{"id":30799,"date":"2021-04-26T13:00:12","date_gmt":"2021-04-26T13:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=30799"},"modified":"2025-01-01T23:39:11","modified_gmt":"2025-01-01T23:39:11","slug":"boltzmann-distribution-in-the-canonical-ensemble","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/boltzmann-distribution-in-the-canonical-ensemble\/","title":{"rendered":"Boltzmann Distribution in the Canonical Ensemble"},"content":{"rendered":"<style>\n\tp, ul, ol {\n\t\ttext-align: justify;\n\t}\n\th1, h2 {\n\t\ttext-align: center;\n\t}\n<\/style>\n<h1>Boltzmann Distribution in the Canonical Ensemble<\/h1>\n<p style=\"text-align:center;\">Thermodynamics reveals how physical systems reach equilibrium and how energy and probability determine their behavior. In this lesson, we will unravel the canonical ensemble and the Boltzmann Distribution, fundamental tools to understand phenomena like chemical reactions and equilibrium in complex systems. You will discover how these ideas connect temperature with order and chaos, enabling the prediction of seemingly unpredictable behavior.<\/p>\n<p style=\"text-align:center;\"><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this lesson, students will be able to:\n<\/p>\n<ol>\n<li><strong>Identify<\/strong> the types of thermodynamic ensembles (microcanonical, canonical, and grand canonical).<\/li>\n<li><strong>Derive<\/strong> the Boltzmann Distribution from thermodynamic principles.<\/li>\n<li><strong>Calculate<\/strong> probabilities associated with microstates using the partition function.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><strong><u>CONTENT INDEX<\/u>:<\/strong><br \/>\n<a href=\"#1\">Thermodynamic Ensembles<\/a><br \/>\n<a href=\"#2\">The Boltzmann Distribution<\/a><br \/>\n<a href=\"#3\">Applications of the Boltzmann Distribution<\/a><br \/>\n<a href=\"#4\">Exercises<\/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>One of the most useful conceptual tools in thermodynamics is the concept of the \u00abensemble.\u00bb Among the variety of ensembles, one of the most widely used is the canonical ensemble, from which the Boltzmann Distribution is derived. Both concepts will be reviewed below.<\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Thermodynamic Ensembles<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=174s\" target=\"_blank\" rel=\"noopener\"><strong>So far, we have used probabilities<\/strong><\/a> to describe thermodynamic systems. Our approach centers on imagining that we can repeat the experiment and measurements an infinite number of times as a way to compensate for our inability to control their microscopic properties (described through microstates). Inspired by these ideas, <a href=\"https:\/\/en.wikipedia.org\/wiki\/Josiah_Willard_Gibbs\" rel=\"noopener, nofollow, noreferrer noopener\" target=\"_blank\">Gibbs<\/a> introduced the concept of \u00abensemble\u00bb in 1878: an idealization in which a large number of \u00abcopies of the system\u00bb are considered, each representing one of its possible states. In thermodynamics, three main types of ensembles are defined:<\/p>\n<ol>\n<li><strong>Microcanonical Ensemble:<\/strong> A collection of systems, all with the same fixed energy.<\/li>\n<li><strong>Canonical Ensemble:<\/strong> A collection of systems, each of which can exchange energy with a large heat reservoir. As we will see later, this establishes (and defines) the temperature of the system.<\/li>\n<li><strong>Grand Canonical Ensemble:<\/strong> A collection of systems where each can exchange both matter (particles) and energy with a large reservoir. Through this, the temperature and chemical potential of the system are defined.<\/li>\n<\/ol>\n<h3>The Canonical Ensemble<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=362s\" target=\"_blank\" rel=\"noopener\"><strong>Let us consider two coupled systems<\/strong><\/a> such that they can exchange energy. This time, however, one of them is enormous compared to the other, and we call it a <strong>reservoir, source,<\/strong> or <strong>heat bath.<\/strong> This reservoir is so vast that we can take large amounts of energy from it without changing its temperature. The number of ways in which the energy quanta rearrange within a reservoir is, consequently, gigantic. The other system is small in comparison, and we simply call it the <strong>system.<\/strong><\/p>\n<p>We will assume that for each allowed energy of the system, there is a unique microstate, and therefore, the system will always have a value of <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega = 1.<\/span> Additionally, we will keep the total energy of the coupled systems fixed at a value of <span class=\"katex-eq\" data-katex-display=\"false\">E.<\/span>\n<p><em>At this point, we can observe that the system and the reservoir form a microcanonical ensemble, where the energy remains constant and all microstates are equally probable.<\/em><\/p>\n<p>In this scenario, if the system&#8217;s energy is <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon,<\/span> then the reservoir&#8217;s energy will be <span class=\"katex-eq\" data-katex-display=\"false\">E - \\varepsilon.<\/span> This situation, where a system is in thermal contact with a large energy reservoir, is what is known as the <strong>Canonical Ensemble.<\/strong><\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Boltzmann Distribution<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=528s\" target=\"_blank\" rel=\"noopener\"><strong>The probability <span class=\"katex-eq\" data-katex-display=\"false\">P(\\varepsilon)<\/span> that the system<\/strong><\/a> has energy <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon<\/span> is proportional to the number of microstates accessible to the reservoir multiplied by the number of microstates accessible to the system. That is:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\varepsilon)\\propto \\Omega(E-\\varepsilon)\\cdot 1.<\/span>\n<p>As we have seen before, temperature can be expressed in terms of the logarithm of <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> through<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{1}{k_B T} = \\frac{d\\ln\\Omega}{dE}<\/span>\n<p>And since <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon \\ll E,<\/span> it is possible to perform a Taylor series expansion of <span class=\"katex-eq\" data-katex-display=\"false\">\\ln\\Omega(E-\\varepsilon)<\/span> around <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon = 0.<\/span> With this, we obtain:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\ln\\Omega(E-\\varepsilon) = \\ln\\Omega(E) - \\frac{d\\ln\\Omega(E)}{dE}\\varepsilon + \\cdots<\/span>\n<p>Using the expressions above, we get:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\ln\\Omega(E-\\varepsilon) = \\ln\\Omega(E) - \\frac{\\varepsilon}{k_B T} + \\cdots<\/span>\n<p>Where <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> is the temperature of the reservoir. At this point, we can neglect higher-order terms in the Taylor series expansion and state the relationship<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\ln \\Omega(E-\\varepsilon) \\approx \\ln\\Omega(E) - \\displaystyle \\frac{\\varepsilon}{k_B T}<\/span>\n<p>Developing this expression further, we find:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E-\\varepsilon) \\approx \\Omega(E) e^{-\\displaystyle \\frac{\\varepsilon}{k_B T}}<\/span>\n<p>Now, comparing this result with the probability <span class=\"katex-eq\" data-katex-display=\"false\">P(\\varepsilon),<\/span> we conclude:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P(\\varepsilon)\\propto e^{-\\varepsilon\/(k_B T)}<\/span>\n<p>Since the system is in thermodynamic equilibrium with the reservoir, they share the same temperature. However, although the temperature <span class=\"katex-eq\" data-katex-display=\"false\">T<\/span> remains constant, the energy <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon<\/span> is not; instead, it follows a probability distribution, which we have just derived. This is known as the <strong>Boltzmann Distribution<\/strong> or the <strong>Canonical Distribution<\/strong> for the canonical ensemble. The term <span class=\"katex-eq\" data-katex-display=\"false\">e^{-\\varepsilon\/(k_B T) }<\/span> is known as the <strong>Boltzmann Factor.<\/strong><\/p>\n<h3>Normalization of the Boltzmann Distribution and the Partition Function<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=ch1bjhU7k70&amp;t=890s\" target=\"_blank\" rel=\"noopener\"><strong>With these developments, we have begun<\/strong><\/a> constructing a probability distribution that describes how a small system behaves when coupled to a large reservoir at temperature <span class=\"katex-eq\" data-katex-display=\"false\">T.<\/span> The system has a reasonable chance of attaining an energy <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon<\/span> less than <span class=\"katex-eq\" data-katex-display=\"false\">k_B T,<\/span> but the exponential term in the Boltzmann distribution decreases rapidly when higher energies are considered. However, we must note that the distribution, as it currently stands, is not strictly a probability distribution; it needs to be normalized. If a system is in contact with a reservoir and has a microstate <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> with energy <span class=\"katex-eq\" data-katex-display=\"false\">E_r,<\/span> then we have:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">P({microstate\\;}r)= \\displaystyle \\frac{e^{-E_r\/(k_B T)}}{\\displaystyle \\sum_{i}e^{-E_i\/(k_B T)}}<\/span>\n<p>The summation in the denominator acts as a normalizing factor that ensures <span class=\"katex-eq\" data-katex-display=\"false\">P<\/span> is a probability distribution. This summation is also known as the <strong>Partition Function<\/strong> and is denoted by <span class=\"katex-eq\" data-katex-display=\"false\">Z<\/span>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">Z = \\displaystyle \\sum_i e^{-E_i\/(k_B T)}<\/span>\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><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Applications of the Boltzmann Distribution<\/h2>\n<p>To illustrate some applications of the canonical ensemble and the Boltzmann Distribution, we will examine how they appear in certain examples. However, before starting, let us introduce a notation for a quantity that frequently appears and can be useful in the future. The factor <span class=\"katex-eq\" data-katex-display=\"false\">\\beta<\/span> is defined through the equality:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta =\\displaystyle  \\frac{1}{k_B T},<\/span>\n<p>so that, based on this, we can write:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\beta = \\displaystyle \\frac{d\\ln\\Omega}{dE},<\/span>\n<h4>The Two-State System Problem<\/h4>\n<p>Imagine the simplest case of all: a system that can only exist in two states\u2014one with energy <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> and the other with energy <span class=\"katex-eq\" data-katex-display=\"false\">\\varepsilon\\gt 0.<\/span> What is the average energy of the system?<\/p>\n<h4>The Isothermal Atmosphere Problem<\/h4>\n<p>A simplified way to study the atmosphere is under the assumption that it is isothermal. Although this assumption is not true, it serves as a first approximation for drawing certain conclusions. For instance, under this assumption, it is possible to estimate the number of particles in the atmosphere as a function of height. How might one make that deduction?<\/p>\n<h4>Explosion Hazard! Relationship Between Chemical Reactions and Temperature<\/h4>\n<p>Many chemical reactions have an activation energy <span class=\"katex-eq\" data-katex-display=\"false\">E_{act}<\/span> that is approximately <span class=\"katex-eq\" data-katex-display=\"false\">1\/2 [eV]<\/span>. At a temperature of <span class=\"katex-eq\" data-katex-display=\"false\">T=300[K],<\/span> which corresponds roughly to room temperature, the probability of a reaction occurring is proportional to:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">e^{-E_{act}\/(k_B T)}<\/span>\n<p>What happens to the reaction probability if the temperature increases by 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><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Exercises<\/h2>\n<ol style=\"text-align: justify; color:#000000;\">\n<li>A system has <span class=\"katex-eq\" data-katex-display=\"false\">N<\/span> states, which can have energy <span class=\"katex-eq\" data-katex-display=\"false\">0<\/span> or <span class=\"katex-eq\" data-katex-display=\"false\">\\Delta.<\/span> Show that the number of configurations <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E)<\/span> of the total system with energy <span class=\"katex-eq\" data-katex-display=\"false\">E=r\\Delta<\/span> (where <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> is an integer) is given by:\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E) =\\displaystyle \\frac{N!}{r!(N-r)!}<\/span>\n<p>Now remove a small amount of energy <span class=\"katex-eq\" data-katex-display=\"false\">s\\Delta<\/span> from the system, where <span class=\"katex-eq\" data-katex-display=\"false\">s\\ll r.<\/span> Show that:<\/p>\n<p style=\"text-align: center;\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Omega(E-\\epsilon) \\approx \\Omega(E)\\displaystyle \\frac{r^s}{(N-r)^s} <\/span>\n<p>and as a consequence, the system has a temperature that can be obtained from the relationship:<\/p>\n<p style=\"text-align: center;\"><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>\n<p>Sketch a graph of <span class=\"katex-eq\" data-katex-display=\"false\">k_B T<\/span> as a function of <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> from <span class=\"katex-eq\" data-katex-display=\"false\">r=0<\/span> to <span class=\"katex-eq\" data-katex-display=\"false\">r=N<\/span> and explain your results.\n<\/li>\n<p><\/br><\/p>\n<li>\n<p style=\"text-align: justify;\">A visible light photon with energy <span class=\"katex-eq\" data-katex-display=\"false\">2[eV]<\/span> is absorbed by a macroscopic body that remains at room temperature.<\/p>\n<p style=\"text-align: justify;\">a) By what factor does <span class=\"katex-eq\" data-katex-display=\"false\">\\Omega<\/span> change for a macroscopic body? <\/p>\n<p style=\"text-align: justify;\">b) Consider a photon emitted by a radio antenna in the FM range (with a typical frequency of <span class=\"katex-eq\" data-katex-display=\"false\">100[MHz]<\/span>). Based on this, repeat the calculations from the previous part when the absorbed photon comes from an FM source. Use the relation <span class=\"katex-eq\" data-katex-display=\"false\">E=hf<\/span>, where <span class=\"katex-eq\" data-katex-display=\"false\">f<\/span> is the wave frequency and <span class=\"katex-eq\" data-katex-display=\"false\">h=4,135\\;667\\;696 \\cdot 10^{-15}[eV \\cdot s]<\/span> is Planck\u2019s constant.<\/p>\n<\/li>\n<p><\/br><\/p>\n<li>\n<p style=\"text-align: justify;\">Find the average energy <span class=\"katex-eq\" data-katex-display=\"false\">\\lt{E}\\gt<\/span> for:<\/p>\n<p style=\"text-align: justify;\">a) A system with <span class=\"katex-eq\" data-katex-display=\"false\">n<\/span> states, where each state can have energies <span class=\"katex-eq\" data-katex-display=\"false\">0, \\varepsilon, 2\\varepsilon, 3\\varepsilon, \\cdots , n\\varepsilon.<\/span>\n<p style=\"text-align: justify;\">b) A harmonic oscillator, where a state can have energies <span class=\"katex-eq\" data-katex-display=\"false\">0, \\varepsilon, 2\\varepsilon, 3\\varepsilon, \\cdots <\/span> (with no upper limit).<\/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>Whiteboard with all calculations<\/strong><\/a><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Boltzmann Distribution in the Canonical Ensemble Thermodynamics reveals how physical systems reach equilibrium and how energy and probability determine their behavior. In this lesson, we will unravel the canonical ensemble and the Boltzmann Distribution, fundamental tools to understand phenomena like chemical reactions and equilibrium in complex systems. You will discover how these ideas connect temperature [&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":95,"footnotes":""},"categories":[635,919],"tags":[],"class_list":["post-30799","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-physics","category-thermodynamics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Boltzmann Distribution in the Canonical Ensemble - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover how the Boltzmann Distribution in the canonical ensemble explains equilibrium and probability in complex thermodynamic systems.\" \/>\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\/en\/boltzmann-distribution-in-the-canonical-ensemble\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Boltzmann Distribution in the Canonical Ensemble\" \/>\n<meta property=\"og:description\" content=\"Discover how the Boltzmann Distribution in the canonical ensemble explains equilibrium and probability in complex thermodynamic systems.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/en\/boltzmann-distribution-in-the-canonical-ensemble\/\" \/>\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=\"2021-04-26T13:00:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-01-01T23:39:11+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=\"Boltzmann Distribution in the Canonical Ensemble\" \/>\n<meta name=\"twitter:description\" content=\"Discover how the Boltzmann Distribution in the canonical ensemble explains equilibrium and probability in complex thermodynamic systems.\" \/>\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\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Boltzmann Distribution in the Canonical Ensemble\",\"datePublished\":\"2021-04-26T13:00:12+00:00\",\"dateModified\":\"2025-01-01T23:39:11+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/\"},\"wordCount\":1606,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2025\\\/01\\\/boltzzmann.jpg\",\"articleSection\":[\"Physics\",\"Thermodynamics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/boltzmann-distribution-in-the-canonical-ensemble\\\/\",\"name\":\"Boltzmann Distribution in the Canonical Ensemble - 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