Mecanică statistică

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class="vector-toc-link" href="#Principiile_mecanicii_statistice_clasice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Principiile mecanicii statistice clasice</span> </div> </a> <button aria-controls="toc-Principiile_mecanicii_statistice_clasice-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 Principiile mecanicii statistice clasice subsection</span> </button> <ul id="toc-Principiile_mecanicii_statistice_clasice-sublist" class="vector-toc-list"> <li id="toc-Stări_microscopice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stări_microscopice"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>Stări microscopice</span> </div> </a> <ul id="toc-Stări_microscopice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Colectiv_statistic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Colectiv_statistic"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.2</span> <span>Colectiv statistic</span> </div> </a> <ul id="toc-Colectiv_statistic-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Valori_medii_și_fluctuații" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Valori_medii_și_fluctuații"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.3</span> <span>Valori medii și fluctuații</span> </div> </a> <ul id="toc-Valori_medii_și_fluctuații-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Distribuții_reprezentative" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Distribuții_reprezentative"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Distribuții reprezentative</span> </div> </a> <button aria-controls="toc-Distribuții_reprezentative-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 Distribuții reprezentative subsection</span> </button> <ul id="toc-Distribuții_reprezentative-sublist" class="vector-toc-list"> <li id="toc-Distribuția_microcanonică" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribuția_microcanonică"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Distribuția microcanonică</span> </div> </a> <ul id="toc-Distribuția_microcanonică-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distribuția_canonică" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribuția_canonică"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Distribuția canonică</span> </div> </a> <ul id="toc-Distribuția_canonică-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distribuția_macrocanonică" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribuția_macrocanonică"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Distribuția macrocanonică</span> </div> </a> <ul id="toc-Distribuția_macrocanonică-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Termodinamică_statistică" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Termodinamică_statistică"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Termodinamică statistică</span> </div> </a> <button aria-controls="toc-Termodinamică_statistică-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 Termodinamică statistică subsection</span> </button> <ul id="toc-Termodinamică_statistică-sublist" class="vector-toc-list"> <li id="toc-Sistem_izolat:_entropie" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sistem_izolat:_entropie"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Sistem izolat: entropie</span> </div> </a> <ul id="toc-Sistem_izolat:_entropie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schimb_de_energie:_energie_liberă" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schimb_de_energie:_energie_liberă"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Schimb de energie: energie liberă</span> </div> </a> <ul id="toc-Schimb_de_energie:_energie_liberă-sublist" class="vector-toc-list"> <li id="toc-Entropia_ca_funcțională_de_densitatea_de_probabilitate" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Entropia_ca_funcțională_de_densitatea_de_probabilitate"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.1</span> <span>Entropia ca funcțională de densitatea de probabilitate</span> </div> </a> <ul id="toc-Entropia_ca_funcțională_de_densitatea_de_probabilitate-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Teorema_echipartiției_energiei" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Teorema_echipartiției_energiei"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2.2</span> <span>Teorema echipartiției energiei</span> </div> </a> <ul id="toc-Teorema_echipartiției_energiei-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Schimb_de_energie_și_substanță:_potențial_macrocanonic" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schimb_de_energie_și_substanță:_potențial_macrocanonic"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Schimb de energie și substanță: potențial macrocanonic</span> </div> </a> <ul id="toc-Schimb_de_energie_și_substanță:_potențial_macrocanonic-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Limitele_mecanicii_statistice_clasice" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Limitele_mecanicii_statistice_clasice"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Limitele mecanicii statistice clasice</span> </div> </a> <ul id="toc-Limitele_mecanicii_statistice_clasice-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Mecanică_statistică_cuantică" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Mecanică_statistică_cuantică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Mecanică statistică cuantică</span> </div> </a> <button aria-controls="toc-Mecanică_statistică_cuantică-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 Mecanică statistică cuantică subsection</span> </button> <ul id="toc-Mecanică_statistică_cuantică-sublist" class="vector-toc-list"> <li id="toc-Stări_staționare_în_mecanica_cuantică" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stări_staționare_în_mecanica_cuantică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Stări staționare în mecanica cuantică</span> </div> </a> <ul id="toc-Stări_staționare_în_mecanica_cuantică-sublist" class="vector-toc-list"> <li id="toc-Nivele_de_energie" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Nivele_de_energie"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.1</span> <span>Nivele de energie</span> </div> </a> <ul id="toc-Nivele_de_energie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spin" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Spin"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1.2</span> <span>Spin</span> </div> </a> <ul id="toc-Spin-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Distribuția_canonică_în_mecanica_statistică_cuantică" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribuția_canonică_în_mecanica_statistică_cuantică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Distribuția canonică în mecanica statistică cuantică</span> </div> </a> <ul id="toc-Distribuția_canonică_în_mecanica_statistică_cuantică-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Sisteme_de_particule_identice" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Sisteme_de_particule_identice"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Sisteme de particule identice</span> </div> </a> <ul id="toc-Sisteme_de_particule_identice-sublist" class="vector-toc-list"> <li id="toc-Relația_dintre_spin_și_statistică" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Relația_dintre_spin_și_statistică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1</span> <span>Relația dintre spin și statistică</span> </div> </a> <ul id="toc-Relația_dintre_spin_și_statistică-sublist" class="vector-toc-list"> <li id="toc-Statistica_Fermi-Dirac_(fermioni)" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Statistica_Fermi-Dirac_(fermioni)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1.1</span> <span>Statistica Fermi-Dirac (fermioni)</span> </div> </a> <ul id="toc-Statistica_Fermi-Dirac_(fermioni)-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Statistica_Bose-Einstein_(bosoni)" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Statistica_Bose-Einstein_(bosoni)"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.1.2</span> <span>Statistica Bose-Einstein (bosoni)</span> </div> </a> <ul id="toc-Statistica_Bose-Einstein_(bosoni)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dependența_de_parametrii_macroscopici" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Dependența_de_parametrii_macroscopici"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2</span> <span>Dependența de parametrii macroscopici</span> </div> </a> <ul id="toc-Dependența_de_parametrii_macroscopici-sublist" class="vector-toc-list"> <li id="toc-Limita_clasică" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Limita_clasică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2.1</span> <span>Limita clasică</span> </div> </a> <ul id="toc-Limita_clasică-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Degenerescență_cuantică" class="vector-toc-list-item vector-toc-level-4"> <a class="vector-toc-link" href="#Degenerescență_cuantică"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3.2.2</span> <span>Degenerescență cuantică</span> </div> </a> <ul id="toc-Degenerescență_cuantică-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Note" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Note"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Note</span> </div> </a> <ul id="toc-Note-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Bibliografie" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Bibliografie"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Bibliografie</span> </div> </a> <ul id="toc-Bibliografie-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vezi_și" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Vezi_și"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Vezi și</span> </div> </a> <ul id="toc-Vezi_și-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Legături_externe" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Legături_externe"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Legături externe</span> </div> </a> <ul id="toc-Legături_externe-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="Cuprins" 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 " 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Disponibil în 59 limbi" > <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-59" 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">59 limbi</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%8A%D9%83%D8%A7%D9%86%D9%8A%D9%83%D8%A7_%D8%A5%D8%AD%D8%B5%D8%A7%D8%A6%D9%8A%D8%A9" title="ميكانيكا إحصائية – arabă" lang="ar" hreflang="ar" data-title="ميكانيكا إحصائية" data-language-autonym="العربية" data-language-local-name="arabă" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-as mw-list-item"><a href="https://as.wikipedia.org/wiki/%E0%A6%AA%E0%A7%B0%E0%A6%BF%E0%A6%B8%E0%A6%BE%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%B2_%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="পৰিসাংখ্যিক বল বিজ্ঞান – asameză" lang="as" hreflang="as" data-title="পৰিসাংখ্যিক বল বিজ্ঞান" data-language-autonym="অসমীয়া" data-language-local-name="asameză" class="interlanguage-link-target"><span>অসমীয়া</span></a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D1%8B%D1%81%D1%82%D1%8B%D1%87%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D1%96%D0%BA%D0%B0" title="Статыстычная механіка – belarusă" lang="be" hreflang="be" data-title="Статыстычная механіка" data-language-autonym="Беларуская" data-language-local-name="belarusă" class="interlanguage-link-target"><span>Беларуская</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Статистическа механика – bulgară" lang="bg" hreflang="bg" data-title="Статистическа механика" data-language-autonym="Български" data-language-local-name="bulgară" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bn mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%AA%E0%A6%B0%E0%A6%BF%E0%A6%B8%E0%A6%82%E0%A6%96%E0%A7%8D%E0%A6%AF%E0%A6%BE%E0%A6%A8%E0%A6%BF%E0%A6%95_%E0%A6%AC%E0%A6%B2%E0%A6%AC%E0%A6%BF%E0%A6%9C%E0%A7%8D%E0%A6%9E%E0%A6%BE%E0%A6%A8" title="পরিসংখ্যানিক বলবিজ্ঞান – bengaleză" lang="bn" hreflang="bn" data-title="পরিসংখ্যানিক বলবিজ্ঞান" data-language-autonym="বাংলা" data-language-local-name="bengaleză" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Mec%C3%A0nica_estad%C3%ADstica" title="Mecànica estadística – catalană" lang="ca" hreflang="ca" data-title="Mecànica estadística" data-language-autonym="Català" data-language-local-name="catalană" 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%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%C4%83_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Статистикăллă механика – ciuvașă" lang="cv" hreflang="cv" data-title="Статистикăллă механика" data-language-autonym="Чӑвашла" data-language-local-name="ciuvașă" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cy mw-list-item"><a href="https://cy.wikipedia.org/wiki/Mecaneg_ystadegol" title="Mecaneg ystadegol – galeză" lang="cy" hreflang="cy" data-title="Mecaneg ystadegol" data-language-autonym="Cymraeg" data-language-local-name="galeză" class="interlanguage-link-target"><span>Cymraeg</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Statistisk_mekanik" title="Statistisk mekanik – daneză" lang="da" hreflang="da" data-title="Statistisk mekanik" data-language-autonym="Dansk" data-language-local-name="daneză" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Statistische_Mechanik" title="Statistische Mechanik – germană" lang="de" hreflang="de" data-title="Statistische Mechanik" data-language-autonym="Deutsch" data-language-local-name="germană" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%84%CE%B1%CF%84%CE%B9%CF%83%CF%84%CE%B9%CE%BA%CE%AE_%CE%BC%CE%B7%CF%87%CE%B1%CE%BD%CE%B9%CE%BA%CE%AE" title="Στατιστική μηχανική – greacă" lang="el" hreflang="el" data-title="Στατιστική μηχανική" data-language-autonym="Ελληνικά" data-language-local-name="greacă" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Statistical_mechanics" title="Statistical mechanics – engleză" lang="en" hreflang="en" data-title="Statistical mechanics" data-language-autonym="English" data-language-local-name="engleză" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Statistika_mekaniko" title="Statistika mekaniko – esperanto" lang="eo" hreflang="eo" data-title="Statistika mekaniko" data-language-autonym="Esperanto" data-language-local-name="esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Mec%C3%A1nica_estad%C3%ADstica" title="Mecánica estadística – spaniolă" lang="es" hreflang="es" data-title="Mecánica estadística" data-language-autonym="Español" data-language-local-name="spaniolă" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Statistiline_mehaanika" title="Statistiline mehaanika – estonă" lang="et" hreflang="et" data-title="Statistiline mehaanika" data-language-autonym="Eesti" data-language-local-name="estonă" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Mekanika_estatistiko" title="Mekanika estatistiko – bască" lang="eu" hreflang="eu" data-title="Mekanika estatistiko" data-language-autonym="Euskara" data-language-local-name="bască" 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%85%DA%A9%D8%A7%D9%86%DB%8C%DA%A9_%D8%A2%D9%85%D8%A7%D8%B1%DB%8C" title="مکانیک آماری – persană" lang="fa" hreflang="fa" data-title="مکانیک آماری" data-language-autonym="فارسی" data-language-local-name="persană" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fi badge-Q70893996 mw-list-item" title=""><a href="https://fi.wikipedia.org/wiki/Statistinen_mekaniikka" title="Statistinen mekaniikka – finlandeză" lang="fi" hreflang="fi" data-title="Statistinen mekaniikka" data-language-autonym="Suomi" data-language-local-name="finlandeză" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-fr badge-Q70893996 mw-list-item" title=""><a href="https://fr.wikipedia.org/wiki/M%C3%A9canique_statistique" title="Mécanique statistique – franceză" lang="fr" hreflang="fr" data-title="Mécanique statistique" data-language-autonym="Français" data-language-local-name="franceză" 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/Meicnic_staitisti%C3%BAil" title="Meicnic staitistiúil – irlandeză" lang="ga" hreflang="ga" data-title="Meicnic staitistiúil" data-language-autonym="Gaeilge" data-language-local-name="irlandeză" 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/Mec%C3%A1nica_estat%C3%ADstica" title="Mecánica estatística – galiciană" lang="gl" hreflang="gl" data-title="Mecánica estatística" data-language-autonym="Galego" data-language-local-name="galiciană" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%BE%E0%A4%82%E0%A4%96%E0%A5%8D%E0%A4%AF%E0%A4%BF%E0%A4%95%E0%A5%80%E0%A4%AF_%E0%A4%AF%E0%A4%BE%E0%A4%82%E0%A4%A4%E0%A5%8D%E0%A4%B0%E0%A4%BF%E0%A4%95%E0%A5%80" title="सांख्यिकीय यांत्रिकी – hindi" lang="hi" hreflang="hi" data-title="सांख्यिकीय यांत्रिकी" 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/Statisti%C4%8Dka_mehanika" title="Statistička mehanika – croată" lang="hr" hreflang="hr" data-title="Statistička mehanika" data-language-autonym="Hrvatski" data-language-local-name="croată" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Statisztikus_mechanika" title="Statisztikus mechanika – maghiară" lang="hu" hreflang="hu" data-title="Statisztikus mechanika" data-language-autonym="Magyar" data-language-local-name="maghiară" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%8E%D5%AB%D5%B3%D5%A1%D5%AF%D5%A1%D5%A3%D6%80%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B4%D5%A5%D5%AD%D5%A1%D5%B6%D5%AB%D5%AF%D5%A1" title="Վիճակագրական մեխանիկա – armeană" lang="hy" hreflang="hy" data-title="Վիճակագրական մեխանիկա" data-language-autonym="Հայերեն" data-language-local-name="armeană" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Mekanika_statistika" title="Mekanika statistika – indoneziană" lang="id" hreflang="id" data-title="Mekanika statistika" data-language-autonym="Bahasa Indonesia" data-language-local-name="indoneziană" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Safne%C3%B0lisfr%C3%A6%C3%B0i" title="Safneðlisfræði – islandeză" lang="is" hreflang="is" data-title="Safneðlisfræði" data-language-autonym="Íslenska" data-language-local-name="islandeză" 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/Meccanica_statistica" title="Meccanica statistica – italiană" lang="it" hreflang="it" data-title="Meccanica statistica" data-language-autonym="Italiano" data-language-local-name="italiană" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%B5%B1%E8%A8%88%E5%8A%9B%E5%AD%A6" title="統計力学 – japoneză" lang="ja" hreflang="ja" data-title="統計力学" data-language-autonym="日本語" data-language-local-name="japoneză" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Статистикалық механика – kazahă" lang="kk" hreflang="kk" data-title="Статистикалық механика" data-language-autonym="Қазақша" data-language-local-name="kazahă" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%86%B5%EA%B3%84%EC%97%AD%ED%95%99" title="통계역학 – coreeană" lang="ko" hreflang="ko" data-title="통계역학" data-language-autonym="한국어" data-language-local-name="coreeană" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Statistin%C4%97_mechanika" title="Statistinė mechanika – lituaniană" lang="lt" hreflang="lt" data-title="Statistinė mechanika" data-language-autonym="Lietuvių" data-language-local-name="lituaniană" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%BA%D0%B0_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Статистичка механика – macedoneană" lang="mk" hreflang="mk" data-title="Статистичка механика" data-language-autonym="Македонски" data-language-local-name="macedoneană" 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%B8%E0%B4%BE%E0%B4%82%E0%B4%96%E0%B5%8D%E0%B4%AF%E0%B4%BF%E0%B4%95%E0%B4%AC%E0%B4%B2%E0%B4%A4%E0%B4%A8%E0%B5%8D%E0%B4%A4%E0%B5%8D%E0%B4%B0%E0%B4%82" title="സാംഖ്യികബലതന്ത്രം – malayalam" lang="ml" hreflang="ml" data-title="സാംഖ്യികബലതന്ത്രം" data-language-autonym="മലയാളം" data-language-local-name="malayalam" class="interlanguage-link-target"><span>മലയാളം</span></a></li><li class="interlanguage-link interwiki-mr mw-list-item"><a href="https://mr.wikipedia.org/wiki/%E0%A4%B8%E0%A4%B0%E0%A4%BE%E0%A4%B8%E0%A4%B0%E0%A5%80_%E0%A4%AF%E0%A4%BE%E0%A4%AE%E0%A4%BF%E0%A4%95%E0%A5%80" title="सरासरी यामिकी – marathi" lang="mr" hreflang="mr" data-title="सरासरी यामिकी" data-language-autonym="मराठी" data-language-local-name="marathi" class="interlanguage-link-target"><span>मराठी</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Mekanik_statistik" title="Mekanik statistik – malaeză" lang="ms" hreflang="ms" data-title="Mekanik statistik" data-language-autonym="Bahasa Melayu" data-language-local-name="malaeză" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Statistisk_mekanikk" title="Statistisk mekanikk – norvegiană nynorsk" lang="nn" hreflang="nn" data-title="Statistisk mekanikk" data-language-autonym="Norsk nynorsk" data-language-local-name="norvegiană nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%B8%E0%A8%9F%E0%A9%88%E0%A8%9F%E0%A8%BF%E0%A8%B8%E0%A8%9F%E0%A9%80%E0%A8%95%E0%A8%B2_%E0%A8%AE%E0%A8%95%E0%A9%88%E0%A8%A8%E0%A8%BF%E0%A8%95%E0%A8%B8" title="ਸਟੈਟਿਸਟੀਕਲ ਮਕੈਨਿਕਸ – punjabi" lang="pa" hreflang="pa" data-title="ਸਟੈਟਿਸਟੀਕਲ ਮਕੈਨਿਕਸ" 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/Mechanika_statystyczna" title="Mechanika statystyczna – poloneză" lang="pl" hreflang="pl" data-title="Mechanika statystyczna" data-language-autonym="Polski" data-language-local-name="poloneză" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ps mw-list-item"><a href="https://ps.wikipedia.org/wiki/%D8%A7%D8%AD%D8%B5%D8%A7%DB%8C%D9%88%D9%8A_%D9%85%DB%8C%D8%AE%D8%A7%D9%86%DB%8C%DA%A9" title="احصایوي میخانیک – paștună" lang="ps" hreflang="ps" data-title="احصایوي میخانیک" data-language-autonym="پښتو" data-language-local-name="paștună" class="interlanguage-link-target"><span>پښتو</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Mec%C3%A2nica_estat%C3%ADstica" title="Mecânica estatística – portugheză" lang="pt" hreflang="pt" data-title="Mecânica estatística" data-language-autonym="Português" data-language-local-name="portugheză" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D1%82%D0%B0%D1%82%D0%B8%D1%81%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%B5%D1%85%D0%B0%D0%BD%D0%B8%D0%BA%D0%B0" title="Статистическая механика – rusă" lang="ru" hreflang="ru" data-title="Статистическая механика" data-language-autonym="Русский" data-language-local-name="rusă" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-scn mw-list-item"><a href="https://scn.wikipedia.org/wiki/Micc%C3%A0nica_stat%C3%ACstica" title="Miccànica statìstica – siciliană" lang="scn" hreflang="scn" data-title="Miccànica statìstica" data-language-autonym="Sicilianu" data-language-local-name="siciliană" class="interlanguage-link-target"><span>Sicilianu</span></a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D8%B4%D9%85%D8%A7%D8%B1%D9%8A%D8%A7%D8%AA%D9%8A_%D9%85%D9%8A%DA%AA%D8%A7%D9%86%DA%AA%D8%B3" title="شمارياتي ميڪانڪس – sindhi" lang="sd" hreflang="sd" data-title="شمارياتي ميڪانڪس" data-language-autonym="سنڌي" data-language-local-name="sindhi" class="interlanguage-link-target"><span>سنڌي</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Statisti%C4%8Dka_mehanika" title="Statistička mehanika – sârbo-croată" lang="sh" hreflang="sh" data-title="Statistička mehanika" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="sârbo-croată" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Statistical_mechanics" title="Statistical mechanics – Simple English" lang="en-simple" hreflang="en-simple" data-title="Statistical mechanics" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li 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Apăsați aici pentru mai multe informații." src="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/20px-Symbol_support_vote.svg.png" decoding="async" width="20" height="21" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/30px-Symbol_support_vote.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/94/Symbol_support_vote.svg/40px-Symbol_support_vote.svg.png 2x" data-file-width="180" data-file-height="185" /></a></span></span></div></div> </div> <div id="siteSub" class="noprint">De la Wikipedia, enciclopedia liberă</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="ro" dir="ltr"><figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Gibbs-Elementary_principles_in_statistical_mechanics.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Gibbs-Elementary_principles_in_statistical_mechanics.png/330px-Gibbs-Elementary_principles_in_statistical_mechanics.png" decoding="async" width="330" height="526" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Gibbs-Elementary_principles_in_statistical_mechanics.png/495px-Gibbs-Elementary_principles_in_statistical_mechanics.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Gibbs-Elementary_principles_in_statistical_mechanics.png/660px-Gibbs-Elementary_principles_in_statistical_mechanics.png 2x" data-file-width="918" data-file-height="1463" /></a><figcaption>Tratatul <i>Principii elementare în mecanica statistică</i>, publicat de Gibbs în 1902, prezintă „fundamentarea rațională a termodinamicii”.</figcaption></figure> <table class="vertical-navbox nowraplinks" style="float:right;clear:right;width:22.0em;margin:0 0 1.0em 1.0em;color:var(--color-base, #000) !important;border:1px solid #aaa;padding:0.2em;border-spacing:0.4em 0;text-align:center;line-height:1.4em;font-size:88%"><tbody><tr><td style="padding-top:0.4em;line-height:1.2em">Parte a seriei de articole despre</td></tr><tr><th style="padding:0.2em 0.4em 0.2em;padding-top:0;font-size:145%;line-height:1.2em;padding-left:0.9em;padding-right:0.9em;"><a href="/wiki/Mecanic%C4%83_clasic%C4%83" title="Mecanică clasică">Mecanică clasică</a></th></tr><tr><td style="padding:0.2em 0 0.4em"><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 {\vec {F}}=m{\vec {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>F</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>a</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {F}}=m{\vec {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b202a8eaba4b424be52bcbaa043727b6ad9860" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.14ex; height:2.843ex;" alt="{\displaystyle {\vec {F}}=m{\vec {a}}}"></span><div style="padding-top:0.2em;line-height:1.2em;font-size:90%;padding:0.6em 0;font-style:italic;"><a href="/wiki/Legile_lui_Newton#Principiul_al_II-lea_al_mecanicii" title="Legile lui Newton">Principiul al II-lea al mecanicii</a></div></td></tr><tr><th style="padding:0.1em;background:#ddf; display:block;margin-bottom:1.0em;"> <div class="hlist"> <ul><li><a href="/wiki/Istoria_mecanicii_clasice" title="Istoria mecanicii clasice">Istorie</a></li> <li><a href="/wiki/Cronologia_mecanicii_clasice" title="Cronologia mecanicii clasice">Cronologie</a></li></ul> </div></th></tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Ramuri</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Cinematic%C4%83" title="Cinematică">Cinematică</a></li> <li><a href="/wiki/Dinamic%C4%83" title="Dinamică">Dinamică</a></li> <li><a href="/wiki/Mecanic%C4%83" title="Mecanică">Mecanică</a> <ul><li><a href="/wiki/Mecanic%C4%83_cereasc%C4%83" title="Mecanică cerească">cerească</a></li> <li><a href="/w/index.php?title=Mecanic%C4%83_aplicat%C4%83&action=edit&redlink=1" class="new" title="Mecanică aplicată — pagină inexistentă">aplicată</a></li> <li><a href="/wiki/Mecanica_mediilor_continue" title="Mecanica mediilor continue">a mediilor continue</a></li> <li><a class="mw-selflink selflink">statistică</a></li></ul></li> <li><a href="/wiki/Static%C4%83" title="Statică">Statică</a></li></ul> </div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Concepte</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Accelera%C8%9Bie" title="Accelerație">Accelerație</a></li> <li><a href="/wiki/Energie" title="Energie">Energie</a> <ul><li><a href="/wiki/Energie_cinetic%C4%83" title="Energie cinetică">cinetică</a></li> <li><a href="/wiki/Energie_poten%C8%9Bial%C4%83" title="Energie potențială">potențială</a></li> <li><a href="/wiki/Energie_cinetic%C4%83_de_rota%C8%9Bie" title="Energie cinetică de rotație">de rotație</a></li></ul></li> <li><a href="/wiki/For%C8%9B%C4%83" title="Forță">Forță</a> <ul><li><a href="/wiki/For%C8%9Ba_aparent%C4%83" title="Forța aparentă">aparentă</a></li> <li><a href="/wiki/For%C8%9B%C4%83_de_contact" title="Forță de contact">contact</a></li> <li><a href="/wiki/Frecare" title="Frecare">frecare</a></li> <li><a href="/wiki/For%C8%9B%C4%83_de_suprafa%C8%9B%C4%83" title="Forță de suprafață">suprafață</a></li> <li><a href="/wiki/For%C8%9B%C4%83_de_volum" title="Forță de volum">volum</a></li></ul></li> <li><a href="/wiki/Impuls" title="Impuls">Impuls</a></li> <li><a href="/wiki/Iner%C8%9Bie_(fizic%C4%83)" title="Inerție (fizică)">Inerție</a></li> <li><a href="/wiki/Lucru_mecanic" title="Lucru mecanic">Lucru mecanic</a> <ul><li><a href="/w/index.php?title=Lucru_mecanic_virtual&action=edit&redlink=1" class="new" title="Lucru mecanic virtual — pagină inexistentă">virtual</a></li></ul></li> <li><a href="/wiki/Mas%C4%83" title="Masă">Masă</a></li> <li><a href="/wiki/Momentul_for%C8%9Bei" title="Momentul forței">Moment</a> <ul><li><a href="/wiki/Moment_cinetic" title="Moment cinetic">cinetic</a></li> <li><a href="/wiki/Moment_de_iner%C8%9Bie" title="Moment de inerție">de inerție</a></li></ul></li> <li><a href="/w/index.php?title=Principiul_lui_D%27Alembert&action=edit&redlink=1" class="new" title="Principiul lui D'Alembert — pagină inexistentă">Principiul lui D'Alembert</a></li> <li><a href="/wiki/Putere" title="Putere">Putere</a></li> <li><a href="/wiki/Randament" title="Randament">Randament</a> <ul><li><a href="/wiki/Randament_mecanic" title="Randament mecanic">mecanic</a></li></ul></li> <li><a href="/wiki/Sistem_de_referin%C8%9B%C4%83" title="Sistem de referință">Sistem de referință</a> <ul><li><a href="/wiki/Sistem_de_referin%C8%9B%C4%83_iner%C8%9Bial" title="Sistem de referință inerțial">inerțial</a></li> <li><a href="/wiki/Sistem_de_referin%C8%9B%C4%83#Sistem_de_referință_neinerțial" title="Sistem de referință">neinerțial</a></li></ul></li> <li><a href="/wiki/Spa%C8%9Biu" title="Spațiu">Spațiu</a></li> <li><a href="/wiki/Timp" title="Timp">Timp</a></li> <li><a href="/wiki/Vitez%C4%83" title="Viteză">Viteză</a> <ul><li><a href="/w/index.php?title=Vitez%C4%83_relativ%C4%83&action=edit&redlink=1" class="new" title="Viteză relativă — pagină inexistentă">relativă</a></li></ul></li></ul> </div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Formulări</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"> <ul><li><div style="padding:0.1em 0;line-height:1.2em;"><b><a href="/wiki/Legile_lui_Newton" title="Legile lui Newton">Legile lui Newton</a></b></div></li> <li><div style="padding:0.1em 0;line-height:1.2em;"><b><a href="/wiki/Mecanic%C4%83_analitic%C4%83" class="mw-redirect" title="Mecanică analitică">Mecanică analitică</a></b> <ul style="list-style:none none; padding:0px; margin:0px"><li><a href="/wiki/Mecanic%C4%83_lagrangian%C4%83" title="Mecanică lagrangiană">Mecanică lagrangiană</a> </li><li> <a href="/wiki/Mecanic%C4%83_hamiltonian%C4%83" title="Mecanică hamiltoniană">Mecanică hamiltoniană</a> </li><li> <a href="/wiki/Ecua%C8%9Bia_Hamilton%E2%80%93Jacobi" title="Ecuația Hamilton–Jacobi">Ecuația Hamilton–Jacobi</a> </li></ul></div></li></ul> </div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Subiecte de bază</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/w/index.php?title=Deplasare&action=edit&redlink=1" class="new" title="Deplasare — pagină inexistentă">Deplasare</a></li> <li><a href="/wiki/Legea_atrac%C8%9Biei_universale" title="Legea atracției universale">Legea atracției universale</a></li> <li><a href="/wiki/Mi%C8%99care_(fizic%C4%83)" title="Mișcare (fizică)">Mișcare</a> <ul><li><a href="/wiki/Ecua%C8%9Bii_de_mi%C8%99care" title="Ecuații de mișcare">ecuații</a></li> <li><a href="/w/index.php?title=Mi%C8%99care_armonic%C4%83_simpl%C4%83&action=edit&redlink=1" class="new" title="Mișcare armonică simplă — pagină inexistentă">armonică</a></li> <li><a href="/wiki/Mi%C8%99care_rectilinie" title="Mișcare rectilinie">rectilinie</a></li></ul></li> <li><a href="/wiki/Solid_rigid" title="Solid rigid">Solid rigid</a></li></ul> </div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;"><a href="/wiki/Rota%C8%9Bie" title="Rotație">Rotație</a></div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Rota%C8%9Bie" title="Rotație">Rotație</a></li> <li><a href="/w/index.php?title=Sistem_de_referin%C8%9B%C4%83_%C3%AEn_rota%C8%9Bie&action=edit&redlink=1" class="new" title="Sistem de referință în rotație — pagină inexistentă">Sistem de referință în rotație</a></li> <li><a href="/wiki/Vitez%C4%83_unghiular%C4%83" title="Viteză unghiulară">Viteză unghiulară</a></li> <li><a href="/wiki/Accelera%C8%9Bie_unghiular%C4%83" title="Accelerație unghiulară">Accelerație unghiulară</a></li> <li><a href="/w/index.php?title=For%C8%9B%C4%83_centripet%C4%83&action=edit&redlink=1" class="new" title="Forță centripetă — pagină inexistentă">Forță centripetă</a> <ul><li><a href="/wiki/For%C8%9Ba_centrifug%C4%83" title="Forța centrifugă">centrifugă</a></li></ul></li> <li><a href="/wiki/For%C8%9Ba_Coriolis" title="Forța Coriolis">Forța Coriolis</a></li> <li><a href="/wiki/Frecven%C8%9B%C4%83" title="Frecvență">Frecvență</a></li> <li><a href="/wiki/Oscila%C8%9Bie" title="Oscilație">Oscilație</a> <ul><li><a href="/wiki/Oscilator_armonic" title="Oscilator armonic">oscilator armonic</a></li> <li><a href="/w/index.php?title=Amortizare&action=edit&redlink=1" class="new" title="Amortizare — pagină inexistentă">amortizare</a></li></ul></li> <li><a href="/wiki/Pendul_gravita%C8%9Bional" title="Pendul gravitațional">Pendul</a> <ul><li><a href="/wiki/Pendul_conic" title="Pendul conic">conic</a></li></ul></li> <li><a href="/wiki/Vibra%C8%9Bie" title="Vibrație">Vibrație</a></li></ul> </div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Oameni de știință</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <ul><li><a href="/wiki/Galileo_Galilei" title="Galileo Galilei">Galileo</a></li> <li><a href="/wiki/Christiaan_Huygens" title="Christiaan Huygens">Huygens</a></li> <li><a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton</a></li> <li><a href="/wiki/Johannes_Kepler" title="Johannes Kepler">Kepler</a></li> <li><a href="/wiki/Jeremiah_Horrocks" title="Jeremiah Horrocks">Horrocks</a></li> <li><a href="/wiki/Edmond_Halley" title="Edmond Halley">Halley</a></li> <li><a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Euler</a></li> <li><a href="/wiki/Jean_le_Rond_d%27Alembert" class="mw-redirect" title="Jean le Rond d'Alembert">d'Alembert</a></li> <li><a href="/wiki/Alexis_Clairaut" class="mw-redirect" title="Alexis Clairaut">Clairaut</a></li> <li><a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Lagrange</a></li> <li><a href="/wiki/Pierre-Simon_Laplace" class="mw-redirect" title="Pierre-Simon Laplace">Laplace</a></li> <li><a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a></li> <li><a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Poisson</a></li> <li><a href="/wiki/Daniel_Bernoulli" title="Daniel Bernoulli">Daniel Bernoulli</a></li> <li><a href="/wiki/Johann_Bernoulli" title="Johann Bernoulli">Johann Bernoulli</a></li> <li><a href="/wiki/Augustin-Louis_Cauchy" class="mw-redirect" title="Augustin-Louis Cauchy">Cauchy</a></li></ul> </div></div></div></td> </tr><tr><td style="padding:0 0.1em 0.4em"> <div class="NavFrame collapsed" style="border:none;padding:0"><div class="NavHead" style="font-size:105%;background:inherit;color:var(--color-base, #000) !important;text-align:left;background:#ddf;text-align:center;">Categorii</div><div class="NavContent plainlist" style="font-size:105%;padding:0.2em 0 0.4em;text-align:center;padding-top:0.35em;"><div class="hlist"> <div class="CategoryTreeTag" data-ct-options="{"mode":0,"hideprefix":20,"showcount":false,"namespaces":false,"notranslations":false}"><div class="CategoryTreeSection"><div class="CategoryTreeItem"><span class="CategoryTreeBullet"><a class="CategoryTreeToggle" data-ct-title="Mecanică_clasică" aria-expanded="false"></a> </span> <bdi dir="ltr"><a href="/wiki/Categorie:Mecanic%C4%83_clasic%C4%83" title="Categorie:Mecanică clasică">Mecanică clasică</a></bdi></div><div class="CategoryTreeChildren" style="display:none"></div></div></div> </div></div></div></td> </tr><tr><td style="text-align:right;font-size:115%;padding-top: 0.6em;padding-top:0.15em;"><div class="plainlinks hlist navbar mini"><ul><li class="nv-view"><a href="/wiki/Format:Mecanic%C4%83_clasic%C4%83" title="Format:Mecanică clasică"><abbr title="Vizualizează acest format">v</abbr></a></li><li class="nv-talk"><a href="/w/index.php?title=Discu%C8%9Bie_Format:Mecanic%C4%83_clasic%C4%83&action=edit&redlink=1" class="new" title="Discuție Format:Mecanică clasică — pagină inexistentă"><abbr title="Discută acest format">d</abbr></a></li><li class="nv-edit"><a class="external text" href="https://ro.wikipedia.org/w/index.php?title=Format:Mecanic%C4%83_clasic%C4%83&action=edit"><abbr title="Modifică acest format">m</abbr></a></li></ul></div></td></tr></tbody></table> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:James_clerk_maxwell.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/James_clerk_maxwell.jpg/160px-James_clerk_maxwell.jpg" decoding="async" width="160" height="215" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ec/James_clerk_maxwell.jpg/240px-James_clerk_maxwell.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ec/James_clerk_maxwell.jpg/320px-James_clerk_maxwell.jpg 2x" data-file-width="591" data-file-height="795" /></a><figcaption>Maxwell</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Boltzmann2.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/160px-Boltzmann2.jpg" decoding="async" width="160" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/240px-Boltzmann2.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ad/Boltzmann2.jpg/320px-Boltzmann2.jpg 2x" data-file-width="490" data-file-height="600" /></a><figcaption>Boltzmann</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Josiah_Willard_Gibbs_-from_MMS-.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Josiah_Willard_Gibbs_-from_MMS-.jpg/160px-Josiah_Willard_Gibbs_-from_MMS-.jpg" decoding="async" width="160" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Josiah_Willard_Gibbs_-from_MMS-.jpg/240px-Josiah_Willard_Gibbs_-from_MMS-.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Josiah_Willard_Gibbs_-from_MMS-.jpg/320px-Josiah_Willard_Gibbs_-from_MMS-.jpg 2x" data-file-width="888" data-file-height="1184" /></a><figcaption>Gibbs</figcaption></figure> <p><b>Mecanica statistică</b>, numită uneori și <i>termodinamică statistică</i>, utilizează <a href="/wiki/Statistic%C4%83" title="Statistică">metode statistice</a> pentru a deduce proprietățile și comportarea sistemelor fizice <a href="/wiki/Scar%C4%83_macroscopic%C4%83" title="Scară macroscopică">macroscopice</a>, la <a href="/wiki/Termodinamic%C4%83#Stări_și_transformări" title="Termodinamică">echilibru termodinamic</a>, pe baza structurii lor <a href="/wiki/Scar%C4%83_microscopic%C4%83" title="Scară microscopică">microscopice</a>. Metodele statistice au fost introduse în acest context de <a href="/wiki/James_Clerk_Maxwell" title="James Clerk Maxwell">Maxwell</a> într-o serie de trei articole (1860–1879) și de <a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann</a> într-o serie de patru articole (1870–1884), care au pus bazele <a href="/wiki/Teoria_cinetic%C4%83_a_gazelor" title="Teoria cinetică a gazelor">teoriei cinetice a gazelor</a>. Mecanica statistică clasică a fost fundamentată de <a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs</a> (1902); ulterior, descrierea stărilor microscopice pe baza <a href="/wiki/Mecanic%C4%83_clasic%C4%83" title="Mecanică clasică">mecanicii clasice</a> a fost corectată și completată conform <a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">mecanicii cuantice</a>. <i><a href="/wiki/Termodinamic%C4%83" title="Termodinamică">Termodinamica</a></i>, <i><a href="/wiki/Teorie_cinetic%C4%83" class="mw-redirect" title="Teorie cinetică">teoria cinetică</a></i> și <i>mecanica statistică</i> sunt discipline înrudite prin obiectul de studiu, dar care diferă prin metodele utilizate; adeseori, ele sunt prezentate împreună, sub denumirea de <a href="/wiki/Fizic%C4%83_statistic%C4%83" title="Fizică statistică">fizică statistică</a>. </p><p><a href="/wiki/Principiile_termodinamicii" title="Principiile termodinamicii">Principiile termodinamicii</a>, rezultate din generalizarea și abstractizarea unor date empirice, exprimă proprietățile aproximative și comportarea probabilă a unor sisteme macroscopice, alcătuite dintr-un număr foarte mare de componente microscopice: <a href="/wiki/Molecul%C4%83" title="Moleculă">molecule</a> și <a href="/wiki/Atom" title="Atom">atomi</a>. Legile mecanicii permit în principiu determinarea completă a stării unui sistem alcătuit din mai multe componente, la orice moment, dacă sunt cunoscute interacțiunile (forțele), precum și starea sistemului (coordonatele și impulsurile componentelor) la un moment anterior. În practică însă, condițiile inițiale nu sunt cunoscute, iar integrarea ecuațiilor de mișcare, pentru un număr foarte mare de componente, se lovește de dificultăți de calcul. Tipic, numărul de molecule dintr-o masă macroscopică de <a href="/wiki/Gaz" title="Gaz">gaz</a>, în <a href="/wiki/Condi%C8%9Bii_standard_de_temperatur%C4%83_%C8%99i_presiune" title="Condiții standard de temperatură și presiune">condiții standard</a>, este de ordinul de mărime al <a href="/wiki/Num%C4%83rul_lui_Avogadro" title="Numărul lui Avogadro">numărului lui Avogadro</a>, adică 10<sup>23</sup>, ceea ce face ca determinarea stării sale mecanice (microscopice) să fie imposibilă. Pe de altă parte, experiența arată că proprietățile termodinamice (macroscopice) ale aceleiași mase de gaz sunt complet determinate de doar doi parametri (de exemplu, este suficientă cunoașterea <a href="/wiki/Energie_liber%C4%83" title="Energie liberă">energiei libere</a> ca funcție de <a href="/wiki/Volum" class="mw-redirect" title="Volum">volum</a> și <a href="/wiki/Temperatur%C4%83" title="Temperatură">temperatură</a>), iar unul dintre aceștia (în acest caz temperatura) nu este de natură mecanică. Legătura dintre aceste două puncte de vedere aparent contradictorii o realizează metodele statistice. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Principiile_mecanicii_statistice_clasice">Principiile mecanicii statistice clasice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=1" title="Modifică secțiunea: Principiile mecanicii statistice clasice" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=1" title="Edit section's source code: Principiile mecanicii statistice clasice"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Stări_microscopice"><span id="St.C4.83ri_microscopice"></span>Stări microscopice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=2" title="Modifică secțiunea: Stări microscopice" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=2" title="Edit section's source code: Stări microscopice"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În mecanica statistică, obiectul de studiu este un sistem (macroscopic) compus dintr-un număr (mare) de subsisteme (microscopice) care interacționează (între ele și cu lumea exterioară) după legi cunoscute. Forțele, atât cele <i>interioare</i> cât și cele <i>exterioare</i>, sunt presupuse <i><a href="/wiki/For%C8%9B%C4%83#Forțe_conservative" title="Forță">conservative</a></i>, adică <a href="/wiki/Energie_mecanic%C4%83" title="Energie mecanică">energia mecanică</a> totală a sistemului (suma dintre <a href="/wiki/Energie_cinetic%C4%83" title="Energie cinetică">energia cinetică</a> și <a href="/wiki/Energie_poten%C8%9Bial%C4%83" title="Energie potențială">energia potențială</a>) rămâne constantă în timpul mișcării. Această ipoteză ilustrează punctul de vedere conform căruia forțele <a href="/wiki/For%C8%9B%C4%83#Forțe_neconservative" title="Forță">neconservative</a>, care produc <a href="/wiki/Disipare" title="Disipare">disiparea</a> energiei sub formă de <a href="/wiki/C%C4%83ldur%C4%83" title="Căldură">căldură</a> (cum sunt forțele de <a href="/wiki/Frecare" title="Frecare">frecare</a>), se manifestă doar la scară macroscopică și sunt consecința interacțiunilor la scară microscopică. </p><p>Este convenabilă scrierea ecuațiilor de mișcare sub <i>forma canonică</i> utilizată în <a href="/wiki/Mecanic%C4%83_hamiltonian%C4%83" title="Mecanică hamiltoniană">mecanica hamiltoniană</a>. Starea unui sistem cu <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"> <mtext> </mtext> <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/eaf8b0f621a23f81aa20d63b5cd59d3dcad83ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ n}"></span> grade de libertate microscopice este caracterizată, la orice moment, prin valorile pe care le iau <i>coordonatele generalizate</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 q=\left(q_{1},\ldots ,q_{n}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q=\left(q_{1},\ldots ,q_{n}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45b4d6eb7b36a9b98e54dbf367671eddf293e6c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.502ex; height:2.843ex;" alt="{\displaystyle q=\left(q_{1},\ldots ,q_{n}\right)}"></span> și <i>impulsurile generalizate</i> conjugate <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=\left(p_{1},\dots ,p_{n}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=\left(p_{1},\dots ,p_{n}\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03d555e87fcada40cf617fb91eba167f502b7e50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:16.99ex; height:2.843ex;" alt="{\displaystyle p=\left(p_{1},\dots ,p_{n}\right).}"></span> Dinamica sistemului este descrisă de <i>ecuațiile canonice</i> ale lui <a href="/wiki/William_Rowan_Hamilton" title="William Rowan Hamilton">Hamilton</a>: </p> <dl><dd><span style="padding-right:4em" id="f1"><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 \left(1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>1</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e7a480f5a268e109a52735e63b1575178435307" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(1\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\dot {q_{i}}}={\frac {\partial H}{\partial p_{i}}},\quad {\dot {p_{i}}}=-{\frac {\partial H}{\partial q_{i}}},\quad \left(i=1,\ldots ,n\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>˙</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {q_{i}}}={\frac {\partial H}{\partial p_{i}}},\quad {\dot {p_{i}}}=-{\frac {\partial H}{\partial q_{i}}},\quad \left(i=1,\ldots ,n\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a851ff9281ce84ed9a5094d17f4aaeaa83cf757" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:41.439ex; height:5.843ex;" alt="{\displaystyle {\dot {q_{i}}}={\frac {\partial H}{\partial p_{i}}},\quad {\dot {p_{i}}}=-{\frac {\partial H}{\partial q_{i}}},\quad \left(i=1,\ldots ,n\right),}"></span></dd></dl> <p>unde punctul deasupra simbolului unei mărimi denotă derivata în raport cu timpul. Funcția <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(p,q\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(p,q\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6da86bad30557fa43bfeb80b3b4380b1bb73971" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.567ex; height:2.843ex;" alt="{\displaystyle H\left(p,q\right),}"></span> numită <i>hamiltoniană</i>, este energia totală a sistemului. În cazul forțelor conservative ea nu depinde explicit de timp, iar din ecuațiile de mișcare rezultă că dependența implicită de timp, prin intermediul variabilelor canonice, este doar aparentă, deci într-adevăr energia totală rămâne constantă: </p> <dl><dd><span style="padding-right:4em" id="f2"><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 \left(2\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>2</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(2\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d123bf4f8d1d5cbdb53228c892027f6ef510b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(2\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(p,q\right)=E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(p,q\right)=E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad31dffc7eb23bf6e903c7696f147fc1b96a5ead" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.054ex; height:2.843ex;" alt="{\displaystyle H\left(p,q\right)=E.}"></span></dd></dl> <p>În terminologia introdusă de Gibbs, o stare microscopică a sistemului se numește <i><a href="/wiki/Faz%C4%83_(mecanic%C4%83_statistic%C4%83)" title="Fază (mecanică statistică)">fază</a></i>; ea poate fi reprezentată geometric printr-un punct de coordonate <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 \left(p,q\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(p,q\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da0f21c9b45a084da5fc082b66fd0531d1460fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle \left(p,q\right)}"></span> într-un spațiu cu <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 \ 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe73c1158ef6b049b32da2687b756dbf79449982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle \ 2n}"></span> dimensiuni, numit <i><a href="/wiki/Spa%C8%9Biul_fazelor" class="mw-redirect" title="Spațiul fazelor">spațiul fazelor</a></i>. Evoluția în timp a sistemului, reprezentată analitic prin dependența de timp a variabilelor canonice, are ca reprezentare geometrică o curbă continuă în spațiul fazelor, numită <i>traiectoria</i> punctului reprezentativ. Întrucât starea sistemului, la orice moment, este complet determinată dacă este cunoscută starea sa la un moment anterior, rezultă că traiectoria în spațiul fazelor este complet determinată de unul din punctele ei și prin fiecare punct din spațiul fazelor trece o singură traiectorie. </p><p>Legea conservării energiei are și ea o reprezentare geometrică simplă: traiectoria punctului reprezentativ este în întregime conținută într-o suprafață de energie constantă, care e o varietate <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 \left(2n-1\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(2n-1\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fda1b7dd109aa9db016c11934f8450f5004b6e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.369ex; height:2.843ex;" alt="{\displaystyle \left(2n-1\right)}"></span>-dimensională în spațiul fazelor <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 \ 2n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe73c1158ef6b049b32da2687b756dbf79449982" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.138ex; height:2.176ex;" alt="{\displaystyle \ 2n}"></span>-dimensional, având ecuația <a href="#f2">(2)</a>. Pentru un sistem în echilibru termodinamic, punctul reprezentativ în spațiul fazelor nu se poate îndepărta la infinit, deci suprafețele de energie constantă nu au pânze care să se îndepărteze la infinit. Fiecare dintre ele e o suprafață închisă, întrucât ecuația <a href="#f2">(2)</a> reprezintă frontiera regiunii în care se află toate stările cu energie mai mică decât sau egală cu <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49028dadfd22a29fe14440e51a55e83cf3d54287" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.003ex; height:2.176ex;" alt="{\displaystyle \ E.}"></span> Volumul acestei regiuni este </p> <dl><dd><span style="padding-right:4em" id="f3"><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 \left(3\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>3</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(3\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5acebf7542a9c2596bd2c8a8c0484ef14c4f7ad2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(3\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \left(E\right)=\int _{H\left(p,q\right)\leq E}dp\,dq,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mi>E</mi> </mrow> </msub> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)=\int _{H\left(p,q\right)\leq E}dp\,dq,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e3d220a9887afdca13e258809ab463e2bc6c46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.678ex; height:6.009ex;" alt="{\displaystyle \Omega \left(E\right)=\int _{H\left(p,q\right)\leq E}dp\,dq,}"></span></dd></dl> <p>unde pentru <a href="/wiki/Element_de_volum" title="Element de volum">elementul de volum</a> în spațiul fazelor s-a folosit notația condensată <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 dp\,dq=dp_{1}\cdots dp_{n}\,dq_{1}\cdots dq_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mo>=</mo> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mi>d</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋯</mo> <mi>d</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp\,dq=dp_{1}\cdots dp_{n}\,dq_{1}\cdots dq_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa0222b73bc2219eefa33417eb70c55fb96c2775" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:30.007ex; height:2.509ex;" alt="{\displaystyle dp\,dq=dp_{1}\cdots dp_{n}\,dq_{1}\cdots dq_{n}.}"></span> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega \left(E\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega \left(E\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab95b1220d150818ca655e3491155e0468ce2923" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.65ex; height:2.843ex;" alt="{\displaystyle \Omega \left(E\right)}"></span> este o funcție monoton crescătoare de <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01358ea1a9419bfbfbaab2467c6db81c1a6b1fb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.356ex; height:2.176ex;" alt="{\displaystyle \ E}"></span>; pentru sisteme cu un număr mare de grade de libertate ea este o funcție foarte rapid crescătoare.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>O consecință importantă a ecuațiilor canonice, numită <i><a href="/wiki/Teorema_lui_Liouville_(mecanic%C4%83_statistic%C4%83)" title="Teorema lui Liouville (mecanică statistică)">teorema lui Liouville</a></i>, poate fi enunțată în modul următor:<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> Fie un domeniu arbitrar <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"></span> în spațiul fazelor; se consideră totalitatea punctelor <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 \left(p,q\right)\in {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(p,q\right)\in {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2c00518097883d8f443ff13199cb17205d3a357" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.715ex; height:2.843ex;" alt="{\displaystyle \left(p,q\right)\in {\mathcal {D}}}"></span> ca reprezentând stări mecanice ale sistemului la un moment inițial <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b2efca4774f7032b3f7ff6aeb1e086457544bd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.42ex; height:2.009ex;" alt="{\displaystyle \ t}"></span>; se urmărește evoluția acestor stări, conform ecuațiilor canonice; fie <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 \left(p^{\prime },q^{\prime }\right)\in {\mathcal {D}}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mo>,</mo> <msup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(p^{\prime },q^{\prime }\right)\in {\mathcal {D}}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/294ade9c490ec29d929b7c203ac74325bfd6b575" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.779ex; height:3.009ex;" alt="{\displaystyle \left(p^{\prime },q^{\prime }\right)\in {\mathcal {D}}^{\prime }}"></span> pozițiile punctelor considerate la un moment ulterior <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ t^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ t^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e03535a46580f40a5ae0c7fb0167ea4fc2689808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.105ex; height:2.509ex;" alt="{\displaystyle \ t^{\prime }}"></span>; atunci volumul domeniului <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 {\mathcal {D}}^{\prime }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}^{\prime }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9aa9604b7a8bf977f2d4e6621c693a95753f3566" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.477ex; height:2.509ex;" alt="{\displaystyle {\mathcal {D}}^{\prime }}"></span> este egal cu volumul domeniului <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 {\mathcal {D}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">D</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {D}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3277962e1959c3241fb1b70c7f0ac6dcefebd966" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle {\mathcal {D}}}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Colectiv_statistic">Colectiv statistic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=3" title="Modifică secțiunea: Colectiv statistic" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=3" title="Edit section's source code: Colectiv statistic"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Starea unui sistem macroscopic în echilibru termodinamic este caracterizată printr-un număr restrâns de parametri, pe când la scară microscopică există un număr enorm de stări mecanice distincte compatibile cu una și aceeași <a href="/wiki/Stare_termodinamic%C4%83" title="Stare termodinamică">stare termodinamică</a>. Gibbs a făcut sugestia că proprietățile termodinamice ale sistemului pot fi calculate, prin metode statistice, pornind de la această mulțime de stări microscopice.<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> Totalitatea stărilor mecanice compatibile cu o stare termodinamică dată alcătuiește un <i>colectiv statistic</i>, sau <i>ansamblu statistic</i>.<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> Întrucât într-o anumită determinare macroscopică doar una dintre aceste stări este efectiv realizată (celelalte reprezentând stări posibile care la rândul lor pot fi efectiv realizate dacă sistemul este readus în starea termodinamică inițială după transformări arbitrare), vorbim despre un colectiv statistic <i>virtual</i>. </p><p>Un colectiv statistic este reprezentat în spațiul fazelor printr-o mulțime de puncte a căror distribuție este descrisă de o <i>densitate de probabilitate</i>, sau <i>funcție de distribuție</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 {\mathcal {P}}\left(p,q\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/546d2cd9638fa1749d9e75920cad3097c99ebe2d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.173ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)}"></span> definită prin aceea că probabilitatea ca punctul reprezentativ al stării sistemului să se afle în interiorul volumului elementar <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 dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/686cf8f21160136aa0f1b492740290bc39c004b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.058ex; height:2.509ex;" alt="{\displaystyle dp\,dq}"></span> situat la coordonate canonice <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 \left(p,q\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(p,q\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6da0f21c9b45a084da5fc082b66fd0531d1460fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle \left(p,q\right)}"></span> este </p> <dl><dd><span style="padding-right:4em" id="f4"><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 \left(4\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>4</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(4\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d14514d5f2f06e22ecceaf0aa5f6eabe5f01128" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(4\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)dp\,dq.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)dp\,dq.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bac02172f71f46b4814c4358e96b2b25b0f292c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.265ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)dp\,dq.}"></span></dd></dl> <p>Densitatea de probabilitate este o funcție în spațiul fazelor, care nu poate lua valori negative și tinde spre zero la infinit. Integrala ei pe întreg spațiul fazelor satisface condiția </p> <dl><dd><span style="padding-right:4em" id="f5"><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 \left(5\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>5</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(5\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/929cf5508f2c9053cf0ea9e76586f1fd6ea2eddc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(5\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int {\mathcal {P}}\left(p,q\right)dp\,dq=1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int {\mathcal {P}}\left(p,q\right)dp\,dq=1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7b210b5460b760f7a816d3cd77eaf8dc032b995" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:20.106ex; height:5.676ex;" alt="{\displaystyle \int {\mathcal {P}}\left(p,q\right)dp\,dq=1,}"></span></dd></dl> <p>care rezultă din regula de sumare a probabilităților și exprimă certitudinea că punctul reprezentativ se află în spațiul fazelor. </p><p>Din teorema lui Liouville rezultă că densitatea de probabilitate este constantă de-a lungul unei traiectorii în spațiul fazelor; se spune că ea e o <i>integrală primă</i> a ecuațiilor canonice. Un sistem hamiltonian admite <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 \ 2n-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2n-1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8c686d586a165eb49da4e9e0d4cfcca1e19e21d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.141ex; height:2.343ex;" alt="{\displaystyle \ 2n-1}"></span> integrale prime care nu depind explicit de timp, una dintre ele fiind energia, adică hamiltoniana <a href="#f2">(2)</a>. Densitatea de probabilitate va fi deci o funcție de hamiltoniana <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(p,q\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(p,q\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3355128975046023a267117d20cb3776b033f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.533ex; height:2.843ex;" alt="{\displaystyle H\left(p,q\right)}"></span> și de alte <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 \ 2n-2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mn>2</mn> <mi>n</mi> <mo>−</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ 2n-2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45cb25f4c7e2ba88fb6950bcf2502cc10985c892" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.141ex; height:2.343ex;" alt="{\displaystyle \ 2n-2}"></span> integrale prime independente de timp. Pentru a reprezenta la scară microscopică stări de echilibru termodinamic, în care proprietățile sistemului sunt independente de timp și depind (la parametri externi constanți) numai de energie, în mecanica statistică se postulează că funcția de distribuție depinde de variabilele canonice numai prin intermediul funcției hamiltoniene:<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> <dl><dd><span style="padding-right:4em" id="f6"><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 \left(6\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>6</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(6\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2f3b13b080d81d56074405a8491b627ec182fa1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(6\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\mathfrak {P}}\left(H\left(p,q\right)\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="fraktur">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\mathfrak {P}}\left(H\left(p,q\right)\right).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/875ba9b2bda9cdd93708c488804b4c44ed4130f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.959ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\mathfrak {P}}\left(H\left(p,q\right)\right).}"></span></dd></dl> <p>Boltzmann a arătat că acest postulat se verifică în cazul sistemelor care posedă proprietatea de <i>ergodicitate</i>: oricare traiectorie în spațiul fazelor se apropie oricât de mult de oricare punct al suprafeței de energie constantă pe care se află în întregime această traiectorie. </p> <div class="mw-heading mw-heading3"><h3 id="Valori_medii_și_fluctuații"><span id="Valori_medii_.C8.99i_fluctua.C8.9Bii"></span>Valori medii și fluctuații</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=4" title="Modifică secțiunea: Valori medii și fluctuații" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=4" title="Edit section's source code: Valori medii și fluctuații"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mecanica statistică reprezintă un punct de vedere diferit, față de termodinamică, asupra valorilor mărimilor mecanice macroscopice la echilibru. În termodinamică, valoarea oricărei mărimi mecanice este univoc determinată dacă sunt cunoscute valorile unui număr restrâns de <a href="/wiki/Parametru_de_stare" class="mw-redirect" title="Parametru de stare">parametri de stare</a> independenți de timp: echilibrul termodinamic este <i>static</i>. În mecanica statistică, starea sistemului este descrisă de un colectiv statistic virtual, iar mărimile mecanice sunt funcții <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\left(p,q\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f\left(p,q\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a4f9932c698195f88ffcf8d479c69f70f28acd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.716ex; height:2.843ex;" alt="{\displaystyle \ f\left(p,q\right)\,}"></span> de variabilele canonice. Readucând sistemul, în mod repetat, în aceeași stare termodinamică, după transformări arbitrare, stările microscopice vor fi diferite, iar mărimea în discuție va avea, în general, valori diferite. La scară microscopică echilibrul termodinamic se manifestă ca o deplasare staționară a colectivului statistic în spațiul fazelor, conform teoremei lui Liouville: el nu este static, ci <i>statistic</i>. </p><p>În statistică, o mărime a cărei valoare numerică nu rezultă în mod univoc din determinarea ei în condiții specificate se numește <a href="/wiki/Variabil%C4%83_aleatoare" class="mw-redirect" title="Variabilă aleatoare">variabilă aleatorie</a>. Variabilei aleatorii <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fb2f6412b09af742fc1ee36c6f8ca23bd0c044" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.506ex; height:2.509ex;" alt="{\displaystyle \ f,}"></span> determinată pe colectivul statistic descris de funcția de distribuție <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 {\mathcal {P}}\left(p,q\right),\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right),\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a307dbefffe1b1a64aeca78d130a66947b7583c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.594ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right),\,}"></span> i se asociază <i>valoarea medie</i> <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> </p> <dl><dd><span style="padding-right:4em" id="f7"><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 \left(7\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>7</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(7\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b911bbe37c248c08d6aad218542bd6ddb13afbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(7\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle f\right\rangle =\int f\left(p,q\right)\,{\mathcal {P}}\left(p,q\right)dp\,dq\;,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mi>f</mi> <mo>⟩</mo> </mrow> <mo>=</mo> <mo>∫</mo> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mspace width="thickmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle f\right\rangle =\int f\left(p,q\right)\,{\mathcal {P}}\left(p,q\right)dp\,dq\;,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ceda1cc083b7f38ee6b2282d1250253057966652" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.199ex; height:5.676ex;" alt="{\displaystyle \left\langle f\right\rangle =\int f\left(p,q\right)\,{\mathcal {P}}\left(p,q\right)dp\,dq\;,}"></span></dd></dl> <p>care depinde de structura sistemului și de condițiile externe. Măsura în care valorile unei variabile aleatorii se îndepărtează de la valoarea medie și între ele este dată de rădăcina pătrată din valoarea medie a pătratului abaterii de la valoarea medie, numită <i>abatere pătratică medie</i>, sau <i>împrăștiere statistică</i>: </p> <dl><dd><span style="padding-right:4em" id="f8"><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 \left(8\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>8</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(8\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e24a2bf3ca53c0658d529c1c1054cdb8759da510" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(8\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta f={\sqrt {\left\langle (f-\left\langle f\right\rangle )^{2}\right\rangle }}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow> <mo>⟨</mo> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>−</mo> <mrow> <mo>⟨</mo> <mi>f</mi> <mo>⟩</mo> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>⟩</mo> </mrow> </msqrt> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta f={\sqrt {\left\langle (f-\left\langle f\right\rangle )^{2}\right\rangle }}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10ddddf2e88a8f7839bd6558f429640e4379e57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.936ex; height:4.843ex;" alt="{\displaystyle \Delta f={\sqrt {\left\langle (f-\left\langle f\right\rangle )^{2}\right\rangle }}\,.}"></span></dd></dl> <p>Determinări experimentale precise au arătat că mărimile mecanice macroscopice din termodinamică pot fi identificate cu valorile medii calculate de mecanica statistică. Ele au detectat și existența unor <i>fluctuații</i> ale acestor mărimi, de ordinul de mărime al abaterilor pătratice medii prezise de mecanica statistică. </p> <div class="mw-heading mw-heading2"><h2 id="Distribuții_reprezentative"><span id="Distribu.C8.9Bii_reprezentative"></span>Distribuții reprezentative</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=5" title="Modifică secțiunea: Distribuții reprezentative" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=5" title="Edit section's source code: Distribuții reprezentative"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Descrierea comportării termodinamice a unui sistem pe baza unui colectiv statistic virtual de stări mecanice microscopice reprezintă un <i>postulat</i> al mecanicii statistice.<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> El este completat prin alegerea <i>a priori</i> a unei anumite distribuții care să fie „reprezentativă”, în sensul ca ea să corespundă gradului de cunoaștere incompletă, din punct de vedere mecanic, a stării sistemului.<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> </p> <div class="mw-heading mw-heading3"><h3 id="Distribuția_microcanonică"><span id="Distribu.C8.9Bia_microcanonic.C4.83"></span>Distribuția microcanonică</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=6" title="Modifică secțiunea: Distribuția microcanonică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=6" title="Edit section's source code: Distribuția microcanonică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În cazul idealizat al unui <a href="/wiki/Sistem_izolat" title="Sistem izolat">sistem izolat</a> de lumea exterioară, energia sistemului este constantă. Funcția de distribuție va fi diferită de zero doar pe suprafața de energie constantă <a href="#f2">(2)</a> unde, pentru a satisface condiția de normare <a href="#f5">(5)</a> ea va fi singulară. Dificultățile matematice legate de caracterul singular al acestei distribuții, numită de Gibbs <i>microcanonică</i>,<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> pot fi ocolite considerând-o drept limită a cazului mai realist în care sunt admise mici fluctuații ale energiei. Densitatea de probabilitate poate fi aleasă constantă în volumul cuprins între suprafețele de energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b4e96ca3ab5d09ee61fc1ae76c27b547b5eb4cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.743ex; height:2.176ex;" alt="{\displaystyle \ E\,}"></span> și <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E+\Delta E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>E</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E+\Delta E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bd4cb41f2505858874d05575bc0db9fb475bd57c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.295ex; height:2.343ex;" alt="{\displaystyle \ E+\Delta E\,}"></span>, unde cantitatea <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 E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \Delta E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55387da607e9762e57117bdfae53cd23dd1888bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.679ex; height:2.176ex;" alt="{\displaystyle \ \Delta E\,}"></span> este de ordinul de mărime al fluctuațiilor de energie, și zero în rest: </p> <dl><dd><span style="padding-right:4em" id="f9"><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 \left(9\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>9</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(9\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/223351a863017775e4d332b1a7eef79c12fdc9f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.972ex; height:2.843ex;" alt="{\displaystyle \left(9\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&{\mbox{pentru }}\ H\left(p,q\right)<E,\\\ C&{\mbox{pentru }}\ E\leq H\left(p,q\right)\leq E+\Delta E,\\\ 0&{\mbox{pentru }}\ E+\Delta E<H\left(p,q\right).\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>pentru </mtext> </mstyle> </mrow> <mtext> </mtext> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>E</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mi>C</mi> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>pentru </mtext> </mstyle> </mrow> <mtext> </mtext> <mi>E</mi> <mo>≤</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mi>E</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>pentru </mtext> </mstyle> </mrow> <mtext> </mtext> <mi>E</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo><</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&{\mbox{pentru }}\ H\left(p,q\right)<E,\\\ C&{\mbox{pentru }}\ E\leq H\left(p,q\right)\leq E+\Delta E,\\\ 0&{\mbox{pentru }}\ E+\Delta E<H\left(p,q\right).\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7529316263e67ebdde4c165d15ded82ac453392" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:50.126ex; height:8.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&{\mbox{pentru }}\ H\left(p,q\right)<E,\\\ C&{\mbox{pentru }}\ E\leq H\left(p,q\right)\leq E+\Delta E,\\\ 0&{\mbox{pentru }}\ E+\Delta E<H\left(p,q\right).\end{cases}}}"></span></dd></dl> <p>Constanta C se determină din condiția <a href="#f5">(5)</a>; pentru valori <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 E\ll E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo>≪</mo> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E\ll E}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/601828bd1e8c7949820bf5d68539e90bec79f0b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.101ex; height:2.176ex;" alt="{\displaystyle \Delta E\ll E}"></span> ea are valoarea </p> <dl><dd><span style="padding-right:4em" id="f10"><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 \left(10\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>10</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(10\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3d0a98b640ebfe02e78f1724d46973d69b774a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(10\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ C={\frac {1}{\Omega ^{\prime }\left(E\right)\Delta E}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>E</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C={\frac {1}{\Omega ^{\prime }\left(E\right)\Delta E}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/247e985178b9d55cdc4f49352ce404542b1724a5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.102ex; height:6.009ex;" alt="{\displaystyle \ C={\frac {1}{\Omega ^{\prime }\left(E\right)\Delta E}}\,}"></span></dd></dl> <p>(apostroful denotă derivata), care devine singulară în limita <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 E\to 0\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>E</mi> <mo stretchy="false">→</mo> <mn>0</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta E\to 0\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b2b9c38639ea4beb458504fe4c9cb4ae7578b446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.522ex; height:2.176ex;" alt="{\displaystyle \Delta E\to 0\,.}"></span> În calculele care utilizează distribuția microcanonică, singularitățile sunt evitate făcând trecerea la limită doar în rezultatul final. </p> <div class="mw-heading mw-heading3"><h3 id="Distribuția_canonică"><span id="Distribu.C8.9Bia_canonic.C4.83"></span>Distribuția canonică</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=7" title="Modifică secțiunea: Distribuția canonică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=7" title="Edit section's source code: Distribuția canonică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pentru un sistem care schimbă energie cu exteriorul în cantități arbitrare, o analiză a modului în care acest proces are loc la scară microscopică duce la concluzia că densitatea de probabilitate depinde exponențial de energia sistemului, adică de hamiltoniană.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> Se obține distribuția <i>canonică</i> <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> </p> <dl><dd><span style="padding-right:4em" id="f11"><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 \left(11\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>11</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(11\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/847aad2c1acc1d3953a35bf5871ebf21a67de390" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(11\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-\beta H\left(p,q\right)}\,,\quad \left(\beta >0\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-\beta H\left(p,q\right)}\,,\quad \left(\beta >0\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a062ef7ec716f852553032e64583a2d69bcd28b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.057ex; height:5.176ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-\beta H\left(p,q\right)}\,,\quad \left(\beta >0\right)\,.}"></span></dd></dl> <p>Pentru a satisface condiția de normare <a href="#f5">(5)</a>, parametrul <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \beta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>β</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \beta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa26e3ee25f8d3e883f0beb3167b89b41e74e1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.3ex; height:2.509ex;" alt="{\displaystyle \ \beta \,}"></span> trebuie să fie pozitiv, iar cantitatea <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,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>Z</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ Z,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bddce4307a04b46a0e434209020b498918bebf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.908ex; height:2.509ex;" alt="{\displaystyle \ Z,}"></span> numită <i>integrală de stare</i> sau <i>funcție de partiție</i>, are valoarea </p> <dl><dd><span style="padding-right:4em" id="f12"><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 \left(12\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>12</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(12\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7c3054f8b575b4e790940d9699c5a3f596e2135" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(12\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\int e^{-\beta H\left(p,q\right)}\,dp\,dq\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\int e^{-\beta H\left(p,q\right)}\,dp\,dq\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d817c35b5cf10f3dccaa6861f0011bcc34278e5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.154ex; height:5.676ex;" alt="{\displaystyle Z=\int e^{-\beta H\left(p,q\right)}\,dp\,dq\,.}"></span></dd></dl> <div class="mw-heading mw-heading3"><h3 id="Distribuția_macrocanonică"><span id="Distribu.C8.9Bia_macrocanonic.C4.83"></span>Distribuția macrocanonică</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=8" title="Modifică secțiunea: Distribuția macrocanonică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=8" title="Edit section's source code: Distribuția macrocanonică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dacă sistemul constă din mai multe componente, între care are loc atât transfer de energie cât și transfer de substanță, este convenabilă descrierea sa printr-un colectiv statistic macrocanonic,<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> care este o colecție ponderată de colective statistice canonice, câte unul pentru fiecare componentă.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> Fie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ c\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>c</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ c\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf47a6b89010654265159bb26496e4cbebc0062d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.975ex; height:1.676ex;" alt="{\displaystyle \ c\,}"></span> numărul de componente și <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_{1},...,N_{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ N_{1},...,N_{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5695a3d82ecd1b514478ed619000516b1452ebd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.868ex; height:2.509ex;" alt="{\displaystyle \ N_{1},...,N_{c}\,}"></span> cantitățile în care sunt prezente aceste componente.<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> Analiza modului în care decurge schimbul de substanță la scară microscopică, similară celei făcute pentru schimbul de energie, arată că densitatea de probabilitate depinde exponențial de fiecare dintre aceste cantități în parte. Distribuția <i>macrocanonică</i> <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> are forma </p> <dl><dd><span style="padding-right:4em" id="f13"><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 \left(13\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>13</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(13\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cac82ccb6327aa21379f3922633396b9b1f8b22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(13\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)},\quad \left(\beta >0\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <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>c</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mo>,</mo> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <mi>β</mi> <mo>></mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)},\quad \left(\beta >0\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d579422b1fb30ca50332081ef30bad26f4b8512a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:44.73ex; height:5.176ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)},\quad \left(\beta >0\right)\,,}"></span></dd></dl> <p>unde </p> <dl><dd><span style="padding-right:4em" id="f14"><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 \left(14\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>14</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(14\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9629e00c8ff6eb082b6c2b3c638b56bae2c6ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(14\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ {\mathcal {Z}}=\int e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>β</mi> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <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>c</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {Z}}=\int e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24f6c65d9676057378ebdf827e0b7057d97f067a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.761ex; height:5.676ex;" alt="{\displaystyle \ {\mathcal {Z}}=\int e^{-\beta \left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)}\,dp\,dq}"></span></dd></dl> <p>este <i>funcția de partiție macrocanonică</i>. Semnificația parametrilor <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cf629bdcc90521bb174119ac00d2f82e66b6858" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.719ex; height:2.509ex;" alt="{\displaystyle \beta \,}"></span> și <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 _{1},...,\mu _{c}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1},...,\mu _{c}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/903c1bb3145f69d146ea9ba760a0a307990a8c2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.359ex; height:2.176ex;" alt="{\displaystyle \mu _{1},...,\mu _{c}\,}"></span> urmează să rezulte din interpretarea termodinamică a distribuțiilor canonică și macrocanonică. </p> <div class="mw-heading mw-heading2"><h2 id="Termodinamică_statistică"><span id="Termodinamic.C4.83_statistic.C4.83"></span>Termodinamică statistică</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=9" title="Modifică secțiunea: Termodinamică statistică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=9" title="Edit section's source code: Termodinamică statistică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Dinamica microscopică a unui sistem este determinată, pe lângă forțele interne, de forțe macroscopice externe, care până acum nu au fost considerate explicit. Fie <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>m</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ m\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05aa6caca8044f6f143efa8fe440d82062abfc21" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.008ex; height:1.676ex;" alt="{\displaystyle \ m\,}"></span> numărul de grade de libertate mecanice macroscopice și <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=\left(x_{1},...,x_{m}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>x</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ x=\left(x_{1},...,x_{m}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d04328f0d654005a303aa6e2bbba0ef9d586592" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.764ex; height:2.843ex;" alt="{\displaystyle \ x=\left(x_{1},...,x_{m}\right)\,}"></span> variabilele de poziție respective. Atât hamiltoniana cât și volumul în spațiul fazelor conținut în interiorul unei suprafețe de energie constantă depind de aceste variabile: </p> <dl><dd><span style="padding-right:4em" id="f15"><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 \left(15\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>15</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(15\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6abf4bdbd0cc18a69e6381fde119f08dc68f97bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(15\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=H\left(p,q\mid x\right)\,;\quad \Omega =\Omega \left(E\mid x\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo>∣</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> <mspace width="1em" /> <mi mathvariant="normal">Ω</mi> <mo>=</mo> <mi mathvariant="normal">Ω</mi> <mrow> <mo>(</mo> <mrow> <mi>E</mi> <mo>∣</mo> <mi>x</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=H\left(p,q\mid x\right)\,;\quad \Omega =\Omega \left(E\mid x\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58da2e7d748f1c4af8dc54dc4d0f69bdafff3b72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.207ex; height:2.843ex;" alt="{\displaystyle H=H\left(p,q\mid x\right)\,;\quad \Omega =\Omega \left(E\mid x\right)\,.}"></span></dd></dl> <p><a href="/wiki/Principiul_%C3%AEnt%C3%A2i_al_termodinamicii" title="Principiul întâi al termodinamicii">Principiul întâi al termodinamicii</a> definește o <a href="/wiki/Func%C8%9Bie_de_stare" title="Funcție de stare">funcție de stare</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>U</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/76d39dc95f2d5fcd55619627007ede8d39767303" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.75ex; height:2.176ex;" alt="{\displaystyle \ U\,}"></span> numită <i><a href="/wiki/Energie_intern%C4%83" title="Energie internă">energie internă</a></i>; mecanica statistică interpretează echilibrul termodinamic ca având caracter statistic, iar energia internă ca valoare medie a energiei microscopice: </p> <dl><dd><span style="padding-right:4em" id="f16"><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 \left(16\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>16</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(16\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a82f7f13cb412d80dd9ca54975f657853491fb3a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(16\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=\left\langle H\right\rangle \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo>=</mo> <mrow> <mo>⟨</mo> <mi>H</mi> <mo>⟩</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U=\left\langle H\right\rangle \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d82b92e0afefa179ea59a6575dfeb20b9793a87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.175ex; height:2.843ex;" alt="{\displaystyle U=\left\langle H\right\rangle \,.}"></span></dd></dl> <p>Fie <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=\left(X_{1},...,X_{m}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>X</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X=\left(X_{1},...,X_{m}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498dea448448fd5ecc37501cf831fe4a54a344f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.022ex; height:2.843ex;" alt="{\displaystyle X=\left(X_{1},...,X_{m}\right)\,}"></span> variabilele de forță asociate cu variabilele de poziție macroscopice; în mecanica statistică și ele sunt considerate valori medii ale unor mărimi aleatorii: </p> <dl><dd><span style="padding-right:4em" id="f17"><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 \left(17\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>17</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(17\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a68ac45650322179dc920a90b838256ea29d5242" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(17\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X_{j}=\left\langle {\frac {\partial H}{\partial x_{j}}}\right\rangle ;\quad \left(j=1,...,m\right)\,.}"> <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> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mo>;</mo> <mspace width="1em" /> <mrow> <mo>(</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>m</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle X_{j}=\left\langle {\frac {\partial H}{\partial x_{j}}}\right\rangle ;\quad \left(j=1,...,m\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/386e7851bc40f7462e5dfdee4aec7cadd7857709" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:33.215ex; height:6.343ex;" alt="{\displaystyle X_{j}=\left\langle {\frac {\partial H}{\partial x_{j}}}\right\rangle ;\quad \left(j=1,...,m\right)\,.}"></span></dd></dl> <p>Lucrul mecanic produs de aceste forțe la deplasări elementare <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 dx=\left(dx_{1},...,dx_{m}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dx=\left(dx_{1},...,dx_{m}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22b2c0745fc9c5c6307cc323bad0572b0be3d8ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.443ex; height:2.843ex;" alt="{\displaystyle dx=\left(dx_{1},...,dx_{m}\right)}"></span> este </p> <dl><dd><span style="padding-right:4em" id="f18"><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 \left(18\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>18</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(18\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a732faf07c5f6def4dc905520adc8ffe9a0c9c97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(18\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta L=\sum _{j=1}^{m}X_{j}\,dx_{j}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>L</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <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 \delta L=\sum _{j=1}^{m}X_{j}\,dx_{j}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c505948eb78985b16dbda1453cc527ed19d37158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.182ex; height:7.176ex;" alt="{\displaystyle \delta L=\sum _{j=1}^{m}X_{j}\,dx_{j}\,.}"></span></dd></dl> <p>Tot conform principiului întâi al termodinamicii, într-o transformare termodinamică elementară diferențiala totală a energiei interne este suma dintre lucrul mecanic efectuat și cantitatea de căldură <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 Q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d64099e28267b81c30ee85e6d72f879208bc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.274ex; height:2.676ex;" alt="{\displaystyle \delta Q\,}"></span> schimbată de sistem: </p> <dl><dd><span style="padding-right:4em" id="f19"><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 \left(19\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>19</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(19\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c89c61f5833abd26cda039baef0565a6f6e2d8c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(19\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle dU=\delta L+\delta Q\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mi>U</mi> <mo>=</mo> <mi>δ</mi> <mi>L</mi> <mo>+</mo> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle dU=\delta L+\delta Q\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d434a3a82fa33baa72663cc454bea834cd66d5e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.49ex; height:2.676ex;" alt="{\displaystyle dU=\delta L+\delta Q\,.}"></span></dd></dl> <p><a href="/wiki/Principiul_al_doilea_al_termodinamicii" title="Principiul al doilea al termodinamicii">Principiul al doilea al termodinamicii</a> definește o funcție de stare <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"> <mtext> </mtext> <mi>S</mi> <mspace width="thinmathspace" /> </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/92371ed59f011bf7d82f2d538d7bcd29f128f2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.176ex;" alt="{\displaystyle \ S\,}"></span> numită <i>entropie</i>; într-o transformare termodinamică elementară <i><a href="/wiki/Proces_reversibil" title="Proces reversibil">reversibilă</a></i> diferențiala totală a entropiei e legată de cantitatea de căldură schimbată de sistem prin relația </p> <dl><dd><span style="padding-right:4em" id="f20"><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 \left(20\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>20</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(20\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab0679cf19948d772b73ec69cb31706cbd6370db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(20\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ dS={\frac {\delta Q}{T}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dS={\frac {\delta Q}{T}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a48ad74b0245a8a2b50789169b0d8a8bded433d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:11.151ex; height:5.343ex;" alt="{\displaystyle \ dS={\frac {\delta Q}{T}}\,.}"></span></dd></dl> <p>Aici <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e698f73ccd6c97095262b954811bca04654f485" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.604ex; height:2.176ex;" alt="{\displaystyle \ T\,}"></span> este <i>temperatura termodinamică</i>, definită de principiul al doilea al termodinamicii, până la un factor constant, ca scară absolută de temperatură, unică printre multele scări de temperatură empirică posibile, definite prin contact termic. </p><p>În rezumat, în mecanica statistică mărimile termodinamice de natură mecanică sunt considerate variabile aleatorii; valorile lor măsurate macroscopic sunt asimilate cu valorile medii ale mărimilor microscopice corespunzătoare, admițându-se existența fluctuațiilor. Mărimile termodinamice <i><a href="/wiki/Temperatur%C4%83" title="Temperatură">temperatură</a></i> și <i><a href="/wiki/Entropie" title="Entropie">entropie</a></i> urmează să fie definite, în cadrul fiecărei distribuții reprezentative, prin parametrii colectivului statistic asociat sistemului. Odată determinat un <a href="/wiki/Poten%C8%9Bial_termodinamic" title="Potențial termodinamic">potențial termodinamic</a> adecvat situației descrise de colectivul statistic, ecuațiile de stare ale sistemului rezultă prin metode termodinamice standard. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Zentralfriedhof_Vienna_-_Boltzmann.JPG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/160px-Zentralfriedhof_Vienna_-_Boltzmann.JPG" decoding="async" width="160" height="213" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/240px-Zentralfriedhof_Vienna_-_Boltzmann.JPG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Zentralfriedhof_Vienna_-_Boltzmann.JPG/320px-Zentralfriedhof_Vienna_-_Boltzmann.JPG 2x" data-file-width="1920" data-file-height="2560" /></a><figcaption>Mormântul lui Boltzmann în Cimitirul Central din Viena, cu formula <i>S = k log W</i> gravată deasupra.</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Sistem_izolat:_entropie">Sistem izolat: entropie</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=10" title="Modifică secțiunea: Sistem izolat: entropie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=10" title="Edit section's source code: Sistem izolat: entropie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Analiza modului în care se stabilește echilibrul termodinamic între două sisteme distribuite microcanonic cu energii <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E_{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E_{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca9b136ecbff782bc09467cba4d1adf4df866908" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.737ex; height:2.509ex;" alt="{\displaystyle \ E_{1}\,}"></span> și <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E_{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E_{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c67b4912f03d0c067aa46f30ae33f938eaa8522" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.737ex; height:2.509ex;" alt="{\displaystyle \ E_{2}\,}"></span>, atunci când sunt aduse în contact termic,<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> arată că produsul <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 \Omega ^{\prime }\left(E_{1}\right)\;\Omega ^{\prime }\left(E_{2}\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Omega ^{\prime }\left(E_{1}\right)\;\Omega ^{\prime }\left(E_{2}\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369203077b500b470df2f0675eb4ddba14dcd700" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.077ex; height:3.009ex;" alt="{\displaystyle \Omega ^{\prime }\left(E_{1}\right)\;\Omega ^{\prime }\left(E_{2}\right)\,}"></span> are un maxim pronunțat pentru o anumită valoare a argumentului (un singur argument independent, întrucât <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E_{1}+E_{2}=constant}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mi>a</mi> <mi>n</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E_{1}+E_{2}=constant}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e5e8f92ea409645a127c1173c455b3c76acff863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.982ex; height:2.509ex;" alt="{\displaystyle \ E_{1}+E_{2}=constant}"></span>) și scade foarte repede de o parte și de alta a acestui maxim. Maximul se realizează atunci când pentru cele două sisteme expresia </p> <dl><dd><span style="padding-right:4em" id="f21"><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 \left(21\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>21</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(21\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49b68ffcc1359fad819072a6508aa68747ce9ee6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(21\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d\,ln\Omega ^{\prime }\left(E\right)}{\ dE}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>E</mi> <mo>)</mo> </mrow> </mrow> <mrow> <mtext> </mtext> <mi>d</mi> <mi>E</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d\,ln\Omega ^{\prime }\left(E\right)}{\ dE}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21b315eff11bf010e97a1d6f5de4ec495a7e519c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.862ex; height:5.843ex;" alt="{\displaystyle {\frac {d\,ln\Omega ^{\prime }\left(E\right)}{\ dE}}}"></span></dd></dl> <p>are aceeași valoare; el indică starea microscopică cea mai probabilă, corespunzătoare stării de <i>echilibru termic</i>, iar valoarea comună este o funcție <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 \phi \left(T\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \left(T\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/935ce5b47cee078f18280f07d94a3957856e5e98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.218ex; height:2.843ex;" alt="{\displaystyle \phi \left(T\right)}"></span> de temperatura la care s-a stabilit acest echilibru. Energia internă este <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ U=E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>U</mi> <mo>=</mo> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ U=E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ad65020f79a26c3a375ede7dc097a272af371ae4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.624ex; height:2.176ex;" alt="{\displaystyle \ U=E\,}"></span>, iar fluctuațiile în jurul acestei stări au loc doar prin schimb de căldură: <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 \ dU=\delta Q\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mi>U</mi> <mo>=</mo> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dU=\delta Q\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b8ad0d52ab694225db981d9969ecb5dbf42730" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.598ex; height:2.676ex;" alt="{\displaystyle \ dU=\delta Q\,.}"></span> Adunând rezultatele, se poate scrie </p> <dl><dd><span style="padding-right:4em" id="f22"><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 \left(22\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>22</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(22\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cd45537adc8a2f5c02ff3179082fa63638ceba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(22\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=\phi \left(T\right)\delta Q\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=\phi \left(T\right)\delta Q\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e281f87de2e0eba58a1ef46df44405c4fe439cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.238ex; height:3.009ex;" alt="{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=\phi \left(T\right)\delta Q\,.}"></span></dd></dl> <p>Prin înmulțirea cantității de căldură <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 Q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta Q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71d64099e28267b81c30ee85e6d72f879208bc27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.274ex; height:2.676ex;" alt="{\displaystyle \delta Q\,}"></span> schimbată reversibil cu funcția <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 \phi \left(T\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \left(T\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4faa0ee3601e83c16ff1a9bdd2ae9dd9475fcbe9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.252ex; height:2.843ex;" alt="{\displaystyle \phi \left(T\right),}"></span> s-a obținut o diferențială totală exactă <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ dS.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mi>S</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dS.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7d56ba203da74bb610e42dbe273ef0c8bb1287f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.943ex; height:2.176ex;" alt="{\displaystyle \ dS.}"></span> Conform principiului al doilea al termodinamicii, funcția <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"> <mtext> </mtext> <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/7bc46c9185d6ff67f4946df6686a8408c9cd1a5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.08ex; height:2.176ex;" alt="{\displaystyle \ S}"></span> este <i><a href="/wiki/Entropie" title="Entropie">entropia</a></i>, iar <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 \phi \left(T\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ϕ</mi> <mrow> <mo>(</mo> <mi>T</mi> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \phi \left(T\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d5489e020ca02b5afd64d60f71eab18cd9ebd1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.605ex; height:2.843ex;" alt="{\displaystyle \phi \left(T\right)\,}"></span> este, până la un factor constant, egală cu inversa temperaturii absolute: </p> <dl><dd><span style="padding-right:4em" id="f23"><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 \left(23\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>23</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(23\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6f2aac4dc5f3da8ee7c8b0645249105d2534716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(23\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=dS\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mo>=</mo> <mi>d</mi> <mi>S</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=dS\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb02219ba47e40645635af9e895fd6e5f1b43744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.461ex; height:3.009ex;" alt="{\displaystyle \ d\,ln\Omega ^{\prime }\left(U\right)=dS\,.}"></span></dd></dl> <p>Prin integrare rezultă </p> <dl><dd><span style="padding-right:4em" id="f24"><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 \left(24\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>24</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(24\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9833a87fcc152968f4e71ba89602c8624ddcaeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(24\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d59baffb95e9293e9162e32b4fa0adcf1e5a817c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.627ex; height:3.009ex;" alt="{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)\,;}"></span></dd></dl> <p>constanta <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c682e4c73906031c21d3ec3951990a6111eeacb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.792ex; height:2.176ex;" alt="{\displaystyle \ k}"></span> a primit numele de <i><a href="/wiki/Constanta_Boltzmann" title="Constanta Boltzmann">constanta Boltzmann</a></i>. Această formulă fundamentală a mecanicii statistice, stabilită de Boltzmann, exprimă legătura dintre entropie și caracteristicile colectivului statistic reprezentat de distribuția microcanonică. </p> <div class="mw-heading mw-heading3"><h3 id="Schimb_de_energie:_energie_liberă"><span id="Schimb_de_energie:_energie_liber.C4.83"></span>Schimb de energie: energie liberă</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=11" title="Modifică secțiunea: Schimb de energie: energie liberă" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=11" title="Edit section's source code: Schimb de energie: energie liberă"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Din relațiile <a href="#f16">(16)</a>–<a href="#f19">(19)</a> și <a href="#f12">(12)</a> rezultă că <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 Q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta Q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f956f67619cbbcaba75d3009a00027908c703e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.855ex; height:2.676ex;" alt="{\displaystyle \ \delta Q\,}"></span>, cantitatea de căldură schimbată de un sistem distribuit canonic într-o transformare elementară reversibilă, satisface egalitatea<sup id="cite_ref-17" class="reference"><a href="#cite_note-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p> <dl><dd><span style="padding-right:4em" id="f25"><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 \left(25\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>25</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(25\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/507731e72f404796e2be45686c7baa837a79924e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(25\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \beta \,\delta Q=d\left(ln\,Z+\beta U\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>β</mi> <mspace width="thinmathspace" /> <mi>δ</mi> <mi>Q</mi> <mo>=</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>l</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mi>Z</mi> <mo>+</mo> <mi>β</mi> <mi>U</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \beta \,\delta Q=d\left(ln\,Z+\beta U\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eca3dde06a5f55b9c0dc6e8864a1e4f98f4f1db6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.229ex; height:2.843ex;" alt="{\displaystyle \ \beta \,\delta Q=d\left(ln\,Z+\beta U\right)\,.}"></span></dd></dl> <p>Argumentul precedent privitor la existența unui factor integrant pentru <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 Q\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>δ</mi> <mi>Q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \delta Q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f956f67619cbbcaba75d3009a00027908c703e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.855ex; height:2.676ex;" alt="{\displaystyle \ \delta Q\,}"></span> duce la concluzia că </p> <dl><dd><span style="padding-right:4em" id="f26"><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 \left(26\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>26</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(26\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60d923e0a22cfb2caeb9fd0a0832d583ab605cf3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(26\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,Z+{\frac {U}{T}}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mi>Z</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>U</mi> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,Z+{\frac {U}{T}}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e32b49c717af265e4444b5a37b5909264de11c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.732ex; height:6.176ex;" alt="{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,Z+{\frac {U}{T}}\right)\,.}"></span></dd></dl> <p>Prin integrare se obțin entropia <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"> <mtext> </mtext> <mi>S</mi> <mspace width="thinmathspace" /> </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/92371ed59f011bf7d82f2d538d7bcd29f128f2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.467ex; height:2.176ex;" alt="{\displaystyle \ S\,}"></span> și apoi <i><a href="/wiki/Energie_liber%C4%83" title="Energie liberă">energia liberă</a></i> (numită și <i>energie liberă Helmholtz</i>) </p> <dl><dd><span style="padding-right:4em" id="f27"><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 \left(27\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>27</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(27\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10593ab4b7b124fbd94e04ecb092a292e443a523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(27\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F=-kT\,lnZ\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mi>Z</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F=-kT\,lnZ\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de927a9a17862aeb88a812aa98790871a9de321c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.265ex; height:2.343ex;" alt="{\displaystyle \ F=-kT\,lnZ\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Entropia_ca_funcțională_de_densitatea_de_probabilitate"><span id="Entropia_ca_func.C8.9Bional.C4.83_de_densitatea_de_probabilitate"></span>Entropia ca funcțională de densitatea de probabilitate</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=12" title="Modifică secțiunea: Entropia ca funcțională de densitatea de probabilitate" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=12" title="Edit section's source code: Entropia ca funcțională de densitatea de probabilitate"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Din relațiile <a href="#f11">(11)</a>, <a href="#f12">(12)</a> și <a href="#f27">(27)</a>, luând logaritmul și apoi valoarea medie, rezultă <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S=-k\left\langle ln\,{\mathcal {P}}\right\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mrow> <mo>⟨</mo> <mrow> <mi>l</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> </mrow> <mo>⟩</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S=-k\left\langle ln\,{\mathcal {P}}\right\rangle }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85fa5a2fd241bb8aa57cce4b9995f0d704ae62f0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.573ex; height:2.843ex;" alt="{\displaystyle \ S=-k\left\langle ln\,{\mathcal {P}}\right\rangle }"></span>, adică </p> <dl><dd><span style="padding-right:4em" id="f28"><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 \left(28\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>28</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(28\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35f55cd144e16eece5540ec7857ad44a3cf0d0fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(28\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S\left[{\mathcal {P}}\right]=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S\left[{\mathcal {P}}\right]=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c581241cdc918182dd560cac3455db63f6ca2793" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.911ex; height:5.676ex;" alt="{\displaystyle \ S\left[{\mathcal {P}}\right]=-k\int {\mathcal {P}}\,ln{\mathcal {P}}\,dp\,dq\,.}"></span></dd></dl> <p>Deși această expresie a fost obținută pe baza distribuției canonice, ea este independentă de caracteristicile vreunui colectiv statistic anumit. Datorită caracterului general al acestei relații, care exprimă entropia ca <a href="/w/index.php?title=Func%C8%9Bional%C4%83&action=edit&redlink=1" class="new" title="Funcțională — pagină inexistentă">funcțională</a><sup><small>⁠(<a href="https://fr.wikipedia.org/wiki/Fonctionnelle" class="extiw" title="fr:Fonctionnelle"><span title="„Fonctionnelle” la Wikipedia în franceză">fr</span></a>)</small></sup><sup class="plainlinks"><small>[<a class="external text" href="https://ro.wikipedia.org/wiki/Special:ContentTranslation?title=Special:ContentTranslation&campaign=contributionsmenu&to=ro&from=fr&page=Fonctionnelle&targettitle=func%C8%9Bional%C4%83">traduceți</a>]</small></sup> de densitatea de probabilitate, ea este adoptată ca definiție a entropiei pentru orice distribuție, chiar în cazul unor distribuții nestaționare.<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> <div class="mw-heading mw-heading4"><h4 id="Teorema_echipartiției_energiei"><span id="Teorema_echiparti.C8.9Biei_energiei"></span>Teorema echipartiției energiei</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=13" title="Modifică secțiunea: Teorema echipartiției energiei" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=13" title="Edit section's source code: Teorema echipartiției energiei"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Distribuția canonică are drept consecință faptul că, pentru oricare dintre variabilele canonice, impuls <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"> <mtext> </mtext> <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/5ed83ecdcd9806460ed61349da6e4904cad281cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.55ex; height:2.009ex;" alt="{\displaystyle \ p_{i}}"></span> sau coordonată <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_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ q_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec3a2921d84774dab117789897367242d6b0ad50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.417ex; height:2.009ex;" alt="{\displaystyle \ q_{i}}"></span>, care figurează explicit în expresia funcției hamiltoniene, există relația<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> </p> <dl><dd><span style="padding-right:4em" id="f29"><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 \left(29\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>29</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(29\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ec65294d2d6324c8a0bb9e5870f95b143dbadc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(29\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle =\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle =kT\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle =\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle =kT\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/107ff8b4b5b0f50f621f840f6f4a560903ef8610" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.292ex; height:6.176ex;" alt="{\displaystyle \left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle =\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle =kT\,.}"></span></dd></dl> <p>Utilitatea acestei teoreme stă în faptul că în general variabila <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"> <mtext> </mtext> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </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/f0a6e2df4323b149b32280467b7a8c7e55f8e692" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.937ex; height:2.009ex;" alt="{\displaystyle \ p_{i}\,}"></span> contribuie la energia cinetică, deci la hamiltoniană, cu un termen <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 {p_{i}^{2}}{2m}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {p_{i}^{2}}{2m}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747f4ed2486521650145facbec99cdce6c742134" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:5.073ex; height:5.843ex;" alt="{\displaystyle {\frac {p_{i}^{2}}{2m}}\,;}"></span> atunci </p> <dl><dd><span style="padding-right:4em" id="f30"><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 \left(30\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>30</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(30\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f48ae889dd1ddf7fdb1f2226b7067fee8777a63d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(30\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle {\frac {p_{i}^{2}}{2m}}\right\rangle ={\frac {1}{2}}\left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle ={\frac {kT}{2}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle {\frac {p_{i}^{2}}{2m}}\right\rangle ={\frac {1}{2}}\left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle ={\frac {kT}{2}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38fb7fbb062bf64b09e57e6d83e9db3146277580" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.759ex; height:7.509ex;" alt="{\displaystyle \left\langle {\frac {p_{i}^{2}}{2m}}\right\rangle ={\frac {1}{2}}\left\langle p_{i}{\frac {\partial H}{\partial p_{i}}}\right\rangle ={\frac {kT}{2}}\,.}"></span></dd></dl> <p>În cazul unui sistem care execută oscilații elastice în coordonata <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_{i}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ q_{i}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b022c93e62fc64b7a08b6d8c05711bcf9b871682" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.451ex; height:2.009ex;" alt="{\displaystyle \ q_{i}\,,}"></span> aceasta contribuie la energia potențială cu un termen <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\,q_{i}^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>c</mi> <mspace width="thinmathspace" /> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ c\,q_{i}^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/414d1019ea7f5c5eeb6c17ac2d8bb1c66933b4fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.108ex; height:3.176ex;" alt="{\displaystyle \ c\,q_{i}^{2}}"></span> și deci </p> <dl><dd><span style="padding-right:4em" id="f31"><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 \left(31\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>31</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(31\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c84b24ca3cdf40fb16b9af4f4475da23899dd5bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(31\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle c\,q_{i}^{2}\right\rangle ={\frac {1}{2}}\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle ={\frac {kT}{2}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <mrow> <mi>c</mi> <mspace width="thinmathspace" /> <msubsup> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow> <mo>⟨</mo> <mrow> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>H</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mrow> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>T</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle c\,q_{i}^{2}\right\rangle ={\frac {1}{2}}\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle ={\frac {kT}{2}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75d6fd3c4b3c078ad0a35fd3712018826e81a2fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.563ex; height:6.176ex;" alt="{\displaystyle \left\langle c\,q_{i}^{2}\right\rangle ={\frac {1}{2}}\left\langle q_{i}{\frac {\partial H}{\partial q_{i}}}\right\rangle ={\frac {kT}{2}}\,.}"></span></dd></dl> <p>Fiecare grad de libertate microscopic contribuie la energia macroscopică, în medie, cu aceeași cantitate <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 {\begin{matrix}{\frac {1}{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1040f72e4158f6fdd3b0d4650280d21acb841d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.968ex; margin-bottom: -0.203ex; width:2.41ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}"></span> kT, pentru fiecare variabilă canonică (impuls sau coordonată) prezentă explicit în hamiltoniană, de unde și numele de <i>teorema echipartiției energiei</i>. </p> <div class="mw-heading mw-heading3"><h3 id="Schimb_de_energie_și_substanță:_potențial_macrocanonic"><span id="Schimb_de_energie_.C8.99i_substan.C8.9B.C4.83:_poten.C8.9Bial_macrocanonic"></span>Schimb de energie și substanță: potențial macrocanonic</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=14" title="Modifică secțiunea: Schimb de energie și substanță: potențial macrocanonic" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=14" title="Edit section's source code: Schimb de energie și substanță: potențial macrocanonic"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Din relațiile <a href="#f16">(16)</a>–<a href="#f19">(19)</a> și <a href="#f14">(14)</a> rezultă, folosind argumentul factorului integrant, că </p> <dl><dd><span style="padding-right:4em" id="f32"><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 \left(32\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>32</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(32\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a2e1d5a7c070dc7b97855d0b6630c604b49be9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(32\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,{\mathcal {Z}}+{\frac {U-\sum _{i=1}^{c}\mu _{i}N_{i}}{T}}\right)\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>k</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>δ</mi> <mi>Q</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>=</mo> <mi>d</mi> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>U</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>c</mi> </mrow> </munderover> <msub> <mi>μ</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> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,{\mathcal {Z}}+{\frac {U-\sum _{i=1}^{c}\mu _{i}N_{i}}{T}}\right)\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af225aeff64ecc8e3eafb2ce02b6d0846ecaebb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.543ex; height:7.509ex;" alt="{\displaystyle \ \beta ={\frac {1}{kT}}\,,\quad dS={\frac {\delta Q}{T}}=d\left(k\,ln\,{\mathcal {Z}}+{\frac {U-\sum _{i=1}^{c}\mu _{i}N_{i}}{T}}\right)\,,}"></span></dd></dl> <p>iar parametrii macrocanonici <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 \ \left(\mu _{1},...,\mu _{c}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow> <mo>(</mo> <mrow> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ \left(\mu _{1},...,\mu _{c}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbe73ce74f3c5204e33c5b4f2d2c574ede2697bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.749ex; height:2.843ex;" alt="{\displaystyle \ \left(\mu _{1},...,\mu _{c}\right)}"></span> sunt identificați cu potențialele chimice din termodinamică.<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> Prin integrare se obține </p> <dl><dd><span style="padding-right:4em" id="f33"><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 \left(33\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>33</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(33\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e313adf41bdf371d491205386408a570049312f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(33\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ TS=kT\,ln\,{\mathcal {Z}}+U-\sum _{i=1}^{c}\mu _{i}N_{i}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>T</mi> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>+</mo> <mi>U</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>c</mi> </mrow> </munderover> <msub> <mi>μ</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> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ TS=kT\,ln\,{\mathcal {Z}}+U-\sum _{i=1}^{c}\mu _{i}N_{i}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c83fc6d78d6f61cb3b81b1af6d60e23c2dd6c7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:31.414ex; height:6.843ex;" alt="{\displaystyle \ TS=kT\,ln\,{\mathcal {Z}}+U-\sum _{i=1}^{c}\mu _{i}N_{i}\,.}"></span></dd></dl> <p>Introducând <a href="/wiki/Poten%C8%9Bial_macrocanonic" title="Potențial macrocanonic">potențialul macrocanonic</a> (numit și <i>energie liberă Landau</i>) </p> <dl><dd><span style="padding-right:4em" id="f34"><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 \left(34\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>34</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(34\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43b1961ac7efc21912b78cb09e4c7dea79f943d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(34\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=U-TS-\sum _{i=1}^{c}\mu _{i}N_{i}=F-G=\sum _{j=1}^{m}X_{j}x_{j}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mi>U</mi> <mo>−</mo> <mi>T</mi> <mi>S</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>c</mi> </mrow> </munderover> <msub> <mi>μ</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> <mo>=</mo> <mi>F</mi> <mo>−</mo> <mi>G</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <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 {\mathcal {F}}=U-TS-\sum _{i=1}^{c}\mu _{i}N_{i}=F-G=\sum _{j=1}^{m}X_{j}x_{j}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a266c8399edfbadf3bd0294ce9ac0f94192a88a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.687ex; height:7.176ex;" alt="{\displaystyle {\mathcal {F}}=U-TS-\sum _{i=1}^{c}\mu _{i}N_{i}=F-G=\sum _{j=1}^{m}X_{j}x_{j}\,,}"></span></dd></dl> <p>rezultatul se scrie într-o formă similară cu <a href="#f27">(27)</a>: </p> <dl><dd><span style="padding-right:4em" id="f35"><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 \left(35\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>35</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(35\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57fcefac1b4d0287db1fac766e26a689d950ec65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(35\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/45128bc679c525b8753b9cc0ef535583abcb81ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.972ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}\,.}"></span></dd></dl> <table class="wikitable"> <caption>Termodinamică statistică </caption> <tbody><tr> <th style="background:#f9f9f9"> </th> <th>Distribuția microcanonică </th> <th>Distribuția canonică </th> <th>Distribuția macrocanonică </th></tr> <tr> <td style="background:#f2f2f2"><b>Densitate de probabilitate</b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&H\left(p,q\right)<U\\\ C&U\leq H\left(p,q\right)\leq U+\Delta U\\\ 0&U+\Delta U<H\left(p,q\right)\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo><</mo> <mi>U</mi> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mi>C</mi> </mtd> <mtd> <mi>U</mi> <mo>≤</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>≤</mo> <mi>U</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mtd> </mtr> <mtr> <mtd> <mtext> </mtext> <mn>0</mn> </mtd> <mtd> <mi>U</mi> <mo>+</mo> <mi mathvariant="normal">Δ</mi> <mi>U</mi> <mo><</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&H\left(p,q\right)<U\\\ C&U\leq H\left(p,q\right)\leq U+\Delta U\\\ 0&U+\Delta U<H\left(p,q\right)\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cde140f8c885fc46a2a9c5770f44e96ee6231846" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:41.612ex; height:8.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\begin{cases}\ 0&H\left(p,q\right)<U\\\ C&U\leq H\left(p,q\right)\leq U+\Delta U\\\ 0&U+\Delta U<H\left(p,q\right)\end{cases}}}"></span> <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 \ C={\frac {1}{\Omega ^{\prime }\left(U\right)\Delta U}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>C</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ C={\frac {1}{\Omega ^{\prime }\left(U\right)\Delta U}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02eb1131e0cd6f13011aad9760053836e55b70fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.116ex; height:6.009ex;" alt="{\displaystyle \ C={\frac {1}{\Omega ^{\prime }\left(U\right)\Delta U}}\,}"></span> </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-H\left(p,q\right)/kT}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-H\left(p,q\right)/kT}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24da8ca3f325f6a1989d8edd77acd03dbbf9b9b3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:23.384ex; height:5.176ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{Z}}\,e^{-H\left(p,q\right)/kT}}"></span> <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 Z=\int e^{-H\left(p,q\right)/kT}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\int e^{-H\left(p,q\right)/kT}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ffb7f97122f06d7132e0cecb86ff350a67cf53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.013ex; height:5.676ex;" alt="{\displaystyle Z=\int e^{-H\left(p,q\right)/kT}\,dp\,dq}"></span> </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <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>c</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fef76cdf15a08b0fa1bdf743d505a17c71cad58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:34.444ex; height:5.176ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)={\frac {1}{\mathcal {Z}}}\,e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}}"></span> <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 \ {\mathcal {Z}}=\int e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}\,dp\,dq}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> <mo>=</mo> <mo>∫</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <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>c</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ {\mathcal {Z}}=\int e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}\,dp\,dq}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/868a9dff2f2344385b7bcdd4f274910243c56ad1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.654ex; height:5.676ex;" alt="{\displaystyle \ {\mathcal {Z}}=\int e^{-\left(H\left(p,q\right)-\sum _{i=1}^{c}\mu _{i}\,N_{i}\right)/kT}\,dp\,dq}"></span> </p> </td></tr> <tr> <td style="background:#f2f2f2"><b>Potențial termodinamic</b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>S</mi> <mo>=</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <msup> <mi mathvariant="normal">Ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-variant" mathvariant="normal">′</mi> </mrow> </msup> <mrow> <mo>(</mo> <mi>U</mi> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81231ec617fd0e97904a5195a1b4b1ff01945e19" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.206ex; height:3.009ex;" alt="{\displaystyle \ S=k\,ln\Omega ^{\prime }\left(U\right)}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ F=-kT\,lnZ}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ F=-kT\,lnZ}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/662ee71c03a3f7e0e7a93aea5bd0bf8e4ceed54a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:14.231ex; height:2.343ex;" alt="{\displaystyle \ F=-kT\,lnZ}"></span> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>T</mi> <mspace width="thinmathspace" /> <mi>l</mi> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">Z</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38641c7b23074ebbc752a56b2315a881556bc1a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.938ex; height:2.343ex;" alt="{\displaystyle {\mathcal {F}}=-kT\,ln{\mathcal {Z}}}"></span> </td></tr> <tr> <td style="background:#f2f2f2"><b>Ecuații de stare</b> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ dS={\frac {dU}{T}}-{\frac {\sum _{j=1}^{m}X_{j}\,dx_{j}}{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mi>S</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>U</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dS={\frac {dU}{T}}-{\frac {\sum _{j=1}^{m}X_{j}\,dx_{j}}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99213795ad86a94c41a9872df9f32cfedbd33060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.433ex; height:6.176ex;" alt="{\displaystyle \ dS={\frac {dU}{T}}-{\frac {\sum _{j=1}^{m}X_{j}\,dx_{j}}{T}}}"></span> <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 \left({\frac {\partial S}{\partial U}}\right)_{x}={\frac {1}{T}}\,,\;\left({\frac {\partial S}{\partial x_{j}}}\right)_{U}=-{\frac {X_{j}}{T}}}"> <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> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>U</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>S</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>T</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial S}{\partial U}}\right)_{x}={\frac {1}{T}}\,,\;\left({\frac {\partial S}{\partial x_{j}}}\right)_{U}=-{\frac {X_{j}}{T}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac5691286ec3cde7a5337af6b3994f72723048a4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.052ex; height:6.509ex;" alt="{\displaystyle \left({\frac {\partial S}{\partial U}}\right)_{x}={\frac {1}{T}}\,,\;\left({\frac {\partial S}{\partial x_{j}}}\right)_{U}=-{\frac {X_{j}}{T}}}"></span> </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ dF=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mi>F</mi> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ dF=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a911fadfca23d1986bad29f09c8786fab1f061" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.441ex; height:7.176ex;" alt="{\displaystyle \ dF=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}}"></span> <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 \left({\frac {\partial F}{\partial T}}\right)_{x}=-S\,,\;\left({\frac {\partial F}{\partial x_{j}}}\right)_{T}=X_{j}}"> <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> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>F</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial F}{\partial T}}\right)_{x}=-S\,,\;\left({\frac {\partial F}{\partial x_{j}}}\right)_{T}=X_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d5dd476febfea9440232dc44fb37116ed70410c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.097ex; height:6.343ex;" alt="{\displaystyle \left({\frac {\partial F}{\partial T}}\right)_{x}=-S\,,\;\left({\frac {\partial F}{\partial x_{j}}}\right)_{T}=X_{j}}"></span> </p> </td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ d{\mathcal {F}}=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}-\sum _{i=1}^{c}N_{i}\,d\mu _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>T</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </munderover> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <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>c</mi> </mrow> </munderover> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ d{\mathcal {F}}=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}-\sum _{i=1}^{c}N_{i}\,d\mu _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b391db803591e0e85dee959bc6f72741bf490863" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:39.68ex; height:7.176ex;" alt="{\displaystyle \ d{\mathcal {F}}=-S\,dT+\sum _{j=1}^{m}X_{j}\,dx_{j}-\sum _{i=1}^{c}N_{i}\,d\mu _{i}}"></span> <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 \left({\frac {\partial {\mathcal {F}}}{\partial T}}\right)_{x,\mu }=-S,\;\left({\frac {\partial {\mathcal {F}}}{\partial x_{j}}}\right)_{\mu ,T}=X_{j},\;\left({\frac {\partial {\mathcal {F}}}{\partial \mu _{i}}}\right)_{T,x}=-N_{i}}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>μ</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi>S</mi> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>μ</mi> <mo>,</mo> <mi>T</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mspace width="thickmathspace" /> <msub> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo>,</mo> <mi>x</mi> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left({\frac {\partial {\mathcal {F}}}{\partial T}}\right)_{x,\mu }=-S,\;\left({\frac {\partial {\mathcal {F}}}{\partial x_{j}}}\right)_{\mu ,T}=X_{j},\;\left({\frac {\partial {\mathcal {F}}}{\partial \mu _{i}}}\right)_{T,x}=-N_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2665048709d91f7b04ccb6f18827c6ca16a6c9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:54.608ex; height:6.676ex;" alt="{\displaystyle \left({\frac {\partial {\mathcal {F}}}{\partial T}}\right)_{x,\mu }=-S,\;\left({\frac {\partial {\mathcal {F}}}{\partial x_{j}}}\right)_{\mu ,T}=X_{j},\;\left({\frac {\partial {\mathcal {F}}}{\partial \mu _{i}}}\right)_{T,x}=-N_{i}}"></span> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Limitele_mecanicii_statistice_clasice">Limitele mecanicii statistice clasice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=15" title="Modifică secțiunea: Limitele mecanicii statistice clasice" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=15" title="Edit section's source code: Limitele mecanicii statistice clasice"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Din teorema echipartiției energiei rezultă că fiecare grad de libertate al unui sistem contribuie la <a href="/wiki/Termodinamic%C4%83#Schimb_de_căldură" title="Termodinamică">capacitatea termică la volum constant</a> pe <a href="/wiki/Mol" title="Mol">mol</a> cu cantitatea <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 {\begin{matrix}{\frac {1}{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1040f72e4158f6fdd3b0d4650280d21acb841d9d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.968ex; margin-bottom: -0.203ex; width:2.41ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}{\frac {1}{2}}\end{matrix}}}"></span> R, independentă de temperatură (R este <a href="/wiki/Constanta_universal%C4%83_a_gazului_ideal" title="Constanta universală a gazului ideal">constanta universală a gazului ideal</a>). Pentru un <a href="/wiki/Gaz_monoatomic" title="Gaz monoatomic">gaz monoatomic</a>, corespunzător celor trei grade de libertate de translație, se obține C<sub>V</sub> = <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 {\begin{matrix}{\frac {3}{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {3}{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa8a9a74a72417c119dc729e9910115d3f2c24cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.986ex; margin-bottom: -0.186ex; width:2.41ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}{\frac {3}{2}}\end{matrix}}}"></span> R. În cazul gazelor biatomice, ținând seama de rotația atomilor constitutivi în jurul centrului de masă, C<sub>V</sub> = <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 {\begin{matrix}{\frac {5}{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {5}{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/555add3556a5b5f32e2f1570e0dbcb23b515fd79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.986ex; margin-bottom: -0.185ex; width:2.41ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}{\frac {5}{2}}\end{matrix}}}"></span> R; iar adăugând contribuția vibrațiilor în lungul axei comune, C<sub>V</sub> = <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 {\begin{matrix}{\frac {7}{2}}\end{matrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>7</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{matrix}{\frac {7}{2}}\end{matrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0453ad254751ccce0665c45b122e7539b0cf2f62" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:2.41ex; height:3.509ex;" alt="{\displaystyle {\begin{matrix}{\frac {7}{2}}\end{matrix}}}"></span> R. Pentru un <a href="/wiki/Corp_(fizic%C4%83)" title="Corp (fizică)">corp</a> <a href="/wiki/Solid" title="Solid">solid</a>, considerat ca format din atomi care vibrează cu amplitudini mici în jurul unor poziții de echilibru stabil (nodurile unei rețele spațiale), C<sub>V</sub> = 3R. Aceste valori sunt confirmate de experiență, la temperatură ordinară, pentru gazele monoatomice și corpurile solide (<a href="/wiki/Legea_Dulong-Petit" title="Legea Dulong-Petit">legea Dulong-Petit</a>), dar nu și pentru vibrațiile moleculelor biatomice. La temperaturi scăzute se constată o dependență de temperatură în toate cazurile: capacitățile termice ale substanțelor tind către zero odată cu temperatura absolută.<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> Rezultatele mecanicii statistice clasice se verifică bine la temperaturi suficient de înalte; dar odată cu descreșterea temperaturii gradele de libertate „îngheață” unul după altul. </p><p>Conform teoremei echipartiției energiei, energia medie a unui oscilator liniar armonic de frecvență <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 \scriptstyle \nu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>ν</mi> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \nu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1823e5e699e243295810ac2b41f1e8f380f3a960" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.871ex; height:1.343ex;" alt="{\displaystyle \scriptstyle \nu }"></span>, în echilibru termic cu un termostat la temperatură T, are valoarea kT, independentă de frecvență.<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> Se obține astfel pentru distribuția spectrală a densității spațiale de energie a radiației termice la temperatură T: </p> <dl><dd><span style="padding-right:4em" id="f36"><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 \left(36\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>36</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(36\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ce9831c0f52a6fbcaf594e7cf4f3ec69ae17758" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(36\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho (\nu ,T)\propto \nu ^{2}\,kT}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>ν</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>k</mi> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\nu ,T)\propto \nu ^{2}\,kT}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fcfa38212aa367a01fe1834dafc1d84d0783829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.558ex; height:3.176ex;" alt="{\displaystyle \rho (\nu ,T)\propto \nu ^{2}\,kT}"></span></dd></dl> <p>(<a href="/wiki/Legea_Rayleigh-Jeans" title="Legea Rayleigh-Jeans">legea Rayleigh-Jeans</a>). Acest rezultat este confirmat de datele experimentale doar la frecvențe joase; creșterea cu pătratul frecvenței se atenuează la frecvențe intermediare, funcția <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 (\nu ,T)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <mo stretchy="false">(</mo> <mi>ν</mi> <mo>,</mo> <mi>T</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho (\nu ,T)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f83bc5d137107312eddd06bc3c2995f0aedae11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.914ex; height:2.843ex;" alt="{\displaystyle \rho (\nu ,T)}"></span> atinge un maxim, iar pentru <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 \scriptstyle \nu \to \infty \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>ν</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \nu \to \infty \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62b0b0f0ae9c65826af8d7fa559eda00b842e2bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.545ex; height:1.509ex;" alt="{\displaystyle \scriptstyle \nu \to \infty \,}"></span> ea tinde asimptotic la zero. Extrapolată la frecvențe înalte, legea Rayleigh-Jeans ar conduce la <i>catastrofa ultravioletă</i>: densitatea totală (integrată peste frecvențe) a energiei radiației termice ar rezulta divergentă.<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p><p><a href="/wiki/%C8%98erban_%C8%9Ai%C8%9Beica" title="Șerban Țițeica">Șerban Țițeica</a> a arătat că mecanica statistică clasică, bazată pe o distribuție continuă a energiei, este incompatibilă cu <a href="/wiki/Principiul_al_treilea_al_termodinamicii" title="Principiul al treilea al termodinamicii">principiul al treilea al termodinamicii</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> </p> <div class="mw-heading mw-heading2"><h2 id="Mecanică_statistică_cuantică"><span id="Mecanic.C4.83_statistic.C4.83_cuantic.C4.83"></span>Mecanică statistică cuantică</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=16" title="Modifică secțiunea: Mecanică statistică cuantică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=16" title="Edit section's source code: Mecanică statistică cuantică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div role="note" class="dezambiguizare rellink boilerplate seealso">Articol principal: <a href="/wiki/Operator_statistic" title="Operator statistic">Operator statistic</a>.</div><style data-mw-deduplicate="TemplateStyles:r16505893">@media screen{html.skin-theme-clientpref-night .mw-parser-output .rellink{display:flex}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .rellink{display:flex}}</style> <p>Mecanica statistică cuantică se bazează pe același postulat conform căruia proprietățile termodinamice ale unui sistem pot fi deduse pe baza unui colectiv statistic reprezentativ de stări microscopice, dar descrierea acestor stări și alcătuirea acestui colectiv diferă față de mecanica clasică. În <a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">mecanica cuantică</a>, o coordonată <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\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e1a86cf57bb32fa93b5761fe63b7fef8c76c889f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.457ex; height:2.009ex;" alt="{\displaystyle q\,}"></span> și impulsul conjugat <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4fa5f88a712eb9b03398066a0577fdcf33e02c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.646ex; height:2.009ex;" alt="{\displaystyle p\,}"></span> nu pot avea simultan valori bine determinate; ele sunt doar statistic determinate, cu abateri pătratice medii care se supun <a href="/wiki/Principiul_incertitudinii" title="Principiul incertitudinii">relației de incertitudine</a> </p> <dl><dd><span style="padding-right:4em" id="f37"><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 \left(37\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>37</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(37\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/70b602d22296a724b5d87aa0694dfcbabf00f358" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(37\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta q\,\Delta p\geq {\frac {\hbar }{2}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>q</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">Δ</mi> <mi>p</mi> <mo>≥</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta q\,\Delta p\geq {\frac {\hbar }{2}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c040976bab02cbf6878632e1d70632ac7f86e659" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.773ex; height:5.343ex;" alt="{\displaystyle \Delta q\,\Delta p\geq {\frac {\hbar }{2}}\,,}"></span></dd></dl> <p>unde <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 \hbar \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b3a426ba16b521a986e6e55512082f80aebe10a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.694ex; height:2.176ex;" alt="{\displaystyle \hbar \,}"></span> este <a href="/wiki/Constanta_Planck" title="Constanta Planck">constanta Planck</a> redusă. Noțiunea clasică de <i>traiectorie</i> (în spațiul configurațiilor sau în spațiul fazelor) își pierde sensul. Spațiul fazelor nu mai e bine definit: el devine o îngrămădire de celule imprecis delimitate, cu volum de ordinul <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 \hbar ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar ^{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0f98d8d617fcc9fb000f643140fb1955189524b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.54ex; height:2.343ex;" alt="{\displaystyle \hbar ^{n}}"></span>, unde <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> <mspace width="thinmathspace" /> </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/205e33e6845813cc72ca346b896a7945f90ca373" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.782ex; height:1.676ex;" alt="{\displaystyle n\,}"></span> este numărul gradelor de libertate.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Preluând și postulatul că probabilitatea unei anumite stări microscopice depinde doar de energia acestei stări (fără argumentarea ergodică, lipsită de sens în context cuantic), descrierea stărilor de energie bine determinată (<a href="/wiki/Stare_sta%C8%9Bionar%C4%83" title="Stare staționară">stări staționare</a>) trebuie să fie cea dată de mecanica cuantică. </p> <div class="mw-heading mw-heading3"><h3 id="Stări_staționare_în_mecanica_cuantică"><span id="St.C4.83ri_sta.C8.9Bionare_.C3.AEn_mecanica_cuantic.C4.83"></span>Stări staționare în mecanica cuantică</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=17" title="Modifică secțiunea: Stări staționare în mecanica cuantică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=17" title="Edit section's source code: Stări staționare în mecanica cuantică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În mecanica cuantică, mărimilor fizice observabile li se asociază operatori. Dinamica e exprimată prin <i><a href="/wiki/Hamiltonian_(mecanic%C4%83_cuantic%C4%83)" title="Hamiltonian (mecanică cuantică)">operatorul hamiltonian</a></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 {\mathcal {H}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/751633b9e4852509dafcc8f38e8a6349e5026dba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.351ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}\,}"></span>, care ia locul <i>funcției hamiltoniene</i> din mecanica clasică. Stările sistemului sunt statistic determinate prin <a href="/wiki/Func%C8%9Bie_de_und%C4%83" title="Funcție de undă">funcția de undă</a>, care satisface <a href="/wiki/Ecua%C8%9Bia_lui_Schr%C3%B6dinger" title="Ecuația lui Schrödinger">ecuația lui Schrödinger</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Nivele_de_energie">Nivele de energie</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=18" title="Modifică secțiunea: Nivele de energie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=18" title="Edit section's source code: Nivele de energie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Atunci când hamiltoniana (operatorul hamiltonian) nu depinde de timp, ea este operatorul asociat observabilei <a href="/wiki/Energie" title="Energie">energie</a>, iar stările se determină rezolvând <i>ecuația lui Schrödinger independentă de timp</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 {\mathcal {H}}u=Eu\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">H</mi> </mrow> </mrow> <mi>u</mi> <mo>=</mo> <mi>E</mi> <mi>u</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}u=Eu\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94d960250d7bbece179a2a6896afa886db128ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.531ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}u=Eu\,.}"></span> Valorile parametrului <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9123abddc2ec35f72035ec59f443c79ee052c9ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.163ex; height:2.176ex;" alt="{\displaystyle E\,}"></span> pentru care această ecuație are soluții <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/880f91e25cd451d89d1f6d0d06852b56a7b74a32" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.717ex; height:1.676ex;" alt="{\displaystyle u\,}"></span> acceptabile fizic reprezintă valorile posibile ale energiei, așa-zise <i>nivele de energie</i>. Este convenabil ca mulțimea nivelelor, numită <i>spectrul</i> energiei, să fie indexată în forma unui șir de valori crescătoare <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0},E_{1},...,E_{j},...\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{0},E_{1},...,E_{j},...\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03a0f15c358b678cb842cf3c4ad601a6c30ce3ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.537ex; height:2.843ex;" alt="{\displaystyle E_{0},E_{1},...,E_{j},...\,;}"></span> indicele de ordine poartă numele de <i>număr cuantic</i>. Soluțiile corespunzătoare descriu stările staționare respective. Unui aceluiași nivel de energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ E_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ E_{j}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94b3076df71934aab078a3b978ae235d7129be47" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.206ex; height:2.843ex;" alt="{\displaystyle \ E_{j}}"></span> îi pot corespunde mai multe stări diferite, descrise de funcții independente <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ u_{j_{1}},...,u_{j_{r}}\,;}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> </msub> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ u_{j_{1}},...,u_{j_{r}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/420a278d0ef8bc28816195ebbaf643e9d9ea4e75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.861ex; height:2.343ex;" alt="{\displaystyle \ u_{j_{1}},...,u_{j_{r}}\,;}"></span> se spune că nivelul respectiv este <i>degenerat de ordin r</i>. În prezența fenomenului de degenerescență trebuie specificate, pe lângă numărul cuantic principal (care indică valoarea energiei), și numere cuantice secundare (care indică valorile altor observabile compatibile, adică măsurabile simultan), necesare pentru a descrie complet starea. În cele ce urmează, se presupune implicit că acest lucru a fost făcut, iar indicele unic reprezintă de fapt un ansamblu complet de numere cuantice <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=\left\{j_{1},...,j_{r}\right\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>j</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j=\left\{j_{1},...,j_{r}\right\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbb8e30a99719dccd18631aa81196ab8d7e37b81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.027ex; width:15.909ex; height:2.843ex;" alt="{\displaystyle j=\left\{j_{1},...,j_{r}\right\}\,}"></span> care caracterizează în întregime starea staționară. </p> <div class="mw-heading mw-heading4"><h4 id="Spin">Spin</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=19" title="Modifică secțiunea: Spin" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=19" title="Edit section's source code: Spin"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Particulele elementare (cum sunt <a href="/wiki/Electron" title="Electron">electronul</a> și <a href="/wiki/Proton" title="Proton">protonul</a>) posedă un <a href="/wiki/Moment_cinetic" title="Moment cinetic">moment cinetic</a> intrinsec (independent de mișcarea orbitală) numit <a href="/wiki/Spin" class="mw-disambig" title="Spin">spin</a>. Mărimea sa este exprimată printr-un <i>număr cuantic de spin</i> care poate lua valori nenegative întregi sau semiîntregi: <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 \scriptstyle s\,=\,0,\,{\frac {1}{2}},\,1,\,{\frac {3}{2}}\,,\,...\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>s</mi> <mspace width="thinmathspace" /> <mo>=</mo> <mspace width="thinmathspace" /> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>,</mo> <mspace width="thinmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle s\,=\,0,\,{\frac {1}{2}},\,1,\,{\frac {3}{2}}\,,\,...\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f558cbaf9f3b09c5ee62a230a9dd10c88911bc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:12.999ex; height:3.176ex;" alt="{\displaystyle \scriptstyle s\,=\,0,\,{\frac {1}{2}},\,1,\,{\frac {3}{2}}\,,\,...\,}"></span> Pentru un sistem de spin s, proiecția spinului pe o direcție dată poate avea 2s + 1 valori, <a href="/wiki/Echidistant" title="Echidistant">echidistante</a> cu pas 1, cuprinse între –s și +s. Pentru electron, ipoteza existenței unui spin <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 \scriptstyle {\frac {1}{2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6365ab33810f89fb39ee1f30dae53207d88718c9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.503ex; height:3.176ex;" alt="{\displaystyle \scriptstyle {\frac {1}{2}}}"></span> a fost formulată de <a href="/wiki/George_Uhlenbeck" title="George Uhlenbeck">Uhlenbeck</a> și <a href="/wiki/Samuel_Goudsmit" title="Samuel Goudsmit">Goudsmit</a>, pentru a explica rezultatele experimentului <a href="/wiki/Spin#Experimentul_Stern-Gerlach" class="mw-disambig" title="Spin">Stern-Gerlach</a>, și dezvoltată teoretic de <a href="/wiki/Pauli" class="mw-redirect" title="Pauli">Pauli</a>. Agregatele de particule (<a href="/wiki/Nucleu_atomic" title="Nucleu atomic">nuclee</a> atomice, <a href="/wiki/Atom" title="Atom">atomi</a>, <a href="/wiki/Molecul%C4%83" title="Moleculă">molecule</a>) pot fi tratate ca particule elementare, dacă structura lor internă rămâne nemodificată în timpul interacției cu alte sisteme; spinul lor este rezultanta momentelor cinetice de spin ale componentelor.<sup id="cite_ref-26" class="reference"><a href="#cite_note-26"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Distribuția_canonică_în_mecanica_statistică_cuantică"><span id="Distribu.C8.9Bia_canonic.C4.83_.C3.AEn_mecanica_statistic.C4.83_cuantic.C4.83"></span>Distribuția canonică în mecanica statistică cuantică</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=20" title="Modifică secțiunea: Distribuția canonică în mecanica statistică cuantică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=20" title="Edit section's source code: Distribuția canonică în mecanica statistică cuantică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Trecând de la o distribuție continuă a energiei <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(p,q)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>p</mi> <mo>,</mo> <mi>q</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(p,q)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32e8192bc4f27a3a2e0be18a2c513b9b30a212df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.533ex; height:2.843ex;" alt="{\displaystyle H(p,q)\,}"></span> la o energie distribuită pe nivele discrete <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 \left\{E_{0},E_{1},...,E_{i},...\right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{E_{0},E_{1},...,E_{i},...\right\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e07583612f946d54cd53556243d65aa3b91320d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.752ex; height:2.843ex;" alt="{\displaystyle \left\{E_{0},E_{1},...,E_{i},...\right\},}"></span> probabilitatea <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 {\mathcal {P}}\left(p,q\right)dp\,dq\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">P</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mi>p</mi> <mo>,</mo> <mi>q</mi> </mrow> <mo>)</mo> </mrow> <mi>d</mi> <mi>p</mi> <mspace width="thinmathspace" /> <mi>d</mi> <mi>q</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {P}}\left(p,q\right)dp\,dq\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed8a839d97453a188b88583a566e256f04f7b63b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.005ex; height:2.843ex;" alt="{\displaystyle {\mathcal {P}}\left(p,q\right)dp\,dq\,}"></span> în spațiul fazelor este înlocuită prin probabilitatea <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> <mspace width="thinmathspace" /> </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/539193d3e3af20578edb578d587218ca8b096df4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.679ex; height:2.509ex;" alt="{\displaystyle P_{i}\,}"></span> de realizare a stării de energie <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{i}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1738302619c665b36b69e6ae72b6ad74ddbd511c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.549ex; height:2.509ex;" alt="{\displaystyle E_{i}\,,}"></span> caracterizată prin numărul cuantic <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> <mspace width="thinmathspace" /> <mo>.</mo> </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/13355bea3eb8e6ff47a92ac595c36bf01c560e82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.836ex; height:2.176ex;" alt="{\displaystyle i\,.}"></span> Echivalentul relațiilor <a href="#f11">(11)</a> și <a href="#f12">(12)</a> în mecanica statistică cuantică este, ținând seama și de <a href="#f26">(26)</a>: </p> <dl><dd><span style="padding-right:4em" id="f38"><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 \left(38\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>38</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(38\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/90ce0a790b95c4f9c30ecb6c5411813db47b96b7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(38\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}={\frac {1}{Z}}\;e^{-E_{i}/kT}\,,}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>Z</mi> </mfrac> </mrow> <mspace width="thickmathspace" /> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}={\frac {1}{Z}}\;e^{-E_{i}/kT}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11ffd1d536a196d598be1623e41f1109bc992fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:16.854ex; height:5.176ex;" alt="{\displaystyle P_{i}={\frac {1}{Z}}\;e^{-E_{i}/kT}\,,}"></span></dd></dl> <dl><dd><span style="padding-right:4em" id="f39"><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 \left(39\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>39</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(39\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c24ca6d7e97deaec422a45d3f9625bbffbb954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(39\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z=\sum _{j}e^{-E_{j}/kT}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z=\sum _{j}e^{-E_{j}/kT}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aca92ea6923e779b4c4cc9b6a96b81cab98201b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:16.912ex; height:5.843ex;" alt="{\displaystyle Z=\sum _{j}e^{-E_{j}/kT}\,.}"></span></dd></dl> <p>Odată cunoscută <i>suma de stare</i> (funcția de partiție) <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\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a0f68bfc7fb1fdc42ee67ee1f50846d8ccfac56" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.714ex; height:2.509ex;" alt="{\displaystyle Z\,,}"></span> proprietățile macroscopice ale sistemului se deduc din energia liberă <a href="#f27">(27)</a> prin metode standard. Determinarea nivelelor de energie pentru un sistem cu un număr foarte mare de grade de libertate este însă o problemă dificilă, chiar dispunând de resurse de calcul moderne. De aceea, o termodinamică statistică bazată pe relațiile <a href="#f38">(38)</a> și <a href="#f39">(39)</a> este greu sau imposibil de construit, în cazul cel mai general. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Enrico_Fermi_1943-49.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Enrico_Fermi_1943-49.jpg/160px-Enrico_Fermi_1943-49.jpg" decoding="async" width="160" height="198" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Enrico_Fermi_1943-49.jpg/240px-Enrico_Fermi_1943-49.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Enrico_Fermi_1943-49.jpg/320px-Enrico_Fermi_1943-49.jpg 2x" data-file-width="2368" data-file-height="2927" /></a><figcaption>Fermi</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Paul_Dirac,_1933.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Paul_Dirac%2C_1933.jpg/160px-Paul_Dirac%2C_1933.jpg" decoding="async" width="160" height="236" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Paul_Dirac%2C_1933.jpg/240px-Paul_Dirac%2C_1933.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/50/Paul_Dirac%2C_1933.jpg/320px-Paul_Dirac%2C_1933.jpg 2x" data-file-width="750" data-file-height="1107" /></a><figcaption>Dirac</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:SatyenBose1925.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/SatyenBose1925.jpg/160px-SatyenBose1925.jpg" decoding="async" width="160" height="214" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/SatyenBose1925.jpg/240px-SatyenBose1925.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/SatyenBose1925.jpg/320px-SatyenBose1925.jpg 2x" data-file-width="372" data-file-height="497" /></a><figcaption>Bose</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Albert_Einstein_(Nobel).png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Albert_Einstein_%28Nobel%29.png/160px-Albert_Einstein_%28Nobel%29.png" decoding="async" width="160" height="226" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/50/Albert_Einstein_%28Nobel%29.png/240px-Albert_Einstein_%28Nobel%29.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/50/Albert_Einstein_%28Nobel%29.png 2x" data-file-width="280" data-file-height="396" /></a><figcaption>Einstein</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/Fi%C8%99ier:Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg/420px-Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg.png" decoding="async" width="420" height="336" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg/630px-Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg/840px-Fermi-Dirac_Bose-Einstein_Maxwell-Boltzmann_statistics.svg.png 2x" data-file-width="600" data-file-height="480" /></a><figcaption>Comparație între statisticile Fermi-Dirac, Bose-Einstein și Maxwell-Boltzmann</figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Sisteme_de_particule_identice">Sisteme de particule identice</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=21" title="Modifică secțiunea: Sisteme de particule identice" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=21" title="Edit section's source code: Sisteme de particule identice"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Problema se simplifică apreciabil dacă sistemul macroscopic considerat constă dintr-un număr mare de subsisteme identice a căror structură internă rămâne practic neafectată de interacțiunile dintre ele; în acest caz se vorbește despre un <i>sistem de <a href="/wiki/Particule_identice" title="Particule identice">particule identice</a></i>. <a href="/wiki/Gaz" title="Gaz">Gazele</a> și electronii din <a href="/wiki/Metal" title="Metal">metale</a> sunt astfel de sisteme. </p><p>Fie un sistem compus dintr-un număr <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> <mspace width="thinmathspace" /> </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/ed4f97d800eaacd982789661ac3896e1181d6137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle N\,}"></span> de particule identice și fie <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 \left\{\epsilon _{0},\epsilon _{1},...,\epsilon _{i},...\right\}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>}</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{\epsilon _{0},\epsilon _{1},...,\epsilon _{i},...\right\}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4de5803d0c6f8ed21b6c72019479c1515ee525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.792ex; height:2.843ex;" alt="{\displaystyle \left\{\epsilon _{0},\epsilon _{1},...,\epsilon _{i},...\right\}\,}"></span> nivelele de energie ale unei particule izolate în condițiile externe date, presupuse cunoscute. Pentru a realiza echilibrul termodinamic, particulele componente trebuie să interacționeze (prin mecanismul „ciocnirilor” din teoria cinetică), dar se presupune că aceste interacțiuni au un efect neglijabil asupra nivelelor de energie. În acest sens, particulele sunt <i>independente</i>, iar nivelele de energie ale sistemului rezultă din însumarea nivelelor de energie ale particulelor componente. Pentru alcătuirea unui colectiv statistic reprezentativ trebuie ținut seama de faptul că, în mecanica cuantică, particulele identice sunt distribuite statistic pe stările uniparticulă, descrierea lor individuală — de genul „particula cu numărul <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> <mspace width="thinmathspace" /> </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/30f268d4be31700461b9f20cabb0724899ad5d27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:1.372ex; height:2.509ex;" alt="{\displaystyle j\,}"></span> se află în starea de energie <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 \epsilon _{j}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{j}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f87f81aedc5e23b5d1f7c1f34054797ad8d5a34f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.241ex; height:2.343ex;" alt="{\displaystyle \epsilon _{j}\,}"></span>” — fiind lipsită de sens. <a href="/wiki/Num%C4%83r_de_particule" title="Număr de particule">Numărul de particule</a> din sistem aflate într-o anumită stare uniparticulă se numește <i>număr de ocupare</i> al acelei stări; exisă deci, în paralel cu șirul nivelelor de energie, șirul numerelor de ocupare <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 \left\{n_{0},n_{1},...,n_{i},...\right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{n_{0},n_{1},...,n_{i},...\right\}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44f92eecd93702f91f130e5daf5ec00ef2fc9d30" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.791ex; height:2.843ex;" alt="{\displaystyle \left\{n_{0},n_{1},...,n_{i},...\right\}.}"></span> Suma energiilor particulelor componente este energia sistemului: </p> <dl><dd><span style="padding-right:4em" id="f40"><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 \left(40\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>40</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(40\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05faac351e388644405fe3f0e85fbc31fb814255" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(40\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\sum _{j}\,n_{j}\,\epsilon _{j}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\sum _{j}\,n_{j}\,\epsilon _{j}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b55b554f23eb5ac82ad9f37a6280f40a0ed6abb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.583ex; height:5.843ex;" alt="{\displaystyle E=\sum _{j}\,n_{j}\,\epsilon _{j}\,.}"></span></dd></dl> <p>Interacțiile dintre particulele componente, fără să modifice nivelele de energie, produc o redistribuire a particulelor pe nivelele existente. Distribuția statistică reprezentativă pentru această situație este distribuția macrocanonică, în care toate componentele intervin cu același potențial chimic, întrucât particulele sunt identice: </p> <dl><dd><span style="padding-right:4em" id="f41"><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 \left(41\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>41</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(41\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f717139c97540f2e4b930603004c6a6346e4be72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(41\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=const\cdot e^{-\left(\sum _{j}n_{j}\epsilon _{j}\,-\sum _{j}\mu \,n_{j}\right)/kT}=const\cdot \prod _{j}\left(e^{-\left(\epsilon _{j}-\mu \right)/kT}\right)^{n_{j}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mo>−</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mi>μ</mi> <mspace width="thinmathspace" /> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=const\cdot e^{-\left(\sum _{j}n_{j}\epsilon _{j}\,-\sum _{j}\mu \,n_{j}\right)/kT}=const\cdot \prod _{j}\left(e^{-\left(\epsilon _{j}-\mu \right)/kT}\right)^{n_{j}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4617cb8ca6e27a95bf115a760997191a11266a99" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:62.797ex; height:6.843ex;" alt="{\displaystyle P=const\cdot e^{-\left(\sum _{j}n_{j}\epsilon _{j}\,-\sum _{j}\mu \,n_{j}\right)/kT}=const\cdot \prod _{j}\left(e^{-\left(\epsilon _{j}-\mu \right)/kT}\right)^{n_{j}}\,.}"></span></dd></dl> <p>Această formulă reprezintă probabilitatea ca cele <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> <mspace width="thinmathspace" /> </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/ed4f97d800eaacd982789661ac3896e1181d6137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.451ex; height:2.176ex;" alt="{\displaystyle N\,}"></span> particule să fie distribuite astfel: în starea <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 \epsilon _{0}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fcf9854c0de85b948ef1f3d7623f529da1f3b0d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.009ex;" alt="{\displaystyle \epsilon _{0}\,}"></span> să se găsească <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}\,}"> <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> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{0}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9e4ba80a6543303d48236039fa4cc128f0cb938" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.009ex;" alt="{\displaystyle n_{0}\,}"></span> particule, în starea <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 \epsilon _{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4997502a22282bdbd6ca78ad358a49a85dcefb6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.385ex; height:2.009ex;" alt="{\displaystyle \epsilon _{1}\,}"></span> să se găsească <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_{1}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n_{1}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82fc209a7c0cd570eba0fa23b5ac94f3c5608b8e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.836ex; height:2.009ex;" alt="{\displaystyle n_{1}\,}"></span> particule, ... etc. Probabilitatea ca în starea de energie <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 \epsilon _{i}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ed9bfb964dd44eef1a6d41ae8c3e9ff60dc53dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.131ex; height:2.009ex;" alt="{\displaystyle \epsilon _{i}\,}"></span> să se găsească <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> <mspace width="thinmathspace" /> </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/d636fa67ff31e695d9568062ea814cf29c47fc35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.581ex; height:2.009ex;" alt="{\displaystyle n_{i}\,}"></span> particule, indiferent de modul în care sunt ocupate celelalte stări, se obține sumând peste celelalte stări, cu rezultatul </p> <dl><dd><span style="padding-right:4em" id="f42"><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 \left(42\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>42</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(42\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/720504ff9aea29974d38db4b2fe713dfba5c3a4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(42\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P_{i}\left(\epsilon _{i},n_{i}\right)=const\cdot \left(e^{-\left(\epsilon _{i}-\mu \right)/kT}\right)^{n_{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> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <msup> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace" /> <mo>;</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{i}\left(\epsilon _{i},n_{i}\right)=const\cdot \left(e^{-\left(\epsilon _{i}-\mu \right)/kT}\right)^{n_{i}}\,;}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9939493d08a67a4095510ee74a9cc331a9fc401c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:35.621ex; height:4.843ex;" alt="{\displaystyle P_{i}\left(\epsilon _{i},n_{i}\right)=const\cdot \left(e^{-\left(\epsilon _{i}-\mu \right)/kT}\right)^{n_{i}}\,;}"></span></dd></dl> <p>factorul constant se determină din condiția de normare a probabilităților </p> <dl><dd><span style="padding-right:4em" id="f43"><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 \left(43\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>43</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(43\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/69db36d1b82e98f7ecf106d0a98fba22623b0585" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(43\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{n_{j}}\,P_{j}\left(\epsilon _{i},n_{j}\right)=1\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{n_{j}}\,P_{j}\left(\epsilon _{i},n_{j}\right)=1\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a47b7db46077ad45e21a2dfab145b1fa07ae2725" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:19.105ex; height:6.009ex;" alt="{\displaystyle \sum _{n_{j}}\,P_{j}\left(\epsilon _{i},n_{j}\right)=1\,.}"></span></dd></dl> <p>Valoarea medie a numărului de ocupare pentru nivelul <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 \epsilon _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9be3140164b763359077d92b2cd33798eb6a488c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.744ex; height:2.009ex;" alt="{\displaystyle \epsilon _{i}}"></span>, care indică distribuția particulelor din sistem pe nivelele de energie uniparticulă, este </p> <dl><dd><span style="padding-right:4em" id="f44"><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 \left(44\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>44</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(44\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce15c7ab4d9cbf74dc2a6e1aa52eff09362f6f20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(44\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle n_{i}\right\rangle =\sum _{n_{j}}\,n_{j}\,P_{j}\left(\epsilon _{i},n_{j}\right)\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> </munder> <mspace width="thinmathspace" /> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mspace width="thinmathspace" /> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle n_{i}\right\rangle =\sum _{n_{j}}\,n_{j}\,P_{j}\left(\epsilon _{i},n_{j}\right)\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822d81ebb9383c69dd2df3017849279cd55caa97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:25.024ex; height:6.009ex;" alt="{\displaystyle \left\langle n_{i}\right\rangle =\sum _{n_{j}}\,n_{j}\,P_{j}\left(\epsilon _{i},n_{j}\right)\,.}"></span></dd></dl> <p>Dacă pentru toate nivelele numărul de ocupare are valoarea <b>1</b>, relația <a href="#f41">(41)</a> se reduce la distribuția canonică, iar relația <a href="#f42">(42)</a> devine <i>distribuția Maxwell-Boltzmann</i> din mecanica statistică clasică.<sup id="cite_ref-27" class="reference"><a href="#cite_note-27"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading4"><h4 id="Relația_dintre_spin_și_statistică"><span id="Rela.C8.9Bia_dintre_spin_.C8.99i_statistic.C4.83"></span>Relația dintre spin și statistică</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=22" title="Modifică secțiunea: Relația dintre spin și statistică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=22" title="Edit section's source code: Relația dintre spin și statistică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Există o relație cu caracter general între tipul de statistică — exprimat prin relațiile <a href="#f42">(42)</a>-<a href="#f44">(44)</a> — de care ascultă un sistem de particule identice și valoarea spinului acestor particule: </p> <ul><li>Pentru particulele de spin semiîntreg <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 \scriptstyle \left(s={\frac {1}{2}},\,{\frac {3}{2}},\,{\frac {5}{2}},\,...\,\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mn>2</mn> </mfrac> </mrow> <mo>,</mo> <mspace width="thinmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left(s={\frac {1}{2}},\,{\frac {3}{2}},\,{\frac {5}{2}},\,...\,\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/204a4e3692bea21fd4864d9fde64186886eade0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.202ex; height:3.509ex;" alt="{\displaystyle \scriptstyle \left(s={\frac {1}{2}},\,{\frac {3}{2}},\,{\frac {5}{2}},\,...\,\right)\,}"></span> numărul de ocupare poate lua numai două valori: <b>0</b> și <b>1</b>. Particulele din această categorie, numite <a href="/wiki/Fermion" title="Fermion">fermioni</a>, se supun statisticii <a href="/wiki/Enrico_Fermi" title="Enrico Fermi">Fermi</a>-<a href="/wiki/Paul_Dirac" title="Paul Dirac">Dirac</a>.</li> <li>Pentru particulele de spin întreg <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 \scriptstyle \left(s=0,\,1,\,2,\,...\,\right)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow> <mo>(</mo> <mrow> <mi>s</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mspace width="thinmathspace" /> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \left(s=0,\,1,\,2,\,...\,\right)\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cbfbf00473cff4c8efa42d9397308bd4b7d914ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.475ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \left(s=0,\,1,\,2,\,...\,\right)\,}"></span> numărul de ocupare poate lua orice valoare întreagă: <b>0</b>, <b>1</b>, <b>2</b>, ... Particulele din această categorie, numite <a href="/wiki/Boson" title="Boson">bosoni</a>, se supun statisticii <a href="/wiki/Satyendra_Nath_Bose" title="Satyendra Nath Bose">Bose</a>-<a href="/wiki/Albert_Einstein" title="Albert Einstein">Einstein</a>.</li></ul> <p>În mecanica cuantică nerelativistă această relație are caracter de <a href="/wiki/Postulat" class="mw-redirect" title="Postulat">postulat</a>, rezultat din analiza datelor experimentale asupra sistemelor de particule identice. O primă formulare, limitată la electroni (care sunt fermioni) e cunoscută ca <a href="/wiki/Principiul_de_excluziune" title="Principiul de excluziune">principiul de excluziune</a> al lui Pauli. Relația dintre spinul semiîntreg/întreg și caracterul de fermion/boson este demonstrată, în ipoteze foarte generale, în cadrul teoriei cuantice relativiste a câmpurilor, sub denumirea de <i>teorema spin-statistică</i>. </p><p>Cu acestea, numărul de ocupare mediu pentru cele două tipuri de statistică se obține din formula <a href="#f44">(44)</a> prin calcul direct: </p> <div class="mw-heading mw-heading5"><h5 id="Statistica_Fermi-Dirac_(fermioni)"><span id="Statistica_Fermi-Dirac_.28fermioni.29"></span>Statistica Fermi-Dirac (fermioni)</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=23" title="Modifică secțiunea: Statistica Fermi-Dirac (fermioni)" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=23" title="Edit section's source code: Statistica Fermi-Dirac (fermioni)"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span style="padding-right:4em" id="f45"><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 \left(45\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>45</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(45\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e5c2e893c6cdfa1b6ee52e798d3f1750f146009" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(45\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>+</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55fb1e8abb18fa7d7e0c38b10ac0a5878c3c4c9e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.968ex; height:5.676ex;" alt="{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}+1}}\,.}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Statistica_Bose-Einstein_(bosoni)"><span id="Statistica_Bose-Einstein_.28bosoni.29"></span>Statistica Bose-Einstein (bosoni)</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=24" title="Modifică secțiunea: Statistica Bose-Einstein (bosoni)" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=24" title="Edit section's source code: Statistica Bose-Einstein (bosoni)"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <dl><dd><span style="padding-right:4em" id="f46"><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 \left(46\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>46</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(46\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f51da309c8b41cbf55d08dab5b99e79b256fdd6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(46\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}-1}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}-1}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06ba4540c8f73551e9ded33d87926f0fd0cb611e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.968ex; height:5.676ex;" alt="{\displaystyle \left\langle n_{i}\right\rangle ={\frac {1}{e^{(\epsilon _{i}-\mu )/kT}-1}}\,.}"></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Dependența_de_parametrii_macroscopici"><span id="Dependen.C8.9Ba_de_parametrii_macroscopici"></span>Dependența de parametrii macroscopici</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=25" title="Modifică secțiunea: Dependența de parametrii macroscopici" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=25" title="Edit section's source code: Dependența de parametrii macroscopici"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Numărul de ocupare mediu depinde de doi parametri macroscopici ai sistemului: temperatura <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/476a8389064c06ab89963a2467aef525838da0cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.023ex; height:2.176ex;" alt="{\displaystyle T\,}"></span> și potențialul chimic <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 \,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eee966f89ad721b5c50cfdab56f0aab6789a1af2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.436ex; height:2.176ex;" alt="{\displaystyle \mu \,.}"></span> Aceștia nu sunt însă independenți, ci sunt legați prin faptul că </p> <dl><dd><span style="padding-right:4em" id="f47"><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 \left(47\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>47</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(47\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67d6913154f9778b31abd76b1b27f4653a0951cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(47\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{j}\left\langle n_{j}\right\rangle =N\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <mi>N</mi> <mspace width="thinmathspace" /> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j}\left\langle n_{j}\right\rangle =N\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f9d7c516f52f952bbfeda8f6d073dd41418eeac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:14.052ex; height:5.843ex;" alt="{\displaystyle \sum _{j}\left\langle n_{j}\right\rangle =N\,.}"></span></dd></dl> <div class="mw-heading mw-heading5"><h5 id="Limita_clasică"><span id="Limita_clasic.C4.83"></span>Limita clasică</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=26" title="Modifică secțiunea: Limita clasică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=26" title="Edit section's source code: Limita clasică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pentru ambele tipuri de statistică, dacă exponențiala de la <a href="/wiki/Numitor" class="mw-redirect" title="Numitor">numitor</a> devine foarte mare în raport cu unitatea, aceasta din urmă poate fi neglijată; se obține </p> <dl><dd><span style="padding-right:4em" id="f48"><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 \left(48\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mn>48</mn> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(48\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a9d95a7ffae7f660568a301ce8b3f362d43d68f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.134ex; height:2.843ex;" alt="{\displaystyle \left(48\right)}"></span></span><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\langle n_{i}\right\rangle =e^{-\left(\epsilon _{i}-\mu \right)/kT}=const\cdot e^{-\epsilon _{i}/kT}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mo>=</mo> <mi>c</mi> <mi>o</mi> <mi>n</mi> <mi>s</mi> <mi>t</mi> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mi>T</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle n_{i}\right\rangle =e^{-\left(\epsilon _{i}-\mu \right)/kT}=const\cdot e^{-\epsilon _{i}/kT}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a412d8e5806bf7b0330b1db3504a11e16839dc5e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.366ex; height:3.343ex;" alt="{\displaystyle \left\langle n_{i}\right\rangle =e^{-\left(\epsilon _{i}-\mu \right)/kT}=const\cdot e^{-\epsilon _{i}/kT}\,,}"></span></dd></dl> <p>adică <i>distribuția Maxwell-Boltzmann</i> din mecanica statistică clasică. Pentru aceasta e necesar ca <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 \epsilon _{i}>\mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ϵ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>></mo> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \epsilon _{i}>\mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96feb8343d3369c570e42e1d07b63eb892743dd9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.244ex; height:2.343ex;" alt="{\displaystyle \epsilon _{i}>\mu }"></span> iar temperatura să fie suficient de înaltă. În acest caz <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 \left\langle n_{i}\right\rangle \ll 1\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>⟨</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>⟩</mo> </mrow> <mo>≪</mo> <mn>1</mn> <mspace width="thinmathspace" /> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\langle n_{i}\right\rangle \ll 1\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d42b2e2c7f939b09084eb91b3cfc33e5390060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.814ex; height:2.843ex;" alt="{\displaystyle \left\langle n_{i}\right\rangle \ll 1\,,}"></span> deci densitatea de particule e foarte mică. Se poate arăta, pe baza relației <a href="#f47">(47)</a>, că această situație se realizează mai ușor în cazul particulelor de masă mare. În aceste condiții, dispar caracteristicile cuantice și proprietățile sistemului sunt cele date de statistica clasică. </p> <div class="mw-heading mw-heading5"><h5 id="Degenerescență_cuantică"><span id="Degenerescen.C8.9B.C4.83_cuantic.C4.83"></span>Degenerescență cuantică</h5><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=27" title="Modifică secțiunea: Degenerescență cuantică" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=27" title="Edit section's source code: Degenerescență cuantică"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>În cazul opus, când exponențiala este de ordinul unității, cele două distribuții duc la rezultate radical diferite de statistica clasică și între ele: apar fenomenele zise de <i>degenerescență cuantică</i>. Evident, aceasta se întâmplă când condițiile din secțiunea precedentă sunt inversate: la temperaturi suficient de scăzute, densități suficient de mari și mase suficient de mici. Mai precis: există o temperatură de prag, cu atât mai ridicată cu cât sistemul este mai dens și masa particulelor e mai mică, sub care apar fenomenele de degenerescență. </p><p>În cazul statisticii Fermi-Dirac, faptul că o particulă ocupă o anumită stare exclude alte particule din această stare, ceea ce echivalează cu o forță repulsivă care se opune condensării sistemului. În cazul electronilor din metale, densitatea este totuși suficient de mare, iar masa foarte mică, ceea ce face ca sistemul să fie degenerat până la temperatura de topire. Din această cauză multe <a href="/wiki/Legea_Dulong-Petit" title="Legea Dulong-Petit">proprietăți ale metalelor</a> la temperatura ordinară nu au putut fi explicate prin statistica clasică. </p><p>Statistica Bose-Einstein, admițând ocuparea unei stări de către un număr foarte mare de particule, echivalează cu o forță atractivă care favorizează condensarea. În cazul unui gaz de atomi de heliu, deși masa este mică, temperatura de prag este foarte scăzută; proprietățile neobișnuite ale <a href="/wiki/Superfluid" title="Superfluid">condensatului de heliu</a> la temperaturi sub 3 K sunt explicate ca fenomene de degenerescență. </p> <div class="mw-heading mw-heading2"><h2 id="Note">Note</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=28" title="Modifică secțiunea: Note" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=28" title="Edit section's source code: Note"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><b><a href="#cite_ref-1">^</a></b> <span class="reference-text">Țițeica (1956), p. 19.</span> </li> <li id="cite_note-2"><b><a href="#cite_ref-2">^</a></b> <span class="reference-text">Țițeica (1956), p. 21; Țițeica (2000), p. 54.</span> </li> <li id="cite_note-3"><b><a href="#cite_ref-3">^</a></b> <span class="reference-text">Gibbs, p. vii.</span> </li> <li id="cite_note-4"><b><a href="#cite_ref-4">^</a></b> <span class="reference-text">În limba română se folosește predominant termenul <i>colectiv statistic</i>, introdus de Țițeica. În engleză s-a impus termenul <i>statistical ensemble</i>, introdus de Gibbs, p. 5.</span> </li> <li id="cite_note-5"><b><a href="#cite_ref-5">^</a></b> <span class="reference-text">Țițeica (1956), pp. 27–30; Țițeica (2000), pp. 60–64.</span> </li> <li id="cite_note-6"><b><a href="#cite_ref-6">^</a></b> <span class="reference-text">Sunt în uz curent două notații standard pentru valoarea medie a unei variabile aleatorii <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 \scriptstyle f:}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mi>f</mi> <mo>:</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle f:}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a0ad6a2258c8accbf48025b1480726564fcc73" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.361ex; height:2.009ex;" alt="{\displaystyle \scriptstyle f:}"></span> cu paranteze unghiulare <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 \scriptstyle \langle f\rangle \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mo fence="false" stretchy="false">⟨</mo> <mi>f</mi> <mo fence="false" stretchy="false">⟩</mo> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle \langle f\rangle \,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7a1dbbb7892fd841740dccb63f7b1650723ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.571ex; height:2.176ex;" alt="{\displaystyle \scriptstyle \langle f\rangle \,}"></span> sau cu bară deasupra <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 \scriptstyle {\overline {f}}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="1"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mspace width="thinmathspace" /> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\overline {f}}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4fd1a165178ea267f03546b015e45aa0db7c20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.449ex; height:2.676ex;" alt="{\displaystyle \scriptstyle {\overline {f}}\,}"></span>.</span> </li> <li id="cite_note-7"><b><a href="#cite_ref-7">^</a></b> <span class="reference-text">Schrödinger, pp. 3–4, argumentează calitativ plauzibilitatea acestui postulat.</span> </li> <li id="cite_note-8"><b><a href="#cite_ref-8">^</a></b> <span class="reference-text">Tolman, pp. 59–63.</span> </li> <li id="cite_note-9"><b><a href="#cite_ref-9">^</a></b> <span class="reference-text">Gibbs, p. 115.</span> </li> <li id="cite_note-10"><b><a href="#cite_ref-10">^</a></b> <span class="reference-text">Țițeica (2000), p. 65–69.</span> </li> <li id="cite_note-11"><b><a href="#cite_ref-11">^</a></b> <span class="reference-text">Gibbs, p. 32.</span> </li> <li id="cite_note-12"><b><a href="#cite_ref-12">^</a></b> <span class="reference-text">În engleză se numește <i>grand canonical ensemble</i>, termen introdus de Gibbs, p. 189.</span> </li> <li id="cite_note-13"><b><a href="#cite_ref-13">^</a></b> <span class="reference-text">Tolman, p. 621.</span> </li> <li id="cite_note-14"><b><a href="#cite_ref-14">^</a></b> <span class="reference-text">Cantitățile pot fi exprimate în unități de masă, <a href="/wiki/Mol" title="Mol">mol</a> sau număr de molecule.</span> </li> <li id="cite_note-15"><b><a href="#cite_ref-15">^</a></b> <span class="reference-text">Gibbs, p. 191.</span> </li> <li id="cite_note-16"><b><a href="#cite_ref-16">^</a></b> <span class="reference-text">Țițeica (1956), pp. 33–37.</span> </li> <li id="cite_note-17"><b><a href="#cite_ref-17">^</a></b> <span class="reference-text">Țițeica (2000), pp. 69–72.</span> </li> <li id="cite_note-18"><b><a href="#cite_ref-18">^</a></b> <span class="reference-text">Țițeica (2000), p. 72.</span> </li> <li id="cite_note-19"><b><a href="#cite_ref-19">^</a></b> <span class="reference-text">Țițeica (1956), pp. 46–49; Țițeica (2000), pp. 72–73.</span> </li> <li id="cite_note-20"><b><a href="#cite_ref-20">^</a></b> <span class="reference-text">Wannier, p. 158; Kittel, p. 64.</span> </li> <li id="cite_note-21"><b><a href="#cite_ref-21">^</a></b> <span class="reference-text">Țițeica (2000), pp. 94–111.</span> </li> <li id="cite_note-22"><b><a href="#cite_ref-22">^</a></b> <span class="reference-text">Țițeica (2000), p. 100.</span> </li> <li id="cite_note-23"><b><a href="#cite_ref-23">^</a></b> <span class="reference-text">Țițeica (2000), pp. 111–113.</span> </li> <li id="cite_note-24"><b><a href="#cite_ref-24">^</a></b> <span class="reference-text">Țițeica, Șerban: <i>Principiul al treilea al termodinamicii și mecanica statistică</i>, Studii și cercetări de fizică, Tomul IV, pp. 7–14 (1953); reprodus în Țițeica (2000), pp. 317–324.</span> </li> <li id="cite_note-25"><b><a href="#cite_ref-25">^</a></b> <span class="reference-text">Țițeica (1956), p. 52.</span> </li> <li id="cite_note-26"><b><a href="#cite_ref-26">^</a></b> <span class="reference-text">Țițeica (1984), pp. 354–355.</span> </li> <li id="cite_note-27"><b><a href="#cite_ref-27">^</a></b> <span class="reference-text">Țițeica (1956), pp. 55–56.</span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Bibliografie">Bibliografie</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=29" title="Modifică secțiunea: Bibliografie" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=29" title="Edit section's source code: Bibliografie"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Ludwig_Boltzmann" title="Ludwig Boltzmann">Boltzmann, Ludwig</a>: <i>Vorlesungen über Gastheorie</i>, I. Theil, Verlag Johann Ambrosius Barth, Leipzig, 1896. <a rel="nofollow" class="external text" href="http://www.archive.org/details/vorlesungenber01bolt">E-book</a>.</li> <li>Boltzmann, Ludwig: <i>Vorlesungen über Gastheorie</i>, II. Theil, Verlag Johann Ambrosius Barth, Leipzig, 1898. <a rel="nofollow" class="external text" href="http://www.archive.org/details/vorlesungenber02bolt">E-book</a>.</li> <li><a href="/wiki/Paul_Ehrenfest" title="Paul Ehrenfest">Ehrenfest, Paul</a> și <a href="/wiki/Tatiana_Afanasieva" title="Tatiana Afanasieva">Tatiana</a>: <i>The Conceptual Foundations of the Statistical Approach in Mechanics</i>, Dover Publications, 2002. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0486495043" class="internal mw-magiclink-isbn">ISBN 0-486-49504-3</a>.</li> <li>Fowler, R.H.: <i>Statistical Mechanics</i>, University Press, Cambridge, 1980, <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0521093775" class="internal mw-magiclink-isbn">ISBN 0 521 09377 5</a>.</li> <li><a href="/wiki/Josiah_Willard_Gibbs" title="Josiah Willard Gibbs">Gibbs, J. Willard</a>: <i>Elementary Principles in Statistical Mechanics</i>, Charles Scribner's Sons, New York, 1902. <a rel="nofollow" class="external text" href="http://www.archive.org/details/elementaryprinci00gibbrich">E-book</a>.</li> <li>Huang, Kerson: <i>Statistical Mechanics</i>, ed. a 2-a, John Wiley & Sons, 1987. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0471815187" class="internal mw-magiclink-isbn">ISBN 0-471-81518-7</a>.</li> <li><a href="/wiki/Charles_Kittel" title="Charles Kittel">Kittel, Charles</a>: <i>Elementary Statistical Physics</i>, Dover Publications, 2004. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0486435148" class="internal mw-magiclink-isbn">ISBN 0-486-43514-8</a>.</li> <li><a href="/wiki/Lev_Landau" title="Lev Landau">Landau, L.D.</a>, <a href="/wiki/Evgheni_Lif%C8%99i%C8%9B" title="Evgheni Lifșiț">Lifshitz, E.M.</a>: <i>Statistical Physics</i>, Pergamon Press, 1980. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0080230385" class="internal mw-magiclink-isbn">ISBN 0-08-023038-5</a>.</li> <li><a href="/wiki/Ilie_G._Murgulescu" title="Ilie G. Murgulescu">Murgulescu, I.G.</a> și <a href="/wiki/Eugen_Segal" title="Eugen Segal">Segal, E.</a>: <i>Introducere în chimia fizică, vol. II, 1-Teoria molecular-cinetică a materiei</i>, București, Editura Academiei RSR, 1979.</li> <li><a href="/wiki/Octav_Onicescu" title="Octav Onicescu">Onicescu, O.</a>, <a href="/wiki/Gheorghe_Mihoc" title="Gheorghe Mihoc">Mihoc, G.</a> și Ionescu-Tulcea, C.T.: <i>Calculul probabilităților și aplicații</i>, Editura Academiei Republicii Populare Romîne, București, 1956.</li> <li><a href="/wiki/Erwin_Schr%C3%B6dinger" title="Erwin Schrödinger">Schrödinger, Erwin</a>: <i>Statistical Thermodynamics</i>, Dover Publications, 1989, <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0486661016" class="internal mw-magiclink-isbn">ISBN 0-486-66101-6</a>.</li> <li><a href="/wiki/Richard_C._Tolman" title="Richard C. Tolman">Tolman, Richard C</a>.: <i>The Principles of Statistical Mechanics</i>, Dover Publications, 1979. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/0486638960" class="internal mw-magiclink-isbn">ISBN 0-486-63896-0</a>.</li> <li><a href="/wiki/%C8%98erban_%C8%9Ai%C8%9Beica" title="Șerban Țițeica">Țițeica, Șerban</a>: <i>Elemente de mecanică statistică</i>, Editura Tehnică, București, 1956.</li> <li>Țițeica, Șerban: <i>Mecanica cuantică</i>, Editura Academiei Republicii Socialiste România, București, 1984.</li> <li>Țițeica, Șerban: <i>Curs de fizică statistică și teoria cuantelor</i>, All Educational, Timișoara, 2000. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/973684319X" class="internal mw-magiclink-isbn">ISBN 973-684-319-X</a></li> <li>Wannier, Gregory H.: <i>Statistical Physics</i>, Dover Publications, 1987. <a href="/wiki/Special:Referin%C8%9Be_%C3%AEn_c%C4%83r%C8%9Bi/048665401X" class="internal mw-magiclink-isbn">ISBN 0-486-65401-X</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Vezi_și"><span id="Vezi_.C8.99i"></span>Vezi și</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=30" title="Modifică secțiunea: Vezi și" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=30" title="Edit section's source code: Vezi și"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Teoria_probabilit%C4%83%C8%9Bilor" title="Teoria probabilităților">Teoria probabilităților</a></li> <li><a href="/wiki/Termodinamic%C4%83" title="Termodinamică">Termodinamică</a></li> <li><a href="/wiki/Teoria_cinetic%C4%83_a_gazelor" title="Teoria cinetică a gazelor">Teoria cinetică a gazelor</a></li> <li><a href="/wiki/Poten%C8%9Bial_termodinamic" title="Potențial termodinamic">Potențial termodinamic</a></li> <li><a href="/wiki/Gaz_perfect" title="Gaz perfect">Gaz perfect</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Legături_externe"><span id="Leg.C4.83turi_externe"></span>Legături externe</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&veaction=edit&section=31" title="Modifică secțiunea: Legături externe" class="mw-editsection-visualeditor"><span>modificare</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Mecanic%C4%83_statistic%C4%83&action=edit&section=31" title="Edit section's source code: Legături externe"><span>modificare sursă</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Loudspeaker.svg" class="mw-file-description" title="Video"><img alt="Video" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/13px-Loudspeaker.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/20px-Loudspeaker.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/26px-Loudspeaker.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> Leonard Susskind: <i><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=H1Zbp6__uNw&p=B72416C707D85AB0&index=1&feature=BF">Modern Physics: Statistical Mechanics</a></i>, <a href="/wiki/Universitatea_Stanford" title="Universitatea Stanford">Stanford University</a>.</li> <li><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Loudspeaker.svg" class="mw-file-description" title="Video"><img alt="Video" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/13px-Loudspeaker.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/20px-Loudspeaker.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/26px-Loudspeaker.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> Moungi Bawendi și Keith Nelson: <i><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=kLqduWF6GXE&p=A62087102CC93765&index=1&feature=BF">Thermodynamics & Kinetics</a></i>, <a href="/wiki/Massachusetts_Institute_of_Technology" title="Massachusetts Institute of Technology">MIT</a>.</li> <li><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Loudspeaker.svg" class="mw-file-description" title="Video"><img alt="Video" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/13px-Loudspeaker.svg.png" decoding="async" width="13" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/20px-Loudspeaker.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8a/Loudspeaker.svg/26px-Loudspeaker.svg.png 2x" data-file-width="20" data-file-height="20" /></a></span> <i><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=DcqfPRqhFVM">Bose-Einstein, Fermi-Dirac, and Maxwell-Boltzmann Statistics</a></i>, <a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com">Wolfram Demonstrations Project</a>.</li> <li>Judith A. McGovern: <i><a rel="nofollow" class="external text" href="http://theory.ph.man.ac.uk/~judith/stat_therm/stat_therm.html">Thermal and Statistical Physics</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120229141255/http://theory.ph.man.ac.uk/~judith/stat_therm/stat_therm.html">Arhivat</a> în <time datetime="2012-02-29">29 februarie 2012</time>, la <a href="/wiki/Wayback_Machine" class="mw-redirect" title="Wayback Machine">Wayback Machine</a>.</i>, University of Manchester.</li> <li>Daniel F. Styer: <i><a rel="nofollow" class="external text" href="http://www.oberlin.edu/physics/dstyer/StatMech/book.pdf">Statistical Mechanics</a></i>, Oberlin College.</li></ul> <div class="noprint tright portal" style="border:solid #aaa 1px; margin:0.5em 0 0.5em 0.5em;"> <table style="background:var(--background-color-interactive-subtle, #f9f9f9); color:inherit; font-size:85%; line-height:110%; max-width:175px;"> <tbody><tr> <td style="text-align: center;"><span typeof="mw:File"><a href="/wiki/Fi%C8%99ier:Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg" class="mw-file-description"><img alt="Portal icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/25px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png" decoding="async" width="25" height="28" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/37px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/6f/Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg/49px-Stylised_atom_with_three_Bohr_model_orbits_and_stylised_nucleus.svg.png 2x" data-file-width="530" data-file-height="600" /></a></span> </td> <td style="padding: 0 0.2em; vertical-align: middle; font-style: italic; font-weight: bold"><b><a href="/wiki/Portal:Fizic%C4%83" title="Portal:Fizică">Portal Fizică </a></b> </td></tr> </tbody></table></div> <table class="navbox" cellspacing="0" style=""> <tbody><tr> <td style="padding:2px;"> <table cellspacing="0" class="nowraplinks collapsible autocollapse" style="width:100%;background:transparent;color:inherit;;"> <tbody><tr> <th style=";" colspan="2" class="navbox-title"><div style="float:left; width:6em;text-align:left;"><div class="noprint plainlinks" style="padding:0; font-size:xx-small; color:var(--color-base, #000); white-space:nowrap; ;"><span style=";;border:none;"><a href="/wiki/Format:Ramurile_fizicii" title="Format:Ramurile fizicii"><span title="Vizualizare format" style=";;border:none;;">v</span></a> <span style="font-size:80%;">•</span> <a href="/wiki/Discu%C8%9Bie_Format:Ramurile_fizicii" title="Discuție Format:Ramurile fizicii"><span title="Discuție format" style=";;border:none;;">d</span></a> <span style="font-size:80%;">•</span> <a class="external text" href="https://ro.wikipedia.org/w/index.php?title=Format:Ramurile_fizicii&action=edit"><span title="Acest format se poate modifica. 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 <span style="font-weight:bold;">·</span>  <a href="/wiki/Mi%C8%99care_(fizic%C4%83)" title="Mișcare (fizică)">Mișcare</a> </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-even"><div style="padding:0em 0.25em"> <a href="/wiki/Termodinamic%C4%83" title="Termodinamică">Termodinamică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Calorimetrie" title="Calorimetrie">Calorimetrie</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Mecanic%C4%83" title="Mecanică">Mecanică</a> (<a href="/wiki/Mecanic%C4%83_clasic%C4%83" title="Mecanică clasică">Mecanică clasică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Mecanica_mediilor_continue" title="Mecanica mediilor continue">Mecanica mediilor continue</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Mecanic%C4%83_cereasc%C4%83" title="Mecanică cerească">Mecanică cerească</a>  <span style="font-weight:bold;">·</span>  <a class="mw-selflink selflink">Mecanică statistică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Mecanica_fluidelor" title="Mecanica fluidelor">Mecanica fluidelor</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Mecanic%C4%83_cuantic%C4%83" title="Mecanică cuantică">Mecanică cuantică</a>)</div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;"><a href="/wiki/Und%C4%83" title="Undă">Unde</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/C%C3%A2mp_(fizic%C4%83)" title="Câmp (fizică)">Câmpuri</a> </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"><div style="padding:0em 0.25em"> <a href="/wiki/Gravita%C8%9Bie" title="Gravitație">Gravitație</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Electromagnetism" title="Electromagnetism">Electromagnetism</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Teoria_cuantic%C4%83_a_c%C3%A2mpurilor" title="Teoria cuantică a câmpurilor">Teoria cuantică a câmpurilor</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Teoria_relativit%C4%83%C8%9Bii" title="Teoria relativității">Relativitate</a> (<a href="/wiki/Teoria_relativit%C4%83%C8%9Bii_restr%C3%A2nse" title="Teoria relativității restrânse">restrânsă</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Teoria_relativit%C4%83%C8%9Bii_generale" title="Teoria relativității generale">generală</a>)</div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">După specialitate </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-even"><div style="padding:0em 0.25em"> <a href="/w/index.php?title=Accelerator_(tehnic%C4%83)&action=edit&redlink=1" class="new" title="Accelerator (tehnică) — pagină inexistentă">Accelerator</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Acustic%C4%83" title="Acustică">Acustică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Astrofizic%C4%83" title="Astrofizică">Astrofizică</a>  <span style="font-weight:bold;">·</span>  <a href="/w/index.php?title=Fizic%C4%83_spa%C8%9Bial%C4%83&action=edit&redlink=1" class="new" title="Fizică spațială — pagină inexistentă">Fizică spațială</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Chimie_fizic%C4%83" title="Chimie fizică">Fizică chimică</a>  <span style="font-weight:bold;">·</span>  <a href="/w/index.php?title=Fizic%C4%83_computa%C8%9Bional%C4%83&action=edit&redlink=1" class="new" title="Fizică computațională — pagină inexistentă">Fizică computațională</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizica_materiei_condensate" title="Fizica materiei condensate">Fizica materiei condensate</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Inginerie_fizic%C4%83" title="Inginerie fizică">Inginerie fizică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizic%C4%83_matematic%C4%83" title="Fizică matematică">Fizică matematică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizic%C4%83_nuclear%C4%83" title="Fizică nucleară">Fizică nucleară</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Optic%C4%83" title="Optică">Optică</a> (<a href="/wiki/Optic%C4%83_geometric%C4%83" title="Optică geometrică">geometrică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Optic%C4%83_cuantic%C4%83" title="Optică cuantică">cuantică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Optic%C4%83_ondulatorie" title="Optică ondulatorie">ondulatorie</a>)  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizica_particulelor_elementare" title="Fizica particulelor elementare">Fizica particulelor elementare</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Plasm%C4%83" title="Plasmă">Fizica plasmei</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizic%C4%83_statistic%C4%83" title="Fizică statistică">Fizică statistică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Spectroscopie" title="Spectroscopie">Spectroscopie</a></div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Combinate cu<br />alte științe </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"><div style="padding:0em 0.25em"> <a href="/wiki/Biofizic%C4%83" title="Biofizică">Biofizică</a> (<a href="/wiki/Bioelectronic%C4%83" title="Bioelectronică">Bioelectronică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Biomagnetism" title="Biomagnetism">Biomagnetism</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Biomecanic%C4%83" title="Biomecanică">Biomecanică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Fizic%C4%83_medical%C4%83" title="Fizică medicală">medicală</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Neurobiofizic%C4%83" title="Neurobiofizică">Neurofizică</a>)  <span style="font-weight:bold;">·</span>  <a href="/wiki/Chimie_biofizic%C4%83" title="Chimie biofizică">Chimie biofizică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Econofizic%C4%83" title="Econofizică">Econofizică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Chimie_fizic%C4%83" title="Chimie fizică">Chimie fizică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Geofizic%C4%83" title="Geofizică">Geofizică</a>  <span style="font-weight:bold;">·</span>  <a href="/wiki/Psihofizic%C4%83" title="Psihofizică">Psihofizică</a></div> </td></tr></tbody></table> </td></tr></tbody></table><style data-mw-deduplicate="TemplateStyles:r16513826">.mw-parser-output .navbox{margin:0 auto 0}@media screen{html.skin-theme-clientpref-night .mw-parser-output .navbox-abovebelow[style],html.skin-theme-clientpref-night .mw-parser-output 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width:6em;text-align:left;"><div class="noprint plainlinks" style="padding:0; font-size:xx-small; color:var(--color-base, #000); white-space:nowrap; ;"><span style=";;border:none;"><a href="/wiki/Format:Fizic%C4%83_statistic%C4%83" title="Format:Fizică statistică"><span title="Vizualizare format" style=";;border:none;;">v</span></a> <span style="font-size:80%;">•</span> <a href="/w/index.php?title=Discu%C8%9Bie_Format:Fizic%C4%83_statistic%C4%83&action=edit&redlink=1" class="new" title="Discuție Format:Fizică statistică — pagină inexistentă"><span title="Discuție format" style=";;border:none;;">d</span></a> <span style="font-size:80%;">•</span> <a class="external text" href="https://ro.wikipedia.org/w/index.php?title=Format:Fizic%C4%83_statistic%C4%83&action=edit"><span title="Acest format se poate modifica. Folosiți butonul de previzualizare înainte de a salva." style=";;border:none;;">m</span></a></span></div></div><span class="" style="font-size: 110%;"><a href="/wiki/Fizic%C4%83_statistic%C4%83" title="Fizică statistică">Fizică statistică</a></span> </th></tr> <tr style="height:2px;"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Termodinamică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"> <div style="padding:0em 0.25em"> <a href="/wiki/Calorimetrie" title="Calorimetrie">Calorimetrie</a> • <a href="/wiki/Capacitate_termic%C4%83" title="Capacitate termică">Capacitate termică</a> • <a href="/wiki/C%C4%83ldur%C4%83_latent%C4%83" title="Căldură latentă">Căldură latentă</a> • <a href="/wiki/Ciclu_termodinamic" title="Ciclu termodinamic">Ciclu termodinamic</a> • <a href="/wiki/Ciclul_Carnot" title="Ciclul Carnot">Ciclul Carnot</a> • <a href="/wiki/Ciclul_Clausius-Rankine" title="Ciclul Clausius-Rankine">Ciclul Clausius-Rankine</a> • <a href="/wiki/Coeficient_de_transformare_adiabatic%C4%83" title="Coeficient de transformare adiabatică">Coeficient de transformare adiabatică</a> • <a href="/wiki/Constanta_universal%C4%83_a_gazului_ideal" title="Constanta universală a gazului ideal">Constanta universală a gazului ideal</a> • <a href="/wiki/Echilibru_termodinamic" title="Echilibru termodinamic">Echilibru termodinamic</a> • <a href="/wiki/Energie_intern%C4%83" title="Energie internă">Energie internă</a> • <a href="/wiki/Energie_liber%C4%83" title="Energie liberă">Energie liberă</a> • <a href="/wiki/Entalpie" title="Entalpie">Entalpie</a> • <a href="/wiki/Entalpie_liber%C4%83" title="Entalpie liberă">Entalpie liberă</a> • <a href="/wiki/Entropia_radia%C8%9Biei_electromagnetice" title="Entropia radiației electromagnetice">Entropia radiației electromagnetice</a> • <a href="/wiki/Entropia_termodinamic%C4%83_(dup%C4%83_Carath%C3%A9odory)" title="Entropia termodinamică (după Carathéodory)">Entropia termodinamică (după Carathéodory)</a> • <a href="/wiki/Entropie" title="Entropie">Entropie</a> • <a href="/wiki/Entropie_termodinamic%C4%83" title="Entropie termodinamică">Entropie termodinamică</a> • <a href="/wiki/Evaporare" title="Evaporare">Evaporare</a> • <a href="/wiki/Faz%C4%83_(termodinamic%C4%83)" title="Fază (termodinamică)">Fază (termodinamică)</a> • <a href="/wiki/Fierbere" title="Fierbere">Fierbere</a> • <a href="/wiki/Formula_lui_Planck" title="Formula lui Planck">Formula lui Planck</a> • <a href="/wiki/Frac%C8%9Bie_molar%C4%83" title="Fracție molară">Fracție molară</a> • <a href="/wiki/Gaz_ideal" title="Gaz ideal">Gaz ideal</a> • <a href="/wiki/Gaz_perfect" title="Gaz perfect">Gaz perfect</a> • <a href="/wiki/Gaz_real" title="Gaz real">Gaz real</a> • <a href="/wiki/Legea_Boyle-Mariotte" title="Legea Boyle-Mariotte">Legea Boyle-Mariotte</a> • <a href="/wiki/Legea_Dulong-Petit" title="Legea Dulong-Petit">Legea Dulong-Petit</a> • <a href="/wiki/Legea_lui_Avogadro" title="Legea lui Avogadro">Legea lui Avogadro</a> • <a href="/wiki/Legea_lui_Dalton" title="Legea lui Dalton">Legea lui Dalton</a> • <a href="/wiki/Legea_lui_Henry" title="Legea lui Henry">Legea lui Henry</a> • <a href="/wiki/Legea_lui_Raoult" title="Legea lui Raoult">Legea lui Raoult</a> • <a href="/wiki/Legile_de_deplasare_ale_lui_Wien" title="Legile de deplasare ale lui Wien">Legile de deplasare ale lui Wien</a> • <a href="/wiki/Legile_lui_Kirchhoff_(radia%C8%9Bie)" title="Legile lui Kirchhoff (radiație)">Legile lui Kirchhoff (radiație)</a> • <a href="/wiki/Lema_lui_Carath%C3%A9odory_(termodinamic%C4%83)" title="Lema lui Carathéodory (termodinamică)">Lema lui Carathéodory (termodinamică)</a> • <a href="/wiki/M%C4%83rimi_molare_de_exces" title="Mărimi molare de exces">Mărimi molare de exces</a> • <a href="/wiki/Paradoxul_lui_Gibbs_(termodinamic%C4%83)" title="Paradoxul lui Gibbs (termodinamică)">Paradoxul lui Gibbs (termodinamică)</a> • <a href="/wiki/Perpetuum_mobile" title="Perpetuum mobile">Perpetuum mobile</a> • <a href="/wiki/Poten%C8%9Bial_chimic" title="Potențial chimic">Potențial chimic</a> • <a href="/wiki/Poten%C8%9Bial_termodinamic" title="Potențial termodinamic">Potențial termodinamic</a> • <a href="/wiki/Presiune_de_vapori" title="Presiune de vapori">Presiune de vapori</a> • <a href="/wiki/Principiile_termodinamicii" title="Principiile termodinamicii">Principiile termodinamicii</a> • <a href="/wiki/Principiul_al_doilea_al_termodinamicii" title="Principiul al doilea al termodinamicii">Principiul al doilea al termodinamicii</a> • <a href="/wiki/Principiul_al_doilea_al_termodinamicii:_Planck_versus_Carath%C3%A9odory" title="Principiul al doilea al termodinamicii: Planck versus Carathéodory">Principiul al doilea al termodinamicii: Planck versus Carathéodory</a> • <a href="/wiki/Principiul_al_treilea_al_termodinamicii" title="Principiul al treilea al termodinamicii">Principiul al treilea al termodinamicii</a> • <a href="/wiki/Principiul_%C3%AEnt%C3%A2i_al_termodinamicii" title="Principiul întâi al termodinamicii">Principiul întâi al termodinamicii</a> • <a href="/wiki/Principiul_zero_al_termodinamicii" title="Principiul zero al termodinamicii">Principiul zero al termodinamicii</a> • <a href="/wiki/Proces_adiabatic" title="Proces adiabatic">Proces adiabatic</a> • <a href="/wiki/Punct_de_fierbere" title="Punct de fierbere">Punct de fierbere</a> • <a href="/wiki/Punct_de_topire" title="Punct de topire">Punct de topire</a> • <a href="/wiki/Radia%C8%9Bie_termic%C4%83" title="Radiație termică">Radiație termică</a> • <a href="/wiki/Rela%C8%9Bia_lui_Mayer" title="Relația lui Mayer">Relația lui Mayer</a> • <a href="/wiki/Rezonatorul_lui_Planck" title="Rezonatorul lui Planck">Rezonatorul lui Planck</a> • <a href="/wiki/Sistem_termodinamic" title="Sistem termodinamic">Sistem termodinamic</a> • <a href="/wiki/Temperatur%C4%83" title="Temperatură">Temperatură</a> • <a href="/wiki/Termochimie" title="Termochimie">Termochimie</a> • <a href="/wiki/Termodinamic%C4%83" title="Termodinamică">Termodinamică</a> • <a href="/wiki/Transformare_Legendre" title="Transformare Legendre">Transformare Legendre</a> • <a href="/wiki/Transformare_termodinamic%C4%83" title="Transformare termodinamică">Transformare termodinamică</a> • <a href="/wiki/Termodinamic%C4%83_chimic%C4%83" title="Termodinamică chimică">Termodinamică chimică</a> •</div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Mecanică statistică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-even"><div style="padding:0em 0.25em"> <a href="/wiki/Entropie_statistic%C4%83" title="Entropie statistică">Entropie statistică</a> • <a href="/wiki/Faz%C4%83_(mecanic%C4%83_statistic%C4%83)" title="Fază (mecanică statistică)">Fază (mecanică statistică)</a> • <a href="/wiki/Grad_de_libertate" title="Grad de libertate">Grad de libertate</a> • <a class="mw-selflink selflink">Mecanică statistică</a> • <a href="/wiki/Operator_statistic" title="Operator statistic">Operator statistic</a></div> </td></tr> <tr style="height:2px"> <td> </td></tr> <tr> <td class="navbox-group" style=";;">Teorie cinetică </td> <td style="text-align:left;border-left:2px solid #fdfdfd;width:100%;padding:0px;;;" class="navbox-list navbox-odd"><div style="padding:0em 0.25em"> <a href="/wiki/Agita%C8%9Bie_termic%C4%83" class="mw-redirect" title="Agitație termică">Agitație termică</a> • <a href="/wiki/Constanta_Boltzmann" title="Constanta Boltzmann">Constanta Boltzmann</a> • <a href="/wiki/Demonul_lui_Maxwell" title="Demonul lui Maxwell">Demonul lui Maxwell</a> • <a href="/wiki/Num%C4%83rul_lui_Avogadro" title="Numărul lui Avogadro">Numărul lui Avogadro</a> • <a href="/wiki/Teoria_cinetic%C4%83_a_gazelor" title="Teoria cinetică a gazelor">Teoria cinetică a gazelor</a> • <a href="/wiki/Teoria_haosului" title="Teoria haosului">Teoria haosului</a></div> </td></tr></tbody></table> </td></tr></tbody></table><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r16513826"> <div role="navigation" class="navbox" aria-label="Navbox" style="padding:3px"><table class="nowraplinks hlist navbox-inner" style="border-spacing:0;color:inherit"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Ajutor:Control_de_autoritate" title="Ajutor:Control de autoritate">Control de autoritate</a></th><td class="navbox-list navbox-odd" style="text-align:left;border-left-width:2px;border-left-style:solid;width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><span class="nowrap"><a href="/wiki/Biblioteca_Nacional_de_Espa%C3%B1a" class="mw-redirect" title="Biblioteca Nacional de España">BNE</a>: <span class="uid"><a rel="nofollow" class="external text" href="http://catalogo.bne.es/uhtbin/authoritybrowse.cgi?action=display&authority_id=XX524922">XX524922</a></span></span></li> <li><span class="nowrap"><a href="/wiki/Biblioth%C3%A8que_nationale_de_France" title="Bibliothèque nationale de France">BNF</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11958255n">cb11958255n</a> <a rel="nofollow" class="external text" href="http://data.bnf.fr/ark:/12148/cb11958255n">(data)</a></span></span></li> <li><span class="nowrap"><a href="/wiki/Integrated_Authority_File" class="mw-redirect" title="Integrated Authority File">GND</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4056999-8">4056999-8</a></span></span></li> <li><span class="nowrap"><a href="/wiki/Library_of_Congress_Control_Number" class="mw-redirect" title="Library of Congress Control Number">LCCN</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/subjects/sh85127571">sh85127571</a></span></span></li> <li><span class="nowrap"><a href="/wiki/National_Diet_Library" class="mw-redirect" title="National Diet Library">NDL</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://id.ndl.go.jp/auth/ndlna/00573177">00573177</a></span></span></li> <li><span class="nowrap"><a href="/wiki/National_Library_of_the_Czech_Republic" class="mw-redirect" title="National Library of the Czech Republic">NKC</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://aleph.nkp.cz/F/?func=find-c&local_base=aut&ccl_term=ica=ph170616&CON_LNG=ENG">ph170616</a></span></span></li> <li><span class="nowrap"><a href="/wiki/Syst%C3%A8me_universitaire_de_documentation" title="Système universitaire de documentation">SUDOC</a>: <span class="uid"><a rel="nofollow" class="external text" href="https://www.idref.fr/027570711">027570711</a></span></span></li></ul> </div></td><td class="navbox-image" rowspan="1" style="width:1px;padding:0px 0px 0px 2px"><div><span class="skin-invert" typeof="mw:File/Frameless"><a 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Mecanică statistică - Wikipedia