A8%A8_%E0%A8%A5%E0%A8%B0%E0%A8%AE%E0%A9%8B%E0%A8%A1%E0%A8%BE%E0%A8%87%E0%A8%A8%E0%A8%BE%E0%A8%AE%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਗੈਰ-ਸੰਤੁਲਨ ਥਰਮੋਡਾਇਨਾਮਿਕਸ – Punjabi" lang="pa" hreflang="pa" data-title="ਗੈਰ-ਸੰਤੁਲਨ ਥਰਮੋਡਾਇਨਾਮਿਕਸ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Termodinamica_de_neechilibru" title="Termodinamica de neechilibru – Romanian" lang="ro" hreflang="ro" data-title="Termodinamica de neechilibru" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9D%D0%B5%D1%80%D0%B0%D0%B2%D0%BD%D0%BE%D0%B2%D0%B5%D1%81%D0%BD%D0%B0%D1%8F_%D1%82%D0%B5%D1%80%D0%BC%D0%BE%D0%B4%D0%B8%D0%BD%D0%B0%D0%BC%D0%B8%D0%BA%D0%B0" title="Неравновесная термодинамика – Russian" lang="ru" hreflang="ru" data-title="Неравновесная термодинамика" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Non-equilibrium thermodynamics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Neravnote%C5%BEna_termodinamika" 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rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><table class="sidebar sidebar-collapse nomobile nowraplinks plainlist"><tbody><tr><th class="sidebar-title" style="padding-bottom:0.3em;border-bottom:1px solid #aaa;"><a href="/wiki/Thermodynamics" title="Thermodynamics">Thermodynamics</a></th></tr><tr><td class="sidebar-image" style="display:block;margin:0.3em 0 0.4em;"><span class="mw-default-size" typeof="mw:File/Frameless"><a href="/wiki/Carnot_heat_engine" title="Carnot heat engine"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/220px-Carnot_heat_engine_2.svg.png" decoding="async" width="220" height="97" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/330px-Carnot_heat_engine_2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Carnot_heat_engine_2.svg/440px-Carnot_heat_engine_2.svg.png 2x" data-file-width="840" data-file-height="370" /></a></span><div class="sidebar-caption">The classical <a href="/wiki/Carnot_heat_engine" title="Carnot heat engine">Carnot heat engine</a></div></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Branches</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamics" title="Thermodynamics">Classical</a></li> <li><a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical</a></li> <li><a href="/wiki/Chemical_thermodynamics" title="Chemical thermodynamics">Chemical</a></li> <li><a href="/wiki/Quantum_thermodynamics" title="Quantum thermodynamics">Quantum thermodynamics</a></li></ul> </div> <ul><li><a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">Equilibrium</a> / <a class="mw-selflink selflink">Non-equilibrium</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Laws_of_thermodynamics" title="Laws of thermodynamics">Laws</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Zeroth_law_of_thermodynamics" title="Zeroth law of thermodynamics">Zeroth</a></li> <li><a href="/wiki/First_law_of_thermodynamics" title="First law of thermodynamics">First</a></li> <li><a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">Second</a></li> <li><a href="/wiki/Third_law_of_thermodynamics" title="Third law of thermodynamics">Third</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_system" title="Thermodynamic system">Systems</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Closed_system" title="Closed system">Closed system</a></li> <li><a href="/wiki/Thermodynamic_system#Open_system" title="Thermodynamic system">Open system</a></li> <li><a href="/wiki/Isolated_system" title="Isolated system">Isolated system</a></li></ul> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_state" title="Thermodynamic state">State</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Equation_of_state" title="Equation of state">Equation of state</a></li> <li><a href="/wiki/Ideal_gas" title="Ideal gas">Ideal gas</a></li> <li><a href="/wiki/Real_gas" title="Real gas">Real gas</a></li> <li><a href="/wiki/State_of_matter" title="State of matter">State of matter</a></li> <li><a href="/wiki/Phase_(matter)" title="Phase (matter)">Phase (matter)</a></li> <li><a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">Equilibrium</a></li> <li><a href="/wiki/Control_volume" title="Control volume">Control volume</a></li> <li><a href="/wiki/Thermodynamic_instruments" title="Thermodynamic instruments">Instruments</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_process" title="Thermodynamic process">Processes</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Isobaric_process" title="Isobaric process">Isobaric</a></li> <li><a href="/wiki/Isochoric_process" title="Isochoric process">Isochoric</a></li> <li><a href="/wiki/Isothermal_process" title="Isothermal process">Isothermal</a></li> <li><a href="/wiki/Adiabatic_process" title="Adiabatic process">Adiabatic</a></li> <li><a href="/wiki/Isentropic_process" title="Isentropic process">Isentropic</a></li> <li><a href="/wiki/Isenthalpic_process" title="Isenthalpic process">Isenthalpic</a></li> <li><a href="/wiki/Quasistatic_process" title="Quasistatic process">Quasistatic</a></li> <li><a href="/wiki/Polytropic_process" title="Polytropic process">Polytropic</a></li> <li><a href="/wiki/Free_expansion" class="mw-redirect" title="Free expansion">Free expansion</a></li> <li><a href="/wiki/Reversible_process_(thermodynamics)" title="Reversible process (thermodynamics)">Reversibility</a></li> <li><a href="/wiki/Irreversible_process" title="Irreversible process">Irreversibility</a></li> <li><a href="/wiki/Endoreversible_thermodynamics" title="Endoreversible thermodynamics">Endoreversibility</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">Cycles</a></th></tr><tr><td class="sidebar-content hlist"> <ul><li><a href="/wiki/Heat_engine" title="Heat engine">Heat engines</a></li> <li><a href="/wiki/Heat_pump_and_refrigeration_cycle" title="Heat pump and refrigeration cycle">Heat pumps</a></li> <li><a href="/wiki/Thermal_efficiency" title="Thermal efficiency">Thermal efficiency</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/List_of_thermodynamic_properties" title="List of thermodynamic properties">System properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div style="font-size:90%;padding-bottom:0.2em;border-bottom:1px solid #aaa;">Note: <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">Conjugate variables</a> in <i>italics</i></div> <table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none;margin-top:0.4em;"><tbody><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_diagrams" title="Thermodynamic diagrams">Property diagrams</a></li> <li><a href="/wiki/Intensive_and_extensive_properties" title="Intensive and extensive properties">Intensive and extensive properties</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Process_function" title="Process function">Process functions</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;;padding-bottom:0.4em;"> <div class="hlist"> <ul><li><a href="/wiki/Work_(thermodynamics)" title="Work (thermodynamics)">Work</a></li> <li><a href="/wiki/Heat" title="Heat">Heat</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/State_function" title="State function">Functions of state</a></th></tr><tr><td class="sidebar-content" style="padding-bottom:0.7em;"> <ul><li><a href="/wiki/Thermodynamic_temperature" title="Thermodynamic temperature">Temperature</a> / <i><a href="/wiki/Entropy" title="Entropy">Entropy</a></i> (<a href="/wiki/Introduction_to_entropy" title="Introduction to entropy">introduction</a>)</li> <li><a href="/wiki/Pressure" title="Pressure">Pressure</a> / <i><a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">Volume</a></i></li> <li><a href="/wiki/Chemical_potential" title="Chemical potential">Chemical potential</a> / <i><a href="/wiki/Particle_number" title="Particle number">Particle number</a></i></li> <li><a href="/wiki/Vapor_quality" title="Vapor quality">Vapor quality</a></li> <li><a href="/wiki/Reduced_properties" title="Reduced properties">Reduced properties</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Material_properties_(thermodynamics)" title="Material properties (thermodynamics)">Material properties</a></div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Thermodynamic_databases_for_pure_substances" title="Thermodynamic databases for pure substances">Property databases</a></li></ul> <div style="font-size:90%;margin-top:0.4em;border-top:1px solid #aaa;text-align:center;"> <table> <tbody><tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Heat_capacity" title="Heat capacity">Specific heat capacity</a> </td> <td style="vertical-align:middle; text-align:left"><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 c=}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c=}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/891d40a9b18752b04065caee655d008b3ec11428" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.46ex; height:1.676ex;" alt="{\displaystyle c=}"></span></td> <td><table><tbody><tr><td><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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span></td><td><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 \partial S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c609f4d3c5692ea4495479ef47594dc67f9fa464" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.817ex; height:2.176ex;" alt="{\displaystyle \partial S}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Compressibility" title="Compressibility">Compressibility</a> </td> <td style="vertical-align:middle; text-align:left"><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 \beta =-}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mo>−</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =-}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01c042bf1456bd4d2a8caed1f4912820a7ecbb3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.239ex; height:2.509ex;" alt="{\displaystyle \beta =-}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebc4a48eb2412f08b54fe438b5139c88f9cfa372" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.487ex; height:2.509ex;" alt="{\displaystyle \partial p}"></span></td></tr></tbody></table></td></tr> <tr><td style="vertical-align:middle; text-align:right"><a href="/wiki/Thermal_expansion" title="Thermal expansion">Thermal expansion</a> </td> <td style="vertical-align:middle; text-align:left"><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 \alpha =}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a92d4583d351f08c1c70985f0c843b2fff1b01e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.941ex; height:1.676ex;" alt="{\displaystyle \alpha =}"></span></td> <td><table><tbody><tr><td><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 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}"></span></td><td><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 \partial V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cecdd9d069fa84159940068fc11a91b6b3b9ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.105ex; height:2.176ex;" alt="{\displaystyle \partial V}"></span></td></tr><tr><td style="border-top:solid 1px black;"><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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span></td><td style="border-top:solid 1px black;"><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 \partial T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/504aa558fff3d00d10b03cadb1085cb0b7bdc631" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.954ex; height:2.176ex;" alt="{\displaystyle \partial T}"></span></td></tr></tbody></table></td></tr> </tbody></table></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_equations" title="Thermodynamic equations">Equations</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Carnot%27s_theorem_(thermodynamics)" title="Carnot's theorem (thermodynamics)">Carnot's theorem</a></li> <li><a href="/wiki/Clausius_theorem" title="Clausius theorem">Clausius theorem</a></li> <li><a href="/wiki/Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">Fundamental relation</a></li> <li><a href="/wiki/Ideal_gas_law" title="Ideal gas law">Ideal gas law</a></li></ul> </div> <ul><li><a href="/wiki/Maxwell_relations" title="Maxwell relations">Maxwell relations</a></li> <li><a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></li> <li><a href="/wiki/Bridgman%27s_thermodynamic_equations" title="Bridgman's thermodynamic equations">Bridgman's equations</a></li> <li><i><a href="/wiki/Table_of_thermodynamic_equations" title="Table of thermodynamic equations">Table of thermodynamic equations</a></i></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><a href="/wiki/Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">Free energy</a></li> <li><a href="/wiki/Free_entropy" title="Free entropy">Free entropy</a></li></ul> </div> <div class="plainlist"><ul><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Internal_energy" title="Internal energy">Internal energy</a><br /><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 U(S,V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(S,V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921f33f9c6551562ec836007b035c2de6323d2d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.912ex; height:2.843ex;" alt="{\displaystyle U(S,V)}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Enthalpy" title="Enthalpy">Enthalpy</a><br /><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 H(S,p)=U+pV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>+</mo> <mi>p</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(S,p)=U+pV}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6407d78e5f39d07f70e2414a92e08e2e068519f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.254ex; height:2.843ex;" alt="{\displaystyle H(S,p)=U+pV}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a><br /><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 A(T,V)=U-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A(T,V)=U-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e93692f031ba6484d82731c54db83a69daed3f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.867ex; height:2.843ex;" alt="{\displaystyle A(T,V)=U-TS}"></span></li><li style="font-size:110%;line-height:1.6em;padding-bottom:0.5em;"><a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a><br /><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 G(T,p)=H-TS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>H</mi> <mo>−</mo> <mi>T</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(T,p)=H-TS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8dd7a8f0b8ae04963da133e3b202432e1b6caed4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.614ex; height:2.843ex;" alt="{\displaystyle G(T,p)=H-TS}"></span></li></ul></div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c"><div class="hlist"><ul><li>History</li><li>Culture</li></ul></div></div></div><div class="sidebar-list-content mw-collapsible-content"><table class="sidebar nomobile nowraplinks" style="background-color: transparent; color: var( --color-base, #202122 ); border-collapse:collapse; border-spacing:0px; border:none; width:100%; margin:0px; font-size:100%; clear:none; float:none"><tbody><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> History</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/History_of_thermodynamics" title="History of thermodynamics">General</a></li> <li><a href="/wiki/History_of_entropy" title="History of entropy">Entropy</a></li> <li><a href="/wiki/Gas_laws" title="Gas laws">Gas laws</a></li></ul> </div> <ul><li><a href="/wiki/History_of_perpetual_motion_machines" title="History of perpetual motion machines">"Perpetual motion" machines</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/Philosophy_of_thermal_and_statistical_physics" class="mw-redirect" title="Philosophy of thermal and statistical physics">Philosophy</a></th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Entropy_(arrow_of_time)" class="mw-redirect" title="Entropy (arrow of time)">Entropy and time</a></li> <li><a href="/wiki/Entropy_and_life" title="Entropy and life">Entropy and life</a></li> <li><a href="/wiki/Brownian_ratchet" title="Brownian ratchet">Brownian ratchet</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Heat_death_paradox" title="Heat death paradox">Heat death paradox</a></li> <li><a href="/wiki/Loschmidt%27s_paradox" title="Loschmidt's paradox">Loschmidt's paradox</a></li> <li><a href="/wiki/Synergetics_(Haken)" title="Synergetics (Haken)">Synergetics</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Theories</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Caloric_theory" title="Caloric theory">Caloric theory</a></li></ul> </div> <ul><li><a href="/wiki/Vis_viva" title="Vis viva"><i>Vis viva</i> <span style="font-size:85%;">("living force")</span></a></li> <li><a href="/wiki/Mechanical_equivalent_of_heat" title="Mechanical equivalent of heat">Mechanical equivalent of heat</a></li> <li><a href="/wiki/Power_(physics)" title="Power (physics)">Motive power</a></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <a href="/wiki/List_of_important_publications_in_physics" title="List of important publications in physics">Key publications</a></th></tr><tr><td class="sidebar-content"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/An_Inquiry_Concerning_the_Source_of_the_Heat_Which_Is_Excited_by_Friction" title="An Inquiry Concerning the Source of the Heat Which Is Excited by Friction">An Inquiry Concerning the<br />Source ... Friction</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/On_the_Equilibrium_of_Heterogeneous_Substances" title="On the Equilibrium of Heterogeneous Substances">On the Equilibrium of<br />Heterogeneous Substances</a></i></div></li> <li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em;"><i><a href="/wiki/Reflections_on_the_Motive_Power_of_Fire" title="Reflections on the Motive Power of Fire">Reflections on the<br />Motive Power of Fire</a></i></div></li></ul></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> Timelines</th></tr><tr><td class="sidebar-content"> <div class="hlist"> <ul><li><a href="/wiki/Timeline_of_thermodynamics" title="Timeline of thermodynamics">Thermodynamics</a></li> <li><a href="/wiki/Timeline_of_heat_engine_technology" title="Timeline of heat engine technology">Heat engines</a></li></ul> </div></td> </tr><tr><th class="sidebar-heading" style="background:#eaeaff;font-style:italic;"> <div class="hlist"><ul><li>Art</li><li>Education</li></ul></div></th></tr><tr><td class="sidebar-content"> <ul><li><a href="/wiki/Maxwell%27s_thermodynamic_surface" title="Maxwell's thermodynamic surface">Maxwell's thermodynamic surface</a></li> <li><a href="/wiki/Entropy_(energy_dispersal)" title="Entropy (energy dispersal)">Entropy as energy dispersal</a></li></ul></td> </tr></tbody></table></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Scientists</div></div><div class="sidebar-list-content mw-collapsible-content"><div class="hlist"> <ul><li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Bernoulli</a></li> <li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li> <li><a href="/wiki/Percy_Williams_Bridgman" title="Percy Williams Bridgman">Bridgman</a></li> <li><a href="/wiki/Constantin_Carath%C3%A9odory" title="Constantin Carathéodory">Carathéodory</a></li> <li><a href="/wiki/Nicolas_L%C3%A9onard_Sadi_Carnot" title="Nicolas Léonard Sadi Carnot">Carnot</a></li> <li><a href="/wiki/Beno%C3%AEt_Paul_%C3%89mile_Clapeyron" class="mw-redirect" title="Benoît Paul Émile Clapeyron">Clapeyron</a></li> <li><a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Clausius</a></li> <li><a href="/wiki/Th%C3%A9ophile_de_Donder" title="Théophile de Donder">de Donder</a></li> <li><a href="/wiki/Pierre_Duhem" title="Pierre Duhem">Duhem</a></li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">von Helmholtz</a></li> <li><a href="/wiki/James_Prescott_Joule" title="James Prescott Joule">Joule</a></li> <li><a href="/wiki/Lord_Kelvin" title="Lord Kelvin">Kelvin</a></li> <li><a href="/wiki/Gilbert_N._Lewis" title="Gilbert N. Lewis">Lewis</a></li> <li><a href="/wiki/Fran%C3%A7ois_Massieu" title="François Massieu">Massieu</a></li> <li><a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li> <li><a href="/wiki/Julius_von_Mayer" title="Julius von Mayer">von Mayer</a></li> <li><a href="/wiki/Walther_Nernst" title="Walther Nernst">Nernst</a></li> <li><a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a></li> <li><a href="/wiki/Max_Planck" title="Max Planck">Planck</a></li> <li><a href="/wiki/William_John_Macquorn_Rankine" class="mw-redirect" title="William John Macquorn Rankine">Rankine</a></li> <li><a href="/wiki/John_Smeaton" title="John Smeaton">Smeaton</a></li> <li><a href="/wiki/Georg_Ernst_Stahl" title="Georg Ernst Stahl">Stahl</a></li> <li><a href="/wiki/Peter_Tait_(physicist)" class="mw-redirect" title="Peter Tait (physicist)">Tait</a></li> <li><a href="/wiki/Benjamin_Thompson" title="Benjamin Thompson">Thompson</a></li> <li><a href="/wiki/Johannes_Diderik_van_der_Waals" title="Johannes Diderik van der Waals">van der Waals</a></li> <li><a href="/wiki/John_James_Waterston" title="John James Waterston">Waterston</a></li></ul> </div></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:#ddf;text-align:center;;color: var(--color-base)"><div class="sidebar-list-title-c">Other</div></div><div class="sidebar-list-content mw-collapsible-content"> <ul><li><a href="/wiki/Nucleation" title="Nucleation">Nucleation</a></li> <li><a href="/wiki/Self-assembly" title="Self-assembly">Self-assembly</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Order_and_disorder" title="Order and disorder">Order and disorder</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below"> <ul><li><span class="noviewer" typeof="mw:File"><span title="Category"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /></span></span> <a href="/wiki/Category:Thermodynamics" title="Category:Thermodynamics">Category</a></li></ul></td></tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Thermodynamics_sidebar" title="Template:Thermodynamics sidebar"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Thermodynamics_sidebar" title="Template talk:Thermodynamics sidebar"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Thermodynamics_sidebar" title="Special:EditPage/Template:Thermodynamics sidebar"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <p><b>Non-equilibrium thermodynamics</b> is a branch of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> that deals with physical systems that are not in <a href="/wiki/Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a> but can be described in terms of macroscopic quantities (non-equilibrium state variables) that represent an extrapolation of the variables used to specify the system in thermodynamic equilibrium. Non-equilibrium thermodynamics is concerned with <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport processes</a> and with the rates of <a href="/wiki/Chemical_reactions" class="mw-redirect" title="Chemical reactions">chemical reactions</a>. </p><p>Almost all systems found in nature are not in thermodynamic equilibrium, for they are changing or can be triggered to change over time, and are continuously and discontinuously subject to flux of matter and energy to and from other systems and to chemical reactions. Many systems and processes can, however, be considered to be in equilibrium locally, thus allowing description by currently known equilibrium thermodynamics.<sup id="cite_ref-:0_1-0" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Nevertheless, some natural systems and processes remain beyond the scope of equilibrium thermodynamic methods due to the existence of non variational dynamics, where the concept of <a href="/wiki/Thermodynamic_free_energy" title="Thermodynamic free energy">free energy</a> is lost.<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>The thermodynamic study of non-equilibrium systems requires more general concepts than are dealt with by <a href="/wiki/Equilibrium_thermodynamics" title="Equilibrium thermodynamics">equilibrium thermodynamics</a>.<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> One fundamental difference between equilibrium thermodynamics and non-equilibrium thermodynamics lies in the behaviour of inhomogeneous systems, which require for their study knowledge of rates of reaction which are not considered in equilibrium thermodynamics of homogeneous systems. This is discussed below. Another fundamental and very important difference is the difficulty, in defining <a href="/wiki/Entropy" title="Entropy">entropy</a> at an instant of time in macroscopic terms for systems not in thermodynamic equilibrium. However, it can be done locally, and the macroscopic entropy will then be given by the integral of the locally defined entropy density. It has been found that many systems far outside global equilibrium still obey the concept of local equilibrium.<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> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Scope">Scope</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=1" title="Edit section: Scope"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Difference_between_equilibrium_and_non-equilibrium_thermodynamics">Difference between equilibrium and non-equilibrium thermodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=2" title="Edit section: Difference between equilibrium and non-equilibrium thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A profound difference separates equilibrium from non-equilibrium thermodynamics. Equilibrium thermodynamics ignores the time-courses of physical processes. In contrast, non-equilibrium thermodynamics attempts to describe their time-courses in continuous detail. </p><p>Equilibrium thermodynamics restricts its considerations to processes that have initial and final states of thermodynamic equilibrium; the time-courses of processes are deliberately ignored. Non-equilibrium thermodynamics, on the other hand, attempting to describe continuous time-courses, needs its <a href="/wiki/State_function" title="State function">state variables</a> to have a very close connection with those of equilibrium thermodynamics.<sup id="cite_ref-Glansdorff&Pr1971_5-0" class="reference"><a href="#cite_note-Glansdorff&Pr1971-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This conceptual issue is overcome under the assumption of local equilibrium, which entails that the relationships that hold between macroscopic state variables at equilibrium hold locally, also outside equilibrium.<sup id="cite_ref-:0_1-1" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Throughout the past decades, the assumption of local equilibrium has been tested, and found to hold, under increasingly extreme conditions, such as in the shock front of violent explosions, on reacting surfaces, and under extreme thermal gradients.<sup id="cite_ref-De_Groot_Mazur_1962_6-0" class="reference"><a href="#cite_note-De_Groot_Mazur_1962-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:1_7-0" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_8-0" class="reference"><a href="#cite_note-:2-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_9-0" class="reference"><a href="#cite_note-:3-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>Thus, non-equilibrium thermodynamics provides a consistent framework for modelling not only the initial and final states of a system, but also the evolution of the system in time. Together with the concept of entropy production, this provides a powerful tool in <a href="/wiki/Process_optimization" title="Process optimization">process optimisation</a>, and provides a theoretical foundation for <a href="/wiki/Exergy_efficiency" title="Exergy efficiency">exergy analysis</a>.<sup id="cite_ref-:0_1-2" class="reference"><a href="#cite_note-:0-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Non-equilibrium_state_variables">Non-equilibrium state variables</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=3" title="Edit section: Non-equilibrium state variables"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The suitable relationship that defines non-equilibrium thermodynamic state variables is as follows. When the system is in local equilibrium, non-equilibrium state variables are such that they can be measured locally with sufficient accuracy by the same techniques as are used to measure thermodynamic state variables, or by corresponding time and space derivatives, including fluxes of matter and energy. In general, non-equilibrium thermodynamic systems are spatially and temporally non-uniform, but their non-uniformity still has a sufficient degree of smoothness to support the existence of suitable time and space derivatives of non-equilibrium state variables. </p><p>Because of the spatial non-uniformity, non-equilibrium state variables that correspond to extensive thermodynamic state variables have to be defined as spatial densities of the corresponding extensive equilibrium state variables. When the system is in local equilibrium, intensive non-equilibrium state variables, for example temperature and pressure, correspond closely with equilibrium state variables. It is necessary that measuring probes be small enough, and rapidly enough responding, to capture relevant non-uniformity. Further, the non-equilibrium state variables are required to be mathematically functionally related to one another in ways that suitably resemble corresponding relations between equilibrium thermodynamic state variables.<sup id="cite_ref-Gyarmati_1970_10-0" class="reference"><a href="#cite_note-Gyarmati_1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In reality, these requirements, although strict, have been shown to be fulfilled even under extreme conditions, such as during phase transitions, at reacting interfaces, and in plasma droplets surrounded by ambient air.<sup id="cite_ref-11" class="reference"><a href="#cite_note-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><sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> There are, however, situations where there are appreciable non-linear effects even at the local scale.<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><sup id="cite_ref-15" class="reference"><a href="#cite_note-15"><span class="cite-bracket">[</span>15<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=Non-equilibrium_thermodynamics&action=edit&section=4" title="Edit section: Overview"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some concepts of particular importance for non-equilibrium thermodynamics include time rate of dissipation of energy (Rayleigh 1873,<sup id="cite_ref-Rayleigh_1873_16-0" class="reference"><a href="#cite_note-Rayleigh_1873-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Lars_Onsager" title="Lars Onsager">Onsager</a> 1931,<sup id="cite_ref-Onsager_1931_I_17-0" class="reference"><a href="#cite_note-Onsager_1931_I-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> also<sup id="cite_ref-Gyarmati_1970_10-1" class="reference"><a href="#cite_note-Gyarmati_1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lavenda_1978_18-0" class="reference"><a href="#cite_note-Lavenda_1978-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup>), time rate of entropy production (Onsager 1931),<sup id="cite_ref-Onsager_1931_I_17-1" class="reference"><a href="#cite_note-Onsager_1931_I-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> thermodynamic fields,<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-Ziegler_1983_20-0" class="reference"><a href="#cite_note-Ziegler_1983-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Balescu_21-0" class="reference"><a href="#cite_note-Balescu-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Dissipative_structure" class="mw-redirect" title="Dissipative structure">dissipative structure</a>,<sup id="cite_ref-Glansdorff&Pr1971_5-1" class="reference"><a href="#cite_note-Glansdorff&Pr1971-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> and non-linear dynamical structure.<sup id="cite_ref-Lavenda_1978_18-1" class="reference"><a href="#cite_note-Lavenda_1978-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </p><p>One problem of interest is the thermodynamic study of non-equilibrium <a href="/wiki/Steady_state" title="Steady state">steady states</a>, in which <a href="/wiki/Entropy" title="Entropy">entropy</a> production and some <a href="/wiki/Flux" title="Flux">flows</a> are non-zero, but there is no <a href="/wiki/Time-variant_system" title="Time-variant system">time variation</a> of physical variables. </p><p>One initial approach to non-equilibrium thermodynamics is sometimes called 'classical irreversible thermodynamics'.<sup id="cite_ref-Lebon_Jou_Casas-Vázquez_2008_22-0" class="reference"><a href="#cite_note-Lebon_Jou_Casas-Vázquez_2008-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> There are other approaches to non-equilibrium thermodynamics, for example <a href="/wiki/Extended_irreversible_thermodynamics" title="Extended irreversible thermodynamics">extended irreversible thermodynamics</a>,<sup id="cite_ref-Lebon_Jou_Casas-Vázquez_2008_22-1" class="reference"><a href="#cite_note-Lebon_Jou_Casas-Vázquez_2008-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JCVL_1993_23-0" class="reference"><a href="#cite_note-JCVL_1993-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> and generalized thermodynamics,<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> but they are hardly touched on in the present article. </p> <div class="mw-heading mw-heading3"><h3 id="Quasi-radiationless_non-equilibrium_thermodynamics_of_matter_in_laboratory_conditions">Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=5" title="Edit section: Quasi-radiationless non-equilibrium thermodynamics of matter in laboratory conditions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to Wildt<sup id="cite_ref-Wildt_1972_25-0" class="reference"><a href="#cite_note-Wildt_1972-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> (see also Essex<sup id="cite_ref-Essex_1984a_26-0" class="reference"><a href="#cite_note-Essex_1984a-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Essex_1984b_27-0" class="reference"><a href="#cite_note-Essex_1984b-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Essex_1984c_28-0" class="reference"><a href="#cite_note-Essex_1984c-28"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup>), current versions of non-equilibrium thermodynamics ignore radiant heat; they can do so because they refer to laboratory quantities of matter under laboratory conditions with temperatures well below those of stars. At laboratory temperatures, in laboratory quantities of matter, thermal radiation is weak and can be practically nearly ignored. But, for example, atmospheric physics is concerned with large amounts of matter, occupying cubic kilometers, that, taken as a whole, are not within the range of laboratory quantities; then thermal radiation cannot be ignored. </p> <div class="mw-heading mw-heading3"><h3 id="Local_equilibrium_thermodynamics">Local equilibrium thermodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=6" title="Edit section: Local equilibrium thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The terms 'classical irreversible thermodynamics'<sup id="cite_ref-Lebon_Jou_Casas-Vázquez_2008_22-2" class="reference"><a href="#cite_note-Lebon_Jou_Casas-Vázquez_2008-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> and 'local equilibrium thermodynamics' are sometimes used to refer to a version of non-equilibrium thermodynamics that demands certain simplifying assumptions, as follows. The assumptions have the effect of making each very small volume element of the system effectively homogeneous, or well-mixed, or without an effective spatial structure. Even within the thought-frame of classical irreversible thermodynamics, care<sup id="cite_ref-Lavenda_1978_18-2" class="reference"><a href="#cite_note-Lavenda_1978-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> is needed in choosing the independent variables<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> for systems. In some writings, it is assumed that the intensive variables of equilibrium thermodynamics are sufficient as the independent variables for the task (such variables are considered to have no 'memory', and do not show hysteresis); in particular, local flow intensive variables are not admitted as independent variables; local flows are considered as dependent on quasi-static local intensive variables. </p><p>Also it is assumed that the local entropy density is the same function of the other local intensive variables as in equilibrium; this is called the local thermodynamic equilibrium assumption<sup id="cite_ref-Gyarmati_1970_10-2" class="reference"><a href="#cite_note-Gyarmati_1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Lavenda_1978_18-3" class="reference"><a href="#cite_note-Lavenda_1978-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Glansdorff&Pr1971_5-2" class="reference"><a href="#cite_note-Glansdorff&Pr1971-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-JCVL_1993_23-1" class="reference"><a href="#cite_note-JCVL_1993-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-De_Groot_Mazur_1962_6-1" class="reference"><a href="#cite_note-De_Groot_Mazur_1962-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Balescu_1975_30-0" class="reference"><a href="#cite_note-Balescu_1975-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mihalas_Mihalas_1984_31-0" class="reference"><a href="#cite_note-Mihalas_Mihalas_1984-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schloegl_1989_32-0" class="reference"><a href="#cite_note-Schloegl_1989-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> (see also Keizer (1987)<sup id="cite_ref-Keizer_1987_33-0" class="reference"><a href="#cite_note-Keizer_1987-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup>). Radiation is ignored because it is transfer of energy between regions, which can be remote from one another. In the classical irreversible thermodynamic approach, there is allowed spatial variation from infinitesimal volume element to adjacent infinitesimal volume element, but it is assumed that the global entropy of the system can be found by simple spatial integration of the local entropy density. This approach assumes spatial and temporal continuity and even differentiability of locally defined intensive variables such as temperature and internal energy density. While these demands may appear severely constrictive, it has been found that the assumptions of local equilibrium hold for a wide variety of systems, including reacting interfaces, on the surfaces of catalysts, in confined systems such as zeolites, under temperature gradients as large as <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 10^{12}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10^{12}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c90fc8874dafeb075a01985c74c1d31f51e7c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.201ex; height:2.676ex;" alt="{\displaystyle 10^{12}}"></span> K m<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 ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d21ba9ae71e157827da571c142bb49c24b78e1a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:2.333ex; height:2.509ex;" alt="{\displaystyle ^{-1}}"></span>, and even in shock fronts moving at up to six times the speed of sound.<sup id="cite_ref-:1_7-1" class="reference"><a href="#cite_note-:1-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:2_8-1" class="reference"><a href="#cite_note-:2-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-:3_9-1" class="reference"><a href="#cite_note-:3-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-34" class="reference"><a href="#cite_note-34"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p><p>In other writings, local flow variables are considered; these might be considered as classical by analogy with the time-invariant long-term time-averages of flows produced by endlessly repeated cyclic processes; examples with flows are in the <a href="/wiki/Thermoelectric_effect" title="Thermoelectric effect">thermoelectric phenomena</a> known as the Seebeck and the Peltier effects, considered by <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" class="mw-redirect" title="William Thomson, 1st Baron Kelvin">Kelvin</a> in the nineteenth century and by <a href="/wiki/Lars_Onsager" title="Lars Onsager">Lars Onsager</a> in the twentieth.<sup id="cite_ref-De_Groot_Mazur_1962_6-2" class="reference"><a href="#cite_note-De_Groot_Mazur_1962-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-35" class="reference"><a href="#cite_note-35"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> These effects occur at metal junctions, which were originally effectively treated as two-dimensional surfaces, with no spatial volume, and no spatial variation. </p> <div class="mw-heading mw-heading4"><h4 id="Local_equilibrium_thermodynamics_with_materials_with_"memory""><span id="Local_equilibrium_thermodynamics_with_materials_with_.22memory.22"></span>Local equilibrium thermodynamics with materials with "memory"</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=7" title="Edit section: Local equilibrium thermodynamics with materials with "memory""><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A further extension of local equilibrium thermodynamics is to allow that materials may have "memory", so that their <a href="/wiki/Constitutive_equation" title="Constitutive equation">constitutive equations</a> depend not only on present values but also on past values of local equilibrium variables. Thus time comes into the picture more deeply than for time-dependent local equilibrium thermodynamics with memoryless materials, but fluxes are not independent variables of state.<sup id="cite_ref-36" class="reference"><a href="#cite_note-36"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Extended_irreversible_thermodynamics">Extended irreversible thermodynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=8" title="Edit section: Extended irreversible thermodynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>Extended irreversible thermodynamics</b> is a branch of non-equilibrium thermodynamics that goes outside the restriction to the local equilibrium hypothesis. The space of state variables is enlarged by including the <a href="/wiki/Flux" title="Flux">fluxes</a> of mass, momentum and energy and eventually higher order fluxes. The formalism is well-suited for describing high-frequency processes and small-length scales materials. </p> <div class="mw-heading mw-heading2"><h2 id="Basic_concepts">Basic concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=9" title="Edit section: Basic concepts"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>There are many examples of stationary non-equilibrium systems, some very simple, like a system confined between two thermostats at different temperatures or the ordinary <a href="/wiki/Couette_flow" title="Couette flow">Couette flow</a>, a fluid enclosed between two flat walls moving in opposite directions and defining non-equilibrium conditions at the walls. <a href="/wiki/Laser" title="Laser">Laser</a> action is also a non-equilibrium process, but it depends on departure from local thermodynamic equilibrium and is thus beyond the scope of classical irreversible thermodynamics; here a strong temperature difference is maintained between two molecular degrees of freedom (with molecular laser, vibrational and rotational molecular motion), the requirement for two component 'temperatures' in the one small region of space, precluding local thermodynamic equilibrium, which demands that only one temperature be needed. Damping of acoustic perturbations or shock waves are non-stationary non-equilibrium processes. Driven <a href="/wiki/Complex_fluids" class="mw-redirect" title="Complex fluids">complex fluids</a>, turbulent systems and glasses are other examples of non-equilibrium systems. </p><p>The mechanics of macroscopic systems depends on a number of extensive quantities. It should be stressed that all systems are permanently interacting with their surroundings, thereby causing unavoidable fluctuations of <a href="/wiki/Extensive_quantity" class="mw-redirect" title="Extensive quantity">extensive quantities</a>. Equilibrium conditions of thermodynamic systems are related to the maximum property of the entropy. If the only extensive quantity that is allowed to fluctuate is the internal energy, all the other ones being kept strictly constant, the temperature of the system is measurable and meaningful. The system's properties are then most conveniently described using the thermodynamic potential <a href="/wiki/Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a> (<i>A</i> = <i>U</i> - <i>TS</i>), a <a href="/wiki/Legendre_transformation" title="Legendre transformation">Legendre transformation</a> of the energy. If, next to fluctuations of the energy, the macroscopic dimensions (volume) of the system are left fluctuating, we use the <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> (<i>G</i> = <i>U</i> + <i>PV</i> - <i>TS</i>), where the system's properties are determined both by the temperature and by the pressure. </p><p>Non-equilibrium systems are much more complex and they may undergo fluctuations of more extensive quantities. The boundary conditions impose on them particular intensive variables, like temperature gradients or distorted collective motions (shear motions, vortices, etc.), often called thermodynamic forces. If free energies are very useful in equilibrium thermodynamics, it must be stressed that there is no general law defining stationary non-equilibrium properties of the energy as is the second law of thermodynamics for the <a href="/wiki/Entropy" title="Entropy">entropy</a> in equilibrium thermodynamics. That is why in such cases a more generalized Legendre transformation should be considered. This is the extended Massieu potential. By definition, the <a href="/wiki/Entropy" title="Entropy">entropy</a> (<i>S</i>) is a function of the collection of <a href="/wiki/Extensive_quantity" class="mw-redirect" title="Extensive quantity">extensive quantities</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 E_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba9f6e3041b052cf13a0ede4ecf35fb4c9cd16c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i}}"></span>. Each extensive quantity has a conjugate intensive variable <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 I_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d896cad5ad71ce7719175ca29d4d109333d4c320" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.823ex; height:2.509ex;" alt="{\displaystyle I_{i}}"></span> (a restricted definition of intensive variable is used here by comparison to the definition given in this link) so that: </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 I_{i}={\frac {\partial {S}}{\partial {E_{i}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I_{i}={\frac {\partial {S}}{\partial {E_{i}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/958ecd58c38e72fb06585bb41b2ddfb87c979b31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:10.237ex; height:5.843ex;" alt="{\displaystyle I_{i}={\frac {\partial {S}}{\partial {E_{i}}}}.}"></span></dd></dl> <p>We then define the extended <a href="/wiki/Massieu_function" title="Massieu function">Massieu function</a> as follows: </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 \ k_{\rm {B}}M=S-\sum _{i}(I_{i}E_{i}),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>M</mi> <mo>=</mo> <mi>S</mi> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k_{\rm {B}}M=S-\sum _{i}(I_{i}E_{i}),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b48faaed34cd46e2b7bde2768a1a1b4c9c38fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:23.217ex; height:5.509ex;" alt="{\displaystyle \ k_{\rm {B}}M=S-\sum _{i}(I_{i}E_{i}),}"></span></dd></dl> <p>where <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 \ k_{\rm {B}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k_{\rm {B}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70e5f7e678a654bc10d1b5f97e81c289a8f578a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.188ex; height:2.509ex;" alt="{\displaystyle \ k_{\rm {B}}}"></span> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, whence </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 \ k_{\rm {B}}\,dM=\sum _{i}(E_{i}\,dI_{i}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <mi>M</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mo stretchy="false">(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k_{\rm {B}}\,dM=\sum _{i}(E_{i}\,dI_{i}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/643ca0c512441542ee5d5e61c90d2d850451a01b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.083ex; height:5.509ex;" alt="{\displaystyle \ k_{\rm {B}}\,dM=\sum _{i}(E_{i}\,dI_{i}).}"></span></dd></dl> <p>The independent variables are the intensities. </p><p>Intensities are global values, valid for the system as a whole. When boundaries impose to the system different local conditions, (e.g. temperature differences), there are intensive variables representing the average value and others representing gradients or higher moments. The latter are the thermodynamic forces driving fluxes of extensive properties through the system. </p><p>It may be shown that the Legendre transformation changes the maximum condition of the entropy (valid at equilibrium) in a minimum condition of the extended Massieu function for stationary states, no matter whether at equilibrium or not. </p> <div class="mw-heading mw-heading2"><h2 id="Stationary_states,_fluctuations,_and_stability"><span id="Stationary_states.2C_fluctuations.2C_and_stability"></span>Stationary states, fluctuations, and stability</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=10" title="Edit section: Stationary states, fluctuations, and stability"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In thermodynamics one is often interested in a stationary state of a process, allowing that the stationary state include the occurrence of unpredictable and experimentally unreproducible fluctuations in the state of the system. The fluctuations are due to the system's internal sub-processes and to exchange of matter or energy with the system's surroundings that create the constraints that define the process. </p><p>If the stationary state of the process is stable, then the unreproducible fluctuations involve local transient decreases of entropy. The reproducible response of the system is then to increase the entropy back to its maximum by irreversible processes: the fluctuation cannot be reproduced with a significant level of probability. Fluctuations about stable stationary states are extremely small except near critical points (Kondepudi and Prigogine 1998, page 323).<sup id="cite_ref-37" class="reference"><a href="#cite_note-37"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> The stable stationary state has a local maximum of entropy and is locally the most reproducible state of the system. There are theorems about the irreversible dissipation of fluctuations. Here 'local' means local with respect to the abstract space of thermodynamic coordinates of state of the system. </p><p>If the stationary state is unstable, then any fluctuation will almost surely trigger the virtually explosive departure of the system from the unstable stationary state. This can be accompanied by increased export of entropy. </p> <div class="mw-heading mw-heading2"><h2 id="Local_thermodynamic_equilibrium">Local thermodynamic equilibrium</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=11" title="Edit section: Local thermodynamic equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The scope of present-day non-equilibrium thermodynamics does not cover all physical processes. A condition for the validity of many studies in non-equilibrium thermodynamics of matter is that they deal with what is known as <i>local thermodynamic equilibrium</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Ponderable_matter">Ponderable matter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=12" title="Edit section: Ponderable matter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i>Local thermodynamic equilibrium of matter</i><sup id="cite_ref-Gyarmati_1970_10-3" class="reference"><a href="#cite_note-Gyarmati_1970-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Glansdorff&Pr1971_5-3" class="reference"><a href="#cite_note-Glansdorff&Pr1971-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Balescu_1975_30-1" class="reference"><a href="#cite_note-Balescu_1975-30"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Mihalas_Mihalas_1984_31-1" class="reference"><a href="#cite_note-Mihalas_Mihalas_1984-31"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Schloegl_1989_32-1" class="reference"><a href="#cite_note-Schloegl_1989-32"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> (see also Keizer (1987)<sup id="cite_ref-Keizer_1987_33-1" class="reference"><a href="#cite_note-Keizer_1987-33"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup> means that conceptually, for study and analysis, the system can be spatially and temporally divided into 'cells' or 'micro-phases' of small (infinitesimal) size, in which classical thermodynamical equilibrium conditions for matter are fulfilled to good approximation. These conditions are unfulfilled, for example, in very rarefied gases, in which molecular collisions are infrequent; and in the boundary layers of a star, where radiation is passing energy to space; and for interacting fermions at very low temperature, where dissipative processes become ineffective. When these 'cells' are defined, one admits that matter and energy may pass freely between contiguous 'cells', slowly enough to leave the 'cells' in their respective individual local thermodynamic equilibria with respect to intensive variables. </p><p>One can think here of two 'relaxation times' separated by order of magnitude.<sup id="cite_ref-Zubarev_1971/1974_38-0" class="reference"><a href="#cite_note-Zubarev_1971/1974-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> The longer relaxation time is of the order of magnitude of times taken for the macroscopic dynamical structure of the system to change. The shorter is of the order of magnitude of times taken for a single 'cell' to reach local thermodynamic equilibrium. If these two relaxation times are not well separated, then the classical non-equilibrium thermodynamical concept of local thermodynamic equilibrium loses its meaning<sup id="cite_ref-Zubarev_1971/1974_38-1" class="reference"><a href="#cite_note-Zubarev_1971/1974-38"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> and other approaches have to be proposed, see for instance <a href="/wiki/Extended_irreversible_thermodynamics" title="Extended irreversible thermodynamics">Extended irreversible thermodynamics</a>. For example, in the atmosphere, the speed of sound is much greater than the wind speed; this favours the idea of local thermodynamic equilibrium of matter for atmospheric heat transfer studies at altitudes below about 60 km where sound propagates, but not above 100 km, where, because of the paucity of intermolecular collisions, sound does not propagate. </p> <div class="mw-heading mw-heading3"><h3 id="Milne's_definition_in_terms_of_radiative_equilibrium"><span id="Milne.27s_definition_in_terms_of_radiative_equilibrium"></span>Milne's definition in terms of radiative equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=13" title="Edit section: Milne's definition in terms of radiative equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Edward_Arthur_Milne" title="Edward Arthur Milne">Edward A. Milne</a>, thinking about stars, gave a definition of 'local thermodynamic equilibrium' in terms of the <a href="/wiki/Thermal_radiation" title="Thermal radiation">thermal radiation</a> of the <a href="/wiki/Matter" title="Matter">matter</a> in each small local 'cell'.<sup id="cite_ref-Milne_1928_39-0" class="reference"><a href="#cite_note-Milne_1928-39"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> He defined 'local thermodynamic equilibrium' in a 'cell' by requiring that it macroscopically absorb and spontaneously emit radiation as if it were in radiative equilibrium in a cavity at the <a href="/wiki/Temperature" title="Temperature">temperature</a> of the matter of the 'cell'. Then it strictly obeys Kirchhoff's law of equality of radiative emissivity and absorptivity, with a black body source function. The key to local thermodynamic equilibrium here is that the rate of collisions of ponderable matter particles such as molecules should far exceed the rates of creation and annihilation of photons. </p> <div class="mw-heading mw-heading2"><h2 id="Entropy_in_evolving_systems">Entropy in evolving systems</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=14" title="Edit section: Entropy in evolving systems"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>It is pointed out by W.T. Grandy Jr,<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> that entropy, though it may be defined for a non-equilibrium system is—when strictly considered—only a macroscopic quantity that refers to the whole system, and is not a dynamical variable and in general does not act as a local potential that describes local physical forces. Under special circumstances, however, one can metaphorically think as if the thermal variables behaved like local physical forces. The approximation that constitutes classical irreversible thermodynamics is built on this metaphoric thinking. </p><p>This point of view shares many points in common with the concept and the use of entropy in continuum thermomechanics,<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> which evolved completely independently of statistical mechanics and maximum-entropy principles. </p> <div class="mw-heading mw-heading3"><h3 id="Entropy_in_non-equilibrium">Entropy in non-equilibrium</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=15" title="Edit section: Entropy in non-equilibrium"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>To describe deviation of the thermodynamic system from equilibrium, in addition to constitutive variables <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 x_{1},x_{2},...,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},...,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67ba2d5eca79eb4f7088ba3bd32c1800779d038a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.52ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},...,x_{n}}"></span> that are used to fix the equilibrium state, as was described above, a set of variables <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 \xi _{1},\xi _{2},\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{1},\xi _{2},\ldots }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a3e189b3fe1a6429f8bbe0416f2f83befa515a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.937ex; height:2.509ex;" alt="{\displaystyle \xi _{1},\xi _{2},\ldots }"></span> that are called <i>internal variables</i> have been introduced. The equilibrium state is considered to be stable and the main property of the internal variables, as measures of non-equilibrium of the system, is their tending to disappear; the local law of disappearing can be written as relaxation equation for each internal variable </p> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </msubsup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31dd3110991e9b30247aca7d92c2a6acc262e76d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:39.186ex; height:5.676ex;" alt="{\displaystyle {\frac {d\xi _{i}}{dt}}=-{\frac {1}{\tau _{i}}}\,\left(\xi _{i}-\xi _{i}^{(0)}\right),\quad i=1,\,2,\ldots ,}"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>where <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 \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74deeeb02f7dea0d346358595241245100ef2f9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.738ex; height:2.843ex;" alt="{\displaystyle \tau _{i}=\tau _{i}(T,x_{1},x_{2},\ldots ,x_{n})}"></span> is a relaxation time of a corresponding variables. It is convenient to consider the initial value <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 \xi _{i}^{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{i}^{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24b8808d0463fedc2e0a15af6c64fb422006e846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.088ex; height:3.176ex;" alt="{\displaystyle \xi _{i}^{0}}"></span> are equal to zero. The above equation is valid for small deviations from equilibrium; The dynamics of internal variables in general case is considered by Pokrovskii.<sup id="cite_ref-dx.doi.org_48-0" class="reference"><a href="#cite_note-dx.doi.org-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p><p>Entropy of the system in non-equilibrium is a function of the total set of variables </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mi>S</mi> <mo stretchy="false">(</mo> <mi>T</mi> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>;</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c2fb5aedb7f035d4f04abe93df29f828fa0c5082" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.965ex; height:2.843ex;" alt="{\displaystyle S=S(T,x_{1},x_{2},,x_{n};\xi _{1},\xi _{2},\ldots )}"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>The essential contribution to the thermodynamics of the non-equilibrium systems was brought by the <a href="/wiki/Nobel_Prize" title="Nobel Prize">Nobel Prize</a> winner Ilya <a href="/wiki/Prigogine" class="mw-redirect" title="Prigogine">Prigogine</a>, when he and his collaborators investigated the systems of chemically reacting substances. The stationary states of such systems exists due to exchange both particles and energy with the environment. In section 8 of the third chapter of his book,<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Prigogine has specified three contributions to the variation of entropy of the considered system at the given volume and constant temperature <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span> . The increment of <a href="/wiki/Entropy" title="Entropy">entropy</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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> can be calculated according to the formula </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038"><table role="presentation" class="numblk" style="margin-left: 1.6em;"><tbody><tr><td class="nowrap"><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 T\,dS=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha =1}^{k}\,\eta _{\alpha }\,\Delta N_{\alpha }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>Q</mi> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munderover> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,dS=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha =1}^{k}\,\eta _{\alpha }\,\Delta N_{\alpha }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/490c3347f6770214cfbba847081320dfe8b83f1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:40.821ex; height:7.676ex;" alt="{\displaystyle T\,dS=\Delta Q-\sum _{j}\,\Xi _{j}\,\Delta \xi _{j}+\sum _{\alpha =1}^{k}\,\eta _{\alpha }\,\Delta N_{\alpha }.}"></span></td> <td></td> <td class="nowrap"><span id="math_1" class="reference nourlexpansion" style="font-weight:bold;">1</span></td></tr></tbody></table> <p>The first term on the right hand side of the equation presents a stream of thermal energy into the system; the last term—a part of a stream of energy <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 h_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a364c19a306fa879b0ac9f394fdf0ce396d700f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.623ex; height:2.509ex;" alt="{\displaystyle h_{\alpha }}"></span> coming into the system with the stream of particles of substances <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 \Delta N_{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta N_{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6283736e515b5edf3db46dbc97769bb95fc4f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.086ex; height:2.509ex;" alt="{\displaystyle \Delta N_{\alpha }}"></span> that can be positive or negative, <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 \eta _{\alpha }=h_{\alpha }-\mu _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta _{\alpha }=h_{\alpha }-\mu _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5ddbc7a99825f0111bb086825c59ba222ef51ec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.687ex; height:2.676ex;" alt="{\displaystyle \eta _{\alpha }=h_{\alpha }-\mu _{\alpha }}"></span>, where <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 \mu _{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0393f09a0a781a6f6867efd4fcf1d3dca3cb1ebe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.686ex; height:2.176ex;" alt="{\displaystyle \mu _{\alpha }}"></span> is <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a> of substance <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span>. The middle term in (1) depicts <a href="/wiki/Energy_dissipation" class="mw-redirect" title="Energy dissipation">energy dissipation</a> (<a href="/wiki/Entropy_production" title="Entropy production">entropy production</a>) due to the relaxation of internal variables <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 \xi _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c210bf2eff47302d0a01f03a56856cdad76dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.928ex; height:2.843ex;" alt="{\displaystyle \xi _{j}}"></span>. In the case of chemically reacting substances, which was investigated by Prigogine, the internal variables appear to be measures of incompleteness of chemical reactions, that is measures of how much the considered system with chemical reactions is out of equilibrium. The theory can be generalised,<sup id="cite_ref-50" class="reference"><a href="#cite_note-50"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-dx.doi.org_48-1" class="reference"><a href="#cite_note-dx.doi.org-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> to consider any deviation from the equilibrium state as an internal variable, so that we consider the set of internal variables <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 \xi _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ξ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \xi _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4c210bf2eff47302d0a01f03a56856cdad76dc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.928ex; height:2.843ex;" alt="{\displaystyle \xi _{j}}"></span> in equation (1) to consist of the quantities defining not only degrees of completeness of all chemical reactions occurring in the system, but also the structure of the system, gradients of temperature, difference of concentrations of substances and so on. </p> <div class="mw-heading mw-heading2"><h2 id="Flows_and_forces">Flows and forces</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=16" title="Edit section: Flows and forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The fundamental relation of classical equilibrium thermodynamics <sup id="cite_ref-W._Greiner_et._al._1997_51-0" class="reference"><a href="#cite_note-W._Greiner_et._al._1997-51"><span class="cite-bracket">[</span>51<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 dS={\frac {1}{T}}dU+{\frac {p}{T}}dV-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}dN_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mi>d</mi> <mi>U</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>p</mi> <mi>T</mi> </mfrac> </mrow> <mi>d</mi> <mi>V</mi> <mo>−</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> <mi>d</mi> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS={\frac {1}{T}}dU+{\frac {p}{T}}dV-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}dN_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd6dd26b4bd6632d3f2633d9bc3d7538f18b58c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:33.102ex; height:6.843ex;" alt="{\displaystyle dS={\frac {1}{T}}dU+{\frac {p}{T}}dV-\sum _{i=1}^{s}{\frac {\mu _{i}}{T}}dN_{i}}"></span></dd></dl> <p>expresses the change in <a href="/wiki/Entropy" title="Entropy">entropy</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 dS}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dS}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66f6ab8a2a8b85df78efbea896fcfeb3d4ea39e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.715ex; height:2.176ex;" alt="{\displaystyle dS}"></span> of a system as a function of the intensive quantities <a href="/wiki/Temperature" title="Temperature">temperature</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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, <a href="/wiki/Pressure" title="Pressure">pressure</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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> and <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 i^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c454bb35f050556d361ee85e06ca923b16a3bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.575ex; height:2.676ex;" alt="{\displaystyle i^{th}}"></span> <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</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 \mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0a0293841cce9eef98b55e53a92b82ae59ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}"></span> and of the differentials of the extensive quantities <a href="/wiki/Energy" title="Energy">energy</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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, <a href="/wiki/Volume_(thermodynamics)" title="Volume (thermodynamics)">volume</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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> and <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 i^{th}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mi>h</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{th}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c454bb35f050556d361ee85e06ca923b16a3bf4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.575ex; height:2.676ex;" alt="{\displaystyle i^{th}}"></span> <a href="/wiki/Particle_number" title="Particle number">particle number</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 N_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef58cebf23adff9199f17325aefb5515fdca99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.666ex; height:2.509ex;" alt="{\displaystyle N_{i}}"></span>. </p><p>Following Onsager (1931,I),<sup id="cite_ref-Onsager_1931_I_17-2" class="reference"><a href="#cite_note-Onsager_1931_I-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> let us extend our considerations to thermodynamically non-equilibrium systems. As a basis, we need locally defined versions of the extensive macroscopic quantities <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 U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span>, <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 V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"></span> and <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 N_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fef58cebf23adff9199f17325aefb5515fdca99d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.666ex; height:2.509ex;" alt="{\displaystyle N_{i}}"></span> and of the intensive macroscopic quantities <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 T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"></span>, <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> and <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 \mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dea0a0293841cce9eef98b55e53a92b82ae59ee4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.201ex; height:2.176ex;" alt="{\displaystyle \mu _{i}}"></span>. </p><p>For classical non-equilibrium studies, we will consider some new locally defined intensive macroscopic variables. We can, under suitable conditions, derive these new variables by locally defining the gradients and flux densities of the basic locally defined macroscopic quantities. </p><p>Such locally defined gradients of intensive macroscopic variables are called 'thermodynamic forces'. They 'drive' flux densities, perhaps misleadingly often called 'fluxes', which are dual to the forces. These quantities are defined in the article on <a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a>. </p><p>Establishing the relation between such forces and flux densities is a problem in statistical mechanics. Flux densities (<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 J_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a98dbd556141d7f0a16eed84fe0ae6467dcd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.09ex; height:2.509ex;" alt="{\displaystyle J_{i}}"></span>) may be coupled. The article on Onsager reciprocal relations considers the stable near-steady thermodynamically non-equilibrium regime, which has dynamics linear in the forces and flux densities. </p><p>In stationary conditions, such forces and associated flux densities are by definition time invariant, as also are the system's locally defined entropy and rate of entropy production. Notably, according to <a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Ilya Prigogine</a> and others, when an open system is in conditions that allow it to reach a stable stationary thermodynamically non-equilibrium state, it organizes itself so as to minimize total entropy production defined locally. This is considered further below. </p><p>One wants to take the analysis to the further stage of describing the behaviour of surface and volume integrals of non-stationary local quantities; these integrals are macroscopic fluxes and production rates. In general the dynamics of these integrals are not adequately described by linear equations, though in special cases they can be so described. </p> <div class="mw-heading mw-heading3"><h3 id="Onsager_reciprocal_relations">Onsager reciprocal relations</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=17" title="Edit section: Onsager reciprocal relations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></div> <p>Following Section III of Rayleigh (1873),<sup id="cite_ref-Rayleigh_1873_16-1" class="reference"><a href="#cite_note-Rayleigh_1873-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> Onsager (1931, I)<sup id="cite_ref-Onsager_1931_I_17-3" class="reference"><a href="#cite_note-Onsager_1931_I-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> showed that in the regime where both the flows (<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 J_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a98dbd556141d7f0a16eed84fe0ae6467dcd23" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.09ex; height:2.509ex;" alt="{\displaystyle J_{i}}"></span>) are small and the thermodynamic forces (<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 F_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18f9a5ed62f131c8ad9acdf7f8539f103f1e0ec9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.294ex; height:2.509ex;" alt="{\displaystyle F_{i}}"></span>) vary slowly, the rate of creation of entropy <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 (\sigma )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>σ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\sigma )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24123087fce149a1c5c8e9ca3e8db078bc97dd76" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.139ex; height:2.843ex;" alt="{\displaystyle (\sigma )}"></span> is <a href="/wiki/Linear_relation" title="Linear relation">linearly related</a> to the flows: </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 \sigma =\sum _{i}J_{i}{\frac {\partial F_{i}}{\partial x_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\sum _{i}J_{i}{\frac {\partial F_{i}}{\partial x_{i}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7275cdbe4b108870f7fb738347177c0b27cdadd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.709ex; height:6.509ex;" alt="{\displaystyle \sigma =\sum _{i}J_{i}{\frac {\partial F_{i}}{\partial x_{i}}}}"></span></dd></dl> <p>and the flows are related to the gradient of the forces, parametrized by a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> of coefficients conventionally denoted <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span>: </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 J_{i}=\sum _{j}L_{ij}{\frac {\partial F_{j}}{\partial x_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{i}=\sum _{j}L_{ij}{\frac {\partial F_{j}}{\partial x_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e97a64431f1a776f876b0241ed148af66a46f43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.549ex; height:7.176ex;" alt="{\displaystyle J_{i}=\sum _{j}L_{ij}{\frac {\partial F_{j}}{\partial x_{j}}}}"></span></dd></dl> <p>from which it follows that: </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 \sigma =\sum _{i,j}L_{ij}{\frac {\partial F_{i}}{\partial x_{i}}}{\frac {\partial F_{j}}{\partial x_{j}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma =\sum _{i,j}L_{ij}{\frac {\partial F_{i}}{\partial x_{i}}}{\frac {\partial F_{j}}{\partial x_{j}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3ba3db8d8f78da84d23c4b063987bb0ef84ae14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:20.237ex; height:7.176ex;" alt="{\displaystyle \sigma =\sum _{i,j}L_{ij}{\frac {\partial F_{i}}{\partial x_{i}}}{\frac {\partial F_{j}}{\partial x_{j}}}}"></span></dd></dl> <p>The <a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">second law of thermodynamics</a> requires that the matrix <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> be <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a>. <a href="/wiki/Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a> considerations involving microscopic reversibility of dynamics imply that the matrix <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"></span> is <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a>. This fact is called the <i>Onsager reciprocal relations</i>. </p><p>The generalization of the above equations for the rate of creation of entropy was given by Pokrovskii.<sup id="cite_ref-dx.doi.org_48-2" class="reference"><a href="#cite_note-dx.doi.org-48"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Speculated_extremal_principles_for_non-equilibrium_processes">Speculated extremal principles for non-equilibrium processes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=18" title="Edit section: Speculated extremal principles for non-equilibrium processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Extremal_principles_in_non-equilibrium_thermodynamics" title="Extremal principles in non-equilibrium thermodynamics">Extremal principles in non-equilibrium thermodynamics</a></div> <p>Until recently, prospects for useful extremal principles in this area have seemed clouded. Nicolis (1999)<sup id="cite_ref-52" class="reference"><a href="#cite_note-52"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> concludes that one model of atmospheric dynamics has an attractor which is not a regime of maximum or minimum dissipation; she says this seems to rule out the existence of a global organizing principle, and comments that this is to some extent disappointing; she also points to the difficulty of finding a thermodynamically consistent form of entropy production. Another top expert offers an extensive discussion of the possibilities for principles of extrema of entropy production and of dissipation of energy: Chapter 12 of Grandy (2008)<sup id="cite_ref-Grandy_2008_53-0" class="reference"><a href="#cite_note-Grandy_2008-53"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> is very cautious, and finds difficulty in defining the 'rate of internal entropy production' in many cases, and finds that sometimes for the prediction of the course of a process, an extremum of the quantity called the rate of dissipation of energy may be more useful than that of the rate of entropy production; this quantity appeared in Onsager's 1931<sup id="cite_ref-Onsager_1931_I_17-4" class="reference"><a href="#cite_note-Onsager_1931_I-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> origination of this subject. Other writers have also felt that prospects for general global extremal principles are clouded. Such writers include Glansdorff and Prigogine (1971), Lebon, Jou and Casas-Vásquez (2008), and Šilhavý (1997). There is good experimental evidence that heat convection does not obey extremal principles for time rate of entropy production.<sup id="cite_ref-Attard_2012_54-0" class="reference"><a href="#cite_note-Attard_2012-54"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> Theoretical analysis shows that chemical reactions do not obey extremal principles for the second differential of time rate of entropy production.<sup id="cite_ref-Keizer_1974_55-0" class="reference"><a href="#cite_note-Keizer_1974-55"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> The development of a general extremal principle seems infeasible in the current state of knowledge. </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=19" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Non-equilibrium thermodynamics has been successfully applied to describe biological processes such as <a href="/wiki/Protein_folding" title="Protein folding">protein folding</a>/unfolding and <a href="/wiki/Membrane_transport" title="Membrane transport">transport through membranes</a>.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup> It is also used to give a description of the dynamics of <a href="/wiki/Nanoparticle" title="Nanoparticle">nanoparticles</a>, which can be out of equilibrium in systems where catalysis and electrochemical conversion is involved.<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup> Also, ideas from non-equilibrium thermodynamics and the informatic theory of entropy have been adapted to describe general economic systems.<sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> <sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>60<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=Non-equilibrium_thermodynamics&action=edit&section=20" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 16em;"> <ul><li><a href="/wiki/Time_crystal" title="Time crystal">Time crystal</a></li> <li><a href="/wiki/Dissipative_system" title="Dissipative system">Dissipative system</a></li> <li><a href="/wiki/Entropy_production" title="Entropy production">Entropy production</a></li> <li><a href="/wiki/Extremal_principles_in_non-equilibrium_thermodynamics" title="Extremal principles in non-equilibrium thermodynamics">Extremal principles in non-equilibrium thermodynamics</a></li> <li><a href="/wiki/Self-organization" title="Self-organization">Self-organization</a></li> <li><a href="/wiki/Autocatalytic_reactions_and_order_creation" class="mw-redirect" title="Autocatalytic reactions and order creation">Autocatalytic reactions and order creation</a></li> <li><a href="/wiki/Self-organizing_criticality" class="mw-redirect" title="Self-organizing criticality">Self-organizing criticality</a></li> <li><a href="/wiki/BBGKY_hierarchy" title="BBGKY hierarchy">Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy of equations</a></li> <li><a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a></li> <li><a href="/wiki/Vlasov_equation" title="Vlasov equation">Vlasov equation</a></li> <li><a href="/wiki/Maxwell%27s_demon" title="Maxwell's demon">Maxwell's demon</a></li> <li><a href="/wiki/Information_entropy" class="mw-redirect" title="Information entropy">Information entropy</a></li> <li><a href="/wiki/Spontaneous_symmetry_breaking" title="Spontaneous symmetry breaking">Spontaneous symmetry breaking</a></li> <li><a href="/wiki/Autopoiesis" title="Autopoiesis">Autopoiesis</a></li> <li><a href="/wiki/Maximum_power_principle" title="Maximum power principle">Maximum power principle</a></li></ul> </div> <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=Non-equilibrium_thermodynamics&action=edit&section=21" 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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-:0-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_1-2"><sup><i><b>c</b></i></sup></a></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="CITEREFKjelstrupBedeauxJohannessenGross2010" class="citation book cs1">Kjelstrup, S; Bedeaux, D; Johannessen, E; Gross, J (June 2010). <a rel="nofollow" class="external text" href="https://www.worldscientific.com/worldscibooks/10.1142/7869"><i>Non-Equilibrium Thermodynamics for Engineers</i></a>. WORLD SCIENTIFIC. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1142%2F7869">10.1142/7869</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-4322-15-7" title="Special:BookSources/978-981-4322-15-7"><bdi>978-981-4322-15-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-Equilibrium+Thermodynamics+for+Engineers&rft.pub=WORLD+SCIENTIFIC&rft.date=2010-06&rft_id=info%3Adoi%2F10.1142%2F7869&rft.isbn=978-981-4322-15-7&rft.aulast=Kjelstrup&rft.aufirst=S&rft.au=Bedeaux%2C+D&rft.au=Johannessen%2C+E&rft.au=Gross%2C+J&rft_id=https%3A%2F%2Fwww.worldscientific.com%2Fworldscibooks%2F10.1142%2F7869&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBodenschatzCannellde_BruynEcke1992" class="citation journal cs1">Bodenschatz, Eberhard; Cannell, David S.; de Bruyn, John R.; Ecke, Robert; Hu, Yu-Chou; <a href="/wiki/Kristina_Lerman" title="Kristina Lerman">Lerman, Kristina</a>; Ahlers, Guenter (December 1992). "Experiments on three systems with non-variational aspects". <i>Physica D: Nonlinear Phenomena</i>. <b>61</b> (<span class="nowrap">1–</span>4): <span class="nowrap">77–</span>93. <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/1992PhyD...61...77B">1992PhyD...61...77B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0167-2789%2892%2990150-L">10.1016/0167-2789(92)90150-L</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+D%3A+Nonlinear+Phenomena&rft.atitle=Experiments+on+three+systems+with+non-variational+aspects&rft.volume=61&rft.issue=%3Cspan+class%3D%22nowrap%22%3E1%E2%80%93%3C%2Fspan%3E4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E77-%3C%2Fspan%3E93&rft.date=1992-12&rft_id=info%3Adoi%2F10.1016%2F0167-2789%2892%2990150-L&rft_id=info%3Abibcode%2F1992PhyD...61...77B&rft.aulast=Bodenschatz&rft.aufirst=Eberhard&rft.au=Cannell%2C+David+S.&rft.au=de+Bruyn%2C+John+R.&rft.au=Ecke%2C+Robert&rft.au=Hu%2C+Yu-Chou&rft.au=Lerman%2C+Kristina&rft.au=Ahlers%2C+Guenter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" 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="CITEREFPokrovskii2020" class="citation book cs1">Pokrovskii, Vladimir (2020). <i>Thermodynamics of Complex Systems: Principles and applications</i>. IOP Publishing, Bristol, UK. <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/2020tcsp.book.....P">2020tcsp.book.....P</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+Complex+Systems%3A+Principles+and+applications.&rft.pub=IOP+Publishing%2C+Bristol%2C+UK.&rft.date=2020&rft_id=info%3Abibcode%2F2020tcsp.book.....P&rft.aulast=Pokrovskii&rft.aufirst=Vladimir&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrootMazur1984" class="citation book cs1">Groot, Sybren Ruurds de; Mazur, Peter (1984). <i>Non-equilibrium thermodynamics</i>. Dover books on physics (Dover ed., 1. publ., unabridged, corr. republ ed.). New York, NY: Dover Publ. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-64741-8" title="Special:BookSources/978-0-486-64741-8"><bdi>978-0-486-64741-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Non-equilibrium+thermodynamics&rft.place=New+York%2C+NY&rft.series=Dover+books+on+physics&rft.edition=Dover+ed.%2C+1.+publ.%2C+unabridged%2C+corr.+republ&rft.pub=Dover+Publ&rft.date=1984&rft.isbn=978-0-486-64741-8&rft.aulast=Groot&rft.aufirst=Sybren+Ruurds+de&rft.au=Mazur%2C+Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-Glansdorff&Pr1971-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-Glansdorff&Pr1971_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Glansdorff&Pr1971_5-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Glansdorff&Pr1971_5-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Glansdorff&Pr1971_5-3"><sup><i><b>d</b></i></sup></a></span> <span class="reference-text">Glansdorff, P., Prigogine, I. 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From Heat Engines to Dissipative Structures</i>, Wiley, Chichester, 1998, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-97394-7" title="Special:BookSources/0-471-97394-7">0-471-97394-7</a>.</span> </li> <li id="cite_note-Zubarev_1971/1974-38"><span class="mw-cite-backlink">^ <a href="#cite_ref-Zubarev_1971/1974_38-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Zubarev_1971/1974_38-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><a href="/wiki/Dmitry_Zubarev" title="Dmitry Zubarev">Zubarev D. N.</a>,(1974). <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=SQy3AAAAIAAJ&hl=ru&source=gbs_ViewAPI">Nonequilibrium Statistical Thermodynamics</a></i>, translated from the Russian by P.J. 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Jr. (2004). <a rel="nofollow" class="external text" href="http://physics.uwyo.edu/~tgrandy/evolve.html">"Time Evolution in Macroscopic Systems. I. Equations of Motion"</a>. <i>Foundations of Physics</i>. <b>34</b> (1): 1. <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/cond-mat/0303290">cond-mat/0303290</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/2004FoPh...34....1G">2004FoPh...34....1G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AFOOP.0000012007.06843.ed">10.1023/B:FOOP.0000012007.06843.ed</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:13096766">13096766</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Time+Evolution+in+Macroscopic+Systems.+I.+Equations+of+Motion&rft.volume=34&rft.issue=1&rft.pages=1&rft.date=2004&rft_id=info%3Aarxiv%2Fcond-mat%2F0303290&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A13096766%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FB%3AFOOP.0000012007.06843.ed&rft_id=info%3Abibcode%2F2004FoPh...34....1G&rft.aulast=Grandy&rft.aufirst=W.T.+Jr.&rft_id=http%3A%2F%2Fphysics.uwyo.edu%2F~tgrandy%2Fevolve.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-41"><span class="mw-cite-backlink"><b><a href="#cite_ref-41">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrandy2004" class="citation journal cs1">Grandy, W.T. Jr. (2004). <a rel="nofollow" class="external text" href="http://physics.uwyo.edu/~tgrandy/entropy.html">"Time Evolution in Macroscopic Systems. II. The Entropy"</a>. <i>Foundations of Physics</i>. <b>34</b> (1): 21. <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/cond-mat/0303291">cond-mat/0303291</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/2004FoPh...34...21G">2004FoPh...34...21G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AFOOP.0000012008.36856.c1">10.1023/B:FOOP.0000012008.36856.c1</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:18573684">18573684</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Time+Evolution+in+Macroscopic+Systems.+II.+The+Entropy&rft.volume=34&rft.issue=1&rft.pages=21&rft.date=2004&rft_id=info%3Aarxiv%2Fcond-mat%2F0303291&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18573684%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FB%3AFOOP.0000012008.36856.c1&rft_id=info%3Abibcode%2F2004FoPh...34...21G&rft.aulast=Grandy&rft.aufirst=W.T.+Jr.&rft_id=http%3A%2F%2Fphysics.uwyo.edu%2F~tgrandy%2Fentropy.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-42"><span class="mw-cite-backlink"><b><a href="#cite_ref-42">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrandy2004" class="citation journal cs1">Grandy, W. 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III: Selected Applications"</a>. <i>Foundations of Physics</i>. <b>34</b> (5): 771. <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/2004FoPh...34..771G">2004FoPh...34..771G</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1023%2FB%3AFOOP.0000022187.45866.81">10.1023/B:FOOP.0000022187.45866.81</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:119406182">119406182</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Foundations+of+Physics&rft.atitle=Time+Evolution+in+Macroscopic+Systems.+III%3A+Selected+Applications&rft.volume=34&rft.issue=5&rft.pages=771&rft.date=2004&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119406182%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1023%2FB%3AFOOP.0000022187.45866.81&rft_id=info%3Abibcode%2F2004FoPh...34..771G&rft.aulast=Grandy&rft.aufirst=W.+T.+Jr&rft_id=http%3A%2F%2Fphysics.uwyo.edu%2F~tgrandy%2Fapplications.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-43"><span class="mw-cite-backlink"><b><a href="#cite_ref-43">^</a></b></span> <span class="reference-text">Grandy 2004 see also <a rel="nofollow" class="external autonumber" href="http://physics.uwyo.edu/~tgrandy/Statistical_Mechanics.html">[1]</a>.</span> </li> <li id="cite_note-44"><span class="mw-cite-backlink"><b><a href="#cite_ref-44">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTruesdell1984" class="citation book cs1">Truesdell, Clifford (1984). <i>Rational Thermodynamics</i> (2 ed.). 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"Entropy production and dynamical complexity in a low-order atmospheric model". <i>Quarterly Journal of the Royal Meteorological Society</i>. <b>125</b> (557): <span class="nowrap">1859–</span>1878. <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/1999QJRMS.125.1859N">1999QJRMS.125.1859N</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fqj.49712555718">10.1002/qj.49712555718</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:121536072">121536072</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Quarterly+Journal+of+the+Royal+Meteorological+Society&rft.atitle=Entropy+production+and+dynamical+complexity+in+a+low-order+atmospheric+model&rft.volume=125&rft.issue=557&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1859-%3C%2Fspan%3E1878&rft.date=1999&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121536072%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1002%2Fqj.49712555718&rft_id=info%3Abibcode%2F1999QJRMS.125.1859N&rft.aulast=Nicolis&rft.aufirst=C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-Grandy_2008-53"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grandy_2008_53-0">^</a></b></span> <span class="reference-text">Grandy, W.T., Jr (2008).</span> </li> <li id="cite_note-Attard_2012-54"><span class="mw-cite-backlink"><b><a href="#cite_ref-Attard_2012_54-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAttard2012" class="citation arxiv cs1">Attard, P. 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"Nonequilibrium thermodynamics of ion transport through membranes". <i>Journal of Colloid and Interface Science</i>. <b>52</b> (3): <span class="nowrap">516–</span>525. <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/1975JCIS...52..516K">1975JCIS...52..516K</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0021-9797%2875%2990276-3">10.1016/0021-9797(75)90276-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Colloid+and+Interface+Science&rft.atitle=Nonequilibrium+thermodynamics+of+ion+transport+through+membranes&rft.volume=52&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E516-%3C%2Fspan%3E525&rft.date=1975-09&rft_id=info%3Adoi%2F10.1016%2F0021-9797%2875%2990276-3&rft_id=info%3Abibcode%2F1975JCIS...52..516K&rft.aulast=Kimizuka&rft.aufirst=Hideo&rft.au=Kaibara%2C+Kozue&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-57"><span class="mw-cite-backlink"><b><a href="#cite_ref-57">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBaranowski1991" class="citation journal cs1">Baranowski, B. (April 1991). "Non-equilibrium thermodynamics as applied to membrane transport". <i>Journal of Membrane Science</i>. <b>57</b> (<span class="nowrap">2–</span>3): <span class="nowrap">119–</span>159. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0376-7388%2800%2980675-4">10.1016/S0376-7388(00)80675-4</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Membrane+Science&rft.atitle=Non-equilibrium+thermodynamics+as+applied+to+membrane+transport&rft.volume=57&rft.issue=%3Cspan+class%3D%22nowrap%22%3E2%E2%80%93%3C%2Fspan%3E3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E119-%3C%2Fspan%3E159&rft.date=1991-04&rft_id=info%3Adoi%2F10.1016%2FS0376-7388%2800%2980675-4&rft.aulast=Baranowski&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-58"><span class="mw-cite-backlink"><b><a href="#cite_ref-58">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBazant2013" class="citation journal cs1">Bazant, Martin Z. (22 March 2013). "Theory of Chemical Kinetics and Charge Transfer based on Nonequilibrium Thermodynamics". <i>Accounts of Chemical Research</i>. <b>46</b> (5): <span class="nowrap">1144–</span>1160. <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/1208.1587">1208.1587</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1021%2Far300145c">10.1021/ar300145c</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/23520980">23520980</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:10827167">10827167</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Accounts+of+Chemical+Research&rft.atitle=Theory+of+Chemical+Kinetics+and+Charge+Transfer+based+on+Nonequilibrium+Thermodynamics&rft.volume=46&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1144-%3C%2Fspan%3E1160&rft.date=2013-03-22&rft_id=info%3Aarxiv%2F1208.1587&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A10827167%23id-name%3DS2CID&rft_id=info%3Apmid%2F23520980&rft_id=info%3Adoi%2F10.1021%2Far300145c&rft.aulast=Bazant&rft.aufirst=Martin+Z.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-59"><span class="mw-cite-backlink"><b><a href="#cite_ref-59">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPokrovskii2011" class="citation book cs1">Pokrovskii, Vladimir (2011). <a rel="nofollow" class="external text" href="https://www.springer.com/physics/complexity/book/978-94-007-2095-4"><i>Econodynamics. The Theory of Social Production</i></a>. Springer, Dordrecht-Heidelberg-London-New York.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Econodynamics.+The+Theory+of+Social+Production.&rft.pub=Springer%2C+Dordrecht-Heidelberg-London-New+York.&rft.date=2011&rft.aulast=Pokrovskii&rft.aufirst=Vladimir&rft_id=https%3A%2F%2Fwww.springer.com%2Fphysics%2Fcomplexity%2Fbook%2F978-94-007-2095-4&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> <li id="cite_note-60"><span class="mw-cite-backlink"><b><a href="#cite_ref-60">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChen2015" class="citation book cs1">Chen, Jing (2015). <a rel="nofollow" class="external text" href="https://www.springer.com/us/book/9781493934645"><i>The Unity of Science and Economics: A New Foundation of Economic Theory</i></a>. Springer.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Unity+of+Science+and+Economics%3A+A+New+Foundation+of+Economic+Theory&rft.pub=Springer&rft.date=2015&rft.aulast=Chen&rft.aufirst=Jing&rft_id=https%3A%2F%2Fwww.springer.com%2Fus%2Fbook%2F9781493934645&rfr_id=info%3Asid%2Fen.wikipedia.org%3ANon-equilibrium+thermodynamics" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading3"><h3 id="Sources">Sources</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=22" title="Edit section: Sources"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><a href="/wiki/Herbert_Callen" title="Herbert Callen">Callen, H.B.</a> (1960/1985). <i>Thermodynamics and an Introduction to Thermostatistics</i>, (1st edition 1960) 2nd edition 1985, Wiley, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-86256-8" title="Special:BookSources/0-471-86256-8">0-471-86256-8</a>.</li> <li>Eu, B.C. (2002). <i>Generalized Thermodynamics. The Thermodynamics of Irreversible Processes and Generalized Hydrodynamics</i>, Kluwer Academic Publishers, Dordrecht, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/1-4020-0788-4" title="Special:BookSources/1-4020-0788-4">1-4020-0788-4</a>.</li> <li>Glansdorff, P., <a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Prigogine, I.</a> (1971). <i>Thermodynamic Theory of Structure, Stability, and Fluctuations</i>, Wiley-Interscience, London, 1971, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-30280-5" title="Special:BookSources/0-471-30280-5">0-471-30280-5</a>.</li> <li>Grandy, W.T. Jr (2008). <i>Entropy and the Time Evolution of Macroscopic Systems</i>. Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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">978-0-19-954617-6</a>.</li> <li>Gyarmati, I. (1967/1970). <i>Non-equilibrium Thermodynamics. Field Theory and Variational Principles</i>, translated from the Hungarian (1967) by E. Gyarmati and W.F. Heinz, Springer, Berlin.</li> <li><a href="/wiki/Elliott_H._Lieb" title="Elliott H. Lieb">Lieb, E.H.</a>, <a href="/wiki/Jakob_Yngvason" title="Jakob Yngvason">Yngvason, J.</a> (1999). 'The physics and mathematics of the second law of thermodynamics', <i>Physics Reports</i>, <b>310</b>: 1–96. <a rel="nofollow" class="external text" href="https://arxiv.org/abs/cond-mat/9708200">See also this.</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Non-equilibrium_thermodynamics&action=edit&section=23" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><a href="/wiki/Hans_Ziegler_(physicist)" title="Hans Ziegler (physicist)">Ziegler, Hans</a> (1977): <i>An introduction to Thermomechanics</i>. North Holland, Amsterdam. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-11080-1" title="Special:BookSources/0-444-11080-1">0-444-11080-1</a>. Second edition (1983) <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-444-86503-9" title="Special:BookSources/0-444-86503-9">0-444-86503-9</a>.</li> <li>Kleidon, A., Lorenz, R.D., editors (2005). <i>Non-equilibrium Thermodynamics and the Production of Entropy</i>, Springer, Berlin. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-22495-5" title="Special:BookSources/3-540-22495-5">3-540-22495-5</a>.</li> <li><a href="/wiki/Ilya_Prigogine" title="Ilya Prigogine">Prigogine, I.</a> (1955/1961/1967). <i>Introduction to Thermodynamics of Irreversible Processes</i>. 3rd edition, Wiley Interscience, New York.</li> <li><a href="/wiki/Dmitry_Zubarev" title="Dmitry Zubarev">Zubarev D. N.</a> (1974): <i><a rel="nofollow" class="external text" href="https://books.google.com/books?id=SQy3AAAAIAAJ&hl=ru&source=gbs_ViewAPI">Nonequilibrium Statistical Thermodynamics</a></i>. New York, Consultants Bureau. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-306-10895-X" title="Special:BookSources/0-306-10895-X">0-306-10895-X</a>; <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-306-10895-2" title="Special:BookSources/978-0-306-10895-2">978-0-306-10895-2</a>.</li> <li>Keizer, J. (1987). <i>Statistical Thermodynamics of Nonequilibrium Processes</i>, Springer-Verlag, New York, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-96501-7" title="Special:BookSources/0-387-96501-7">0-387-96501-7</a>.</li> <li><a href="/wiki/Dmitry_Zubarev" title="Dmitry Zubarev">Zubarev D. N.</a>, Morozov V., Ropke G. (1996): <i>Statistical Mechanics of Nonequilibrium Processes: Basic Concepts, Kinetic Theory</i>. John Wiley & Sons. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-05-501708-0" title="Special:BookSources/3-05-501708-0">3-05-501708-0</a>.</li> <li><a href="/wiki/Dmitry_Zubarev" title="Dmitry Zubarev">Zubarev D. N.</a>, Morozov V., Ropke G. (1997): <i>Statistical Mechanics of Nonequilibrium Processes: Relaxation and Hydrodynamic Processes</i>. John Wiley & Sons. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-527-40084-2" title="Special:BookSources/3-527-40084-2">3-527-40084-2</a>.</li> <li>Tuck, Adrian F. (2008). <i>Atmospheric turbulence : a molecular dynamics perspective</i>. Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-923653-4" title="Special:BookSources/978-0-19-923653-4">978-0-19-923653-4</a>.</li> <li>Grandy, W.T. Jr (2008). <i>Entropy and the Time Evolution of Macroscopic Systems</i>. Oxford University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><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">978-0-19-954617-6</a>.</li> <li>Kondepudi, D., Prigogine, I. (1998). <i>Modern Thermodynamics: From Heat Engines to Dissipative Structures</i>. John Wiley & Sons, Chichester. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-97393-9" title="Special:BookSources/0-471-97393-9">0-471-97393-9</a>.</li> <li>de Groot S.R., <a href="/wiki/Peter_Mazur" title="Peter Mazur">Mazur P.</a> (1984). <i>Non-Equilibrium Thermodynamics</i> (Dover). <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-64741-2" title="Special:BookSources/0-486-64741-2">0-486-64741-2</a></li> <li>Ramiro Augusto Salazar La Rotta. (2011). <i>The Non-Equilibrium Thermodynamics, Perpetual</i></li></ul> </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=Non-equilibrium_thermodynamics&action=edit&section=24" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20110406071945/http://www-dcf.ds.mpg.de/build.php/Titel/Research_english.html?sub=1&ver=en">Stephan Herminghaus' Dynamics of Complex Fluids Department at the Max Planck Institute for Dynamics and Self Organization</a></li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20120519172014/http://www.worldscibooks.com/physics/1622.html">Non-equilibrium Statistical Thermodynamics applied to Fluid Dynamics and Laser Physics</a> - 1992- book by Xavier de Hemptinne.</li> <li><a rel="nofollow" class="external text" href="https://dx.doi.org/10.1063/1.2012462">Nonequilibrium Thermodynamics of Small Systems</a> - PhysicsToday.org</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20050308053832/http://www.intothecool.com/energetic.php">Into the Cool</a> - 2005 book by Dorion Sagan and Eric D. 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