Diffusion - Wikipedia

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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Diffusion_vs._bulk_flow" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diffusion_vs._bulk_flow"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Diffusion vs. bulk flow</span> </div> </a> <ul id="toc-Diffusion_vs._bulk_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_in_the_context_of_different_disciplines" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diffusion_in_the_context_of_different_disciplines"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Diffusion in the context of different disciplines</span> </div> </a> <ul id="toc-Diffusion_in_the_context_of_different_disciplines-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History_of_diffusion_in_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History_of_diffusion_in_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>History of diffusion in physics</span> </div> </a> <ul id="toc-History_of_diffusion_in_physics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_models_of_diffusion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Basic_models_of_diffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Basic models of diffusion</span> </div> </a> <button aria-controls="toc-Basic_models_of_diffusion-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 Basic models of diffusion subsection</span> </button> <ul id="toc-Basic_models_of_diffusion-sublist" class="vector-toc-list"> <li id="toc-Definition_of_diffusion_flux" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Definition_of_diffusion_flux"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Definition of diffusion flux</span> </div> </a> <ul id="toc-Definition_of_diffusion_flux-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Normal_single_component_concentration_gradient" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Normal_single_component_concentration_gradient"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Normal single component concentration gradient</span> </div> </a> <ul id="toc-Normal_single_component_concentration_gradient-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Multicomponent_diffusion_and_thermodiffusion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Multicomponent_diffusion_and_thermodiffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Multicomponent diffusion and thermodiffusion</span> </div> </a> <ul id="toc-Multicomponent_diffusion_and_thermodiffusion-sublist" class="vector-toc-list"> <li id="toc-Nondiagonal_diffusion_must_be_nonlinear" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Nondiagonal_diffusion_must_be_nonlinear"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>Nondiagonal diffusion must be nonlinear</span> </div> </a> <ul id="toc-Nondiagonal_diffusion_must_be_nonlinear-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applied_forces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applied_forces"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.4</span> <span>Applied forces</span> </div> </a> <ul id="toc-Applied_forces-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_across_a_membrane" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_across_a_membrane"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.5</span> <span>Diffusion across a membrane</span> </div> </a> <ul id="toc-Diffusion_across_a_membrane-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Ballistic_time_scale" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Ballistic_time_scale"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.6</span> <span>Ballistic time scale</span> </div> </a> <ul id="toc-Ballistic_time_scale-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Jumps_on_the_surface_and_in_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Jumps_on_the_surface_and_in_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.7</span> <span>Jumps on the surface and in solids</span> </div> </a> <ul id="toc-Jumps_on_the_surface_and_in_solids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Porous_media" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Porous_media"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.8</span> <span>Porous media</span> </div> </a> <ul id="toc-Porous_media-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Diffusion_in_physics" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Diffusion_in_physics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Diffusion in physics</span> </div> </a> <button aria-controls="toc-Diffusion_in_physics-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 Diffusion in physics subsection</span> </button> <ul id="toc-Diffusion_in_physics-sublist" class="vector-toc-list"> <li id="toc-Diffusion_coefficient_in_kinetic_theory_of_gases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_coefficient_in_kinetic_theory_of_gases"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Diffusion coefficient in kinetic theory of gases</span> </div> </a> <ul id="toc-Diffusion_coefficient_in_kinetic_theory_of_gases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>The theory of diffusion in gases based on Boltzmann's equation</span> </div> </a> <ul id="toc-The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_of_electrons_in_solids" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_of_electrons_in_solids"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Diffusion of electrons in solids</span> </div> </a> <ul id="toc-Diffusion_of_electrons_in_solids-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_in_geophysics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_in_geophysics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>Diffusion in geophysics</span> </div> </a> <ul id="toc-Diffusion_in_geophysics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Dialysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Dialysis"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Dialysis</span> </div> </a> <ul id="toc-Dialysis-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Random_walk_(random_motion)" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Random_walk_(random_motion)"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Random walk (random motion)</span> </div> </a> <button aria-controls="toc-Random_walk_(random_motion)-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 Random walk (random motion) subsection</span> </button> <ul id="toc-Random_walk_(random_motion)-sublist" class="vector-toc-list"> <li id="toc-Separation_of_diffusion_from_convection_in_gases" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Separation_of_diffusion_from_convection_in_gases"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.1</span> <span>Separation of diffusion from convection in gases</span> </div> </a> <ul id="toc-Separation_of_diffusion_from_convection_in_gases-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_types_of_diffusion" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_types_of_diffusion"> <div class="vector-toc-text"> <span class="vector-toc-numb">6.2</span> <span>Other types of diffusion</span> </div> </a> <ul id="toc-Other_types_of_diffusion-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Diffusion</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 79 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-79" 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">79 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Diffusie" title="Diffusie – Afrikaans" lang="af" hreflang="af" data-title="Diffusie" data-language-autonym="Afrikaans" data-language-local-name="Afrikaans" class="interlanguage-link-target"><span>Afrikaans</span></a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D8%B4%D8%A7%D8%B1" title="انتشار – Arabic" lang="ar" hreflang="ar" data-title="انتشار" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Diffuziya" title="Diffuziya – Azerbaijani" lang="az" hreflang="az" data-title="Diffuziya" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AC%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%AA%E0%A6%A8" title="ব্যাপন – Bangla" lang="bn" hreflang="bn" data-title="ব্যাপন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%94%D1%8B%D1%84%D1%83%D0%B7%D1%96%D1%8F" title="Дыфузія – Belarusian" lang="be" hreflang="be" data-title="Дыфузія" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%94%D1%8B%D1%84%D1%83%D0%B7%D1%96%D1%8F" title="Дыфузія – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Дыфузія" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target"><span>Беларуская (тарашкевіца)</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%83%D0%B7%D0%B8%D1%8F" title="Дифузия – Bulgarian" lang="bg" hreflang="bg" data-title="Дифузия" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bs mw-list-item"><a href="https://bs.wikipedia.org/wiki/Difuzija" title="Difuzija – Bosnian" lang="bs" hreflang="bs" data-title="Difuzija" data-language-autonym="Bosanski" data-language-local-name="Bosnian" class="interlanguage-link-target"><span>Bosanski</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Difusi%C3%B3" title="Difusió – Catalan" lang="ca" hreflang="ca" data-title="Difusió" 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%94%D0%B8%D1%84%D1%84%D1%83%D0%B7%D0%B8" title="Диффузи – Chuvash" lang="cv" hreflang="cv" data-title="Диффузи" 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/Difuze" title="Difuze – Czech" lang="cs" hreflang="cs" data-title="Difuze" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Diffusion" title="Diffusion – Danish" lang="da" hreflang="da" data-title="Diffusion" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de badge-Q17437798 badge-goodarticle mw-list-item" title="good article badge"><a href="https://de.wikipedia.org/wiki/Diffusion" title="Diffusion – German" lang="de" hreflang="de" data-title="Diffusion" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Difusioon" title="Difusioon – Estonian" lang="et" hreflang="et" data-title="Difusioon" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CE%B9%CE%AC%CF%87%CF%85%CF%83%CE%B7" title="Διάχυση – Greek" lang="el" hreflang="el" data-title="Διάχυση" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Difusi%C3%B3n_(f%C3%ADsica)" title="Difusión (física) – Spanish" lang="es" hreflang="es" data-title="Difusión (física)" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Difuzo" title="Difuzo – Esperanto" lang="eo" hreflang="eo" data-title="Difuzo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Difusio" title="Difusio – Basque" lang="eu" hreflang="eu" data-title="Difusio" 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/%D9%BE%D8%AF%DB%8C%D8%AF%D9%87_%D8%A7%D9%86%D8%AA%D8%B4%D8%A7%D8%B1" title="پدیده انتشار – Persian" lang="fa" hreflang="fa" data-title="پدیده انتشار" 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/Diffusion_de_la_mati%C3%A8re" title="Diffusion de la matière – French" lang="fr" hreflang="fr" data-title="Diffusion de la matière" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Idirleathadh" title="Idirleathadh – Irish" lang="ga" hreflang="ga" data-title="Idirleathadh" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Difusi%C3%B3n" title="Difusión – Galician" lang="gl" hreflang="gl" data-title="Difusión" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%99%95%EC%82%B0" title="확산 – Korean" lang="ko" hreflang="ko" data-title="확산" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D6%86%D5%B8%D6%82%D5%A6%D5%AB%D5%A1" title="Դիֆուզիա – Armenian" lang="hy" hreflang="hy" data-title="Դիֆուզիա" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%BF%E0%A4%B8%E0%A4%B0%E0%A4%A3" title="विसरण – Hindi" lang="hi" hreflang="hi" data-title="विसरण" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Difuzija" title="Difuzija – Croatian" lang="hr" hreflang="hr" data-title="Difuzija" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Difusi" title="Difusi – Indonesian" lang="id" hreflang="id" data-title="Difusi" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Diffusion" title="Diffusion – Interlingua" lang="ia" hreflang="ia" data-title="Diffusion" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target"><span>Interlingua</span></a></li><li class="interlanguage-link interwiki-xh mw-list-item"><a href="https://xh.wikipedia.org/wiki/I-diffusion" title="I-diffusion – Xhosa" lang="xh" hreflang="xh" data-title="I-diffusion" data-language-autonym="IsiXhosa" data-language-local-name="Xhosa" class="interlanguage-link-target"><span>IsiXhosa</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Sveim" title="Sveim – Icelandic" lang="is" hreflang="is" data-title="Sveim" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target"><span>Íslenska</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Diffusione_di_materia" title="Diffusione di materia – Italian" lang="it" hreflang="it" data-title="Diffusione di materia" 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%A4%D7%A2%D7%A4%D7%95%D7%A2" title="פעפוע – Hebrew" lang="he" hreflang="he" data-title="פעפוע" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-jv mw-list-item"><a href="https://jv.wikipedia.org/wiki/Dhifusi" title="Dhifusi – Javanese" lang="jv" hreflang="jv" data-title="Dhifusi" data-language-autonym="Jawa" data-language-local-name="Javanese" class="interlanguage-link-target"><span>Jawa</span></a></li><li class="interlanguage-link interwiki-kn mw-list-item"><a href="https://kn.wikipedia.org/wiki/%E0%B2%AA%E0%B3%8D%E0%B2%B0%E0%B2%B8%E0%B2%B0%E0%B2%A3%E0%B3%86" title="ಪ್ರಸರಣೆ – Kannada" lang="kn" hreflang="kn" data-title="ಪ್ರಸರಣೆ" data-language-autonym="ಕನ್ನಡ" data-language-local-name="Kannada" class="interlanguage-link-target"><span>ಕನ್ನಡ</span></a></li><li class="interlanguage-link interwiki-ka mw-list-item"><a href="https://ka.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A4%E1%83%A3%E1%83%96%E1%83%98%E1%83%90" title="დიფუზია – Georgian" lang="ka" hreflang="ka" data-title="დიფუზია" data-language-autonym="ქართული" data-language-local-name="Georgian" class="interlanguage-link-target"><span>ქართული</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D1%83%D0%B7%D0%B8%D1%8F" title="Диффузия – Kazakh" lang="kk" hreflang="kk" data-title="Диффузия" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Difizyon" title="Difizyon – Haitian Creole" lang="ht" hreflang="ht" data-title="Difizyon" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%84%D1%83%D0%B7%D0%B8%D1%8F" title="Диффузия – Kyrgyz" lang="ky" hreflang="ky" data-title="Диффузия" data-language-autonym="Кыргызча" data-language-local-name="Kyrgyz" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-la mw-list-item"><a href="https://la.wikipedia.org/wiki/Diffusio" title="Diffusio – Latin" lang="la" hreflang="la" data-title="Diffusio" data-language-autonym="Latina" data-language-local-name="Latin" class="interlanguage-link-target"><span>Latina</span></a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/Dif%C5%ABzija" title="Difūzija – Latvian" lang="lv" hreflang="lv" data-title="Difūzija" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target"><span>Latviešu</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Difuzija" title="Difuzija – Lithuanian" lang="lt" hreflang="lt" data-title="Difuzija" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Diff%C3%BAzi%C3%B3" title="Diffúzió – Hungarian" lang="hu" hreflang="hu" data-title="Diffúzió" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A0%D0%B0%D1%81%D0%B5%D1%98%D1%83%D0%B2%D0%B0%D1%9A%D0%B5" title="Расејување – Macedonian" lang="mk" hreflang="mk" data-title="Расејување" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-ml mw-list-item"><a href="https://ml.wikipedia.org/wiki/%E0%B4%85%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%BC%E0%B4%B5%E0%B5%8D%E0%B4%AF%E0%B4%BE%E0%B4%AA%E0%B4%A8%E0%B4%82" title="അന്തർവ്യാപനം – Malayalam" lang="ml" hreflang="ml" data-title="അന്തർവ്യാപനം" data-language-autonym="മലയാളം" data-language-local-name="Malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-xmf mw-list-item"><a href="https://xmf.wikipedia.org/wiki/%E1%83%93%E1%83%98%E1%83%A4%E1%83%A3%E1%83%96%E1%83%98%E1%83%90" title="დიფუზია – Mingrelian" lang="xmf" hreflang="xmf" data-title="დიფუზია" data-language-autonym="მარგალური" data-language-local-name="Mingrelian" class="interlanguage-link-target"><span>მარგალური</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Peresapan" title="Peresapan – Malay" lang="ms" hreflang="ms" data-title="Peresapan" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Diffusie" title="Diffusie – Dutch" lang="nl" hreflang="nl" data-title="Diffusie" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%8B%A1%E6%95%A3" title="拡散 – Japanese" lang="ja" hreflang="ja" data-title="拡散" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Diffusjon" title="Diffusjon – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Diffusjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Diffusjon" title="Diffusjon – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Diffusjon" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-om mw-list-item"><a href="https://om.wikipedia.org/wiki/Qirimbisa_(diffusion)" title="Qirimbisa (diffusion) – Oromo" lang="om" hreflang="om" data-title="Qirimbisa (diffusion)" data-language-autonym="Oromoo" data-language-local-name="Oromo" class="interlanguage-link-target"><span>Oromoo</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Diffuziya" title="Diffuziya – Uzbek" lang="uz" hreflang="uz" data-title="Diffuziya" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%AA%E0%A9%8D%E0%A8%B0%E0%A8%B8%E0%A8%B0%E0%A8%A3_(%E0%A8%B5%E0%A8%BF%E0%A8%97%E0%A8%BF%E0%A8%86%E0%A8%A8)" title="ਪ੍ਰਸਰਣ (ਵਿਗਿਆਨ) – Punjabi" lang="pa" hreflang="pa" data-title="ਪ੍ਰਸਰਣ (ਵਿਗਿਆਨ)" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target"><span>ਪੰਜਾਬੀ</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dyfuzja" title="Dyfuzja – Polish" lang="pl" hreflang="pl" data-title="Dyfuzja" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Difus%C3%A3o_molecular" title="Difusão molecular – Portuguese" lang="pt" hreflang="pt" data-title="Difusão molecular" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Difuziune" title="Difuziune – Romanian" lang="ro" hreflang="ro" data-title="Difuziune" 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%94%D0%B8%D1%84%D1%84%D1%83%D0%B7%D0%B8%D1%8F" title="Диффузия – Russian" lang="ru" hreflang="ru" data-title="Диффузия" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Difuzion" title="Difuzion – Albanian" lang="sq" hreflang="sq" data-title="Difuzion" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target"><span>Shqip</span></a></li><li class="interlanguage-link interwiki-si mw-list-item"><a href="https://si.wikipedia.org/wiki/%E0%B7%80%E0%B7%92%E0%B7%83%E0%B6%BB%E0%B6%AB%E0%B6%BA" title="විසරණය – Sinhala" lang="si" hreflang="si" data-title="විසරණය" data-language-autonym="සිංහල" data-language-local-name="Sinhala" class="interlanguage-link-target"><span>සිංහල</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Diffusion" title="Diffusion – Simple English" lang="en-simple" hreflang="en-simple" data-title="Diffusion" 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-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Dif%C3%BAzia_(fyzika)" title="Difúzia (fyzika) – Slovak" lang="sk" hreflang="sk" data-title="Difúzia (fyzika)" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Difuzija" title="Difuzija – Slovenian" lang="sl" hreflang="sl" data-title="Difuzija" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-so mw-list-item"><a href="https://so.wikipedia.org/wiki/Diffusion" title="Diffusion – Somali" lang="so" hreflang="so" data-title="Diffusion" data-language-autonym="Soomaaliga" data-language-local-name="Somali" class="interlanguage-link-target"><span>Soomaaliga</span></a></li><li class="interlanguage-link interwiki-ckb mw-list-item"><a href="https://ckb.wikipedia.org/wiki/%D8%A8%DA%B5%D8%A7%D9%88%D8%A8%D9%88%D9%88%D9%86%DB%95%D9%88%DB%95" title="بڵاوبوونەوە – Central Kurdish" lang="ckb" hreflang="ckb" data-title="بڵاوبوونەوە" data-language-autonym="کوردی" data-language-local-name="Central Kurdish" class="interlanguage-link-target"><span>کوردی</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%83%D0%B7%D0%B8%D1%98%D0%B0" title="Дифузија – Serbian" lang="sr" hreflang="sr" data-title="Дифузија" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Difuzija" title="Difuzija – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Difuzija" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Diffuusio" title="Diffuusio – Finnish" lang="fi" hreflang="fi" data-title="Diffuusio" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Diffusion" title="Diffusion – Swedish" lang="sv" hreflang="sv" data-title="Diffusion" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%AA%E0%AE%B0%E0%AE%B5%E0%AE%B2%E0%AF%8D" title="பரவல் – Tamil" lang="ta" hreflang="ta" data-title="பரவல்" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-te mw-list-item"><a href="https://te.wikipedia.org/wiki/%E0%B0%B5%E0%B1%8D%E0%B0%AF%E0%B0%BE%E0%B0%AA%E0%B0%A8%E0%B0%AE%E0%B1%81" title="వ్యాపనము – Telugu" lang="te" hreflang="te" data-title="వ్యాపనము" data-language-autonym="తెలుగు" data-language-local-name="Telugu" class="interlanguage-link-target"><span>తెలుగు</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B8%81%E0%B8%B2%E0%B8%A3%E0%B9%81%E0%B8%9E%E0%B8%A3%E0%B9%88" title="การแพร่ – Thai" lang="th" hreflang="th" data-title="การแพร่" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dif%C3%BCzyon" title="Difüzyon – Turkish" lang="tr" hreflang="tr" data-title="Difüzyon" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%94%D0%B8%D1%84%D1%83%D0%B7%D1%96%D1%8F" title="Дифузія – Ukrainian" lang="uk" hreflang="uk" data-title="Дифузія" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%A7%D9%86%D8%AA%D8%B4%D8%A7%D8%B1_(%D8%B7%D8%A8%DB%8C%D8%B9%DB%8C%D8%A7%D8%AA)" title="انتشار (طبیعیات) – Urdu" lang="ur" hreflang="ur" data-title="انتشار (طبیعیات)" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/Khu%E1%BA%BFch_t%C3%A1n" title="Khuếch tán – Vietnamese" lang="vi" hreflang="vi" data-title="Khuếch tán" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-war mw-list-item"><a href="https://war.wikipedia.org/wiki/Difusyon" title="Difusyon – Waray" lang="war" hreflang="war" data-title="Difusyon" data-language-autonym="Winaray" data-language-local-name="Waray" class="interlanguage-link-target"><span>Winaray</span></a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E6%89%A9%E6%95%A3%E4%BD%9C%E7%94%A8" title="扩散作用 – Wu" lang="wuu" hreflang="wuu" data-title="扩散作用" data-language-autonym="吴语" data-language-local-name="Wu" class="interlanguage-link-target"><span>吴语</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E6%93%B4%E6%95%A3" title="擴散 – Cantonese" lang="yue" hreflang="yue" data-title="擴散" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E6%89%A9%E6%95%A3%E4%BD%9C%E7%94%A8" title="扩散作用 – Chinese" lang="zh" hreflang="zh" data-title="扩散作用" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q163214#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div 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searchaux" style="display:none">Transport of dissolved species from the highest to the lowest concentration region</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">This article is about the generic concept of the time-dependent process. For other uses, see <a href="/wiki/Diffusion_(disambiguation)" class="mw-disambig" title="Diffusion (disambiguation)">Diffusion (disambiguation)</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffusion.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Diffusion.svg/220px-Diffusion.svg.png" decoding="async" width="220" height="134" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Diffusion.svg/330px-Diffusion.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/12/Diffusion.svg/440px-Diffusion.svg.png 2x" data-file-width="410" data-file-height="250" /></a><figcaption> Some particles are <a href="/wiki/Dissolution_(chemistry)" class="mw-redirect" title="Dissolution (chemistry)">dissolved</a> in a glass of water. At first, the particles are all near one top corner of the glass. If the particles randomly move around ("diffuse") in the water, they eventually become distributed randomly and uniformly from an area of high concentration to an area of low, and organized (diffusion continues, but with no net <a href="/wiki/Flux" title="Flux">flux</a>).</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_0" poster="//upload.wikimedia.org/wikipedia/commons/thumb/d/d5/Diffusion_v2_20101120.ogv/220px--Diffusion_v2_20101120.ogv.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="123" data-durationhint="70" data-mwtitle="Diffusion_v2_20101120.ogv" data-mwprovider="wikimediacommons" resource="/wiki/File:Diffusion_v2_20101120.ogv"><source src="//upload.wikimedia.org/wikipedia/commons/d/d5/Diffusion_v2_20101120.ogv" type="video/ogg; codecs="theora"" data-width="720" data-height="404" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d5/Diffusion_v2_20101120.ogv/Diffusion_v2_20101120.ogv.144p.mjpeg.mov" type="video/quicktime" data-transcodekey="144p.mjpeg.mov" data-width="256" data-height="144" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d5/Diffusion_v2_20101120.ogv/Diffusion_v2_20101120.ogv.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="240" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d5/Diffusion_v2_20101120.ogv/Diffusion_v2_20101120.ogv.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="360" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/d/d5/Diffusion_v2_20101120.ogv/Diffusion_v2_20101120.ogv.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="360" /></video></span><figcaption>Time lapse video of diffusion a dye dissolved in water into a gel.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:DiffusionMicroMacro.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4d/DiffusionMicroMacro.gif/250px-DiffusionMicroMacro.gif" decoding="async" width="250" height="208" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4d/DiffusionMicroMacro.gif 1.5x" data-file-width="360" data-file-height="300" /></a><figcaption>Diffusion from a microscopic and b macroscopic point of view. Initially, there are <a href="/wiki/Solute" class="mw-redirect" title="Solute">solute</a> molecules on the left side of a barrier (purple line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. <u>Top:</u> A single molecule moves around randomly. <u>Middle:</u> With more molecules, there is a statistical trend that the solute fills the container more and more uniformly. <u>Bottom:</u> With an enormous number of solute molecules, all randomness is gone: The solute appears to move smoothly and deterministically from high-concentration areas to low-concentration areas. There is no microscopic <a href="/wiki/Force" title="Force">force</a> pushing molecules rightward, but there <i>appears</i> to be one in the bottom panel. This apparent force is called an <i><a href="/wiki/Entropic_force" title="Entropic force">entropic force</a></i>.</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Blausen_0315_Diffusion.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Blausen_0315_Diffusion.png/220px-Blausen_0315_Diffusion.png" decoding="async" width="220" height="103" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Blausen_0315_Diffusion.png/330px-Blausen_0315_Diffusion.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Blausen_0315_Diffusion.png/440px-Blausen_0315_Diffusion.png 2x" data-file-width="3200" data-file-height="1500" /></a><figcaption>Three-dimensional rendering of diffusion of purple dye in water.</figcaption></figure> <p><b>Diffusion</b> is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher <a href="/wiki/Concentration" title="Concentration">concentration</a> to a region of lower concentration. Diffusion is driven by a gradient in <a href="/wiki/Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a> or <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a>. It is possible to diffuse "uphill" from a region of lower concentration to a region of higher concentration, as in <a href="/wiki/Spinodal_decomposition" title="Spinodal decomposition">spinodal decomposition</a>. Diffusion is a stochastic process due to the inherent randomness of the diffusing entity and can be used to model many real-life stochastic scenarios. Therefore, diffusion and the corresponding mathematical models are used in several fields beyond physics, such as <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Probability_theory" title="Probability theory">probability theory</a>, <a href="/wiki/Information_theory" title="Information theory">information theory</a>, <a href="/wiki/Neural_networks" class="mw-redirect" title="Neural networks">neural networks</a>, <a href="/wiki/Finance" title="Finance">finance</a>, and <a href="/wiki/Marketing" title="Marketing">marketing</a>. </p><p>The concept of diffusion is widely used in many fields, including <a href="/wiki/Physics" title="Physics">physics</a> (<a href="/wiki/Molecular_diffusion" title="Molecular diffusion">particle diffusion</a>), <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>, <a href="/wiki/Biology" title="Biology">biology</a>, <a href="/wiki/Sociology" title="Sociology">sociology</a>, <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Statistics" title="Statistics">statistics</a>, <a href="/wiki/Data_science" title="Data science">data science</a>, and <a href="/wiki/Finance" title="Finance">finance</a> (diffusion of people, ideas, data and price values). The central idea of diffusion, however, is common to all of these: a substance or collection undergoing diffusion spreads out from a point or location at which there is a higher concentration of that substance or collection. </p><p>A <a href="/wiki/Gradient" title="Gradient">gradient</a> is the change in the value of a quantity; for example, concentration, <a href="/wiki/Pressure" title="Pressure">pressure</a>, or <a href="/wiki/Temperature" title="Temperature">temperature</a> with the change in another variable, usually <a href="/wiki/Distance" title="Distance">distance</a>. A change in concentration over a distance is called a <a href="/wiki/Molecular_diffusion" title="Molecular diffusion">concentration gradient</a>, a change in pressure over a distance is called a <a href="/wiki/Pressure_gradient" title="Pressure gradient">pressure gradient</a>, and a change in temperature over a distance is called a <a href="/wiki/Temperature_gradient" title="Temperature gradient">temperature gradient</a>. </p><p>The word <i>diffusion</i> derives from the <a href="/wiki/Latin" title="Latin">Latin</a> word, <i>diffundere</i>, which means "to spread out". </p><p>A distinguishing feature of diffusion is that it depends on particle <a href="/wiki/Random_walk" title="Random walk">random walk</a>, and results in mixing or mass transport without requiring directed bulk motion. Bulk motion, or bulk flow, is the characteristic of <a href="/wiki/Advection" title="Advection">advection</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The term <a href="/wiki/Convection" title="Convection">convection</a> is used to describe the combination of both <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport phenomena</a>. </p><p>If a diffusion process can be described by <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws</a>, it is called a normal diffusion (or Fickian diffusion); Otherwise, it is called an <a href="/wiki/Anomalous_diffusion" title="Anomalous diffusion">anomalous diffusion</a> (or non-Fickian diffusion). </p><p>When talking about the extent of diffusion, two length scales are used in two different scenarios: </p> <ol><li><a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> of an <a href="/wiki/Impulse_response" title="Impulse response">impulsive</a> point source (for example, one single spray of perfume)—the square root of the <a href="/wiki/Mean_squared_displacement" title="Mean squared displacement">mean squared displacement</a> from this point. In Fickian diffusion, this is <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 {\sqrt {2nDt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> <mi>n</mi> <mi>D</mi> <mi>t</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2nDt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea8f794e1c49c674621d2d215146303cb7d5c113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.257ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2nDt}}}"></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 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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> is the <a href="/wiki/Dimension" title="Dimension">dimension</a> of this Brownian motion;</li> <li><a href="/wiki/Fick%27s_laws_of_diffusion#Example_solutions_and_generalization" title="Fick's laws of diffusion">Constant concentration source</a> in one dimension—the diffusion length. In Fickian diffusion, this is <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 2{\sqrt {Dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>D</mi> <mi>t</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2{\sqrt {Dt}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76beaf1fb4136b1aa5841c63ad61272a80d6690c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.862ex; height:3.009ex;" alt="{\displaystyle 2{\sqrt {Dt}}}"></span>.</li></ol> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Diffusion_vs._bulk_flow">Diffusion vs. bulk flow</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=1" title="Edit section: Diffusion vs. bulk flow"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>"Bulk flow" is the movement/flow of an entire body due to a pressure gradient (for example, water coming out of a tap). "Diffusion" is the gradual movement/dispersion of concentration within a body with no net movement of matter. An example of a process where both <a href="/wiki/Mass_flow_(life_sciences)" title="Mass flow (life sciences)">bulk motion</a> and diffusion occur is human breathing.<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>First, there is a "bulk flow" process. The <a href="/wiki/Lungs" class="mw-redirect" title="Lungs">lungs</a> are located in the <a href="/wiki/Thoracic_cavity" title="Thoracic cavity">thoracic cavity</a>, which expands as the first step in external respiration. This expansion leads to an increase in volume of the <a href="/wiki/Pulmonary_alveolus" title="Pulmonary alveolus">alveoli</a> in the lungs, which causes a decrease in pressure in the alveoli. This creates a pressure gradient between the <a href="/wiki/Air" class="mw-redirect" title="Air">air</a> outside the body at relatively high pressure and the alveoli at relatively low pressure. The air moves down the pressure gradient through the airways of the lungs and into the alveoli until the pressure of the air and that in the alveoli are equal, that is, the movement of air by bulk flow stops once there is no longer a pressure gradient. </p><p>Second, there is a "diffusion" process. The air arriving in the alveoli has a higher concentration of oxygen than the "stale" air in the alveoli. The increase in oxygen concentration creates a concentration gradient for oxygen between the air in the alveoli and the blood in the <a href="/wiki/Capillaries" class="mw-redirect" title="Capillaries">capillaries</a> that surround the alveoli. Oxygen then moves by diffusion, down the concentration gradient, into the blood. The other consequence of the air arriving in alveoli is that the concentration of <a href="/wiki/Carbon_dioxide" title="Carbon dioxide">carbon dioxide</a> in the alveoli decreases. This creates a concentration gradient for carbon dioxide to diffuse from the blood into the alveoli, as fresh air has a very low concentration of carbon dioxide compared to the <a href="/wiki/Blood" title="Blood">blood</a> in the body. </p><p>Third, there is another "bulk flow" process. The pumping action of the <a href="/wiki/Heart" title="Heart">heart</a> then transports the blood around the body. As the left ventricle of the heart contracts, the volume decreases, which increases the pressure in the ventricle. This creates a pressure gradient between the heart and the capillaries, and blood moves through <a href="/wiki/Blood_vessel" title="Blood vessel">blood vessels</a> by bulk flow down the pressure gradient. </p> <div class="mw-heading mw-heading2"><h2 id="Diffusion_in_the_context_of_different_disciplines">Diffusion in the context of different disciplines</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=2" title="Edit section: Diffusion in the context of different disciplines"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg/200px-Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg" decoding="async" width="200" height="299" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg/300px-Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1e/Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg/400px-Centrotherm_diffusion_furnaces_at_LAAS_0481.jpg 2x" data-file-width="2592" data-file-height="3872" /></a><figcaption>Diffusion furnaces used for <a href="/wiki/Thermal_oxidation" title="Thermal oxidation">thermal oxidation</a></figcaption></figure> <p>There are two ways to introduce the notion of <i>diffusion</i>: either a <a href="https://en.wiktionary.org/wiki/phenomenon" class="extiw" title="wikt:phenomenon">phenomenological approach</a> starting with <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a> and their mathematical consequences, or a physical and atomistic one, by considering the <i><a href="/wiki/Random_walk" title="Random walk">random walk</a> of the diffusing particles</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the phenomenological approach, <i>diffusion is the movement of a substance from a region of high concentration to a region of low concentration without bulk motion</i>. According to Fick's laws, the diffusion <a href="/wiki/Flux#Flux_as_flow_rate_per_unit_area" title="Flux">flux</a> is proportional to the negative <a href="/wiki/Gradient" title="Gradient">gradient</a> of concentrations. It goes from regions of higher concentration to regions of lower concentration. Sometime later, various generalizations of Fick's laws were developed in the frame of <a href="/wiki/Thermodynamics" title="Thermodynamics">thermodynamics</a> and <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">non-equilibrium thermodynamics</a>.<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><p>From the <i>atomistic point of view</i>, diffusion is considered as a result of the random walk of the diffusing particles. In <a href="/wiki/Molecular_diffusion" title="Molecular diffusion">molecular diffusion</a>, the moving molecules in a gas, liquid, or solid are self-propelled by kinetic energy. Random walk of small particles in suspension in a fluid was discovered in 1827 by <a href="/wiki/Robert_Brown_(botanist,_born_1773)" title="Robert Brown (botanist, born 1773)">Robert Brown</a>, who found that minute particle suspended in a liquid medium and just large enough to be visible under an optical microscope exhibit a rapid and continually irregular motion of particles known as Brownian movement. The theory of the <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> and the atomistic backgrounds of diffusion were developed by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The concept of diffusion is typically applied to any subject matter involving random walks in <a href="/wiki/Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">ensembles</a> of individuals. </p><p>In <a href="/wiki/Chemistry" title="Chemistry">chemistry</a> and <a href="/wiki/Materials_science" title="Materials science">materials science</a>, diffusion also refers to the movement of fluid molecules in porous solids.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Different types of diffusion are distinguished in porous solids. <a href="/wiki/Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a> occurs when the collision with another molecule is more likely than the collision with the pore walls. Under such conditions, the diffusivity is similar to that in a non-confined space and is proportional to the mean free path. <a href="/wiki/Knudsen_diffusion" title="Knudsen diffusion">Knudsen diffusion</a> occurs when the pore diameter is comparable to or smaller than the mean free path of the molecule diffusing through the pore. Under this condition, the collision with the pore walls becomes gradually more likely and the diffusivity is lower. Finally there is configurational diffusion, which happens if the molecules have comparable size to that of the pore. Under this condition, the diffusivity is much lower compared to molecular diffusion and small differences in the kinetic diameter of the molecule cause large differences in <a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusivity</a>. </p><p><a href="/wiki/Biologist" title="Biologist">Biologists</a> often use the terms "net movement" or "net diffusion" to describe the movement of ions or molecules by diffusion. For example, oxygen can diffuse through cell membranes so long as there is a higher concentration of oxygen outside the cell. However, because the movement of molecules is random, occasionally oxygen molecules move out of the cell (against the concentration gradient). Because there are more oxygen molecules outside the cell, the <a href="/wiki/Probability" title="Probability">probability</a> that oxygen molecules will enter the cell is higher than the probability that oxygen molecules will leave the cell. Therefore, the "net" movement of oxygen molecules (the difference between the number of molecules either entering or leaving the cell) is into the cell. In other words, there is a <i>net movement</i> of oxygen molecules down the concentration gradient. </p> <div class="mw-heading mw-heading2"><h2 id="History_of_diffusion_in_physics">History of diffusion in physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=3" title="Edit section: History of diffusion in physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In the scope of time, diffusion in solids was used long before the theory of diffusion was created. For example, <a href="/wiki/Pliny_the_Elder" title="Pliny the Elder">Pliny the Elder</a> had previously described the <a href="/wiki/Cementation_process" title="Cementation process">cementation process</a>, which produces steel from the element <a href="/wiki/Iron" title="Iron">iron</a> (Fe) through carbon diffusion. Another example is well known for many centuries, the diffusion of colors of <a href="/wiki/Stained_glass" title="Stained glass">stained glass</a> or <a href="/wiki/Earthenware" title="Earthenware">earthenware</a> and <a href="/wiki/Chinese_ceramics" title="Chinese ceramics">Chinese ceramics</a>. </p><p>In modern science, the first systematic experimental study of diffusion was performed by <a href="/wiki/Thomas_Graham_(chemist)" title="Thomas Graham (chemist)">Thomas Graham</a>. He studied diffusion in gases, and the main phenomenon was described by him in 1831–1833:<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <blockquote><p>"...gases of different nature, when brought into contact, do not arrange themselves according to their density, the heaviest undermost, and the lighter uppermost, but they spontaneously diffuse, mutually and equally, through each other, and so remain in the intimate state of mixture for any length of time." </p></blockquote> <p>The measurements of Graham contributed to <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> deriving, in 1867, the coefficient of diffusion for CO<sub>2</sub> in the air. The error rate is less than 5%. </p><p>In 1855, <a href="/wiki/Adolf_Fick" class="mw-redirect" title="Adolf Fick">Adolf Fick</a>, the 26-year-old anatomy demonstrator from Zürich, proposed <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">his law of diffusion</a>. He used Graham's research, stating his goal as "the development of a fundamental law, for the operation of diffusion in a single element of space". He asserted a deep analogy between diffusion and conduction of heat or electricity, creating a formalism similar to <a href="/wiki/Thermal_conduction" title="Thermal conduction">Fourier's law for heat conduction</a> (1822) and <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a> for electric current (1827). </p><p><a href="/wiki/Robert_Boyle" title="Robert Boyle">Robert Boyle</a> demonstrated diffusion in solids in the 17th century<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> by penetration of zinc into a copper coin. Nevertheless, diffusion in solids was not systematically studied until the second part of the 19th century. <a href="/wiki/William_Chandler_Roberts-Austen" title="William Chandler Roberts-Austen">William Chandler Roberts-Austen</a>, the well-known British metallurgist and former assistant of Thomas Graham studied systematically solid state diffusion on the example of gold in lead in 1896. :<sup id="cite_ref-Mehrer2009_9-0" class="reference"><a href="#cite_note-Mehrer2009-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p> <blockquote> <p>"... My long connection with Graham's researches made it almost a duty to attempt to extend his work on liquid diffusion to metals." </p> </blockquote> <p>In 1858, <a href="/wiki/Rudolf_Clausius" title="Rudolf Clausius">Rudolf Clausius</a> introduced the concept of the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a>. In the same year, <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">James Clerk Maxwell</a> developed the first atomistic theory of transport processes in gases. The modern atomistic theory of diffusion and <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> was developed by <a href="/wiki/Albert_Einstein" title="Albert Einstein">Albert Einstein</a>, <a href="/wiki/Marian_Smoluchowski" title="Marian Smoluchowski">Marian Smoluchowski</a> and <a href="/wiki/Jean-Baptiste_Perrin" class="mw-redirect" title="Jean-Baptiste Perrin">Jean-Baptiste Perrin</a>. <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Ludwig Boltzmann</a>, in the development of the atomistic backgrounds of the macroscopic <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport processes</a>, introduced the <a href="/wiki/Boltzmann_equation" title="Boltzmann equation">Boltzmann equation</a>, which has served mathematics and physics with a source of transport process ideas and concerns for more than 140 years.<sup id="cite_ref-ChapmanCowling_10-0" class="reference"><a href="#cite_note-ChapmanCowling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p><p>In 1920–1921, <a href="/wiki/George_de_Hevesy" title="George de Hevesy">George de Hevesy</a> measured <a href="/wiki/Self-diffusion" title="Self-diffusion">self-diffusion</a> using <a href="/wiki/Radioisotope" class="mw-redirect" title="Radioisotope">radioisotopes</a>. He studied self-diffusion of radioactive isotopes of lead in the liquid and solid lead. </p><p><a href="/wiki/Yakov_Frenkel" title="Yakov Frenkel">Yakov Frenkel</a> (sometimes, Jakov/Jacob Frenkel) proposed, and elaborated in 1926, the idea of diffusion in crystals through local defects (vacancies and <a href="/wiki/Interstitial_defect" title="Interstitial defect">interstitial</a> atoms). He concluded, the diffusion process in condensed matter is an ensemble of elementary jumps and quasichemical interactions of particles and defects. He introduced several mechanisms of diffusion and found rate constants from experimental data. </p><p>Sometime later, <a href="/wiki/Carl_Wagner" title="Carl Wagner">Carl Wagner</a> and <a href="/wiki/Walter_H._Schottky" title="Walter H. Schottky">Walter H. Schottky</a> developed Frenkel's ideas about mechanisms of diffusion further. Presently, it is universally recognized that atomic defects are necessary to mediate diffusion in crystals.<sup id="cite_ref-Mehrer2009_9-1" class="reference"><a href="#cite_note-Mehrer2009-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/Henry_Eyring_(chemist)" title="Henry Eyring (chemist)">Henry Eyring</a>, with co-authors, applied his theory of <a href="/wiki/Transition_state_theory" title="Transition state theory">absolute reaction rates</a> to Frenkel's quasichemical model of diffusion.<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> The analogy between <a href="/wiki/Chemical_kinetics" title="Chemical kinetics">reaction kinetics</a> and diffusion leads to various nonlinear versions of Fick's law.<sup id="cite_ref-GorbanMMNP2011_12-0" class="reference"><a href="#cite_note-GorbanMMNP2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Basic_models_of_diffusion">Basic models of diffusion</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=4" title="Edit section: Basic models of diffusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Definition_of_diffusion_flux">Definition of diffusion flux</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=5" title="Edit section: Definition of diffusion flux"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Each model of diffusion expresses the <b>diffusion flux</b> with the use of concentrations, densities and their derivatives. Flux is a vector <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 \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }"></span> representing the quantity and direction of transfer. Given a small <a href="/wiki/Area" title="Area">area</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 \Delta S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a2efec755f08f95da4e8d1f7f2682861fb59be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.176ex;" alt="{\displaystyle \Delta S}"></span> with normal <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 {\boldsymbol {\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d030f5358ad62e959e7907ec1508746a563160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\nu }}}"></span>, the transfer of a <a href="/wiki/Physical_quantity" title="Physical quantity">physical quantity</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}"> <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> through the area <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 S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15a2efec755f08f95da4e8d1f7f2682861fb59be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.435ex; height:2.176ex;" alt="{\displaystyle \Delta S}"></span> per time <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 t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}"></span> is </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 \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>N</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>S</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/128ef623007330633b78107463ea2dfcd7136425" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.129ex; height:2.843ex;" alt="{\displaystyle \Delta N=(\mathbf {J} ,{\boldsymbol {\nu }})\,\Delta S\,\Delta t+o(\Delta S\,\Delta t)\,,}"></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 (\mathbf {J} ,{\boldsymbol {\nu }})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cf65696f51b30c5c6459fecc4b514f122acb7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.637ex; height:2.843ex;" alt="{\displaystyle (\mathbf {J} ,{\boldsymbol {\nu }})}"></span> is the <a href="/wiki/Inner_product" class="mw-redirect" title="Inner product">inner product</a> 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 o(\cdots )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>o</mi> <mo stretchy="false">(</mo> <mo>⋯</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle o(\cdots )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e7f4091204b3105d76efe55de1b966e968e083d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.66ex; height:2.843ex;" alt="{\displaystyle o(\cdots )}"></span> is the <a href="/wiki/Little-o_notation" class="mw-redirect" title="Little-o notation">little-o notation</a>. If we use the notation of <a href="/wiki/Vector_area" title="Vector area">vector area</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 \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b0a29b1a4cd2027f316f1b8eda89e3b1bd451b9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.755ex; height:2.176ex;" alt="{\displaystyle \Delta \mathbf {S} ={\boldsymbol {\nu }}\,\Delta S}"></span> then </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 \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>N</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>,</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">S</mi> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67eb487a1cf501db6387cbf586cbf29df6345c2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.301ex; height:2.843ex;" alt="{\displaystyle \Delta N=(\mathbf {J} ,\Delta \mathbf {S} )\,\Delta t+o(\Delta \mathbf {S} \,\Delta t)\,.}"></span></dd></dl> <p>The <a href="/wiki/Dimensional_analysis" title="Dimensional analysis">dimension</a> of the diffusion flux is [flux] = [quantity]/([time]·[area]). The diffusing physical quantity <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> may be the number of particles, mass, energy, electric charge, or any other scalar <a href="/wiki/Extensive_quantity" class="mw-redirect" title="Extensive quantity">extensive quantity</a>. For its density, <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/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span>, the diffusion equation has the form </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 {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>+</mo> <mi>W</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8121584b68d0e12291d92fb0ad2fc83cdf9aad4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:19.761ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=-\nabla \cdot \mathbf {J} +W\,,}"></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 W}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54a9c4c547f4d6111f81946cad242b18298d70b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}"></span> is intensity of any local source of this quantity (for example, the rate of a chemical reaction). For the diffusion equation, the <b>no-flux boundary conditions</b> can be formulated 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 (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9825b82ee811b0ba1e906fede9b0aab4f797d9ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.176ex; height:2.843ex;" alt="{\displaystyle (\mathbf {J} (x),{\boldsymbol {\nu }}(x))=0}"></span> on the boundary, 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 {\boldsymbol {\nu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">ν</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\nu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d030f5358ad62e959e7907ec1508746a563160" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.413ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\nu }}}"></span> is the normal to the boundary at point <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Normal_single_component_concentration_gradient">Normal single component concentration gradient</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=6" title="Edit section: Normal single component concentration gradient"><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/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's laws of diffusion</a></div> <p>Fick's first law: The diffusion flux, <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 \mathbf {J} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7686846b1a6b756cb514954000004ab5e7b2a5ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.381ex; height:2.176ex;" alt="{\displaystyle \mathbf {J} }"></span>, is proportional to the negative gradient of spatial concentration, <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(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83e53b78791849dd11cf02b8d0f9af829fb7bbc5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.407ex; height:2.843ex;" alt="{\displaystyle n(x,t)}"></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 \mathbf {J} =-D(x)\,\nabla n(x,t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =-D(x)\,\nabla n(x,t),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e077043433b8454791868f07fa25257610f54f69" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.728ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} =-D(x)\,\nabla n(x,t),}"></span></dd></dl> <p>where <i>D</i> is the <a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusion coefficient</a>. The corresponding <a href="/wiki/Diffusion_equation" title="Diffusion equation">diffusion equation</a> (Fick's second law) is </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 {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86e9ef052a5407d0903c28b414a2cfa377553176" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.912ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=\nabla \cdot (D(x)\,\nabla n(x,t))\,.}"></span></dd></dl> <p>In case the diffusion coefficient is independent of <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"></span>, Fick's second law can be simplified to </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 {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63835ecba3329656c052314b7c969562e345c3eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.542ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial n(x,t)}{\partial t}}=D\,\Delta n(x,t)\ ,}"></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 \Delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }"></span> is the <a href="/wiki/Laplace_operator" title="Laplace operator">Laplace operator</a>, </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 \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>n</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adec6221c21e124e481d225ffce1561975e7069f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.052ex; height:7.009ex;" alt="{\displaystyle \Delta n(x,t)=\sum _{i}{\frac {\partial ^{2}n(x,t)}{\partial x_{i}^{2}}}\ .}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Multicomponent_diffusion_and_thermodiffusion">Multicomponent diffusion and thermodiffusion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=7" title="Edit section: Multicomponent diffusion and thermodiffusion"><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/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a></div> <p>Fick's law describes diffusion of an admixture in a medium. The concentration of this admixture should be small and the gradient of this concentration should be also small. The driving force of diffusion in Fick's law is the antigradient of concentration, <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 -\nabla n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\nabla n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b86681c20965fdd279bcab10e6e123a9b83b8a36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.139ex; height:2.343ex;" alt="{\displaystyle -\nabla n}"></span>. </p><p>In 1931, <a href="/wiki/Lars_Onsager" title="Lars Onsager">Lars Onsager</a><sup id="cite_ref-Onsager1931_13-0" class="reference"><a href="#cite_note-Onsager1931-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> included the multicomponent transport processes in the general context of linear non-equilibrium thermodynamics. For multi-component transport, </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 \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8536ffece049a8b71606cea568ed594be02762f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.949ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{i}=\sum _{j}L_{ij}X_{j}\,,}"></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 \mathbf {J} _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e487b57ee92e16a526d0da3cc0cb73eccb5554ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.18ex; height:2.509ex;" alt="{\displaystyle \mathbf {J} _{i}}"></span> is the flux of the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th physical quantity (component), <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_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca3cb1ef7c9f25e85e1957e4eb58a72fa16a0066" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle X_{j}}"></span> is the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"></span>th <a href="/wiki/Conjugate_variables_(thermodynamics)" title="Conjugate variables (thermodynamics)">thermodynamic force</a> 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 L_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a310c7a19bc362591bb6809bb95b7e67baa59965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.06ex; height:2.843ex;" alt="{\displaystyle L_{ij}}"></span> is Onsager's matrix of <i>kinetic <a href="/wiki/Transport_coefficient" title="Transport coefficient">transport coefficients</a></i>. </p><p>The thermodynamic forces for the transport processes were introduced by Onsager as the space gradients of the derivatives of the <a href="/wiki/Entropy" title="Entropy">entropy</a> density <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/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> (he used the term "force" in quotation marks or "driving force"): </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 X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{i}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4da389d3c04f44f892763fbb650c0e5a16a42907" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:15.241ex; height:6.176ex;" alt="{\displaystyle X_{i}=\nabla {\frac {\partial s(n)}{\partial n_{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 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/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span> are the "thermodynamic coordinates". For the heat and mass transfer one can take <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_{0}=u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}=u}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9095fc3d8cc58402ff93e16cbd8d294f94338c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.877ex; height:2.009ex;" alt="{\displaystyle n_{0}=u}"></span> (the density of internal energy) 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/57f87f905ba5a4d8c691ccaecd65fc47bd007ba4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.194ex; height:2.009ex;" alt="{\displaystyle n_{i}}"></span> is the concentration of the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th component. The corresponding driving forces are the space vectors </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 X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mtext> </mtext> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <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> <mspace width="thickmathspace" /> <mo stretchy="false">(</mo> <mi>i</mi> <mo>></mo> <mn>0</mn> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3b0840ce38c3ab2c02127d6648bb2395452e946" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.804ex; height:5.176ex;" alt="{\displaystyle X_{0}=\nabla {\frac {1}{T}}\ ,\;\;\;X_{i}=-\nabla {\frac {\mu _{i}}{T}}\;(i>0),}"></span> because <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 \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>u</mi> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <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> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> </mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45a9ca71a6f5a299402ef594207736e64235ce20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.457ex; height:6.509ex;" alt="{\displaystyle \mathrm {d} s={\frac {1}{T}}\,\mathrm {d} u-\sum _{i\geq 1}{\frac {\mu _{i}}{T}}\,{\rm {d}}n_{i}}"></span></dd></dl> <p>where <i>T</i> is the absolute temperature 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> is the chemical potential of the <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"></span>th component. It should be stressed that the separate diffusion equations describe the mixing or mass transport without bulk motion. Therefore, the terms with variation of the total pressure are neglected. It is possible for diffusion of small admixtures and for small gradients. </p><p>For the linear Onsager equations, we must take the thermodynamic forces in the linear approximation near equilibrium: </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 X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> <mi mathvariant="normal">∇</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ceb3ac1de281751b802241370c33fc69d4bd764" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:28.71ex; height:7.343ex;" alt="{\displaystyle X_{i}=\sum _{k\geq 0}\left.{\frac {\partial ^{2}s(n)}{\partial n_{i}\,\partial n_{k}}}\right|_{n=n^{*}}\nabla n_{k}\ ,}"></span></dd></dl> <p>where the derivatives of <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/01d131dfd7673938b947072a13a9744fe997e632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:1.676ex;" alt="{\displaystyle s}"></span> are calculated at equilibrium <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"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </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/9615e0726d9d6cc58d117a844dca1ca84bb81f7e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.343ex;" alt="{\displaystyle n^{*}}"></span>. The matrix of the <i>kinetic coefficients</i> <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_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a310c7a19bc362591bb6809bb95b7e67baa59965" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.06ex; height:2.843ex;" alt="{\displaystyle L_{ij}}"></span> should be symmetric (<a href="/wiki/Onsager_reciprocal_relations" title="Onsager reciprocal relations">Onsager reciprocal relations</a>) and <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive definite</a> (<a href="/wiki/Second_law_of_thermodynamics" title="Second law of thermodynamics">for the entropy growth</a>). </p><p>The transport equations are </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 {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>div</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mi>div</mi> <mo>⁡</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <mrow> <mo>[</mo> <mrow> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≥</mo> <mn>0</mn> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>s</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> </mrow> <mo>]</mo> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mtext> </mtext> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f61d376b495038f57128d2c6ea83f733b7ae0b83" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:72.826ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial n_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=-\sum _{j\geq 0}L_{ij}\operatorname {div} X_{j}=\sum _{k\geq 0}\left[-\sum _{j\geq 0}L_{ij}\left.{\frac {\partial ^{2}s(n)}{\partial n_{j}\,\partial n_{k}}}\right|_{n=n^{*}}\right]\,\Delta n_{k}\ .}"></span></dd></dl> <p>Here, all the indexes <span class="nowrap"><i>i</i>, <i>j</i>, <i>k</i> = 0, 1, 2, ...</span> are related to the internal energy (0) and various components. The expression in the square brackets is 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 D_{ik}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{ik}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec9de6defea9f68c3dc1f888bb1023881b3e4123" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.58ex; height:2.509ex;" alt="{\displaystyle D_{ik}}"></span> of the diffusion (<i>i</i>,<i>k</i> > 0), thermodiffusion (<i>i</i> > 0, <i>k</i> = 0 or <i>k</i> > 0, <i>i</i> = 0) and <a href="/wiki/Thermal_conductivity" class="mw-redirect" title="Thermal conductivity">thermal conductivity</a> (<span class="nowrap"><i>i</i> = <i>k</i> = 0</span>) coefficients. </p><p>Under <a href="/wiki/Isothermal_process" title="Isothermal process">isothermal conditions</a> <i>T</i> = constant. The relevant thermodynamic potential is the free energy (or the <a href="/wiki/Free_entropy" title="Free entropy">free entropy</a>). The thermodynamic driving forces for the isothermal diffusion are antigradients of chemical potentials, <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/T)\,\nabla \mu _{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -(1/T)\,\nabla \mu _{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2ccce0811144693833fd033a88dad7078d20c6a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.213ex; height:3.009ex;" alt="{\displaystyle -(1/T)\,\nabla \mu _{j}}"></span>, and the matrix of diffusion coefficients is </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 D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>≥</mo> <mn>1</mn> </mrow> </munder> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cec19e6740db6f14803d8a1b700dda3e25c1a5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:31.642ex; height:7.176ex;" alt="{\displaystyle D_{ik}={\frac {1}{T}}\sum _{j\geq 1}L_{ij}\left.{\frac {\partial \mu _{j}(n,T)}{\partial n_{k}}}\right|_{n=n^{*}}}"></span></dd></dl> <p>(<i>i,k</i> > 0). </p><p>There is intrinsic arbitrariness in the definition of the thermodynamic forces and kinetic coefficients because they are not measurable separately and only their combinations <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="{\textstyle \sum _{j}L_{ij}X_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <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> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{j}L_{ij}X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc6370157ef78650d7f028264530ab8a28ae6802" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:9.645ex; height:3.343ex;" alt="{\textstyle \sum _{j}L_{ij}X_{j}}"></span> can be measured. For example, in the original work of Onsager<sup id="cite_ref-Onsager1931_13-1" class="reference"><a href="#cite_note-Onsager1931-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> the thermodynamic forces include additional multiplier <i>T</i>, whereas in the <a href="/wiki/Course_of_Theoretical_Physics" title="Course of Theoretical Physics">Course of Theoretical Physics</a><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> this multiplier is omitted but the sign of the thermodynamic forces is opposite. All these changes are supplemented by the corresponding changes in the coefficients and do not affect the measurable quantities. </p> <div class="mw-heading mw-heading4"><h4 id="Nondiagonal_diffusion_must_be_nonlinear">Nondiagonal diffusion must be nonlinear</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=8" title="Edit section: Nondiagonal diffusion must be nonlinear"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The formalism of linear irreversible thermodynamics (Onsager) generates the systems of linear diffusion equations in the form </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 {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81e382a1947aa1dc8735d23b59404e384e29367b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.089ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}\,\Delta c_{j}.}"></span></dd></dl> <p>If the matrix of diffusion coefficients is diagonal, then this system of equations is just a collection of decoupled Fick's equations for various components. Assume that diffusion is non-diagonal, for example, <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 D_{12}\neq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>≠</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{12}\neq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a36dd99272e3a8ccd32ec7c26d33ff92009f1e97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.061ex; height:2.676ex;" alt="{\displaystyle D_{12}\neq 0}"></span>, and consider the state with <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_{2}=\cdots =c_{n}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>⋯</mo> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{2}=\cdots =c_{n}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56bec79d38fa6f8540601b5e1d3aa9a295b907c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.467ex; height:2.509ex;" alt="{\displaystyle c_{2}=\cdots =c_{n}=0}"></span>. At this state, <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 c_{2}/\partial t=D_{12}\,\Delta c_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38d40e9fcdf6d5ab081f5f6bf200bf6944cc59fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.982ex; height:2.843ex;" alt="{\displaystyle \partial c_{2}/\partial t=D_{12}\,\Delta c_{1}}"></span>. If <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 D_{12}\,\Delta c_{1}(x)<0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{12}\,\Delta c_{1}(x)<0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee9ccf9ce4d7b3f01f4e5c74368f9049cd83ffbb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.584ex; height:2.843ex;" alt="{\displaystyle D_{12}\,\Delta c_{1}(x)<0}"></span> at some points, then <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_{2}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{2}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7750489f279606a91bf8b550c637b087efc6b822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.2ex; height:2.843ex;" alt="{\displaystyle c_{2}(x)}"></span> becomes negative at these points in a short time. Therefore, linear non-diagonal diffusion does not preserve positivity of concentrations. Non-diagonal equations of multicomponent diffusion must be non-linear.<sup id="cite_ref-GorbanMMNP2011_12-1" class="reference"><a href="#cite_note-GorbanMMNP2011-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Applied_forces">Applied forces</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=9" title="Edit section: Applied forces"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The <a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a> connects the diffusion coefficient and the mobility (the ratio of the particle's terminal <a href="/wiki/Drift_velocity" title="Drift velocity">drift velocity</a> to an applied <a href="/wiki/Force" title="Force">force</a>).<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> For charged particles: </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 D={\frac {\mu \,k_{\text{B}}T}{q}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>B</mtext> </mrow> </msub> <mi>T</mi> </mrow> <mi>q</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/666dd11f3911d5e87d853377d393cc02a595db3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:12.538ex; height:5.843ex;" alt="{\displaystyle D={\frac {\mu \,k_{\text{B}}T}{q}},}"></span></dd></dl> <p>where <i>D</i> is the <a href="/wiki/Fick%27s_law_of_diffusion" class="mw-redirect" title="Fick's law of diffusion">diffusion constant</a>, <i>μ</i> is the "mobility", <i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <i>T</i> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>, and <i>q</i> is the <a href="/wiki/Elementary_charge" title="Elementary charge">elementary charge</a>, that is, the charge of one electron. </p><p>Below, to combine in the same formula the chemical potential <i>μ</i> and the mobility, we use for mobility the notation <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 {\mathfrak {m}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathfrak {m}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc0e9162e96758157a34a6e44967288b481a7cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:1.676ex;" alt="{\displaystyle {\mathfrak {m}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion_across_a_membrane">Diffusion across a membrane</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=10" title="Edit section: Diffusion across a membrane"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The mobility-based approach was further applied by T. Teorell.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> In 1935, he studied the diffusion of ions through a membrane. He formulated the essence of his approach in the formula: </p> <dl><dd><b>the flux is equal to mobility × concentration × force per gram-ion</b>.</dd></dl> <p>This is the so-called <i>Teorell formula</i>.<sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (December 2023)">citation needed</span></a></i>]</sup> The term "gram-ion" ("gram-particle") is used for a quantity of a substance that contains the <a href="/wiki/Avogadro_constant" title="Avogadro constant">Avogadro number</a> of ions (particles). The common modern term is <a href="/wiki/Mole_(unit)" title="Mole (unit)">mole</a>. </p><p>The force under isothermal conditions consists of two parts: </p> <ol><li>Diffusion force caused by concentration gradient: <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 -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>R</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>eq</mtext> </mrow> </msup> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ee12d5ba735b50f1ef17e9d113c18d58015fe2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:33.127ex; height:5.176ex;" alt="{\displaystyle -RT{\frac {1}{n}}\,\nabla n=-RT\,\nabla (\ln(n/n^{\text{eq}}))}"></span>.</li> <li>Electrostatic force caused by electric potential gradient: <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 q\,\nabla \varphi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\,\nabla \varphi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/320618130d088ba650502730a8b41fada726e296" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.913ex; height:2.676ex;" alt="{\displaystyle q\,\nabla \varphi }"></span>.</li></ol> <p>Here <i>R</i> is the gas constant, <i>T</i> is the absolute temperature, <i>n</i> is the concentration, the equilibrium concentration is marked by a superscript "eq", <i>q</i> is the charge and <i>φ</i> is the electric potential. </p><p>The simple but crucial difference between the Teorell formula and the Onsager laws is the concentration factor in the Teorell expression for the flux. In the Einstein–Teorell approach, if for the finite force the concentration tends to zero then the flux also tends to zero, whereas the Onsager equations violate this simple and physically obvious rule. </p><p>The general formulation of the Teorell formula for non-perfect systems under isothermal conditions is<sup id="cite_ref-GorbanMMNP2011_12-2" class="reference"><a href="#cite_note-GorbanMMNP2011-12"><span class="cite-bracket">[</span>12<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 \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>μ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>external force per mole</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e2e9d45e2b2f4e941883907d1062a9902443872" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:56.225ex; height:6.176ex;" alt="{\displaystyle \mathbf {J} ={\mathfrak {m}}\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)(-\nabla \mu +({\text{external force per mole}})),}"></span></dd></dl> <p>where <i>μ</i> is the <a href="/wiki/Chemical_potential" title="Chemical potential">chemical potential</a>, <i>μ</i><sub>0</sub> is the standard value of the chemical potential. The expression <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=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>exp</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>μ</mi> <mo>−</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow> <mi>R</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebf9dd6bce92dc028a635d05d2443ca38096b561" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.836ex; height:6.176ex;" alt="{\displaystyle a=\exp \left({\frac {\mu -\mu _{0}}{RT}}\right)}"></span> is the so-called <a href="/wiki/Activity_(chemistry)" class="mw-redirect" title="Activity (chemistry)">activity</a>. It measures the "effective concentration" of a species in a non-ideal mixture. In this notation, the Teorell formula for the flux has a very simple form<sup id="cite_ref-GorbanMMNP2011_12-3" class="reference"><a href="#cite_note-GorbanMMNP2011-12"><span class="cite-bracket">[</span>12<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 \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>μ</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>external force per mole</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aac0ac115a8bf9d09a24b4f9dc68f639ce057a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.56ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} ={\mathfrak {m}}a(-\nabla \mu +({\text{external force per mole}})).}"></span></dd></dl> <p>The standard derivation of the activity includes a normalization factor and for small concentrations <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=n/n^{\ominus }+o(n/n^{\ominus })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo>+</mo> <mi>o</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be70d2c715bd1f9a904bcc9b3797094d5360100d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.031ex; height:3.009ex;" alt="{\displaystyle a=n/n^{\ominus }+o(n/n^{\ominus })}"></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 n^{\ominus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n^{\ominus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f8aee2b1a42d52398634a97fc56108b2cc88e24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.906ex; height:2.509ex;" alt="{\displaystyle n^{\ominus }}"></span> is the standard concentration. Therefore, this formula for the flux describes the flux of the normalized dimensionless quantity <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/n^{\ominus }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n/n^{\ominus }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc988f8e1d07ca158fd134d7b8aa9b85c18ff714" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.463ex; height:3.009ex;" alt="{\displaystyle n/n^{\ominus }}"></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 {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>⊖</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">m</mi> </mrow> </mrow> <mi>a</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mi>μ</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>external force per mole</mtext> </mrow> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f8ae11dd009457b8fd39d1a583ed5d4b3e30ab5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:53.706ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial (n/n^{\ominus })}{\partial t}}=\nabla \cdot [{\mathfrak {m}}a(\nabla \mu -({\text{external force per mole}}))].}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Ballistic_time_scale">Ballistic time scale</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=11" title="Edit section: Ballistic time scale"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Einstein model neglects the inertia of the diffusing partial. The alternative <a href="/wiki/Langevin_equation" title="Langevin equation">Langevin equation</a> starts with Newton's second law of motion:<sup id="cite_ref-Bian_SoftMatt_17-0" class="reference"><a href="#cite_note-Bian_SoftMatt-17"><span class="cite-bracket">[</span>17<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 m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>μ</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4d030b5fedbb12b8da476401ff31a6d10f77c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.235ex; height:6.176ex;" alt="{\displaystyle m{\frac {d^{2}x}{dt^{2}}}=-{\frac {1}{\mu }}{\frac {dx}{dt}}+F(t)}"></span></dd></dl> <p>where </p> <ul><li><i>x</i> is the position.</li> <li><i>μ</i> is the mobility of the particle in the fluid or gas, which can be calculated using the <a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a>.</li> <li><i>m</i> is the mass of the particle.</li> <li><i>F</i> is the random force applied to the particle.</li> <li><i>t</i> is time.</li></ul> <p>Solving this equation, one obtained the time-dependent diffusion constant in the long-time limit and when the particle is significantly denser than the surrounding fluid,<sup id="cite_ref-Bian_SoftMatt_17-1" class="reference"><a href="#cite_note-Bian_SoftMatt-17"><span class="cite-bracket">[</span>17<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 D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mo stretchy="false">(</mo> <mi>m</mi> <mi>μ</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e93db9ffc3b562d266044a6ffd6b2502c854291a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.239ex; height:3.343ex;" alt="{\displaystyle D(t)=\mu \,k_{\rm {B}}T(1-e^{-t/(m\mu )})}"></span></dd></dl> <p>where </p> <ul><li><i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>;</li> <li><i>T</i> is the <a href="/wiki/Absolute_temperature" class="mw-redirect" title="Absolute temperature">absolute temperature</a>.</li> <li><i>μ</i> is the mobility of the particle in the fluid or gas, which can be calculated using the <a href="/wiki/Einstein_relation_(kinetic_theory)" title="Einstein relation (kinetic theory)">Einstein relation (kinetic theory)</a>.</li> <li><i>m</i> is the mass of the particle.</li> <li><i>t</i> is time.</li></ul> <p>At long time scales, Einstein's result is recovered, but short time scales, the <i>ballistic regime</i> are also explained. Moreover, unlike the Einstein approach, a velocity can be defined, leading to the <a href="/wiki/Fluctuation-dissipation_theorem" class="mw-redirect" title="Fluctuation-dissipation theorem">Fluctuation-dissipation theorem</a>, connecting the competition between friction and random forces in defining the temperature.<sup id="cite_ref-Bian_SoftMatt_17-2" class="reference"><a href="#cite_note-Bian_SoftMatt-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: 3.2">: 3.2 </span></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Jumps_on_the_surface_and_in_solids">Jumps on the surface and in solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=12" title="Edit section: Jumps on the surface and in solids"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Chemical_surface_diffusion_slow.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/5/5f/Chemical_surface_diffusion_slow.gif" decoding="async" width="288" height="216" class="mw-file-element" data-file-width="288" data-file-height="216" /></a><figcaption>Diffusion in the monolayer: oscillations near temporary equilibrium positions and jumps to the nearest free places.</figcaption></figure> <p><a href="/wiki/Surface_diffusion" title="Surface diffusion">Diffusion of reagents on the surface</a> of a <a href="/wiki/Catalyst" class="mw-redirect" title="Catalyst">catalyst</a> may play an important role in heterogeneous catalysis. The model of diffusion in the ideal monolayer is based on the jumps of the reagents on the nearest free places. This model was used for CO on Pt oxidation under low gas pressure. </p><p>The system includes several reagents <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_{1},A_{2},\ldots ,A_{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1},A_{2},\ldots ,A_{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53bd9d8024bb7542f39ed6446c586b8ee2da8404" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.225ex; height:2.509ex;" alt="{\displaystyle A_{1},A_{2},\ldots ,A_{m}}"></span> on the surface. Their surface concentrations are <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_{1},c_{2},\ldots ,c_{m}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b092ce106163df27010a07e86eb61ab78af862e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.663ex; height:2.009ex;" alt="{\displaystyle c_{1},c_{2},\ldots ,c_{m}.}"></span> The surface is a lattice of the adsorption places. Each reagent molecule fills a place on the surface. Some of the places are free. The concentration of the free places is <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 z=c_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59b0bf39018e7d0e04947194fc5195aae843336e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.248ex; height:2.009ex;" alt="{\displaystyle z=c_{0}}"></span>. The sum of all <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01acb7953ba52c2aa44264b5d0f8fd223aa178a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.807ex; height:2.009ex;" alt="{\displaystyle c_{i}}"></span> (including free places) is constant, the density of adsorption places <i>b</i>. </p><p>The jump model gives for the diffusion flux of <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1aed3b5def921afbe6cc48aaf8f9b11c6f1c1e2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.543ex; height:2.509ex;" alt="{\displaystyle A_{i}}"></span> (<i>i</i> = 1, ..., <i>n</i>): </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 \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi mathvariant="normal">∇</mi> <mi>z</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e62d2876591f6c0c24854dc77bd002742a487757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.027ex; height:2.843ex;" alt="{\displaystyle \mathbf {J} _{i}=-D_{i}[z\,\nabla c_{i}-c_{i}\nabla z]\,.}"></span></dd></dl> <p>The corresponding diffusion equation is:<sup id="cite_ref-GorbanMMNP2011_12-4" class="reference"><a href="#cite_note-GorbanMMNP2011-12"><span class="cite-bracket">[</span>12<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 {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>div</mi> <mo>⁡</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>z</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59acf7a4d07ec81e21aec16a7dd999c091b60b79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:36.415ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=-\operatorname {div} \mathbf {J} _{i}=D_{i}[z\,\Delta c_{i}-c_{i}\,\Delta z]\,.}"></span></dd></dl> <p>Due to the conservation law, <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 z=b-\sum _{i=1}^{n}c_{i}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mi>b</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>n</mi> </mrow> </munderover> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f20c8bb1cb3749064c232c430293810b2f0fa0c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.607ex; height:6.843ex;" alt="{\displaystyle z=b-\sum _{i=1}^{n}c_{i}\,,}"></span> and we have the system of <i>m</i> diffusion equations. For one component we get Fick's law and linear equations because <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 (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>c</mi> <mo>−</mo> <mi>c</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>b</mi> <mo>−</mo> <mi>c</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>b</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2d67337c71b81e1cb225e5fe89e7a8a530b24af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.234ex; height:2.843ex;" alt="{\displaystyle (b-c)\,\nabla c-c\,\nabla (b-c)=b\,\nabla c}"></span>. For two and more components the equations are nonlinear. </p><p>If all particles can exchange their positions with their closest neighbours then a simple generalization gives </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 \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aba83bc12bd5bab3419c70e17305975783df881d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.844ex; height:5.843ex;" alt="{\displaystyle \mathbf {J} _{i}=-\sum _{j}D_{ij}[c_{j}\,\nabla c_{i}-c_{i}\,\nabla c_{j}]}"></span></dd> <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 {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo stretchy="false">[</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63b9c514f4d2f44400b6315598831cacc5edaee9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:30.429ex; height:6.843ex;" alt="{\displaystyle {\frac {\partial c_{i}}{\partial t}}=\sum _{j}D_{ij}[c_{j}\,\Delta c_{i}-c_{i}\,\Delta c_{j}]}"></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 D_{ij}=D_{ji}\geq 0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{ij}=D_{ji}\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6755d8cbb508049601a971b6711f5f411601c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.162ex; height:2.843ex;" alt="{\displaystyle D_{ij}=D_{ji}\geq 0}"></span> is a symmetric matrix of coefficients that characterize the intensities of jumps. The free places (vacancies) should be considered as special "particles" with concentration <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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1882ba8f1dc60f0c68a642abb5af093c73910921" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.061ex; height:2.009ex;" alt="{\displaystyle c_{0}}"></span>. </p><p>Various versions of these jump models are also suitable for simple diffusion mechanisms in solids. </p> <div class="mw-heading mw-heading3"><h3 id="Porous_media">Porous media</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=13" title="Edit section: Porous media"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For diffusion in porous media the basic equations are (if Φ is constant):<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup> </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 \mathbf {J} =-\phi D\,\nabla n^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">J</mi> </mrow> <mo>=</mo> <mo>−</mo> <mi>ϕ</mi> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3fa5a3ffd20c85fa14169bee4378e8ea92b3f48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.99ex; height:2.676ex;" alt="{\displaystyle \mathbf {J} =-\phi D\,\nabla n^{m}}"></span></dd> <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 {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b7dc637c003fe9d293aa9e6fac9748d978f516a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.998ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\,,}"></span></dd></dl> <p>where <i>D</i> is the diffusion coefficient, Φ is porosity, <i>n</i> is the concentration, <i>m</i> > 0 (usually <i>m</i> > 1, the case <i>m</i> = 1 corresponds to Fick's law). </p><p>Care must be taken to properly account for the porosity (Φ) of the porous medium in both the flux terms and the accumulation terms.<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> For example, as the porosity goes to zero, the molar flux in the porous medium goes to zero for a given concentration gradient. Upon applying the divergence of the flux, the porosity terms cancel out and the second equation above is formed. </p><p>For diffusion of gases in porous media this equation is the formalization of <a href="/wiki/Darcy%27s_law" title="Darcy's law">Darcy's law</a>: the <a href="/wiki/Volumetric_flux" title="Volumetric flux">volumetric flux</a> of a gas in the porous media is </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 q=-{\frac {k}{\mu }}\,\nabla p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>k</mi> <mi>μ</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/317c8e7284e54ff00f6d5abacd717fa12da05029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.706ex; height:5.843ex;" alt="{\displaystyle q=-{\frac {k}{\mu }}\,\nabla p}"></span></dd></dl> <p>where <i>k</i> is the <a href="/wiki/Permeation" title="Permeation">permeability</a> of the medium, <i>μ</i> is the <a href="/wiki/Viscosity" title="Viscosity">viscosity</a> and <i>p</i> is the pressure. </p><p>The advective molar flux is given as </p><p><i>J</i> = <i>nq</i> </p><p>and for <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\sim n^{\gamma }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>∼</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>γ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\sim n^{\gamma }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d5f663de35f7e79b9471d9d5a10cf8409a4956a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.877ex; height:2.676ex;" alt="{\displaystyle p\sim n^{\gamma }}"></span> Darcy's law gives the equation of diffusion in porous media with <i>m</i> = <i>γ</i> + 1. </p><p>In porous media, the average linear velocity (ν), is related to the volumetric flux as: </p><p><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 \upsilon =q/\phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>υ</mi> <mo>=</mo> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ϕ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \upsilon =q/\phi }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e3b633c1887f8d521153c3aab108b81b75a819f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.971ex; height:2.843ex;" alt="{\displaystyle \upsilon =q/\phi }"></span> </p><p>Combining the advective molar flux with the diffusive flux gives the advection dispersion equation </p><p><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 {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mtext> </mtext> <mo>−</mo> <mi>ν</mi> <mo>⋅</mo> <mi mathvariant="normal">∇</mi> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fc7493e64e704687a9390a12557545fed63defc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.949ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial n}{\partial t}}=D\,\Delta n^{m}\ -\nu \cdot \nabla n^{m},}"></span> </p><p>For underground water infiltration, the <a href="/wiki/Boussinesq_approximation_(buoyancy)" title="Boussinesq approximation (buoyancy)">Boussinesq approximation</a> gives the same equation with <i>m</i> = 2. </p><p>For plasma with the high level of radiation, the <a href="/wiki/Yakov_Borisovich_Zel%27dovich" class="mw-redirect" title="Yakov Borisovich Zel'dovich">Zeldovich</a>–Raizer equation gives <i>m</i> > 4 for the heat transfer. </p> <div class="mw-heading mw-heading2"><h2 id="Diffusion_in_physics">Diffusion in physics</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=14" title="Edit section: Diffusion in physics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Diffusion_coefficient_in_kinetic_theory_of_gases">Diffusion coefficient in kinetic theory of gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=15" title="Edit section: Diffusion coefficient in kinetic theory of gases"><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">See also: <a href="/wiki/Kinetic_theory_of_gases#Diffusion_coefficient_and_diffusion_flux" title="Kinetic theory of gases">Kinetic theory of gases § Diffusion coefficient and diffusion flux</a></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Translational_motion.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6d/Translational_motion.gif" decoding="async" width="300" height="263" class="mw-file-element" data-file-width="300" data-file-height="263" /></a><figcaption>Random collisions of particles in a gas.</figcaption></figure> <p>The diffusion coefficient <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 D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"></span> is the coefficient in the <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's first law</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 J=-D\,\partial n/\partial x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D\,\partial n/\partial x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/389e8c8237d7c26d83c9b4cd3aba65df2b0ca1e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.212ex; height:2.843ex;" alt="{\displaystyle J=-D\,\partial n/\partial x}"></span>, where <i>J</i> is the diffusion flux (<a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a>) per unit area per unit time, <i>n</i> (for ideal mixtures) is the concentration, <i>x</i> is the position [length]. </p><p>Consider two gases with molecules of the same diameter <i>d</i> and mass <i>m</i> (<a href="/wiki/Self-diffusion" title="Self-diffusion">self-diffusion</a>). In this case, the elementary mean free path theory of diffusion gives for the diffusion coefficient </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 D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>ℓ</mi> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <mrow> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mrow> <mi>P</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44b8ae29eec5ca6b6acd318062260c58b6cdfab1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:29.482ex; height:7.676ex;" alt="{\displaystyle D={\frac {1}{3}}\ell v_{T}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}m}}}{\frac {T^{3/2}}{Pd^{2}}}\,,}"></span></dd></dl> <p>where <i>k</i><sub>B</sub> is the <a href="/wiki/Boltzmann_constant" title="Boltzmann constant">Boltzmann constant</a>, <i>T</i> is the <a href="/wiki/Temperature" title="Temperature">temperature</a>, <i>P</i> is the <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 \ell }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f066e981e530bacc07efc6a10fa82deee985929e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.97ex; height:2.176ex;" alt="{\displaystyle \ell }"></span> is the <a href="/wiki/Mean_free_path" title="Mean free path">mean free path</a>, and <i>v<sub>T</sub></i> is the mean thermal speed: </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 \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ℓ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> <mi>π</mi> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>P</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mrow> <mn>8</mn> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> </mrow> <mrow> <mi>π</mi> <mi>m</mi> </mrow> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2517767774f635a8f988833518c3b57530e634f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:31.924ex; height:6.676ex;" alt="{\displaystyle \ell ={\frac {k_{\rm {B}}T}{{\sqrt {2}}\pi d^{2}P}}\,,\;\;\;v_{T}={\sqrt {\frac {8k_{\rm {B}}T}{\pi m}}}\,.}"></span></dd></dl> <p>We can see that the diffusion coefficient in the mean free path approximation grows with <i>T</i> as <i>T</i><sup>3/2</sup> and decreases with <i>P</i> as 1/<i>P</i>. If we use for <i>P</i> the <a href="/wiki/Ideal_gas_law" title="Ideal gas law">ideal gas law</a> <i>P</i> = <i>RnT</i> with the total concentration <i>n</i>, then we can see that for given concentration <i>n</i> the diffusion coefficient grows with <i>T</i> as <i>T</i><sup>1/2</sup> and for given temperature it decreases with the total concentration as 1/<i>n</i>. </p><p>For two different gases, A and B, with molecular masses <i>m</i><sub>A</sub>, <i>m</i><sub>B</sub> and molecular diameters <i>d</i><sub>A</sub>, <i>d</i><sub>B</sub>, the mean free path estimate of the diffusion coefficient of A in B and B in A is: </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 D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mn>3</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <msubsup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msubsup> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <msup> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">A</mi> </mrow> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24885a7b46490e7ba358100addd7f4ff2737f4a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:46.117ex; height:8.009ex;" alt="{\displaystyle D_{\rm {AB}}={\frac {2}{3}}{\sqrt {\frac {k_{\rm {B}}^{3}}{\pi ^{3}}}}{\sqrt {{\frac {1}{2m_{\rm {A}}}}+{\frac {1}{2m_{\rm {B}}}}}}{\frac {4T^{3/2}}{P(d_{\rm {A}}+d_{\rm {B}})^{2}}}\,,}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="The_theory_of_diffusion_in_gases_based_on_Boltzmann's_equation"><span id="The_theory_of_diffusion_in_gases_based_on_Boltzmann.27s_equation"></span>The theory of diffusion in gases based on Boltzmann's equation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=16" title="Edit section: The theory of diffusion in gases based on Boltzmann's equation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In Boltzmann's kinetics of the mixture of gases, each gas has its own distribution function, <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}(x,c,t)}"> <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> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{i}(x,c,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07158d06bb6c24df9108c0c0bab3a9cf0f5f3ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.992ex; height:2.843ex;" alt="{\displaystyle f_{i}(x,c,t)}"></span>, where <i>t</i> is the time moment, <i>x</i> is position and <i>c</i> is velocity of molecule of the <i>i</i>th component of the mixture. Each component has its mean velocity <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="{\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mi>c</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d4ec36bb9ab506319f6f8377f8f7b3d867b504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:27.784ex; height:3.676ex;" alt="{\textstyle C_{i}(x,t)={\frac {1}{n_{i}}}\int _{c}cf(x,c,t)\,dc}"></span>. If the velocities <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_{i}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{i}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0c58249cad8365aeae99216253f71a939f928d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.474ex; height:2.843ex;" alt="{\displaystyle C_{i}(x,t)}"></span> do not coincide then there exists <i>diffusion</i>. </p><p>In the <a href="/wiki/Chapman%E2%80%93Enskog_theory" title="Chapman–Enskog theory">Chapman–Enskog</a> approximation, all the distribution functions are expressed through the densities of the conserved quantities:<sup id="cite_ref-ChapmanCowling_10-1" class="reference"><a href="#cite_note-ChapmanCowling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <ul><li>individual concentrations of particles, <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="{\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28122674d0592ffe3a0099f076538bb87e12bb54" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.336ex; height:3.176ex;" alt="{\textstyle n_{i}(x,t)=\int _{c}f_{i}(x,c,t)\,dc}"></span> (particles per volume),</li> <li>density of momentum <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="{\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/992e531d2741bde5bb0e780ba6ed32759fdf2330" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.149ex; height:3.009ex;" alt="{\textstyle \sum _{i}m_{i}n_{i}C_{i}(x,t)}"></span> (<i>m<sub>i</sub></i> is the <i>i</i>th particle mass),</li> <li>density of kinetic energy <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <mrow> <mo>(</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msubsup> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>c</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4de0c8d71d84cae2b15fa8c67a95ef1e7134fb0c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:57.919ex; height:7.509ex;" alt="{\displaystyle \sum _{i}\left(n_{i}{\frac {m_{i}C_{i}^{2}(x,t)}{2}}+\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc\right).}"></span></li></ul> <p>The kinetic temperature <i>T</i> and pressure <i>P</i> are defined in 3D space as </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 {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>T</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mn>2</mn> </mfrac> </mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>c</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>c</mi> <mo>;</mo> <mspace width="1em" /> <mi>P</mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">B</mi> </mrow> </mrow> </msub> <mi>n</mi> <mi>T</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3110a480cc2ed2918f0cd0244666c7d5270de029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:59.33ex; height:6.343ex;" alt="{\displaystyle {\frac {3}{2}}k_{\rm {B}}T={\frac {1}{n}}\int _{c}{\frac {m_{i}(c_{i}-C_{i}(x,t))^{2}}{2}}f_{i}(x,c,t)\,dc;\quad P=k_{\rm {B}}nT,}"></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="{\textstyle n=\sum _{i}n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle n=\sum _{i}n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88c9fe3a77fe6965716762412bb7d08d0b6a84e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.328ex; height:3.009ex;" alt="{\textstyle n=\sum _{i}n_{i}}"></span> is the total density. </p><p>For two gases, the difference between velocities, <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_{1}-C_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}-C_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ca5f4da98421fcd9a64a93e80aea6dc87e11638" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.273ex; height:2.509ex;" alt="{\displaystyle C_{1}-C_{2}}"></span> is given by the expression:<sup id="cite_ref-ChapmanCowling_10-2" class="reference"><a href="#cite_note-ChapmanCowling-10"><span class="cite-bracket">[</span>10<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 C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mrow> <mo>{</mo> <mrow> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>P</mi> <mi>n</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>P</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>T</mi> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ad79bb78267c350d78ae002648e97b8b376e686" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:104.631ex; height:6.509ex;" alt="{\displaystyle C_{1}-C_{2}=-{\frac {n^{2}}{n_{1}n_{2}}}D_{12}\left\{\nabla \left({\frac {n_{1}}{n}}\right)+{\frac {n_{1}n_{2}(m_{2}-m_{1})}{Pn(m_{1}n_{1}+m_{2}n_{2})}}\nabla P-{\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})+k_{T}{\frac {1}{T}}\nabla T\right\},}"></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 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> is the force applied to the molecules of the <i>i</i>th component 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 k_{T}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{T}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58133b8078ef48c84ccc76dc178c14614a13d455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.601ex; height:2.509ex;" alt="{\displaystyle k_{T}}"></span> is the thermodiffusion ratio. </p><p>The coefficient <i>D</i><sub>12</sub> is positive. This is the diffusion coefficient. Four terms in the formula for <i>C</i><sub>1</sub>−<i>C</i><sub>2</sub> describe four main effects in the diffusion of gases: </p> <ol><li><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 \nabla \,\left({\frac {n_{1}}{n}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>n</mi> </mfrac> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f396e2b9d57a58447b2a1de259c62cb8fc4eb0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.771ex; height:4.843ex;" alt="{\displaystyle \nabla \,\left({\frac {n_{1}}{n}}\right)}"></span> describes the flux of the first component from the areas with the high ratio <i>n</i><sub>1</sub>/<i>n</i> to the areas with lower values of this ratio (and, analogously the flux of the second component from high <i>n</i><sub>2</sub>/<i>n</i> to low <i>n</i><sub>2</sub>/<i>n</i> because <i>n</i><sub>2</sub>/<i>n</i> = 1 – <i>n</i><sub>1</sub>/<i>n</i>);</li> <li><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 {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04f6eec1b3de6082e4e673390831337171aba06f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:21.649ex; height:6.509ex;" alt="{\displaystyle {\frac {n_{1}n_{2}(m_{2}-m_{1})}{n(m_{1}n_{1}+m_{2}n_{2})}}\nabla P}"></span> describes the flux of the heavier molecules to the areas with higher pressure and the lighter molecules to the areas with lower pressure, this is <a href="/w/index.php?title=Barodiffusion&action=edit&redlink=1" class="new" title="Barodiffusion (page does not exist)">barodiffusion</a>;</li> <li><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 {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7613b9c6b92a333c2df2dd93586c77b10bc829d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.066ex; height:5.509ex;" alt="{\displaystyle {\frac {m_{1}n_{1}m_{2}n_{2}}{P(m_{1}n_{1}+m_{2}n_{2})}}(F_{1}-F_{2})}"></span> describes diffusion caused by the difference of the forces applied to molecules of different types. For example, in the Earth's gravitational field, the heavier molecules should go down, or in electric field the charged molecules should move, until this effect is not equilibrated by the sum of other terms. This effect should not be confused with barodiffusion caused by the pressure gradient.</li> <li><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_{T}{\frac {1}{T}}\nabla T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mi mathvariant="normal">∇</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{T}{\frac {1}{T}}\nabla T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d512540d66035966ea079093816eebf95b62102" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:8.645ex; height:5.176ex;" alt="{\displaystyle k_{T}{\frac {1}{T}}\nabla T}"></span> describes <a href="/wiki/Thermodiffusion" class="mw-redirect" title="Thermodiffusion">thermodiffusion</a>, the diffusion flux caused by the temperature gradient.</li></ol> <p>All these effects are called <i>diffusion</i> because they describe the differences between velocities of different components in the mixture. Therefore, these effects cannot be described as a <i>bulk</i> transport and differ from advection or convection. </p><p>In the first approximation,<sup id="cite_ref-ChapmanCowling_10-3" class="reference"><a href="#cite_note-ChapmanCowling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>2</mn> <mi>n</mi> <mo stretchy="false">(</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f17effad1f63d0da95fb3082d73481f845e1785" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:40.198ex; height:7.009ex;" alt="{\displaystyle D_{12}={\frac {3}{2n(d_{1}+d_{2})^{2}}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}}"></span> for rigid spheres;</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>8</mn> <mi>n</mi> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mo stretchy="false">)</mo> <mi mathvariant="normal">Γ</mi> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> <mo stretchy="false">(</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>k</mi> <mi>T</mi> </mrow> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mrow> <mi>ν</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a21bffa231a21e8104224bb96f51c7a59685b908" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:58.628ex; height:8.176ex;" alt="{\displaystyle D_{12}={\frac {3}{8nA_{1}({\nu })\Gamma (3-{\frac {2}{\nu -1}})}}\left[{\frac {kT(m_{1}+m_{2})}{2\pi m_{1}m_{2}}}\right]^{1/2}\left({\frac {2kT}{\kappa _{12}}}\right)^{\frac {2}{\nu -1}}}"></span> for repulsing force <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 \kappa _{12}r^{-\nu }.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> <msup> <mi>r</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>ν</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa _{12}r^{-\nu }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e70da5e66512b6ce6005c4d8c50730a62bd31d42" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.293ex; height:2.843ex;" alt="{\displaystyle \kappa _{12}r^{-\nu }.}"></span></li></ul> <p>The number <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_{1}({\nu })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>ν</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A_{1}({\nu })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c936753fcd4212799e795205102391c4e7c1b475" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.839ex; height:2.843ex;" alt="{\displaystyle A_{1}({\nu })}"></span> is defined by quadratures (formulas (3.7), (3.9), Ch. 10 of the classical Chapman and Cowling book<sup id="cite_ref-ChapmanCowling_10-4" class="reference"><a href="#cite_note-ChapmanCowling-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup>) </p><p>We can see that the dependence on <i>T</i> for the rigid spheres is the same as for the simple mean free path theory but for the power repulsion laws the exponent is different. Dependence on a total concentration <i>n</i> for a given temperature has always the same character, 1/<i>n</i>. </p><p>In applications to gas dynamics, the diffusion flux and the bulk flow should be joined in one system of transport equations. The bulk flow describes the mass transfer. Its velocity <i>V</i> is the mass average velocity. It is defined through the momentum density and the mass concentrations: </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 V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mi>ρ</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1338f199241559cc69498082d839d657e12b14b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:14.859ex; height:6.176ex;" alt="{\displaystyle V={\frac {\sum _{i}\rho _{i}C_{i}}{\rho }}\,.}"></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 \rho _{i}=m_{i}n_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho _{i}=m_{i}n_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2b99a8f27d66a325df4109d279429c80612512e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.135ex; height:2.176ex;" alt="{\displaystyle \rho _{i}=m_{i}n_{i}}"></span> is the mass concentration of the <i>i</i>th species, <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="{\textstyle \rho =\sum _{i}\rho _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>ρ</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \rho =\sum _{i}\rho _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e896c6d06097eee40e53c24b97734f8aec4359" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.943ex; height:3.009ex;" alt="{\textstyle \rho =\sum _{i}\rho _{i}}"></span> is the mass density. </p><p>By definition, the diffusion velocity of the <i>i</i>th component is <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_{i}=C_{i}-V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=C_{i}-V}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5334ac70a0df91ac78d451d57312273ab3df86c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.115ex; height:2.509ex;" alt="{\displaystyle v_{i}=C_{i}-V}"></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="{\textstyle \sum _{i}\rho _{i}v_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}\rho _{i}v_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fca540e312885c99be5304d1c3bbab7872209b44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.83ex; height:3.009ex;" alt="{\textstyle \sum _{i}\rho _{i}v_{i}=0}"></span>. The mass transfer of the <i>i</i>th component is described by the <a href="/wiki/Continuity_equation" title="Continuity equation">continuity equation</a> </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 {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>V</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{i}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13efa65115eb15709d0f0e07e788afc526ef2f38" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:32.171ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial \rho _{i}}{\partial t}}+\nabla (\rho _{i}V)+\nabla (\rho _{i}v_{i})=W_{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 W_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7301a4cfd04d4f5db4549fdf23746a0d2ce9f387" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.993ex; height:2.509ex;" alt="{\displaystyle W_{i}}"></span> is the net mass production rate in chemical reactions, <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="{\textstyle \sum _{i}W_{i}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>W</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}W_{i}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fd533319d12f9950eeb4d6040d9ba0db1680471" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.895ex; height:3.009ex;" alt="{\textstyle \sum _{i}W_{i}=0}"></span>. </p><p>In these equations, the term <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 \nabla (\rho _{i}V)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>V</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (\rho _{i}V)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9678db80562a408fa692f3ae43db50ce4bb93ab4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.534ex; height:2.843ex;" alt="{\displaystyle \nabla (\rho _{i}V)}"></span> describes advection of the <i>i</i>th component and the term <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 \nabla (\rho _{i}v_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla (\rho _{i}v_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494ed640f1181691c279b703690d7865fa5c3aab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.674ex; height:2.843ex;" alt="{\displaystyle \nabla (\rho _{i}v_{i})}"></span> represents diffusion of this component. </p><p>In 1948, <a href="/wiki/Wendell_H._Furry" title="Wendell H. Furry">Wendell H. Furry</a> proposed to use the <i>form</i> of the diffusion rates found in kinetic theory as a framework for the new phenomenological approach to diffusion in gases. This approach was developed further by F.A. Williams and S.H. Lam.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> For the diffusion velocities in multicomponent gases (<i>N</i> components) they used </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 v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7aeac7715f2a0564fbd425e275c3ca71af7fff16" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:37.39ex; height:7.676ex;" alt="{\displaystyle v_{i}=-\left(\sum _{j=1}^{N}D_{ij}\mathbf {d} _{j}+D_{i}^{(T)}\,\nabla (\ln T)\right)\,;}"></span></dd> <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 \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi mathvariant="normal">∇</mi> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi mathvariant="normal">∇</mi> <mo stretchy="false">(</mo> <mi>ln</mi> <mo>⁡</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bd0fa2de03db870590d4864c0121ff0be93baac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:37.173ex; height:3.176ex;" alt="{\displaystyle \mathbf {d} _{j}=\nabla X_{j}+(X_{j}-Y_{j})\,\nabla (\ln P)+\mathbf {g} _{j}\,;}"></span></dd> <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 \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">g</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ρ</mi> <mi>P</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff8231f63097a42d71a702dd5c77aa1a0bdffe2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.171ex; height:7.509ex;" alt="{\displaystyle \mathbf {g} _{j}={\frac {\rho }{P}}\left(Y_{j}\sum _{k=1}^{N}Y_{k}(f_{k}-f_{j})\right)\,.}"></span></dd></dl> <p>Here, <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 D_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88badd2360f6d55865a1e8f1b4e3994451bcc075" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.401ex; height:2.843ex;" alt="{\displaystyle D_{ij}}"></span> is the diffusion coefficient 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 D_{i}^{(T)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{i}^{(T)}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08d19cbfd375c42bbc48bddecd4b0a240edbd98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.593ex; height:3.676ex;" alt="{\displaystyle D_{i}^{(T)}}"></span> is the thermal diffusion coefficient, <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/65da883ca3d16b461e46c94777b0d9c4aa010e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.509ex;" alt="{\displaystyle f_{i}}"></span> is the body force per unit mass acting on the <i>i</i>th species, <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_{i}=P_{i}/P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{i}=P_{i}/P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c269a5ba2076f26f51956549e257425ad91d212" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.022ex; height:2.843ex;" alt="{\displaystyle X_{i}=P_{i}/P}"></span> is the partial pressure fraction of the <i>i</i>th species (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 P_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ba1396129f7be3c7f828a571b6649e6807d10d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.292ex; height:2.509ex;" alt="{\displaystyle P_{i}}"></span> is the partial pressure), <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 Y_{i}=\rho _{i}/\rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ρ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y_{i}=\rho _{i}/\rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8af34b032213668c3113d079d6f71e024e0f6635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.615ex; height:2.843ex;" alt="{\displaystyle Y_{i}=\rho _{i}/\rho }"></span> is the mass fraction of the <i>i</i>th species, 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="{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msub> <mi>Y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18c246dd5e2ad7eb4b6e1c93392c40bd402f26f3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.162ex; height:3.009ex;" alt="{\textstyle \sum _{i}X_{i}=\sum _{i}Y_{i}=1.}"></span> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Diffusion_center.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/Diffusion_center.gif/350px-Diffusion_center.gif" decoding="async" width="350" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/15/Diffusion_center.gif 1.5x" data-file-width="473" data-file-height="356" /></a><figcaption>As carriers are generated (green:electrons and purple:holes) due to light shining at the center of an intrinsic semiconductor, they diffuse towards two ends. Electrons have higher diffusion constant than holes leading to fewer excess electrons at the center as compared to holes.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Diffusion_of_electrons_in_solids">Diffusion of electrons in solids</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=17" title="Edit section: Diffusion of electrons in solids"><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/Diffusion_current" title="Diffusion current">Diffusion current</a></div> <p>When the density of electrons in solids is not in equilibrium, diffusion of electrons occurs. For example, when a bias is applied to two ends of a chunk of semiconductor, or a light shines on one end (see right figure), electrons diffuse from high density regions (center) to low density regions (two ends), forming a gradient of electron density. This process generates current, referred to as <a href="/wiki/Diffusion_current" title="Diffusion current">diffusion current</a>. </p><p>Diffusion current can also be described by <a href="/wiki/Fick%27s_laws_of_diffusion" title="Fick's laws of diffusion">Fick's first law</a> </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=-D\,\partial n/\partial x\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>J</mi> <mo>=</mo> <mo>−</mo> <mi>D</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J=-D\,\partial n/\partial x\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7686fa789286940651b862b24dda1dcc13250e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.246ex; height:2.843ex;" alt="{\displaystyle J=-D\,\partial n/\partial x\,,}"></span></dd></dl> <p>where <i>J</i> is the diffusion current density (<a href="/wiki/Amount_of_substance" title="Amount of substance">amount of substance</a>) per unit area per unit time, <i>n</i> (for ideal mixtures) is the electron density, <i>x</i> is the position [length]. </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion_in_geophysics">Diffusion in geophysics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=18" title="Edit section: Diffusion in geophysics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analytical and numerical models that solve the diffusion equation for different initial and boundary conditions have been popular for studying a wide variety of changes to the Earth's surface. Diffusion has been used extensively in erosion studies of hillslope retreat, bluff erosion, fault scarp degradation, wave-cut terrace/shoreline retreat, alluvial channel incision, coastal shelf retreat, and delta progradation.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup> Although the Earth's surface is not literally diffusing in many of these cases, the process of diffusion effectively mimics the holistic changes that occur over decades to millennia. Diffusion models may also be used to solve inverse boundary value problems in which some information about the depositional environment is known from paleoenvironmental reconstruction and the diffusion equation is used to figure out the sediment influx and time series of landform changes.<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Dialysis">Dialysis</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=19" title="Edit section: Dialysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Semipermeable_membrane_(svg).svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Semipermeable_membrane_%28svg%29.svg/280px-Semipermeable_membrane_%28svg%29.svg.png" decoding="async" width="280" height="212" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Semipermeable_membrane_%28svg%29.svg/420px-Semipermeable_membrane_%28svg%29.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Semipermeable_membrane_%28svg%29.svg/560px-Semipermeable_membrane_%28svg%29.svg.png 2x" data-file-width="948" data-file-height="719" /></a><figcaption>Schematic of semipermeable membrane during <a href="/wiki/Hemodialysis" title="Hemodialysis">hemodialysis</a>, where blood is red, dialysing fluid is blue, and the membrane is yellow.</figcaption></figure> <p>Dialysis works on the principles of the diffusion of solutes and <a href="/wiki/Ultrafiltration" title="Ultrafiltration">ultrafiltration</a> of fluid across a <a href="/wiki/Semi-permeable_membrane" class="mw-redirect" title="Semi-permeable membrane">semi-permeable membrane</a>. Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration.<sup id="cite_ref-Mosby_23-0" class="reference"><a href="#cite_note-Mosby-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> Blood flows by one side of a semi-permeable membrane, and a dialysate, or special dialysis fluid, flows by the opposite side. A semipermeable membrane is a thin layer of material that contains holes of various sizes, or pores. Smaller solutes and fluid pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells and large proteins). This replicates the filtering process that takes place in the kidneys when the blood enters the kidneys and the larger substances are separated from the smaller ones in the <a href="/wiki/Glomerulus_(kidney)" title="Glomerulus (kidney)">glomerulus</a>.<sup id="cite_ref-Mosby_23-1" class="reference"><a href="#cite_note-Mosby-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Random_walk_(random_motion)"><span id="Random_walk_.28random_motion.29"></span>Random walk (random motion)</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=20" title="Edit section: Random walk (random motion)"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><span><video id="mwe_player_1" poster="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Random_motion.webm/220px--Random_motion.webm.jpg" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="220" height="91" data-durationhint="6" data-mwtitle="Random_motion.webm" data-mwprovider="wikimediacommons" resource="/wiki/File:Random_motion.webm"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Random_motion.webm/Random_motion.webm.480p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="480p.vp9.webm" data-width="854" data-height="352" /><source src="//upload.wikimedia.org/wikipedia/commons/3/30/Random_motion.webm" type="video/webm; codecs="vp8, vorbis"" data-width="900" data-height="372" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Random_motion.webm/Random_motion.webm.240p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="240p.vp9.webm" data-width="426" data-height="176" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Random_motion.webm/Random_motion.webm.360p.vp9.webm" type="video/webm; codecs="vp9, opus"" data-transcodekey="360p.vp9.webm" data-width="640" data-height="264" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/3/30/Random_motion.webm/Random_motion.webm.360p.webm" type="video/webm; codecs="vp8, vorbis"" data-transcodekey="360p.webm" data-width="640" data-height="264" /></video></span><figcaption>The apparent random motion of atoms, ions or molecules explained. Substances appear to move randomly due to collisions with other substances. From the iBook <i>Cell Membrane Transport</i>, free license granted by IS3D, LLC, 2014.</figcaption></figure><p> One common misconception is that individual atoms, ions or molecules move randomly, which they do not. In the animation on the right, the ion in the left panel appears to have "random" motion in the absence of other ions. As the right panel shows, however, this motion is not random but is the result of "collisions" with other ions. As such, the movement of a single atom, ion, or molecule within a mixture just appears random when viewed in isolation. The movement of a substance within a mixture by "random walk" is governed by the kinetic energy within the system that can be affected by changes in concentration, pressure or temperature. (This is a classical description. At smaller scales, quantum effects will be non-negligible, in general. Thus, the study of the movement of a single atom becomes more subtle since particles at such small scales are described by probability amplitudes rather than deterministic measures of position and velocity.) </p><div class="mw-heading mw-heading3"><h3 id="Separation_of_diffusion_from_convection_in_gases">Separation of diffusion from convection in gases</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=21" title="Edit section: Separation of diffusion from convection in gases"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>While Brownian motion of multi-molecular mesoscopic particles (like pollen grains studied by Brown) is observable under an optical microscope, molecular diffusion can only be probed in carefully controlled experimental conditions. Since Graham experiments, it is well known that avoiding of convection is necessary and this may be a non-trivial task. </p><p>Under normal conditions, molecular diffusion dominates only at lengths in the nanometre-to-millimetre range. On larger length scales, transport in liquids and gases is normally due to another <a href="/wiki/Transport_phenomena" title="Transport phenomena">transport phenomenon</a>, <a href="/wiki/Convection" title="Convection">convection</a>. To separate diffusion in these cases, special efforts are needed. </p><p>In contrast, <a href="/wiki/Heat_conduction" class="mw-redirect" title="Heat conduction">heat conduction</a> through solid media is an everyday occurrence (for example, a metal spoon partly immersed in a hot liquid). This explains why the diffusion of heat was explained mathematically before the diffusion of mass. </p> <div class="mw-heading mw-heading3"><h3 id="Other_types_of_diffusion">Other types of diffusion</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Diffusion&action=edit&section=22" title="Edit section: Other types of diffusion"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Anisotropic_diffusion" title="Anisotropic diffusion">Anisotropic diffusion</a>, also known as the Perona–Malik equation, enhances high gradients</li> <li><a href="/wiki/Atomic_diffusion" title="Atomic diffusion">Atomic diffusion</a>, in solids</li> <li><a href="/wiki/Bohm_diffusion" title="Bohm diffusion">Bohm diffusion</a>, spread of plasma across magnetic fields</li> <li><a href="/wiki/Eddy_diffusion" title="Eddy diffusion">Eddy diffusion</a>, in coarse-grained description of turbulent flow</li> <li><a href="/wiki/Effusion" title="Effusion">Effusion</a> of a gas through small holes</li> <li><a href="/wiki/Electronics" title="Electronics">Electronic</a> diffusion, resulting in an <a href="/wiki/Current_(electricity)" class="mw-redirect" title="Current (electricity)">electric current</a> called the <a href="/wiki/Diffusion_current" title="Diffusion current">diffusion current</a></li> <li><a href="/wiki/Facilitated_diffusion" title="Facilitated diffusion">Facilitated diffusion</a>, present in some organisms</li> <li><a href="/wiki/Gaseous_diffusion" title="Gaseous diffusion">Gaseous diffusion</a>, used for <a href="/wiki/Isotope_separation" title="Isotope separation">isotope separation</a></li> <li><a href="/wiki/Heat_equation" title="Heat equation">Heat equation</a>, diffusion of thermal energy</li> <li><a href="/wiki/It%C5%8D_diffusion" class="mw-redirect" title="Itō diffusion">Itō diffusion</a>, mathematisation of Brownian motion, continuous stochastic process.</li> <li><a href="/wiki/Knudsen_diffusion" title="Knudsen diffusion">Knudsen diffusion</a> of gas in long pores with frequent wall collisions</li> <li><a href="/wiki/L%C3%A9vy_flight" title="Lévy flight">Lévy flight</a></li> <li><a href="/wiki/Molecular_diffusion" title="Molecular diffusion">Molecular diffusion</a>, diffusion of molecules from more dense to less dense areas</li> <li><a href="/wiki/Momentum_diffusion" title="Momentum diffusion">Momentum diffusion</a> ex. the diffusion of the <a href="/wiki/Hydrodynamic" class="mw-redirect" title="Hydrodynamic">hydrodynamic</a> velocity field</li> <li><a href="/wiki/Photon_diffusion" title="Photon diffusion">Photon diffusion</a></li> <li><a href="/wiki/Plasma_diffusion" title="Plasma diffusion">Plasma diffusion</a></li> <li><a href="/wiki/Random_walk" title="Random walk">Random walk</a>,<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> model for diffusion</li> <li><a href="/wiki/Reverse_diffusion" title="Reverse diffusion">Reverse diffusion</a>, against the concentration gradient, in phase separation</li> <li><a href="/wiki/Rotational_diffusion" title="Rotational diffusion">Rotational diffusion</a>, random reorientation of molecules</li> <li><a href="/wiki/Spin_diffusion" title="Spin diffusion">Spin diffusion</a>, diffusion of <a href="/wiki/Spin_magnetic_moment" class="mw-redirect" title="Spin magnetic moment">spin magnetic moments</a> in solids</li> <li><a href="/wiki/Surface_diffusion" title="Surface diffusion">Surface diffusion</a>, diffusion of adparticles on a surface</li> <li><a href="/wiki/Taxis" title="Taxis">Taxis</a> is an animal's directional movement activity in response to a stimulus <ul><li><a href="/wiki/Kinesis_(biology)" title="Kinesis (biology)">Kinesis</a> is an animal's non-directional movement activity in response to a stimulus</li></ul></li> <li><a href="/wiki/Trans-cultural_diffusion" class="mw-redirect" title="Trans-cultural diffusion">Trans-cultural diffusion</a>, diffusion of cultural traits across geographical area</li> <li><a href="/wiki/Turbulent_diffusion" title="Turbulent diffusion">Turbulent diffusion</a>, transport of mass, heat, or momentum within a turbulent fluid</li></ul> <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=Diffusion&action=edit&section=23" 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"> <ul><li><a href="/wiki/Anomalous_diffusion" title="Anomalous diffusion">Anomalous diffusion</a> – Diffusion process with a non-linear relationship to time</li> <li><a href="/wiki/Convection%E2%80%93diffusion_equation" title="Convection–diffusion equation">Convection–diffusion equation</a> – Combination of the diffusion and convection (advection) equations</li> <li><a href="/wiki/Diffusion-limited_aggregation" title="Diffusion-limited aggregation">Diffusion-limited aggregation</a></li> <li><a href="/wiki/Darken%27s_equations" title="Darken's equations">Darken's equations</a></li> <li><a href="/wiki/Isobaric_counterdiffusion" title="Isobaric counterdiffusion">Isobaric counterdiffusion</a> – Gaseous diffusion through body tissue at constant total pressure</li> <li><a href="/wiki/Sorption" title="Sorption">Sorption</a> – Physical or chemical process by which one substance becomes attached to another</li> <li><a href="/wiki/Osmosis" title="Osmosis">Osmosis</a> – Migration of molecules to a region of lower solute concentration</li> <li><a href="/wiki/Percolation_theory" title="Percolation theory">Percolation theory</a> – Mathematical theory on behavior of connected clusters in a random graph</li> <li><a href="/wiki/Social_Networks" class="mw-redirect" title="Social Networks">Social Networks</a> – journal<span style="display:none" class="category-wikidata-fallback-annotation">Pages displaying wikidata descriptions as a fallback</span><span style="display:none" class="category-spaceless-annotation">Pages displaying short descriptions with no spaces</span></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=Diffusion&action=edit&section=24" 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 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