{"id":34734,"date":"2021-05-26T13:00:19","date_gmt":"2021-05-26T13:00:19","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34734"},"modified":"2025-09-20T06:58:10","modified_gmt":"2025-09-20T06:58:10","slug":"distributio-boltzmanniana-in-coetu-canonico","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/distributio-boltzmanniana-in-coetu-canonico\/","title":{"rendered":"Distributio Boltzmanniana in coetu canonico"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\ttext-align:center;\n\t}\n<\/style>\n<h1>Distributio Boltzmanniana in coetu canonico<\/h1>\n<p style=\"text-align:center;\">Thermodynamica nobis ostendit quomodo systemata physica aequilibrium consequantur et quomodo energia et probabilitas eorum mores determinent. In hac lectione, coetum canonicum atque Distributio Boltzmanniana explicabuntur, instrumenta fundamentalia ad intellegendos phaenomena sicut reactiones chemicas et aequilibrium in systematibus complexis. Cognosces quomodo hae notiones temperaturam cum ordine et inordinatione conectant, sinentes praedicere mores eorum quae videantur impraedicibilia.<\/p>\n<p style=\"text-align:center;\"><strong>Proposita Discendi:<\/strong><br \/>\nExpleta hac lectione discipulus poterit\n<\/p>\n<ol>\n<li><strong>Identificare<\/strong> genera coetuum thermodynamicorum (microcanonicum, canonicum, et magnum canonicum).<\/li>\n<li><strong>Deducere<\/strong> Distributionem Boltzmannianam ex principiis thermodynamicis.<\/li>\n<li><strong>Computare<\/strong> probabilitates microstatibus associatas utens functione partitionis.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>INDEX CONTENTORUM<\/u>:<\/strong><br \/>\n<a href=\"#1\">Coetus in thermodynamica<\/a><br \/>\n<a href=\"#2\">Distributio Boltzmanniana<\/a><br \/>\n<a href=\"#3\">Applicationes Distributionis Boltzmannianae<\/a><br \/>\n<a href=\"#4\">Exercitia<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/ch1bjhU7k70\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p>Unum ex instrumentis notionum utilissimis thermodynamicae est notio \u00abcoetus\u00bb. Ex varietate existentium unus ex frequentissime adhibitis est coetus canonicus, et ex eo derivatur distributio Boltzmanniana. Utraque notio infra recognoscetur.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Coetus in thermodynamica<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=174s\" target=\"_blank\" rel=\"noopener\"><strong>Adhuc probabilitates usi sumus<\/strong><\/a> ad describenda systemata thermodynamica, et ratio nostra in eo consistit ut fingeamus nos experimentum ac mensuras infinitas vicibus repetere posse velut modum supplendi incapacitatem nostram ad proprietates microscopicas (per microstatus descriptas) moderandas. His notionibus inspiratus, <a href=\"https:\/\/es.wikipedia.org\/wiki\/Josiah_Willard_Gibbs\" rel=\"noopener, nofollow, noreferrer noopener\" target=\"_blank\">Gibbs<\/a> anno 1878 introducit notionem \u00abcollectivitatis\u00bb sive \u00abcoetuum\u00bb: est idealizatio in qua consideratur magnus numerus \u00abexemplaribus systematis\u00bb, quarum unaquaeque unum ex statibus eius possibilibus repraesentat. In thermodynamica tres species coetuum principales dantur.<\/p>\n<ol>\n<li><strong>Coetus Microcanonicus:<\/strong> Est collectio systematum quae omnes eandem energiam fixam possident.<\/li>\n<li><strong>Coetus Canonicus:<\/strong> Est collectio systematum, quorum unumquodque potest energiam cum magna copia caloris commutare. Ut postea videbimus, hoc constituit (atque definit) temperaturam systematis.<\/li>\n<li><strong>Coetus Magnus Canonicus:<\/strong> Est collectio systematum ubi unumquodque potest et materiam (particulas) et energiam cum magna horreo eorum commutare. Per hoc definiuntur temperatura et potentia chemica systematis.<\/li>\n<\/ol>\n<h3>Coetus Canonicus<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=362s\" target=\"_blank\" rel=\"noopener\"><strong>Duo systemata coniuncta consideremus<\/strong><\/a> ita ut energiam commutare possint. Hoc autem tempore unum ex his immensum respectu alterius faciemus atque <strong>reservam, fontem<\/strong> vel <strong>balneum caloris<\/strong> appellabimus. Haec reserva tam immensa est ut magnas copias energiae sumi possimus sine mutatione temperaturae eius. Numerus modorum quibus quanta energiae in reserva disponuntur est, proinde, ingens. Alterum systema parvum est comparatione et simpliciter <strong>systema<\/strong> vocabitur.<\/p>\n<p>Assumemus quod pro qualibet energia systematis permissa unus tantum microstatus existat, atque ideo systema semper habebit valorem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega=1<\/span>.<\/span> Praeterea, energiam totalem systematum coniunctarum figemus cum valore <span class=\"katex-eq\" data-katex-display=\"false\">E<\/span>.<\/p>\n<p><em>Si hoc loco consistamus videbimus systema et reservam formare coetum microcanonicum, ubi energia constans manet et omnes microstatus aequiprobabiles sunt.<\/em><\/p>\n<p>In hoc casu, si energia systematis est <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon<\/span>, ita ut energia reservae sit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E - \\epsilon<\/span>.<\/span> Haec condicio in qua systema in contactu thermico cum magna copia energiae est, id est quod vocatur <strong>Coetus Canonicus.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Distributio Boltzmanniana<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=528s\" target=\"_blank\" rel=\"noopener\"><strong>Probabilitas <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\epsilon)<\/span><\/span> ut systema<\/strong><\/a> energiam <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> habeat est proportionalis numero microstatuum qui reservae sunt accessibiles multiplicato per numerum microstatuum qui systemati sunt accessibiles. Id est:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\epsilon)\\propto \\Omega(E-\\epsilon)\\cdot 1<\/span>.<\/span><\/p>\n<p>Ut iam antea vidimus, temperatura exprimi potest in terminis logarithmi <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> per<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{k_B T} = \\frac{d\\ln\\Omega}{dE}<\/span><\/span><\/p>\n<p>Et cum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon \\ll E<\/span>,<\/span> fieri potest expansio in seriebus Taylor de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln\\Omega(E-\\epsilon)<\/span><\/span> circa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon = 0<\/span>.<\/span> Ex hoc habebitur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\ln\\Omega(E-\\epsilon) = \\ln\\Omega(E) - \\frac{d\\ln\\Omega(E)}{dE}\\epsilon + \\cdots<\/span><\/span><\/p>\n<p>Deinde, ex duabus ultimis expressionibus habetur:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\ln\\Omega(E-\\epsilon) = \\ln\\Omega(E) - \\frac{\\epsilon}{k_B T} + \\cdots<\/span><\/span><\/p>\n<p>Ubi <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> est temperatura reservae. Hoc loco possumus reliquos terminos expansionis in seriebus Taylor neglegere et dicere valere relationem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln \\Omega(E-\\epsilon) \\approx \\ln\\Omega(E) - \\displaystyle \\frac{\\epsilon}{k_B T}<\/span><\/span><\/p>\n<p>Si hanc ultimam expressionem explicemus ad hoc perveniemus<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E-\\epsilon) \\approx \\Omega(E) e^{-\\displaystyle \\frac{\\epsilon}{k_B T}}<\/span><\/span><\/p>\n<p>Nunc, si hoc ultimum cum probabilitate <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\epsilon)<\/span><\/span> comparemus concluditur<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\epsilon)\\propto e^{-\\epsilon\/(k_B T)}<\/span><\/span><\/p>\n<p>Cum systema in aequilibrio thermodynamico cum reserva sit, eandem temperaturam habent. Attamen, quamvis temperatura <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> constans maneat, energia <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> non est; immo, ad distributionem probabilitatis alligatur, quam modo obtinuimus. Hoc appellatur <strong>Distributio Boltzmanniana,<\/strong> sive <strong>Canonica<\/strong> pro coetu canonico. Terminus <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-\\epsilon\/(k_B T) }<\/span><\/span> notus est ut <strong>Factor Boltzmannianus.<\/strong><\/p>\n<h3>Normalizatio Distributionis Boltzmannianae et Functio Partitionis<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=890s\" target=\"_blank\" rel=\"noopener\"><strong>His evolutionibus coepimus<\/strong><\/a> componere distributionem probabilitatum quae describit quomodo parvum systema se gerat cum coniunctum sit magnae reservae cum temperatura <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span>. Systema habet opportunitatem rationabilem obtinendi energiam <span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon<\/span> minorem quam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k_B T<\/span><\/span>, sed exponentialem in distributione Boltzmanniana celeriter decrescere cum agitur de energia maiore obtinenda. Nunc autem notandum est distributionem talem qualem habemus non esse stricte distributionem probabilitatis, sed prius normalizandam esse. Si systema in contactum cum reserva ponitur et habet microstatum <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> cum energia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_r<\/span><\/span>, tunc habebimus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">P({microestado\\;}r)= \\displaystyle \\frac{e^{-E_r\/(k_B T)}}{\\displaystyle \\sum_{i}e^{-E_i\/(k_B T)}}<\/span><\/span><\/p>\n<p>Suma quae in denominatore posita est munus normalizandi implet quod sinit <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> esse distributionem probabilitatis. Suma denominatori inscripta etiam nota est ut <strong>Functio Partitionis<\/strong> et denotatur per <span class=\"katex-eq\" data-katex-display=\"false\">Z<\/span>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">Z = \\displaystyle \\sum_i e^{-E_i\/(k_B T)}<\/span><\/span><\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/lDtYzvXiNb4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h2>Applicationes Distributionis Boltzmannianae<\/h2>\n<p>Ad illustrandas nonnullas applicationes coetus canonici et distributionis Boltzmannianae videbimus quomodo apparent cum quaedam exempla considerantur. Antequam vero incipiamus notationem introducemus pro quantitate quae satis frequenter apparet et quae in futuro utilis esse potest. Definitor factor <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> per aequalitatem<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta =\\displaystyle  \\frac{1}{k_B T}<\/span>,<\/span><\/p>\n<p>ita ut, ex hoc, scribere possimus:<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta = \\displaystyle \\frac{d\\ln\\Omega}{dE}<\/span>,<\/span><\/p>\n<h4>Problema systematis cum solum duobus statibus possibilibus<\/h4>\n<p>Fingamus casum omnium simplicissimum, systema quod solum duobus statibus esse potest: uno cum energia <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> et altero cum energia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\epsilon\\gt 0<\/span>.<\/span> Quaenam est energia media systematis?<\/p>\n<h4>Problema atmosphaerae isothermicae<\/h4>\n<p>Modus simplicatus ad atmosphaeram investigandam est sub suppositione quod haec est isothermica. Quamvis talis suppositio falsa sit, valet ut prima appropinquatio ad aliquas conclusiones obtinendas. Exempli gratia: sub hac suppositione possibile est aestimare numerum particularum quas componunt tamquam functionem altitudinis. Quomodo putas talem deductionem fieri posse?<\/p>\n<h4>Periculum explosionis! Relatio inter reactiones chemicas et temperaturam<\/h4>\n<p>Multae reactiones chemicae habent quandam energiam activationis <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E_{act}<\/span><\/span> quae circa <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1\/2 [eV]<\/span><\/span> invenitur. Ad temperaturam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">T=300[K]<\/span>,<\/span> quae fere cum temperatura ambienti congruit, probabilitas ut reactio fiat est proportionalis ad<\/p>\n<p style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-E_{act}\/(k_B T)}<\/span><\/span><\/p>\n<p>Quid fiet de probabilitate reactionis si temperatura augeatur 10[K]?<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/Pmthx9bQdO0\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><br \/>\n<a name=\"4\"><\/a><\/p>\n<h2>Exercitia<\/h2>\n<ol style=\"text-align: justify;\">\n<li>Systema habet <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> status, qui possunt habere energiam <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> vel <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta<\/span>. Ostende numerum modorum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E)<\/span><\/span> configurationum totius systematis cum energia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E=r\\Delta<\/span><\/span> (cum <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> sit integer) datum esse per\n<p  style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E) =\\displaystyle \\frac{N!}{r!(N-r)!}<\/span><\/span><\/p>\n<p>Nunc remove parvam quantitatem energiae <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">s\\Delta<\/span><\/span> e systemate, ubi <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">s\\ll r<\/span>.<\/span> Ostende:<\/p>\n<p  style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E-\\epsilon) \\approx \\Omega(E)\\displaystyle \\frac{r^s}{(N-r)^s} <\/span><\/span><\/p>\n<p>et ex hoc consequitur systema habere temperaturam quae ex relatione obtineri potest<\/p>\n<p  style=\"text-align: center;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{k_B T}  = \\frac{1}{\\Delta}\\ln \\left(\\frac{N-r}{r} \\right) <\/span><\/span><\/p>\n<p>Designa graphice <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k_B T<\/span><\/span> tamquam functionem <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> ab <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span><\/span> usque ad <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r=N<\/span><\/span> et explica eventus tuos.\n<\/li>\n<p><\/br><\/p>\n<li>\n<p>Photon lucis visibilis cum energia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">2[eV]<\/span><\/span> absorbetur a corpore macroscopico quod manet ad temperaturam ambientem.<\/p>\n<p>a) Quo facto mutatur <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> pro corpore macroscopico? <\/p>\n<p>b) Considera photon emissum ab antenna radiophonica in campo FM (cum frequentia typica <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">100[MHz]<\/span><\/span>). Ex hoc repete calculos prioris partis cum photon absorptus sit ex fonte FM. Ad hoc utere relatione <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">E=hf<\/span><\/span> ubi <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> est frequentia undae et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">h=4,135\\;667\\;696 \\cdot 10^{-15}[eV \\cdot s]<\/span><\/span> est constans Planckiana.<\/p>\n<\/li>\n<p><\/br><\/p>\n<li>\n<p>Inveni energiam mediam <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\lt{E}\\gt<\/span><\/span> pro:<\/p>\n<p>a) Systemate cum <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> statibus, ubi quisque status potest habere energias <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0, \\epsilon, 2\\epsilon, 3\\epsilon, \\cdots , n\\epsilon<\/span>.<\/span><\/p>\n<p>b) Oscillatore harmonico, ubi status potest habere energias <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">0, \\epsilon, 2\\epsilon, 3\\epsilon, \\cdots <\/span><\/span> (sine limite superiore).<\/p>\n<\/li>\n<\/ol>\n<p><a href=\"https:\/\/drive.google.com\/file\/d\/1YU3OV7gGxLo-mINPLSMcSc0U7ki3zaqX\/view?usp=sharing\" rel=\"noopener\" target=\"_blank\"><strong>Tabula cum omnibus calculis<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Distributio Boltzmanniana in coetu canonico Thermodynamica nobis ostendit quomodo systemata physica aequilibrium consequantur et quomodo energia et probabilitas eorum mores determinent. In hac lectione, coetum canonicum atque Distributio Boltzmanniana explicabuntur, instrumenta fundamentalia ad intellegendos phaenomena sicut reactiones chemicas et aequilibrium in systematibus complexis. Cognosces quomodo hae notiones temperaturam cum ordine et inordinatione conectant, sinentes praedicere [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":30796,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[1250,1292],"tags":[],"class_list":["post-34734","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physica","category-thermodynamica"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Distributio Boltzmanniana in coetu canonico - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo Distributio Boltzmanniana in coetu canonico aequilibrium et probabilitatem in systematibus thermodynamicis complexis explicet.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/la\/distributio-boltzmanniana-in-coetu-canonico\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Distributio Boltzmanniana in coetu canonico\" \/>\n<meta property=\"og:description\" content=\"Disce quomodo Distributio Boltzmanniana in coetu canonico aequilibrium et probabilitatem in systematibus thermodynamicis complexis explicet.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/la\/distributio-boltzmanniana-in-coetu-canonico\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-05-26T13:00:19+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-20T06:58:10+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/boltzzmann-1024x585.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Distributio Boltzmanniana in coetu canonico\" \/>\n<meta name=\"twitter:description\" content=\"Disce quomodo Distributio Boltzmanniana in coetu canonico aequilibrium et probabilitatem in systematibus thermodynamicis complexis explicet.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/toposuranos.com\/material\/wp-content\/uploads\/2025\/01\/boltzzmann.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Distributio Boltzmanniana in coetu canonico\",\"datePublished\":\"2021-05-26T13:00:19+00:00\",\"dateModified\":\"2025-09-20T06:58:10+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/\"},\"wordCount\":1344,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/boltzzmann.jpg\",\"articleSection\":[\"Physica\",\"Thermodynamica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/la\\\/distributio-boltzmanniana-in-coetu-canonico\\\/\",\"name\":\"Distributio Boltzmanniana in coetu canonico - 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