<!DOCTYPE html>
<html class="client-nojs vector-feature-night-mode-disabled vector-feature-language-in-header-enabled vector-feature-language-in-main-page-header-disabled vector-feature-page-tools-pinned-disabled vector-feature-toc-pinned-clientpref-1 vector-feature-main-menu-pinned-disabled vector-feature-limited-width-clientpref-1 vector-feature-limited-width-content-enabled vector-feature-custom-font-size-clientpref-1 vector-feature-appearance-pinned-clientpref-1 vector-sticky-header-enabled" lang="en" dir="ltr"><head>
<meta charset="UTF-8">
<title>Grand potential</title>
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<link rel="canonical" href="https://en.wikipedia.org/wiki/Grand_potential"> <link href="./mw/ext.cite.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/ext.math.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.icons.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.search.codex.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/skins.vector.styles.css" rel="stylesheet" type="text/css">
<link href="./mw/user.styles.css" rel="stylesheet" type="text/css">
<meta name="ResourceLoaderDynamicStyles" content="">
<link rel="stylesheet" type="text/css" href="./mw/site.styles.css">
<link rel="stylesheet" type="text/css" href="./mw/noscript.css">
<link rel="stylesheet" type="text/css" href="./footer.css">
<link rel="stylesheet" type="text/css" href="./vector-2022.css">
</head>
<body class="skin--responsive skin-vector skin-vector-search-vue mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Grand_potential rootpage-Grand_potential skin-vector-2022 action-view">
<div class="mw-page-container">
<div class="mw-page-container-inner">
<div class="mw-content-container">
<main id="content" class="mw-body">
<header class="mw-body-header vector-page-titlebar">
<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Grand potential</span></span>
</h1>
</header>
<a id="top"></a>
<div id="bodyContent" class="vector-body ve-init-mw-desktopArticleTarget-targetContainer" aria-labelledby="firstHeading" data-mw-ve-target-container="">
<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">
<style data-mw-deduplicate="TemplateStyles:r1129693374">
/* start https://en.wikipedia.org/ */
.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}
/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1126788409">
/* start https://en.wikipedia.org/ */
.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}
/* end https://en.wikipedia.org/ */
</style><style data-mw-deduplicate="TemplateStyles:r1246091330">
/* start https://en.wikipedia.org/ */
.mw-parser-output .sidebar{width:22em;float:right;clear:right;margin:0.5em 0 1em 1em;background:var(--background-color-neutral-subtle,#f8f9fa);border:1px solid var(--border-color-base,#a2a9b1);padding:0.2em;text-align:center;line-height:1.4em;font-size:88%;border-collapse:collapse;display:table}body.skin-minerva .mw-parser-output .sidebar{display:table!important;float:right!important;margin:0.5em 0 1em 1em!important}.mw-parser-output .sidebar-subgroup{width:100%;margin:0;border-spacing:0}.mw-parser-output .sidebar-left{float:left;clear:left;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-none{float:none;clear:both;margin:0.5em 1em 1em 0}.mw-parser-output .sidebar-outer-title{padding:0 0.4em 0.2em;font-size:125%;line-height:1.2em;font-weight:bold}.mw-parser-output .sidebar-top-image{padding:0.4em}.mw-parser-output .sidebar-top-caption,.mw-parser-output .sidebar-pretitle-with-top-image,.mw-parser-output .sidebar-caption{padding:0.2em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-pretitle{padding:0.4em 0.4em 0;line-height:1.2em}.mw-parser-output .sidebar-title,.mw-parser-output .sidebar-title-with-pretitle{padding:0.2em 0.8em;font-size:145%;line-height:1.2em}.mw-parser-output .sidebar-title-with-pretitle{padding:0.1em 0.4em}.mw-parser-output .sidebar-image{padding:0.2em 0.4em 0.4em}.mw-parser-output .sidebar-heading{padding:0.1em 0.4em}.mw-parser-output .sidebar-content{padding:0 0.5em 0.4em}.mw-parser-output .sidebar-content-with-subgroup{padding:0.1em 0.4em 0.2em}.mw-parser-output .sidebar-above,.mw-parser-output .sidebar-below{padding:0.3em 0.8em;font-weight:bold}.mw-parser-output .sidebar-collapse .sidebar-above,.mw-parser-output .sidebar-collapse .sidebar-below{border-top:1px solid #aaa;border-bottom:1px solid #aaa}.mw-parser-output .sidebar-navbar{text-align:right;font-size:115%;padding:0 0.4em 0.4em}.mw-parser-output .sidebar-list-title{padding:0 0.4em;text-align:left;font-weight:bold;line-height:1.6em;font-size:105%}.mw-parser-output .sidebar-list-title-c{padding:0 0.4em;text-align:center;margin:0 3.3em}@media(max-width:640px){body.mediawiki .mw-parser-output .sidebar{width:100%!important;clear:both;float:none!important;margin-left:0!important;margin-right:0!important}}body.skin--responsive .mw-parser-output .sidebar a>img{max-width:none!important}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}
/* end https://en.wikipedia.org/ */
</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><th class="sidebar-title"><a href="Statistical_mechanics" title="Statistical mechanics">Statistical mechanics</a></th></tr><tr><td class="sidebar-image"></td></tr><tr><td class="sidebar-content plainlist">
<ul><li><a href="Thermodynamics" title="Thermodynamics">Thermodynamics</a></li>
<li><a href="Kinetic_theory_of_gases" title="Kinetic theory of gases">Kinetic theory</a></li></ul></td>
</tr><tr><td class="sidebar-content plainlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Particle_statistics" title="Particle statistics">Particle statistics</a></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Spin%E2%80%93statistics_theorem" title="Spin–statistics theorem">Spin–statistics theorem</a></li>
<li><a href="Indistinguishable_particles" title="Indistinguishable particles">Indistinguishable particles</a></li>
<li><a href="Maxwell%E2%80%93Boltzmann_statistics" title="Maxwell–Boltzmann statistics">Maxwell–Boltzmann</a></li>
<li><a href="Bose%E2%80%93Einstein_statistics" title="Bose–Einstein statistics">Bose–Einstein</a></li>
<li><a href="Fermi%E2%80%93Dirac_statistics" title="Fermi–Dirac statistics">Fermi–Dirac</a></li>
<li><a href="Parastatistics" title="Parastatistics">Parastatistics</a></li>
<li><a href="Anyon" title="Anyon">Anyonic statistics</a></li>
<li><a href="Braid_statistics" title="Braid statistics">Braid statistics</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content plainlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Statistical_ensemble_(mathematical_physics)" class="mw-redirect" title="Statistical ensemble (mathematical physics)">Thermodynamic ensembles</a></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><i>NVE</i> <a href="Microcanonical_ensemble" title="Microcanonical ensemble">Microcanonical</a></li>
<li><i>NVT</i> <a href="Canonical_ensemble" title="Canonical ensemble">Canonical</a></li>
<li><i>µVT</i> <a href="Grand_canonical_ensemble" title="Grand canonical ensemble">Grand canonical</a></li>
<li><i>NPH</i> <a href="Isoenthalpic%E2%80%93isobaric_ensemble" title="Isoenthalpic–isobaric ensemble">Isoenthalpic–isobaric</a></li>
<li><i>NPT</i> <a href="Isothermal%E2%80%93isobaric_ensemble" title="Isothermal–isobaric ensemble">Isothermal–isobaric</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content plainlist">
<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Models</div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Debye_model" title="Debye model">Debye</a></li>
<li><a href="Einstein_solid" title="Einstein solid">Einstein</a></li>
<li><a href="Ising_model" title="Ising model">Ising</a></li>
<li><a href="Potts_model" title="Potts model">Potts</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-content plainlist">
<div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)"><a href="Thermodynamic_potential" title="Thermodynamic potential">Potentials</a></div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="Internal_energy" title="Internal energy">Internal energy</a></li>
<li><a href="Enthalpy" title="Enthalpy">Enthalpy</a></li>
<li><a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz free energy</a></li>
<li><a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a></li>
</ul></div></div></td>
</tr><tr><td class="sidebar-content plainlist">
<div class="sidebar-list mw-collapsible mw-collapsed hlist"><div class="sidebar-list-title" style="background:transparent;border-top:1px solid #aaa;text-align:center;color: var(--color-base)">Scientists</div><div class="sidebar-list-content mw-collapsible-content">
<ul><li><a href="James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a></li>
<li><a href="Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a></li>
<li><a href="Hermann_von_Helmholtz" title="Hermann von Helmholtz">Helmholtz</a></li>
<li><a href="Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a></li>
<li><a href="Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a></li>
<li><a href="Albert_Einstein" title="Albert Einstein">Einstein</a></li>
<li><a href="Paul_Dirac" title="Paul Dirac">Dirac</a></li>
<li><a href="Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest</a></li>
<li><a href="John_von_Neumann" title="John von Neumann">von Neumann</a></li>
<li><a href="Richard_C._Tolman" title="Richard C. Tolman">Tolman</a></li>
<li><a href="Peter_Debye" title="Peter Debye">Debye</a></li>
<li><a href="Enrico_Fermi" title="Enrico Fermi">Fermi</a></li>
<li><a href="John_Lighton_Synge" title="John Lighton Synge">Synge</a></li>
<li><a href="Ernst_Ising" title="Ernst Ising">Ising</a></li>
<li><a href="Lev_Landau" title="Lev Landau">Landau</a></li></ul></div></div></td>
</tr><tr><td class="sidebar-navbar"><style data-mw-deduplicate="TemplateStyles:r1239400231">
/* start https://en.wikipedia.org/ */
.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}
/* end https://en.wikipedia.org/ */
</style></td></tr></tbody></table>
<p>The <b>grand potential</b> or <b>Landau potential</b> or <b>Landau free energy</b> is a quantity used in <a href="Statistical_mechanics" title="Statistical mechanics">statistical mechanics</a>, especially for <a href="Irreversible_process" title="Irreversible process">irreversible processes</a> in <a href="Open_system_(systems_theory)" title="Open system (systems theory)">open systems</a>.
The grand potential is the characteristic state function for the <a href="Grand_canonical_ensemble" title="Grand canonical ensemble">grand canonical ensemble</a>.
</p>
<meta property="mw:PageProp/toc">
<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>
<p>The grand potential is defined by
<span class="mwe-math-element mwe-math-element-block"><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 \Phi _{\text{G}}{\stackrel {\mathrm {def} }{{}={}}}U-TS-\mu N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">f</mi>
</mrow>
</mrow>
</mover>
</mrow>
</mrow>
<mi>U</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{\text{G}}{\stackrel {\mathrm {def} }{{}={}}}U-TS-\mu N}</annotation>
</semantics>
</math></span></span>
where <i>U</i> is the <a href="Internal_energy" title="Internal energy">internal energy</a>, <i>T</i> is the <a href="Temperature" title="Temperature">temperature</a> of the system, <i>S</i> is the <a href="Entropy" title="Entropy">entropy</a>, <i>μ</i> is the <a href="Chemical_potential" title="Chemical potential">chemical potential</a>, and <i>N</i> is the number of particles in the system.
</p><p>The change in the grand potential is given by
<span class="mwe-math-element mwe-math-element-block"><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 {\begin{aligned}d\Phi _{\text{G}}&=dU-T\,dS-S\,dT-\mu d\,N-N\,d\mu \\&=-P\,dV-S\,dT-N\,d\mu \end{aligned}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true">
<mtr>
<mtd>
<mi>d</mi>
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
</mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mi>d</mi>
<mi>U</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>S</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>T</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>d</mi>
<mspace width="thinmathspace"></mspace>
<mi>N</mi>
<mo>−<!-- − --></mo>
<mi>N</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>μ<!-- μ --></mi>
</mtd>
</mtr>
<mtr>
<mtd></mtd>
<mtd>
<mi></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>P</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>V</mi>
<mo>−<!-- − --></mo>
<mi>S</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>T</mi>
<mo>−<!-- − --></mo>
<mi>N</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>μ<!-- μ --></mi>
</mtd>
</mtr>
</mtable>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}d\Phi _{\text{G}}&=dU-T\,dS-S\,dT-\mu d\,N-N\,d\mu \\&=-P\,dV-S\,dT-N\,d\mu \end{aligned}}}</annotation>
</semantics>
</math></span></span>
where <i>P</i> is <a href="Pressure" title="Pressure">pressure</a> and <i>V</i> is <a href="Volume" title="Volume">volume</a>, using the <a href="Fundamental_thermodynamic_relation" title="Fundamental thermodynamic relation">fundamental thermodynamic relation</a> (combined <a href="First_law_of_thermodynamics" title="First law of thermodynamics">first</a> and <a href="First_law_of_thermodynamics" title="First law of thermodynamics">second</a> <a href="Thermodynamic_laws" class="mw-redirect" title="Thermodynamic laws">thermodynamic laws</a>);
</p><p><span class="mwe-math-element mwe-math-element-block"><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 dU=T\,dS-P\,dV+\mu \,dN}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>d</mi>
<mi>U</mi>
<mo>=</mo>
<mi>T</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>P</mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>V</mi>
<mo>+</mo>
<mi>μ<!-- μ --></mi>
<mspace width="thinmathspace"></mspace>
<mi>d</mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle dU=T\,dS-P\,dV+\mu \,dN}</annotation>
</semantics>
</math></span></span>
</p><p>When the system is in <a href="Thermodynamic_equilibrium" title="Thermodynamic equilibrium">thermodynamic equilibrium</a>, Φ<sub>G</sub> is a minimum. This can be seen by considering that <i>d</i>Φ<sub>G</sub> is zero if the volume is fixed and the temperature and chemical potential have stopped evolving.
</p>
<div class="mw-heading mw-heading3"><h3 id="Landau_free_energy">Landau free energy</h3></div>
<p>Some authors refer to the grand potential as the <i>Landau free energy</i> or <b>Landau potential</b> and write its definition as:<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><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><span class="mwe-math-element mwe-math-element-block"><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 \Omega {\stackrel {\mathrm {def} }{{}={}}}F-\mu N=U-TS-\mu N}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-REL">
<mover>
<mrow class="MJX-TeXAtom-OP MJX-fixedlimits">
<mrow class="MJX-TeXAtom-ORD">
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
</mrow>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mi mathvariant="normal">d</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">f</mi>
</mrow>
</mrow>
</mover>
</mrow>
</mrow>
<mi>F</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>N</mi>
<mo>=</mo>
<mi>U</mi>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi>S</mi>
<mo>−<!-- − --></mo>
<mi>μ<!-- μ --></mi>
<mi>N</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega {\stackrel {\mathrm {def} }{{}={}}}F-\mu N=U-TS-\mu N}</annotation>
</semantics>
</math></span></span>
</p><p>named after Russian physicist <a href="Lev_Landau" title="Lev Landau">Lev Landau</a>, which may be a synonym for the grand potential, depending on system stipulations. For homogeneous systems, one obtains <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega =-PV}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Ω<!-- Ω --></mi>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mi>P</mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Omega =-PV}</annotation>
</semantics>
</math></span><img src="./dfd8dfea4ccac57602fa8e172a6604e433f5ff67.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.117ex; height:2.343ex;" alt="{\displaystyle \Omega =-PV}" loading="lazy"></span>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
</p>
<div class="mw-heading mw-heading2"><h2 id="Homogeneous_systems_(vs._inhomogeneous_systems)">Homogeneous systems (vs. inhomogeneous systems)</h2></div>
<p>In the case of a scale-invariant type of system (where a system of volume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda V}</annotation>
</semantics>
</math></span><img src="./79a720a73909c3d407646d0d7327b805f0db34b4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.142ex; height:2.176ex;" alt="{\displaystyle \lambda V}" loading="lazy"></span> has exactly the same set of microstates as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>λ<!-- λ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>
</semantics>
</math></span><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"></span> systems of volume <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle V}</annotation>
</semantics>
</math></span><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"></span>), then when the system expands new particles and energy will flow in from the reservoir to fill the new volume with a homogeneous extension of the original system.
The pressure, then, must be constant with respect to changes in volume:
</p><p><span class="mwe-math-element mwe-math-element-block"><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 \left({\frac {\partial \langle P\rangle }{\partial V}}\right)_{\mu ,T}=0,}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>P</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>T</mi>
</mrow>
</msub>
<mo>=</mo>
<mn>0</mn>
<mo>,</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \langle P\rangle }{\partial V}}\right)_{\mu ,T}=0,}</annotation>
</semantics>
</math></span></span>
</p><p>and all extensive quantities (particle number, energy, entropy, potentials, ...) must grow linearly with volume, e.g.
</p><p><span class="mwe-math-element mwe-math-element-block"><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 \left({\frac {\partial \langle N\rangle }{\partial V}}\right)_{\mu ,T}={\frac {N}{V}}.}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mrow>
<mo>(</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>N</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
</mrow>
<mrow>
<mi mathvariant="normal">∂<!-- ∂ --></mi>
<mi>V</mi>
</mrow>
</mfrac>
</mrow>
<mo>)</mo>
</mrow>
<mrow class="MJX-TeXAtom-ORD">
<mi>μ<!-- μ --></mi>
<mo>,</mo>
<mi>T</mi>
</mrow>
</msub>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>N</mi>
<mi>V</mi>
</mfrac>
</mrow>
<mo>.</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial \langle N\rangle }{\partial V}}\right)_{\mu ,T}={\frac {N}{V}}.}</annotation>
</semantics>
</math></span></span>
</p><p>In this case we simply have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>P</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}</annotation>
</semantics>
</math></span><img src="./cea7d8463d60ed03db479eea53d9f2e56897d539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.449ex; height:2.843ex;" alt="{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}" loading="lazy"></span>, as well as the familiar relationship <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G=\langle N\rangle \mu }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>G</mi>
<mo>=</mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>N</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mi>μ<!-- μ --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle G=\langle N\rangle \mu }</annotation>
</semantics>
</math></span><img src="./865336f4de8b2f845e56204abcf36ad21ecb6636.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.2ex; height:2.843ex;" alt="{\displaystyle G=\langle N\rangle \mu }" loading="lazy"></span> for the <a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs free energy</a>.
The value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\text{G}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{\text{G}}}</annotation>
</semantics>
</math></span><img src="./0226578737e0aaa90a4e31f17d906a5787bf0f82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.2ex; height:2.509ex;" alt="{\displaystyle \Phi _{\text{G}}}" loading="lazy"></span> can be understood as the work that can be extracted from the system by shrinking it down to nothing (putting all the particles and energy back into the reservoir). The fact that <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
<mo>=</mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>P</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}</annotation>
</semantics>
</math></span><img src="./cea7d8463d60ed03db479eea53d9f2e56897d539.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.449ex; height:2.843ex;" alt="{\displaystyle \Phi _{\text{G}}=-\langle P\rangle V}" loading="lazy"></span> is negative implies that the extraction of particles from the system to the reservoir requires energy input.
</p><p>Such homogeneous scaling does not exist in many systems. For example, when analyzing the ensemble of electrons in a single molecule or even a piece of metal floating in space, doubling the volume of the space does double the number of electrons in the material.<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>
The problem here is that, although electrons and energy are exchanged with a reservoir, the material host is not allowed to change.
Generally in small systems, or systems with long range interactions (those outside the <a href="Thermodynamic_limit" title="Thermodynamic limit">thermodynamic limit</a>), <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi _{\text{G}}\neq -\langle P\rangle V}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msub>
<mi mathvariant="normal">Φ<!-- Φ --></mi>
<mrow class="MJX-TeXAtom-ORD">
<mtext>G</mtext>
</mrow>
</msub>
<mo>≠<!-- ≠ --></mo>
<mo>−<!-- − --></mo>
<mo fence="false" stretchy="false">⟨<!-- ⟨ --></mo>
<mi>P</mi>
<mo fence="false" stretchy="false">⟩<!-- ⟩ --></mo>
<mi>V</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Phi _{\text{G}}\neq -\langle P\rangle V}</annotation>
</semantics>
</math></span><img src="./dd3dc46e71c455c6ecafcf8463a0f14038fb78e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.449ex; height:2.843ex;" alt="{\displaystyle \Phi _{\text{G}}\neq -\langle P\rangle V}" loading="lazy"></span>.<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Gibbs_free_energy" title="Gibbs free energy">Gibbs energy</a></li>
<li><a href="Helmholtz_free_energy" title="Helmholtz free energy">Helmholtz energy</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<style data-mw-deduplicate="TemplateStyles:r1239543626">
/* start https://en.wikipedia.org/ */
.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}
/* end https://en.wikipedia.org/ */
</style><div class="reflist">
<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
/* start https://en.wikipedia.org/ */
.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}
/* end https://en.wikipedia.org/ */
</style><cite id="CITEREFLee,_J._Chang2002" class="citation book cs1">Lee, J. Chang (2002). "5". <i>Thermal Physics - Entropy and Free Energies</i>. New Jersey: World Scientific.</cite><span class="cs1-maint citation-comment"><code class="cs1-code">{{cite book}}</code>: CS1 maint: publisher location (link)</span></span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Reference on "Landau potential" is found in the book: <cite id="CITEREFD._Goodstein" class="citation book cs1">D. Goodstein. <i>States of Matter</i>. p. 19.</cite></span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFMcGovern" class="citation web cs1">McGovern, Judith. <a rel="nofollow" class="external text" href="http://theory.physics.manchester.ac.uk/~judith/stat_therm/node88.html">"The Grand Potential"</a>. <i>PHYS20352 Thermal and Statistical Physics</i>. University of Manchester<span class="reference-accessdate">. Retrieved <span class="nowrap">5 December</span> 2016</span>.</cite></span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><cite id="CITEREFBrachman1954" class="citation journal cs1">Brachman, M. K. (1954). "Fermi Level, Chemical Potential, and Gibbs Free Energy". <i>The Journal of Chemical Physics</i>. <b>22</b> (6): 1152. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1954JChPh..22.1152B">1954JChPh..22.1152B</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.1740312">10.1063/1.1740312</a>.</cite></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFHill2002" class="citation book cs1">Hill, Terrell L. (2002). <i>Thermodynamics of Small Systems</i>. Courier Dover Publications. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780486495095</bdi>.</cite></span>
</li>
</ol></div></div>
<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>
<ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060909024313/http://theory.ph.man.ac.uk/~judith/stat_therm/node88.html">Grand Potential (Manchester University)</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
This article is issued from <a class="external text" title="Last edited on 2025-04-08" href="https://en.wikipedia.org/wiki/?title=Grand_potential&oldid=1284596692">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.
</div>
</div><!--/htdig_noindex--></div>
</div>
</main>
</div>
</div>
</div>
</body></html>