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dbpedia:Satyendra_Nath_Bose dbpedia-owl:knownFor .
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dbpedia:Satyendra_Nath_Bose dbpprop:knownFor .
dbpedia:Bose_Distribution dbpprop:redirect .
dbpedia:Bose_distribution dbpprop:redirect .
dbpedia:Bose-Einstein_Distribution dbpprop:redirect .
dbpedia:Bose-einstein_distribution dbpprop:redirect .
dbpedia:Albert_Einstein dbpedia-owl:knownFor ;
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rdf:type ns5:FundamentalPhysicsConcepts .
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rdfs:label "Statystyka Bosego-Einsteina"@pl ,
"Statistique de Bose-Einstein"@fr ,
"Statistica di Bose-Einstein"@it ,
"Estad\u00EDstica de Bose-Einstein"@es ,
"Bose\u2013Einstein statistics"@en ,
"\u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1"@zh ,
"Estat\u00EDstica de Bose-Einstein"@pt ,
"\u30DC\u30FC\u30B9\u5206\u5E03\u95A2\u6570"@ja ,
"Bosen\u2013Einsteinin statistiikka"@fi ,
"Bose-Einstein-statistik"@sv ,
"Bose-Einsteinstatistiek"@nl ,
"\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435 \u2014 \u042D\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430"@ru ,
"Boseho-Einsteinovo rozd\u011Blen\u00ED"@cs ,
"\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435-\u0415\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430"@uk ,
"Bose-Einstein-Statistik"@de ,
"Estad\u00EDstica de Bose-Einstein"@ca ;
dbpprop:abstract "A la f\u00EDsica estad\u00EDstica, l'estad\u00EDstica de Bose\u2013Einstein (o m\u00E9s col\u00B7loquialment estad\u00EDstica B-E) determina la distribuci\u00F3 estad\u00EDstica d'un conjunt de bosons indistingibles en equilibri t\u00E8rmic sobre un conjunt d'estats d'energia. Els bosons, a difer\u00E8ncia dels fermions, no estan subjectes al principi d'exclusi\u00F3 de Pauli: un nombre il\u00B7limitat d'ells poden ocupar el mateix estat qu\u00E0ntic a la vegada. Aix\u00F2 explica perqu\u00E8, a baixes temperatures, el seu comportament difereix notablement del dels fermions, ja que tots els bosons tendiran a aplegar-se en l'estat de m\u00EDnima energia, formant el que s'anomena condensat de Bose\u2013Einstein. L'estad\u00EDsitca de Bose-Einstein va ser introduida per descriure la distribuci\u00F3 de fotons en la llei de Planck de la radiaci\u00F3 del cos negre, per Satyendra Nath Bose el 1924, i posteriorment va ser generalitzada pel cas de part\u00EDcules amb massa per Einstein. D'acord amb aquesta estad\u00EDstica, el nombre mitj\u00E0 de bosons en un estat i n_i \u00E9s: n_i = \\frac {g_i} {e^{(\\varepsilon_i-\\mu)/kT} - 1} on \\varepsilon_i \\geq \\mu i: g_i \u00E9s la degeneraci\u00F3 de l'estat i \\varepsilon_i \u00E9s l'energia de l'estat i \\mu \u00E9s el potencial qu\u00EDmic del sistema k_B \u00E9s la constant de Boltzmann T \u00E9s la temperatura absoluta Aquesta expressi\u00F3 es redueix a la corresponent a l'estad\u00EDstica de Maxwell-Boltzmann per a energies grans (\\varepsilon_i-\\mu \\gg k_BT)."@ca ,
"Em mec\u00E2nica estat\u00EDstica, a estat\u00EDstica de Bose\u2013Einstein (ou mais coloquialmente estat\u00EDstica B-E) determina a distribui\u00E7\u00E3o estat\u00EDstica de b\u00F3sons id\u00EAnticos indistingu\u00EDveis sobre os estados de energia em equil\u00EDbrio t\u00E9rmico."@pt ,
"Boseho-Einsteinovo rozd\u011Blen\u00ED popisuje ve statistick\u00E9 fyzice syst\u00E9my slo\u017Een\u00E9 z boson\u016F, tedy \u010D\u00E1stic se symetrickou vlnovou funkc\u00ED a celo\u010D\u00EDseln\u00FDm spinem. Bose-Einsteinov\u00FDm rozd\u011Blen\u00EDm se \u0159\u00EDd\u00ED nap\u0159\u00EDklad fotony, je z n\u011Bj tedy mo\u017En\u00E9 odvodit nap\u0159\u00EDklad Planck\u016Fv vyza\u0159ovac\u00ED z\u00E1kon. Rozd\u011Blen\u00ED poprv\u00E9 popsal Satyendra Nath Bose, roku 1924 ho pak zobecnil Albert Einstein. N\u00E1zev bosony je pr\u00E1v\u011B podle Boseho."@cs ,
"\u0421\u0442\u0430\u0442\u0438\u0301\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435\u2014\u0415\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430 \u2014 \u0446\u0435 \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u0438\u0439 \u0432\u0438\u0434 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0437\u0430 \u0435\u043D\u0435\u0440\u0433\u0456\u0454\u044E \u0447\u0430\u0441\u0442\u043E\u043A, \u044F\u043A\u0456 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u043E \u0431\u043E\u0437\u043E\u043D\u0456\u0432. \u0417\u0433\u0456\u0434\u043D\u043E \u0437 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u043E\u043C \u0411\u043E\u0437\u0435-\u0415\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430 \u0439\u043C\u043E\u0432\u0456\u0440\u043D\u0456\u0441\u0442\u044C, \u0449\u043E \u0432 \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u043E\u043C\u0435\u0445\u0430\u043D\u0456\u0447\u043D\u0456\u0439 \u0431\u0430\u0433\u0430\u0442\u043E\u0447\u0430\u0441\u0442\u0438\u043D\u043A\u043E\u0432\u0456\u0439 \u0441\u0438\u0441\u0442\u0435\u043C\u0456 \u0456\u0441\u043D\u0443\u0454 \u0431\u043E\u0437\u043E\u043D \u0443 \u043E\u0434\u043D\u043E\u0447\u0430\u0441\u0442\u0438\u043D\u043A\u043E\u0432\u043E\u043C\u0443 \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u043E\u043C\u0443 \u0441\u0442\u0430\u043D\u0456 <math> |n\\rangle </math> \u0456\u0437 \u0435\u043D\u0435\u0440\u0433\u0456\u0454\u044E <math> \\varepsilon_n </math> \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0444\u043E\u0440\u043C\u0443\u043B\u043E\u044E <math> f(\\varepsilon_n) = \\frac{1}{e^{(\\varepsilon_n - \\mu)/k_B T} -1} </math>, \u0434\u0435 <math> \\mu </math> \u2014 \u0445\u0456\u043C\u0456\u0447\u043D\u0438\u0439 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B, <math> k_B </math> \u2014 \u0441\u0442\u0430\u043B\u0430 \u0411\u043E\u043B\u044C\u0446\u043C\u0430\u043D\u0430, T \u2014 \u0442\u0435\u043C\u043F\u0435\u0440\u0430\u0442\u0443\u0440\u0430. \u041E\u0441\u043A\u0456\u043B\u044C\u043A\u0438 \u0439\u043C\u043E\u0432\u0456\u0440\u043D\u0456\u0441\u0442\u044C \u043F\u043E\u0432\u0438\u043D\u043D\u0430 \u0431\u0443\u0442\u0438 \u0434\u043E\u0434\u0430\u0442\u043D\u0438\u043C \u0447\u0438\u0441\u043B\u043E\u043C, \u0437\u043D\u0430\u0447\u0435\u043D\u043D\u044F \u0445\u0456\u043C\u0456\u0447\u043D\u043E\u0433\u043E \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B\u0443 \u0437\u0430\u0432\u0436\u0434\u0438 \u043C\u0435\u043D\u0448\u0435 \u0437\u0430 \u0435\u043D\u0435\u0440\u0433\u0456\u044E \u043E\u0441\u043D\u043E\u0432\u043D\u043E\u0433\u043E \u0441\u0442\u0430\u043D\u0443 \u0431\u043E\u0437\u043E\u043D\u0456\u0432. \u042F\u043A\u0449\u043E \u043A\u0456\u043B\u044C\u043A\u0456\u0441\u0442\u044C \u0431\u043E\u0437\u043E\u043D\u0456\u0432 \u0441\u0442\u0440\u043E\u0433\u043E \u0432\u0438\u0437\u043D\u0430\u0447\u0435\u043D\u0430 (N), \u0442\u043E \u0445\u0456\u043C\u0456\u0447\u043D\u0438\u0439 \u043F\u043E\u0442\u0435\u043D\u0446\u0456\u0430\u043B \u0432\u0438\u0437\u043D\u0430\u0447\u0430\u0454\u0442\u044C\u0441\u044F \u0456\u0437 \u0443\u043C\u043E\u0432\u0438 \u043D\u043E\u0440\u043C\u0443\u0432\u0430\u043D\u043D\u044F \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443. <math> N = \\sum_n \\frac{1}{e^{(\\varepsilon_n - \\mu)/k_B T} -1}. </math>"@uk ,
"In meccanica statistica, la statistica di Bose-Einstein (indicata anche come statistica B-E) determina la distribuzione statistica relativa agli stati energetici, all\u2019equilibrio termico, dei bosoni, nell\u2019ipotesi che siano identici e indistinguibili tra loro. La statistica di Fermi-Dirac e quella di Bose-Einstein vengono applicate quando si devono considerare gli effetti quantistici e le particelle sono considerate indistinguibili. Gli effetti quantistici si manifestano quando la concentrazione di particelle (N/V) \u00E8 maggiore della concentrazione quantistica nq . La concentrazione quantistica si ha quando la distanza tra le particelle si avvicina alla loro lunghezza d'onda termica di de Broglie, cio\u00E8 quando le funzioni d\u2019onda associate alle particelle si incontrano in zone nelle quali hanno valori non trascurabili, ma non si sovrappongono. Poich\u00E9 la concentrazione quantistica dipende dalla temperatura, le alte temperature fanno in modo che la maggior parte dei sistemi si collochi entro i limiti classici, a meno che essi abbiano una densit\u00E0 molto alta, come ad esempio in una stella nana bianca. La statistica di Fermi-Dirac si applica ai fermioni, particelle che rispettano il principio di esclusione di Pauli; quella di Bose-Einstein si applica ai bosoni. Entrambe si confondono con la statistica di Maxwell-Boltzmann nel caso in cui siano coinvolte alte temperature o basse concentrazioni. La statistica di Maxwell-Boltzmann \u00E8 spesso descritta come la statistica delle particelle classiche e distinguibili. Per capire quest\u2019ultimo concetto pensiamo di considerare la particella A nella posizione 1 e la particella B nella posizione 2. Se le particelle sono distinguibili questa configurazione \u00E8 diversa da quella in cui, ciascuna particella occupa la posizione dell\u2019altra (A in 2 e B in 1). Quando questa idea venne compresa a fondo, contribu\u00EC a determinare la giusta distribuzione delle particelle negli stati energetici (distribuzione di Boltzmann), ma condusse anche a risultati non fisicamente accettabili per quanto riguarda l\u2019entropia, come mostrato nel paradosso di Gibbs. Questi problemi sparirono quando si comprese che tutte le particelle sono tra loro indistinguibili. Ribadiamo che entrambe queste distribuzioni di probabilit\u00E0 approssimano la distribuzione di Maxwell-Boltzmann nel limite di alte temperature e basse densit\u00E0, senza il bisogno di nessuna ulteriore assunzione. La statistica di Maxwell-Boltzmann \u00E8 particolarmente utile nello studio dei gas, mentre quella di Fermi-Dirac \u00E8 utilizzata pi\u00F9 spesso nello studio degli elettroni nei solidi. Per questi motivi esse costituiscono la base della teoria dei semiconduttori e dell\u2019elettronica. I bosoni, contrariamente ai fermioni, non seguono il principio di esclusione di Pauli: infatti un numero illimitato di particelle potrebbe occupare lo stesso stato energetico (livello di energia), allo stesso tempo. Questo spiega perch\u00E9 a basse temperature i bosoni possono diventare molto diversi dai fermioni; infatti essi tendono ad ammassarsi nello stesso livello di bassa energia, formando ci\u00F2 che \u00E8 noto come condensato di Bose-Einstein. La statistica di Bose-Einstein \u00E8 stata introdotta nel 1920 da Satyendra Nath Bose per i fotoni ed \u00E8 stata estesa agli atomi da Albert Einstein nel 1924. Il numero di particelle, occupanti l\u2019i-imo livello di energia, previsto dalla statistica di Bose-Einstein \u00E8: n_i = \\frac {g_i} {e^{(\\epsilon_i-\\mu)/kT} - 1} con <math>\\epsilon_i > \\mu e dove: ni \u00E8 il numero di particelle nello stato i, gi esprime la degenerazione dello stato i, εi \u00E8 l'energia dell'i-imo stato, μ \u00E8 il potenziale chimico, k \u00E8 la costante di Boltzmann, T \u00E8 la temperatura assoluta. Ci\u00F2 si riduce alla statistica di Maxwell-Boltzmann per energie (\u03B5i-\u03BC) >> kT."@it ,
"In statistical mechanics, Bose\u2013Einstein statistics (or more colloquially B\u2013E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Concept Fermi\u2013Dirac and Bose\u2013Einstein statistics apply when quantum effects are important and the particles are \"indistinguishable\". Quantum effects appear if the concentration of particles (N/V) ≥ nq. Here nq is the quantum concentration, for which the interparticle distance is equal to the thermal de Broglie wavelength, so that the wavefunctions of the particles are touching but not overlapping. Fermi\u2013Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), and Bose\u2013Einstein statistics apply to bosons. As the quantum concentration depends on temperature; most systems at high temperatures obey the classical (Maxwell\u2013Boltzmann) limit unless they have a very high density, as for a white dwarf. Both Fermi\u2013Dirac and Bose\u2013Einstein become Maxwell\u2013Boltzmann statistics at high temperature or at low concentration. Bosons, unlike fermions, are not subject to the Pauli exclusion principle an unlimited number of particles may occupy the same state at the same time. This explains why, at low temperatures, bosons can behave very differently from fermions; all the particles will tend to congregate together at the same lowest-energy state, forming what is known as a Bose\u2013Einstein condensate. B\u2013E statistics was introduced for photons in 1920 by Bose and generalized to atoms by Einstein in 192. The expected number of particles in an energy state i for B\u2013E statistics is n_i \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}-1} with \\varepsilon_i > \\mu and where ni is the number of particles in state i gi is the degeneracy of state i \u03B5i is the energy of the ith state \u03BC is the chemical potential k is Boltzmann's constant T is absolute temperature This reduces to Maxwell\u2013Boltzmann statistics for energies \\varepsilon_i-\\mu \\gg kT . History In the early 1920s Satyendra Nath Bose, a professor of University of Dhaka was intrigued by Einstein's theory of light waves being made of particles called photons. Bose was interested in deriving Planck's radiation formula, which Planck obtained largely by guessing. In 1900 Max Planck had derived his formula by manipulating the math to fit the empirical evidence. Using the particle picture of Einstein, Bose was able to derive the radiation formula by systematically developing a statistics of massless particles without the constraint of particle number conservation. Bose derived Planck's Law of Radiation by proposing different states for the photon. Instead of statistical independence of particles, Bose put particles into cells and described statistical independence of cells of phase space. Such systems allow two polarization states, and exhibit totally symmetric wavefunctions. He developed a statistical law governing the behaviour pattern of photons quite successfully. However, he was not able to publish his work; no journals in Europe would accept his paper, being unable to understand it. Bose sent his paper to Einstein, who saw the significance of it and used his influence to get it published. A derivation of the Bose\u2013Einstein distribution Suppose we have a number of energy levels, labeled by index \\displaystyle i, each level having energy \\displaystyle \\varepsilon_i and containing a total of \\displaystyle n_i particles. Suppose each level contains \\displaystyle g_i distinct sublevels, all of which have the same energy, and which are distinguishable. For example, two particles may have different momenta, in which case they are distinguishable from each other, yet they can still have the same energy. The value of \\displaystyle g_i associated with level \\displaystyle i is called the \"degeneracy\" of that energy level. Any number of bosons can occupy the same sublevel. Let \\displaystyle w(n,g) be the number of ways of distributing \\displaystyle n particles among the \\displaystyle g sublevels of an energy level. There is only one way of distributing \\displaystyle n particles with one sublevel, therefore \\displaystyle w(n,1)1. It is easy to see that there are \\displaystyle (n1) ways of distributing \\displaystyle n particles in two sublevels which we will write as w(n,2)\\frac{(n1)!}{n!1!}. With a little thought it can be seen that the number of ways of distributing \\displaystyle n particles in three sublevels is w(n) w(n,2) w(n-1,2) \\cdots w(1,2) w(0,2) so that w(n)\\sum_{k0}^n w(n-k,2) \\sum_{k0}^n\\frac{(n-k1)!}{(n-k)!1!}\\frac{(n2)!}{n!2!} where we have used the following theorem involving binomial coefficients \\sum_{k0}^n\\frac{(ka)!}{k!a!}\\frac{(na1)!}{n!(a1)!}. Continuing this process, we can see that \\displaystyle w(n,g) is just a binomial coefficient w(n,g)\\frac{(ng-1)!}{n!(g-1)!}. The number of ways that a set of occupation numbers \\displaystyle n_i can be realized is the product of the ways that each individual energy level can be populated W \\prod_i w(n_i,g_i) \\prod_i \\frac{(n_ig_i-1)!}{n_i!(g_i-1)!} \\approx\\prod_i \\frac{(n_ig_i)!}{n_i!(g_i)!} where the approximation assumes that g_i \\gg 1. Following the same procedure used in deriving the Maxwell\u2013Boltzmann statistics, we wish to find the set of \\displaystyle n_i for which \\displaystyle W is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of \\displaystyle W and \\displaystyle \\ln(W) occur at the value of \\displaystyle N_i and, since it is easier to accomplish mathematically, we will maximise the latter function instead. We constrain our solution using Lagrange multipliers forming the function f(n_i)\\ln(W)\\alpha(N-\\sum n_i)\\beta(E-\\sum n_i \\varepsilon_i) Using the g_i \\gg 1 approximation and using Stirling's approximation for the factorials \\left(\\ln\\approx x\\ln-x\\right) gives f(n_i)\\sum_i (n_i g_i) \\ln(n_i g_i) - n_i \\ln(n_i) - g_i \\ln (g_i) \\alpha\\left(N-\\sum n_i\\right)\\beta\\left(E-\\sum n_i \\varepsilon_i\\right). Taking the derivative with respect to \\displaystyle n_i, and setting the result to zero and solving for \\displaystyle n_i, yields the Bose\u2013Einstein population numbers n_i \\frac{g_i}{e^{\\alpha\\beta \\varepsilon_i}-1}. It can be shown thermodynamically that \\displaystyle \\beta \\frac{1}{kT}, where \\displaystyle k is Boltzmann's constant and \\displaystyle T is the temperature. It can also be shown that \\displaystyle \\alpha - \\frac{\\mu}{kT}, where \\displaystyle \\mu is the chemical potential, so that finally n_i \\frac{g_i}{e^{(\\varepsilon_i-\\mu)/kT}-1}. Note that the above formula is sometimes written n_i \\frac{g_i}{e^{\\varepsilon_i/kT}/z-1}, where \\displaystyle z\\exp(\\mu/kT) is the absolute activity. Notes The purpose of these notes is to clarify some aspects of the derivation of the Bose\u2013Einstein (B\u2013E) distribution for beginners. The enumeration of cases (or ways) in the B\u2013E distribution can be recast as follows. Consider a game of dice throwing in which there are \\displaystyle n dice, with each die taking values in the set \\displaystyle \\left\\{ 1, \\dots, g \\right\\}, for g \\ge 1. The constraints of the game are that the value of a die \\displaystyle i, denoted by \\displaystyle m_i, has to be greater than or equal to the value of die \\displaystyle (i-1), denoted by \\displaystyle m_{i-1}, in the previous throw, i.e. , m_i \\ge m_{i-1}. Thus a valid sequence of die throws can be described by an n-tuple \\displaystyle \\left(m_1, m_2, \\dots, m_n \\right), such that m_i \\ge m_{i-1}. Let \\displaystyle S(n,g) denote the set of these valid n-tuples Then the quantity \\displaystyle w(n,g) (defined above as the number of ways to distribute \\displaystyle n particles among the \\displaystyle g sublevels of an energy level) is the cardinality of \\displaystyle S(n,g), i.e. , the number of elements (or valid n-tuples) in \\displaystyle S(n,g). Thus the problem of finding and expression for \\displaystyle w(n,g) becomes the problem of counting the elements in \\displaystyle S(n,g). Example n, g S \\left\\{ \\underbrace{ (111)}_{(a)}, \\underbrace{ (11)}_{(b)}, \\underbrace{ (1)}_{(c)}, \\right. \\left. \\underbrace{,,,, }_{(d)} \\right\\} \\displaystyle w 15 (there are \\displaystyle 15 elements in \\displaystyle S) Subset \\displaystyle (a) is obtained by fixing all indices \\displaystyle m_i to \\displaystyle 1, except for the last index, \\displaystyle m_n, which is incremented from \\displaystyle 1 to \\displaystyle g. Subset \\displaystyle (b) is obtained by fixing \\displaystyle m_1 m_2 1, and incrementing \\displaystyle m_ from \\displaystyle 2 to \\displaystyle g. Due to the constraint \\displaystyle m_i \\ge m_{i-1} on the indices in \\displaystyle S(n,g), the index \\displaystyle m_ must automatically take values in \\displaystyle \\left\\{ 2, \\right\\}. The construction of subsets \\displaystyle (c) and \\displaystyle (d) follows in the same manner. Each element of \\displaystyle S can be thought of as a multiset of cardinality \\displaystyle n; the elements of such multiset are taken from the set \\displaystyle \\left\\{ 1, 2, \\right\\} of cardinality \\displaystyle g, and the number of such multisets is the multiset coefficient \\displaystyle \\left\\langle \\begin{matrix} \t \\end{matrix} \\right\\rangle { - 1 \\choose -1} { - 1 \\choose } \\frac {! 2!} 15 More generally, each element of \\displaystyle S(n,g) is a multiset of cardinality \\displaystyle n (number of dice) with elements taken from the set \\displaystyle \\left\\{ 1, \\dots, g \\right\\} of cardinality \\displaystyle g (number of possible values of each die), and the number of such multisets, i.e. , \\displaystyle w(n,g) is the multiset coefficient which is exactly the same as the formula for \\displaystyle w(n,g), as derived above with the aid of a theorem involving binomial coefficients, namely To understand the decomposition or for example, \\displaystyle n and \\displaystyle g \\displaystyle w w(2) w(2) w(2,2) w(1,2) w(0,2), let us rearrange the elements of \\displaystyle S as follows S \\left\\{ \\underbrace{ \t \t }_{(\\alpha)}, \\underbrace{ \t (111{\\color{Red}\\underset{}{}}), \t (112{\\color{Red}\\underset{}{}}), \t (122{\\color{Red}\\underset{}{}}), \t (222{\\color{Red}\\underset{}{}}) }_{(\\beta)}, \\right. \\left. \\underbrace{ \t (11{\\color{Red}\\underset{}{}}), \t (12{\\color{Red}\\underset{}{}}), \t (22{\\color{Red}\\underset{}{}}) }_{(\\gamma)}, \\underbrace{ \t (1{\\color{Red}\\underset{}{}}), \t (2{\\color{Red}\\underset{}{}}) }_{(\\delta)} \\underbrace{ \t ({\\color{Red}\\underset{}{}}) }_{(\\omega)} \\right\\} . Clearly, the subset \\displaystyle (\\alpha) of \\displaystyle S is the same as the set \\displaystyle S(2) \\left\\{ \t \t \\right\\} . By deleting the index \\displaystyle m_ (shown in red with double underline) in the subset \\displaystyle (\\beta) of \\displaystyle S, one obtains the set \\displaystyle S(2) \\left\\{ \t \t \\right\\} . In other words, there is a one-to-one correspondence between the subset \\displaystyle (\\beta) of \\displaystyle S and the set \\displaystyle S(2). We write \\displaystyle (\\beta) \\longleftrightarrow S(2) . Similarly, it is easy to see that \\displaystyle (\\gamma) \\longleftrightarrow S(2,2) \\left\\{ \t \\right\\} \\displaystyle (\\delta) \\longleftrightarrow S(1,2) \\left\\{ \t \\right\\} \\displaystyle (\\omega) \\longleftrightarrow S(0,2) \\phi (empty set). Thus we can write \\displaystyle S \\bigcup_{k0}^{} S(-k,2) or more generally, and since the sets \\displaystyle S(i,g-1) \\, \\ {\\rm for} \\ i 0, \\dots, n are non-intersecting, we thus have with the convention that Continuing the process, we arrive at the following formula \\displaystyle w(n,g) \\sum_{k_10}^{n} \\sum_{k_20}^{n-k_1} w(n - k_1 - k_2, g-2) \\sum_{k_10}^{n} \\sum_{k_20}^{n-k_1} \\cdots \\sum_{k_g0}^{n-\\sum_{j1}^{g-1} k_j} w(n - \\sum_{i1}^{g} k_i, 0). Using the convention (7)2 above, we obtain the formula keeping in mind that for \\displaystyle q and \\displaystyle p being constants, we have It can then be verified that (8) and give the same result for \\displaystyle w, \\displaystyle w, \\displaystyle w(2), etc. Information Retrieval In recent years, Bose Einstein statistics have also been used as a method for term weighting in information retrieval. The method is one of a collection of DFR (\"Divergence From Randomness\") models, the basic notion being that Bose Einstein statistics may be a useful indicator in cases where a particular term and a particular document have a significant relationship that would not have occurred purely by chance. Source code for implementing this model is available from the [http//ir. dcs. gla. ac. uk/terrier/doc/dfr_description. html Terrier project] at the University of Glasgow. See also Bose-Einstein correlations Maxwell\u2013Boltzmann statistics Fermi\u2013Dirac statistics Parastatistics Planck's law of black body radiation Notes References . . . CategoryFundamental physics concepts CategoryStatistical mechanics CategoryQuantum field theory CategoryParticle statistics CategoryAlbert Einstein ar\u0625\u062D\u0635\u0627\u0621 \u0628\u0648\u0632-\u0623\u064A\u0646\u0634\u062A\u0627\u064A\u0646 bsBose-Einstein statistika caEstad\u00EDstica de Bose-Einstein csBoseho-Einsteinovo rozd\u011Blen\u00ED deBose-Einstein-Statistik etBose-Einsteini statistika esEstad\u00EDstica de Bose-Einstein eoStatistiko de Bose-Einstein frStatistique de Bose-Einstein ko\uBCF4\uC988-\uC544\uC778\uC288\uD0C0\uC778 \uD1B5\uACC4 hrBose-Einsteinova statistika itStatistica di Bose-Einstein he\u05D4\u05EA\u05E4\u05DC\u05D2\u05D5\u05EA \u05D1\u05D5\u05D6-\u05D0\u05D9\u05D9\u05E0\u05E9\u05D8\u05D9\u05D9\u05DF nlBose-Einsteinstatistiek ja\u30DC\u30FC\u30B9\u5206\u5E03\u95A2\u6570 plStatystyka Bosego-Einsteina ptEstat\u00EDstica de Bose-Einstein ru\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435 \u2014 \u042D\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430 simpleBose-Einstein statistics skBoseho-Einsteinove rozdelenie slBose-Einsteinova statistika fiBosen\u2013Einsteinin statistiikka svBose-Einstein-statistik uk\u0421\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435-\u0415\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430 zh\u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1"@en ,
"En m\u00E9canique quantique et en physique statistique, la statistique de Bose-Einstein d\u00E9signe la distribution statistique de bosons indiscernables (tous similaires) sur les \u00E9tats d'\u00E9nergie d'un syst\u00E8me \u00E0 l'\u00E9quilibre thermodynamique. La distribution en question r\u00E9sulte d'une particularit\u00E9 des bosons : les particules de spin entier ne sont pas assujetties au principe d'exclusion de Pauli, \u00E0 savoir que plusieurs bosons peuvent occuper simultan\u00E9ment un m\u00EAme \u00E9tat quantique."@fr ,
"\u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1\u662F\u73BB\u8272\u5B50\u6240\u4F9D\u4ECE\u7684\u7EDF\u8BA1\u89C4\u5F8B\u3002 \u6839\u636E\u91CF\u5B50\u529B\u5B66\uFF0C\u73BB\u8272\u5B50\u662F\u81EA\u65CB\u4E3A\u6574\u6570\u7684\u7C92\u5B50\uFF0C\u5176\u672C\u5F81\u6CE2\u51FD\u6570\u5BF9\u79F0\uFF0C\u5728\u73BB\u8272\u5B50\u7684\u67D0\u4E00\u4E2A\u80FD\u7EA7\u4E0A\uFF0C\u53EF\u4EE5\u5BB9\u7EB3\u65E0\u9650\u4E2A\u7C92\u5B50\u3002\u56E0\u800C\u7B26\u5408\u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1\u5206\u5E03\u7684\u7C92\u5B50\uFF0C\u5F53\u4ED6\u4EEC\u5904\u4E8E\u67D0\u4E00\u5206\u5E03<math>\\left\\{ n_j \\right\\}\uFF08\u201C\u67D0\u4E00\u5206\u5E03\u201D\u6307\u8FD9\u6837\u4E00\u79CD\u72B6\u6001\uFF1A\u5373\u5728\u80FD\u91CF\u4E3A<math>\\left\\{ \\epsilon_j \\right\\}\u7684\u80FD\u7EA7\u4E0A\u540C\u65F6\u6709<math>n_j\u4E2A\u7C92\u5B50\u5B58\u5728\u7740\uFF0C\u4E0D\u96BE\u60F3\u8C61\uFF0C\u5F53\u4ECE\u5B8F\u89C2\u89C2\u5BDF\u4F53\u7CFB\u80FD\u91CF\u4E00\u5B9A\u7684\u65F6\u5019\uFF0C\u4ECE\u5FAE\u89C2\u89D2\u5EA6\u89C2\u5BDF\u4F53\u7CFB\u53EF\u80FD\u6709\u5F88\u591A\u79CD\u4E0D\u540C\u7684\u5206\u5E03\u72B6\u6001\uFF0C\u800C\u4E14\u5728\u8FD9\u4E9B\u4E0D\u540C\u7684\u5206\u5E03\u72B6\u6001\u4E2D\uFF0C\u603B\u6709\u4E00\u4E9B\u72B6\u6001\u51FA\u73B0\u7684\u51E0\u7387\u7279\u522B\u7684\u5927\uFF0C\u800C\u5176\u4E2D\u51FA\u73B0\u51E0\u7387\u6700\u5927\u7684\u5206\u5E03\u72B6\u6001\u88AB\u79F0\u4E3A\u6700\u53EF\u51E0\u5206\u5E03\uFF09\u65F6\uFF0C\u4F53\u7CFB\u603B\u72B6\u6001\u6570\u4E3A\uFF1A \\Omega_j=\\frac{(g_j+n_j-1)!}{n_j!(g_j-1)!} \u5BF9\u8FD9\u4E00\u516C\u5F0F\u7684\u7406\u89E3\u662F\u8FD9\u6837\u7684\uFF1A\u628Ag_j\u4E2A\u7B80\u5E76\u80FD\u7EA7\u770B\u4F5C\u4E00\u4E2A\u62E5\u6709g_j\u4E2A\u9694\u5BA4\u7684\u5927\u76D2\u5B50\uFF0C\u628An_j\u4E2A\u7C92\u5B50\u770B\u4F5C\u51C6\u5907\u653E\u5165\u76D2\u5B50\u4E2D\u7684n_j\u4E2A\u4E0D\u53EF\u533A\u5206\u7684\u5C0F\u7403\uFF0C\u5219\u53EF\u4EE5\u628A\u8FD9\u4E2A\u5411\u76D2\u5B50\u91CC\u9762\u653E\u5C0F\u7403\u7684\u8FC7\u7A0B\u770B\u4F5Cn_j\u4E2A\u5C0F\u7403\u548C\u76D2\u5B50\u4E2Dg_j-1\u4E2A\u9694\u5BA4\u58C1\u7684\u968F\u673A\u6392\u5217\u8FC7\u7A0B\uFF0C\u5219\u8FD9\u6837\u7684\u6392\u5217\u4E00\u5171\u6709(g_j+n_j-1)!\u79CD\u53EF\u80FD\u51FA\u73B0\u7684\u72B6\u6001\uFF1B\u53E6\u4E00\u65B9\u9762\uFF0C\u5C0F\u7403\u548C\u5C0F\u7403\u662F\u4E0D\u53EF\u533A\u5206\u7684\uFF0C\u9694\u5BA4\u58C1\u548C\u9694\u5BA4\u58C1\u4E5F\u662F\u4E0D\u53EF\u533A\u5206\u7684\uFF0C\u56E0\u6B64\u5BF9\u5C0F\u7403\u548C\u9694\u5BA4\u58C1\u7684\u8BA1\u6570\u90FD\u6709\u91CD\u590D\uFF0C\u9700\u8981\u9664\u4EE5\u8FD9\u79CD\u91CD\u590D\u8BA1\u6570(g_j-1)!\u548C(n_j)!\uFF0C\u6700\u7EC8\u5F97\u5230\u7684\u7ED3\u679C\u5C31\u662F\u4E0A\u8FF0\u7ED3\u679C\u3002 \\Omega_j=\\frac{(g_j+n_j-1)!}{n_j!(g_j-1)!} g_j=3;n_j=2;\\Omega_j=6 \u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1\u7684\u6700\u53EF\u51E0\u5206\u5E03\u7684\u6570\u5B66\u8868\u8FBE\u5F0F\u4E3A\uFF1A \\left\\{ n_j^{BE} \\right\\}=\\frac{g_j e^\\alpha e^{\\beta\\epsilon_j}}{1 - e^\\alpha e^{\\beta\\epsilon_j}} \u7531\u4E8E\u91CF\u5B50\u7EDF\u8BA1\u5728\u6570\u5B66\u5904\u7406\u4E0A\u975E\u5E38\u56F0\u96BE\uFF0C\u56E0\u6B64\u5728\u5904\u7406\u5B9E\u9645\u95EE\u9898\u65F6\u7ECF\u5E38\u5F15\u5165\u4E00\u4E9B\u8FD1\u4F3C\u6761\u4EF6\uFF0C\u4F7F\u8D39\u7C73-\u72C4\u62C9\u514B\u7EDF\u8BA1\u548C\u73BB\u8272-\u7231\u56E0\u65AF\u5766\u7EDF\u8BA1\u9000\u5316\u6210\u4E3A\u7ECF\u5178\u7684\u9EA6\u514B\u65AF\u97E6-\u73BB\u5C14\u5179\u66FC\u7EDF\u8BA1\u3002"@zh ,
"La estad\u00EDstica de Bose-Einstein es un tipo de mec\u00E1nica estad\u00EDstica aplicable a la determinaci\u00F3n de las propiedades estad\u00EDsticas de conjuntos grandes de part\u00EDculas indistinguibles capaces de coexistir en el mismo estado cu\u00E1ntico en equilibrio t\u00E9rmico. A bajas temperaturas los bosones tienden a tener un comportamiento cu\u00E1ntico similar que puede llegar a ser id\u00E9ntico a temperaturas cercanas al cero absoluto en un estado de la materia conocido como condensado de Bose-Einstein y producido por primera vez en laboratorio en el a\u00F1o 1995. El condensador Bose-Einstein funciona a temperaturas cercanas al cero absoluto, -273,16\u00B0C(0 Kelvin). La estad\u00EDstica de Bose-Einstein fue introducida para estudiar las propiedades estad\u00EDsticas de los fotones en 1920 por el f\u00EDsico hind\u00FA Satyendra Nath Bose y generalizada para \u00E1tomos y otros bosones por Albert Einstein en 1924. Este tipo de estad\u00EDstica est\u00E1 \u00EDntimamente relacionada con la estad\u00EDstica de Maxwell-Boltzmann (derivada inicialmente para gases) y a las estad\u00EDsticas de Fermi-Dirac (aplicables a part\u00EDculas denominadas fermiones sobre las que rige el principio de exclusi\u00F3n de Pauli que impide que dos fermiones compartan el mismo estado cu\u00E1ntico). La estad\u00EDstica de Bose-Einstein se reduce a la estad\u00EDstica de Maxwell-Boltzmann para energ\u00EDas suficientemente elevadas."@es ,
"\u30DC\u30FC\u30B9\u5206\u5E03\u95A2\u6570 (Bose distribution function) \u306F\u3001\u30DC\u30FC\u30B9=\u30A2\u30A4\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u5206\u5E03\u95A2\u6570 (Bose=Einstein distribution function) \u3068\u3082\u547C\u3070\u308C\u3001\u30DC\u30FC\u30B9\uFF1D\u30A2\u30A4\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u7D71\u8A08\u306B\u5F93\u3046\u7C92\u5B50\uFF08\u30DC\u30FC\u30B9\u7C92\u5B50\uFF09\u306E\u5206\u5E03\u95A2\u6570\u3067\u3042\u308B\u3002 \u30DC\u30FC\u30B9\u5206\u5E03\u95A2\u6570\u306F\u30A8\u30CD\u30EB\u30AE\u30FC <math>\\varepsilon</math> \u306E\u95A2\u6570 <math>f</math> \u3068\u3057\u3066\u3001\u4EE5\u4E0B\u306E\u5F0F\u3067\u8868\u3055\u308C\u308B\u3002 <math> f(\\varepsilon) = \\, {1 \\over {\\exp\\left\\{ {1 \\over {k_B T} } (\\varepsilon - \\mu)\\right\\} - 1 } } </math> <math> k_B </math>\u3000\uFF1A\u3000\u30DC\u30EB\u30C4\u30DE\u30F3\u5B9A\u6570 <math> \\mu </math>\u3000\uFF1A\u3000\u5316\u5B66\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\uFF08\u30B1\u30DF\u30AB\u30EB\u30DD\u30C6\u30F3\u30B7\u30E3\u30EB\uFF09 <math>\\mu\\leq0</math> \u3067\u3042\u308B\u3002<math>\\mu=0\\,</math> \u3068\u306A\u308B\u306E\u306F\u751F\u6210\u304A\u3088\u3073\u6D88\u6EC5\u304C\u8D77\u3053\u308B\u5149\u5B50\u3084\u30D5\u30A9\u30CE\u30F3\u306A\u3069\u306E\u7C92\u5B50\u7CFB\u304B\u3001 \u30DC\u30FC\u30B9\uFF1D\u30A2\u30A4\u30F3\u30B7\u30E5\u30BF\u30A4\u30F3\u51DD\u7E2E\u3092\u8D77\u3053\u3057\u3066\u3044\u308B\u7C92\u5B50\u7CFB\u3067\u3042\u308B\u3002"@ja ,
"\u0421\u0442\u0430\u0442\u0438\u0301\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0301\u0437\u0435\u00A0\u2014 \u042D\u0439\u043D\u0448\u0442\u0435\u0301\u0439\u043D\u0430\u00A0\u2014 \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u0430\u044F \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430, \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u043C\u0430\u044F \u043A \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u043C \u0447\u0430\u0441\u0442\u0438\u0446 \u0441 \u043D\u0443\u043B\u0435\u0432\u044B\u043C \u0438\u043B\u0438 \u0446\u0435\u043B\u043E\u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044B\u043C \u0441\u043F\u0438\u043D\u043E\u043C; \u043F\u0440\u0435\u0434\u043B\u043E\u0436\u0435\u043D\u0430 \u0432 1924 \u0433\u043E\u0434\u0443 \u0438\u043D\u0434\u0438\u0439\u0441\u043A\u0438\u043C \u0444\u0438\u0437\u0438\u043A\u043E\u043C \u0428. \u0411\u043E\u0437\u0435 \u0434\u043B\u044F \u043A\u0432\u0430\u043D\u0442\u043E\u0432 \u0441\u0432\u0435\u0442\u0430; \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0430 \u0410. \u042D\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u043E\u043C \u0434\u043B\u044F \u043C\u043E\u043B\u0435\u043A\u0443\u043B \u0438\u0434\u0435\u0430\u043B\u044C\u043D\u044B\u0445 \u0433\u0430\u0437\u043E\u0432. \u0425\u0430\u0440\u0430\u043A\u0442\u0435\u0440\u043D\u0430\u044F \u043E\u0441\u043E\u0431\u0435\u043D\u043D\u043E\u0441\u0442\u044C\u00A0\u2014 \u0432 \u043E\u0434\u043D\u043E\u043C \u0438 \u0442\u043E\u043C \u0436\u0435 \u0441\u043E\u0441\u0442\u043E\u044F\u043D\u0438\u0438 \u043C\u043E\u0436\u0435\u0442 \u043D\u0430\u0445\u043E\u0434\u0438\u0442\u044C\u0441\u044F \u043B\u044E\u0431\u043E\u0435 \u0447\u0438\u0441\u043B\u043E \u043E\u0434\u0438\u043D\u0430\u043A\u043E\u0432\u044B\u0445 \u0447\u0430\u0441\u0442\u0438\u0446 (\u0432 \u043F\u0440\u043E\u0442\u0438\u0432\u043E\u043F\u043E\u043B\u043E\u0436\u043D\u043E\u0441\u0442\u044C \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0435 \u0424\u0435\u0440\u043C\u0438\u00A0\u2014 \u0414\u0438\u0440\u0430\u043A\u0430 \u0434\u043B\u044F \u0447\u0430\u0441\u0442\u0438\u0446 \u0441 \u043F\u043E\u043B\u0443\u0446\u0435\u043B\u044B\u043C \u0441\u043F\u0438\u043D\u043E\u043C, \u0441\u043E\u0433\u043B\u0430\u0441\u043D\u043E \u043A\u043E\u0442\u043E\u0440\u043E\u0439 \u043A\u0430\u0436\u0434\u043E\u0435 \u0441\u043E\u0441\u0442\u043E\u044F\u043D\u0438\u0435 \u043C\u043E\u0436\u0435\u0442 \u0431\u044B\u0442\u044C \u0437\u0430\u043D\u044F\u0442\u043E \u043D\u0435 \u0431\u043E\u043B\u0435\u0435 \u0447\u0435\u043C \u043E\u0434\u043D\u043E\u0439 \u0447\u0430\u0441\u0442\u0438\u0446\u0435\u0439). \u0414\u043B\u044F \u0441\u0438\u043B\u044C\u043D\u043E \u0440\u0430\u0437\u0440\u0435\u0436\u0451\u043D\u043D\u044B\u0445 \u0433\u0430\u0437\u043E\u0432 (\u043A\u0430\u043A \u0438 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430 \u0424\u0435\u0440\u043C\u0438\u00A0\u2014 \u0414\u0438\u0440\u0430\u043A\u0430) \u043F\u0435\u0440\u0435\u0445\u043E\u0434\u0438\u0442 \u0432 \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0443 \u041C\u0430\u043A\u0441\u0432\u0435\u043B\u043B\u0430\u00A0\u2014 \u0411\u043E\u043B\u044C\u0446\u043C\u0430\u043D\u0430."@ru ,
"Statystyka Bosego-Einsteina \u2013 statystyka dotycz\u0105ca bozon\u00F3w traktowanych jako gaz bozonowy, cz\u0105stek o spinie ca\u0142kowitym, kt\u00F3rych nie obowi\u0105zuje zakaz Pauliego. Zgodnie z rozk\u0142adem Bosego-Einsteina \u015Brednia ilo\u015B\u0107 cz\u0105stek w danym stanie kwantowym jest r\u00F3wna <math>\\langle n_i \\rangle=\\frac n Z \\frac{g_i}{e^{\\beta (E_i-\\mu)}-1}</math> gdzie: <math>n_i</math> \u2013 \u015Brednia liczba cz\u0105steczek w i-tym stanie, <math>E_i</math> \u2013 energia i-tego stanu, <math>g_i</math> \u2013 degeneracja i-tego stanu, <math>n</math> \u2013 ca\u0142kowita liczba cz\u0105stek, <math>\\mu </math> \u2013 potencja\u0142 chemiczny, <math>\\beta = \\frac1{k_BT}</math>, gdzie <math>k_B</math> jest sta\u0142\u0105 Boltzmanna, T \u2013 temperatura w skali Kelvina, <math>Z=\\sum_i \\frac{g_i}{e^{\\beta (E_i-\\mu)}-1}</math> suma statystyczna. Potencja\u0142 chemiczny w tym rozk\u0142adzie jest zawsze ujemny lub r\u00F3wny zeru. Gdy temperatura jest wysoka, mo\u017Cna zaniedba\u0107 czynnik \u20131 i rozk\u0142ad przechodzi w rozk\u0142ad fizyki klasycznej, klasyczny rozk\u0142ad Boltzmanna <math>\\langle n_i\\rangle =\\frac n Z g_i e^{-\\beta (E_i-\\mu)}</math> Rozk\u0142adowi Bosego-Einsteina podlegaj\u0105 oczywi\u015Bcie fotony (o spinie 1) \u2013 nosi on wtedy nazw\u0119 rozk\u0142adu Plancka, kt\u00F3ry t\u0142umaczy promieniowanie cia\u0142a doskonale czarnego. Jego wprowadzenie przez Plancka zapocz\u0105tkowa\u0142o mechanik\u0119 kwantow\u0105. Zakaz Pauliego nie dotyczy bozon\u00F3w, umo\u017Cliwia to ich kondensacj\u0119."@pl ,
"Die Bose-Einstein-Statistik, benannt nach Satyendranath Bose und Albert Einstein, ist eine Verteilung in der Quantenstatistik. Sie beschreibt die mittlere Besetzungszahl <math> \\langle n(E) \\rangle </math> eines Quantenzustands der Energie E im thermodynamischen Gleichgewicht bei der absoluten Temperatur <math>T</math> f\u00FCr identische Bosonen als besetzende Teilchen. Sie ist analog zur Fermi-Dirac-Statistik, bei der statt Bosonen Fermionen betrachtet werden. Bei Wechselwirkungsfreiheit ergibt sich f\u00FCr Bosonen die folgende Formel: <math> \\langle n(E) \\rangle = \\frac {1}{e^{\\beta (E - \\mu)} - 1} </math> Hierbei ist \u03BC das chemische Potential und <math>\\beta</math> \u00FCblicherweise gleich <math>1/(k_B T)</math> (mit der Boltzmann-Konstanten kB und der absoluten Temperatur T). Die Wahl des Faktors <math>\\beta</math> h\u00E4ngt von der verwendeten Temperaturskala ab. Wird die Temperatur in Energieeinheiten, etwa Joule, gemessen, so betr\u00E4gt er <math>1/T</math>. Dies geschieht, wenn der Faktor kB auch in der Definition der Entropie - welche dann einheitenlos ist - nicht auftaucht. Man beachte, dass es sich um die Besetzungszahl eines Quantenzustandes handelt. Ben\u00F6tigt man die Besetzungszahl eines entarteten Energieniveaus, so ist obiger Ausdruck mit dem Entartungsgrad gi desselben zu multiplizieren. Im Falle der Fermi-Dirac-Statistik erh\u00E4lt man im Nenner +1 anstelle von -1. Unterhalb einer sehr tiefen kritischen Temperatur <math>T_\\lambda</math> erh\u00E4lt man im Spezialfall der Wechselwirkungsfreiheit - unter der Annahme, dass \u03BC gegen das Energie-Minimum strebt - die Bose-Einstein-Kondensation. F\u00FCr Fermionen existiert analog die Fermi-Dirac-Statistik, die ebenso wie die Bose-Einstein-Statistik im Grenzfall gro\u00DFer Energie E in die Boltzmann-Statistik \u00FCbergeht."@de ,
"Bosen\u2013Einsteinin statistiikka (BE-statistiikka) on statististessa fysiikassa jakaumalaki, joka osoittaa samanlaatuisten bosonien energiatilojen jakauma termodynaamisessa tasapainotilassa. Bosonit ovat alkeishiukkasia, jotka toisin kuin fermionit eiv\u00E4t noudata Paulin kieltos\u00E4\u00E4nt\u00F6\u00E4 ja joita n\u00E4in ollen voi olla rajoittamaton m\u00E4\u00E4r\u00E4 samassa energiatilassa. Sellaisia ovat esimerkiksi fotonit ja erilaiset mesonit. Bosen\u2013Einsteinin jakauman johti ensimm\u00E4isen\u00E4 fotoneille Satyendra Nath Bose vuonna 1920, ja sen yleisti atomeille Albert Einstein vuonna 1924."@fi ,
"Bose-Einstein-statistik, (B\u2013E-statistik) \u00E4r inom statistisk mekanik, liksom Fermi-Dirac-statistiken, f\u00F6rdelningar inom kvantstatistiken."@sv ,
"De Bose-Einsteinstatistiek beschrijft de gemiddelde bezetting voor de energieniveaus van ononderscheidbare bosonen in thermisch evenwicht. Een boson is een deeltje met een spin die een gehele waarde heeft, en dat daarom niet aan het uitsluitingsprincipe van Pauli voldoet. De Bose-Einsteinstatistiek werd ontwikkeld door Satyendra Nath Bose voor fotonen en gegeneraliseerd tot atomen door Albert Einstein. Volgens de Bose-Einsteinverdeling is het aantal deeltjes in een bepaalde energietoestand gelijk aan <math>\\langle n \\rangle=\\frac{1}{e^{\\beta (E-\\mu)}-1}</math> waar <math>E</math> de energie van die toestand is, <math>\\mu </math> de chemische potentiaal is, en <math>\\beta = 1/(k_BT)</math>, waarin <math>k_B</math> de Boltzmannconstante en <math>T</math> - de temperatuur in kelvin. De chemische potentiaal in deze vergelijking is altijd negatief of nul. Bij hogere temperaturen (<math>E - \\mu \\gg k_B T</math>) kan de term '-1' verwaarloosd worden, waardoor de vergelijking gelijk wordt aan de Maxwell-Boltzmann-verdeling uit de klassieke fysica. <math>\\langle n\\rangle =e^{-\\beta (E-\\mu)}</math> Aangezien fotonen spin 1 hebben, en dus bosonen zijn, voldoen zij aan de Bose-Einsteinstatistiek. Toegepast op licht staat deze vergelijking ook bekend als de Wet van Planck. Deze vergelijking verklaart het gedrag van zwarte lichamen. De afleiding van deze vergelijking door Max Planck vormde de start van het wetenschapsgebied van de kwantummechanica. Het feit dat het uitsluitingsprincipe van Pauli niet opgaat voor bosonen leidt tot de mogelijkheid van het vormen van een Bose-Einsteincondensaat, waarbij een aanzienlijke fractie van de deeeltjes zich in de grondtoestand bevindt."@nl ;
rdfs:comment "A la f\u00EDsica estad\u00EDstica, l'estad\u00EDstica de Bose\u2013Einstein (o m\u00E9s col\u00B7loquialment estad\u00EDstica B-E) determina la distribuci\u00F3 estad\u00EDstica d'un conjunt de bosons indistingibles en equilibri t\u00E8rmic sobre un conjunt d'estats d'energia. Els bosons, a difer\u00E8ncia dels fermions, no estan subjectes al principi d'exclusi\u00F3 de Pauli: un nombre il\u00B7limitat d'ells poden ocupar el mateix estat qu\u00E0ntic a la vegada."@ca ,
"La estad\u00EDstica de Bose-Einstein es un tipo de mec\u00E1nica estad\u00EDstica aplicable a la determinaci\u00F3n de las propiedades estad\u00EDsticas de conjuntos grandes de part\u00EDculas indistinguibles capaces de coexistir en el mismo estado cu\u00E1ntico en equilibrio t\u00E9rmico."@es ,
"De Bose-Einsteinstatistiek beschrijft de gemiddelde bezetting voor de energieniveaus van ononderscheidbare bosonen in thermisch evenwicht. Een boson is een deeltje met een spin die een gehele waarde heeft, en dat daarom niet aan het uitsluitingsprincipe van Pauli voldoet. De Bose-Einsteinstatistiek werd ontwikkeld door Satyendra Nath Bose voor fotonen en gegeneraliseerd tot atomen door Albert Einstein."@nl ,
"En m\u00E9canique quantique et en physique statistique, la statistique de Bose-Einstein d\u00E9signe la distribution statistique de bosons indiscernables (tous similaires) sur les \u00E9tats d'\u00E9nergie d'un syst\u00E8me \u00E0 l'\u00E9quilibre thermodynamique. La distribution en question r\u00E9sulte d'une particularit\u00E9 des bosons : les particules de spin entier ne sont pas assujetties au principe d'exclusion de Pauli, \u00E0 savoir que plusieurs bosons peuvent occuper simultan\u00E9ment un m\u00EAme \u00E9tat quantique."@fr ,
"In meccanica statistica, la statistica di Bose-Einstein (indicata anche come statistica B-E) determina la distribuzione statistica relativa agli stati energetici, all\u2019equilibrio termico, dei bosoni, nell\u2019ipotesi che siano identici e indistinguibili tra loro. La statistica di Fermi-Dirac e quella di Bose-Einstein vengono applicate quando si devono considerare gli effetti quantistici e le particelle sono considerate indistinguibili."@it ,
"Die Bose-Einstein-Statistik, benannt nach Satyendranath Bose und Albert Einstein, ist eine Verteilung in der Quantenstatistik. Sie beschreibt die mittlere Besetzungszahl <math> \\langle n(E) \\rangle </math> eines Quantenzustands der Energie E im thermodynamischen Gleichgewicht bei der absoluten Temperatur <math>T</math> f\u00FCr identische Bosonen als besetzende Teilchen. Sie ist analog zur Fermi-Dirac-Statistik, bei der statt Bosonen Fermionen betrachtet werden."@de ,
""@ja ,
""@zh ,
"In statistical mechanics, Bose\u2013Einstein statistics (or more colloquially B\u2013E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Concept Fermi\u2013Dirac and Bose\u2013Einstein statistics apply when quantum effects are important and the particles are \"indistinguishable\". Quantum effects appear if the concentration of particles (N/V) ≥ nq."@en ,
"Statystyka Bosego-Einsteina \u2013 statystyka dotycz\u0105ca bozon\u00F3w traktowanych jako gaz bozonowy, cz\u0105stek o spinie ca\u0142kowitym, kt\u00F3rych nie obowi\u0105zuje zakaz Pauliego."@pl ,
"Bose-Einstein-statistik, (B\u2013E-statistik) \u00E4r inom statistisk mekanik, liksom Fermi-Dirac-statistiken, f\u00F6rdelningar inom kvantstatistiken."@sv ,
"Boseho-Einsteinovo rozd\u011Blen\u00ED popisuje ve statistick\u00E9 fyzice syst\u00E9my slo\u017Een\u00E9 z boson\u016F, tedy \u010D\u00E1stic se symetrickou vlnovou funkc\u00ED a celo\u010D\u00EDseln\u00FDm spinem. Bose-Einsteinov\u00FDm rozd\u011Blen\u00EDm se \u0159\u00EDd\u00ED nap\u0159\u00EDklad fotony, je z n\u011Bj tedy mo\u017En\u00E9 odvodit nap\u0159\u00EDklad Planck\u016Fv vyza\u0159ovac\u00ED z\u00E1kon. Rozd\u011Blen\u00ED poprv\u00E9 popsal Satyendra Nath Bose, roku 1924 ho pak zobecnil Albert Einstein. N\u00E1zev bosony je pr\u00E1v\u011B podle Boseho."@cs ,
"\u0421\u0442\u0430\u0442\u0438\u0301\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0437\u0435\u2014\u0415\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u0430 \u2014 \u0446\u0435 \u043E\u0441\u043E\u0431\u043B\u0438\u0432\u0438\u0439 \u0432\u0438\u0434 \u0440\u043E\u0437\u043F\u043E\u0434\u0456\u043B\u0443 \u0437\u0430 \u0435\u043D\u0435\u0440\u0433\u0456\u0454\u044E \u0447\u0430\u0441\u0442\u043E\u043A, \u044F\u043A\u0456 \u043D\u0430\u043B\u0435\u0436\u0430\u0442\u044C \u0434\u043E \u0431\u043E\u0437\u043E\u043D\u0456\u0432."@uk ,
"Bosen\u2013Einsteinin statistiikka (BE-statistiikka) on statististessa fysiikassa jakaumalaki, joka osoittaa samanlaatuisten bosonien energiatilojen jakauma termodynaamisessa tasapainotilassa. Bosonit ovat alkeishiukkasia, jotka toisin kuin fermionit eiv\u00E4t noudata Paulin kieltos\u00E4\u00E4nt\u00F6\u00E4 ja joita n\u00E4in ollen voi olla rajoittamaton m\u00E4\u00E4r\u00E4 samassa energiatilassa. Sellaisia ovat esimerkiksi fotonit ja erilaiset mesonit."@fi ,
"\u0421\u0442\u0430\u0442\u0438\u0301\u0441\u0442\u0438\u043A\u0430 \u0411\u043E\u0301\u0437\u0435\u00A0\u2014 \u042D\u0439\u043D\u0448\u0442\u0435\u0301\u0439\u043D\u0430\u00A0\u2014 \u043A\u0432\u0430\u043D\u0442\u043E\u0432\u0430\u044F \u0441\u0442\u0430\u0442\u0438\u0441\u0442\u0438\u043A\u0430, \u043F\u0440\u0438\u043C\u0435\u043D\u044F\u0435\u043C\u0430\u044F \u043A \u0441\u0438\u0441\u0442\u0435\u043C\u0430\u043C \u0447\u0430\u0441\u0442\u0438\u0446 \u0441 \u043D\u0443\u043B\u0435\u0432\u044B\u043C \u0438\u043B\u0438 \u0446\u0435\u043B\u043E\u0447\u0438\u0441\u043B\u0435\u043D\u043D\u044B\u043C \u0441\u043F\u0438\u043D\u043E\u043C; \u043F\u0440\u0435\u0434\u043B\u043E\u0436\u0435\u043D\u0430 \u0432 1924 \u0433\u043E\u0434\u0443 \u0438\u043D\u0434\u0438\u0439\u0441\u043A\u0438\u043C \u0444\u0438\u0437\u0438\u043A\u043E\u043C \u0428. \u0411\u043E\u0437\u0435 \u0434\u043B\u044F \u043A\u0432\u0430\u043D\u0442\u043E\u0432 \u0441\u0432\u0435\u0442\u0430; \u0438\u0441\u043F\u043E\u043B\u044C\u0437\u043E\u0432\u0430\u043D\u0430 \u0410. \u042D\u0439\u043D\u0448\u0442\u0435\u0439\u043D\u043E\u043C \u0434\u043B\u044F \u043C\u043E\u043B\u0435\u043A\u0443\u043B \u0438\u0434\u0435\u0430\u043B\u044C\u043D\u044B\u0445 \u0433\u0430\u0437\u043E\u0432."@ru ,
"Em mec\u00E2nica estat\u00EDstica, a estat\u00EDstica de Bose\u2013Einstein (ou mais coloquialmente estat\u00EDstica B-E) determina a distribui\u00E7\u00E3o estat\u00EDstica de b\u00F3sons id\u00EAnticos indistingu\u00EDveis sobre os estados de energia em equil\u00EDbrio t\u00E9rmico."@pt .
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