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- In statistical mechanics, Bose–Einstein statistics (or more colloquially B–E statistics) determines the statistical distribution of identical indistinguishable bosons over the energy states in thermal equilibrium. Concept Fermi–Dirac and Bose–Einstein 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–Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), and Bose–Einstein statistics apply to bosons. As the quantum concentration depends on temperature; most systems at high temperatures obey the classical (Maxwell–Boltzmann) limit unless they have a very high density, as for a white dwarf. Both Fermi–Dirac and Bose–Einstein become Maxwell–Boltzmann 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–Einstein condensate. B–E 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–E 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 εi is the energy of the ith state μ is the chemical potential k is Boltzmann's constant T is absolute temperature This reduces to Maxwell–Boltzmann 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–Einstein 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–Boltzmann 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–Einstein 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–Einstein (B–E) distribution for beginners. The enumeration of cases (or ways) in the B–E 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} \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{ }_{(\alpha)}, \underbrace{ (111{\color{Red}\underset{}{}}), (112{\color{Red}\underset{}{}}), (122{\color{Red}\underset{}{}}), (222{\color{Red}\underset{}{}}) }_{(\beta)}, \right. \left. \underbrace{ (11{\color{Red}\underset{}{}}), (12{\color{Red}\underset{}{}}), (22{\color{Red}\underset{}{}}) }_{(\gamma)}, \underbrace{ (1{\color{Red}\underset{}{}}), (2{\color{Red}\underset{}{}}) }_{(\delta)} \underbrace{ ({\color{Red}\underset{}{}}) }_{(\omega)} \right\} . Clearly, the subset \displaystyle (\alpha) of \displaystyle S is the same as the set \displaystyle S(2) \left\{ \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\{ \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\{ \right\} \displaystyle (\delta) \longleftrightarrow S(1,2) \left\{ \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–Boltzmann statistics Fermi–Dirac statistics Parastatistics Planck's law of black body radiation Notes References . . . 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- 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> für identische Bosonen als besetzende Teilchen. Sie ist analog zur Fermi-Dirac-Statistik, bei der statt Bosonen Fermionen betrachtet werden. Bei Wechselwirkungsfreiheit ergibt sich für Bosonen die folgende Formel: <math> \langle n(E) \rangle = \frac {1}{e^{\beta (E - \mu)} - 1} </math> Hierbei ist μ das chemische Potential und <math>\beta</math> üblicherweise gleich <math>1/(k_B T)</math> (mit der Boltzmann-Konstanten kB und der absoluten Temperatur T). Die Wahl des Faktors <math>\beta</math> hängt von der verwendeten Temperaturskala ab. Wird die Temperatur in Energieeinheiten, etwa Joule, gemessen, so beträgt 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. Benötigt man die Besetzungszahl eines entarteten Energieniveaus, so ist obiger Ausdruck mit dem Entartungsgrad gi desselben zu multiplizieren. Im Falle der Fermi-Dirac-Statistik erhält man im Nenner +1 anstelle von -1. Unterhalb einer sehr tiefen kritischen Temperatur <math>T_\lambda</math> erhält man im Spezialfall der Wechselwirkungsfreiheit - unter der Annahme, dass μ gegen das Energie-Minimum strebt - die Bose-Einstein-Kondensation. Für Fermionen existiert analog die Fermi-Dirac-Statistik, die ebenso wie die Bose-Einstein-Statistik im Grenzfall großer Energie E in die Boltzmann-Statistik übergeht.
- A la física estadística, l'estadística de Bose–Einstein (o més col·loquialment estadística B-E) determina la distribució estadística d'un conjunt de bosons indistingibles en equilibri tèrmic sobre un conjunt d'estats d'energia. Els bosons, a diferència dels fermions, no estan subjectes al principi d'exclusió de Pauli: un nombre il·limitat d'ells poden ocupar el mateix estat quàntic a la vegada. Això explica perquè, a baixes temperatures, el seu comportament difereix notablement del dels fermions, ja que tots els bosons tendiran a aplegar-se en l'estat de mínima energia, formant el que s'anomena condensat de Bose–Einstein. L'estadísitca de Bose-Einstein va ser introduida per descriure la distribució de fotons en la llei de Planck de la radiació del cos negre, per Satyendra Nath Bose el 1924, i posteriorment va ser generalitzada pel cas de partícules amb massa per Einstein. D'acord amb aquesta estadística, el nombre mitjà de bosons en un estat i n_i és: n_i = \frac {g_i} {e^{(\varepsilon_i-\mu)/kT} - 1} on \varepsilon_i \geq \mu i: g_i és la degeneració de l'estat i \varepsilon_i és l'energia de l'estat i \mu és el potencial químic del sistema k_B és la constant de Boltzmann T és la temperatura absoluta Aquesta expressió es redueix a la corresponent a l'estadística de Maxwell-Boltzmann per a energies grans (\varepsilon_i-\mu \gg k_BT).
- Boseho-Einsteinovo rozdělení popisuje ve statistické fyzice systémy složené z bosonů, tedy částic se symetrickou vlnovou funkcí a celočíselným spinem. Bose-Einsteinovým rozdělením se řídí například fotony, je z něj tedy možné odvodit například Planckův vyzařovací zákon. Rozdělení poprvé popsal Satyendra Nath Bose, roku 1924 ho pak zobecnil Albert Einstein. Název bosony je právě podle Boseho.
- La estadística de Bose-Einstein es un tipo de mecánica estadística aplicable a la determinación de las propiedades estadísticas de conjuntos grandes de partículas indistinguibles capaces de coexistir en el mismo estado cuántico en equilibrio térmico. A bajas temperaturas los bosones tienden a tener un comportamiento cuántico similar que puede llegar a ser idéntico 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 año 1995. El condensador Bose-Einstein funciona a temperaturas cercanas al cero absoluto, -273,16°C(0 Kelvin). La estadística de Bose-Einstein fue introducida para estudiar las propiedades estadísticas de los fotones en 1920 por el físico hindú Satyendra Nath Bose y generalizada para átomos y otros bosones por Albert Einstein en 1924. Este tipo de estadística está íntimamente relacionada con la estadística de Maxwell-Boltzmann (derivada inicialmente para gases) y a las estadísticas de Fermi-Dirac (aplicables a partículas denominadas fermiones sobre las que rige el principio de exclusión de Pauli que impide que dos fermiones compartan el mismo estado cuántico). La estadística de Bose-Einstein se reduce a la estadística de Maxwell-Boltzmann para energías suficientemente elevadas.
- Bosen–Einsteinin statistiikka (BE-statistiikka) on statististessa fysiikassa jakaumalaki, joka osoittaa samanlaatuisten bosonien energiatilojen jakauma termodynaamisessa tasapainotilassa. Bosonit ovat alkeishiukkasia, jotka toisin kuin fermionit eivät noudata Paulin kieltosääntöä ja joita näin ollen voi olla rajoittamaton määrä samassa energiatilassa. Sellaisia ovat esimerkiksi fotonit ja erilaiset mesonit. Bosen–Einsteinin jakauman johti ensimmäisenä fotoneille Satyendra Nath Bose vuonna 1920, ja sen yleisti atomeille Albert Einstein vuonna 1924.
- En mécanique quantique et en physique statistique, la statistique de Bose-Einstein désigne la distribution statistique de bosons indiscernables (tous similaires) sur les états d'énergie d'un système à l'équilibre thermodynamique. La distribution en question résulte d'une particularité des bosons : les particules de spin entier ne sont pas assujetties au principe d'exclusion de Pauli, à savoir que plusieurs bosons peuvent occuper simultanément un même état quantique.
- In meccanica statistica, la statistica di Bose-Einstein (indicata anche come statistica B-E) determina la distribuzione statistica relativa agli stati energetici, all’equilibrio termico, dei bosoni, nell’ipotesi 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) è 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, cioè quando le funzioni d’onda associate alle particelle si incontrano in zone nelle quali hanno valori non trascurabili, ma non si sovrappongono. Poiché 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 densità 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 è spesso descritta come la statistica delle particelle classiche e distinguibili. Per capire quest’ultimo concetto pensiamo di considerare la particella A nella posizione 1 e la particella B nella posizione 2. Se le particelle sono distinguibili questa configurazione è diversa da quella in cui, ciascuna particella occupa la posizione dell’altra (A in 2 e B in 1). Quando questa idea venne compresa a fondo, contribuì 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’entropia, 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 probabilità approssimano la distribuzione di Maxwell-Boltzmann nel limite di alte temperature e basse densità, senza il bisogno di nessuna ulteriore assunzione. La statistica di Maxwell-Boltzmann è particolarmente utile nello studio dei gas, mentre quella di Fermi-Dirac è utilizzata più spesso nello studio degli elettroni nei solidi. Per questi motivi esse costituiscono la base della teoria dei semiconduttori e dell’elettronica. 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 perché a basse temperature i bosoni possono diventare molto diversi dai fermioni; infatti essi tendono ad ammassarsi nello stesso livello di bassa energia, formando ciò che è noto come condensato di Bose-Einstein. La statistica di Bose-Einstein è stata introdotta nel 1920 da Satyendra Nath Bose per i fotoni ed è stata estesa agli atomi da Albert Einstein nel 1924. Il numero di particelle, occupanti l’i-imo livello di energia, previsto dalla statistica di Bose-Einstein è: n_i = \frac {g_i} {e^{(\epsilon_i-\mu)/kT} - 1} con <math>\epsilon_i > \mu e dove: ni è il numero di particelle nello stato i, gi esprime la degenerazione dello stato i, εi è l'energia dell'i-imo stato, μ è il potenziale chimico, k è la costante di Boltzmann, T è la temperatura assoluta. Ciò si riduce alla statistica di Maxwell-Boltzmann per energie (εi-μ) >> kT.
- ボース分布関数 (Bose distribution function) は、ボース=アインシュタイン分布関数 (Bose=Einstein distribution function) とも呼ばれ、ボース＝アインシュタイン統計に従う粒子（ボース粒子）の分布関数である。 ボース分布関数はエネルギー <math>\varepsilon</math> の関数 <math>f</math> として、以下の式で表される。 <math> f(\varepsilon) = \, {1 \over {\exp\left\{ {1 \over {k_B T} } (\varepsilon - \mu)\right\} - 1 } } </math> <math> k_B </math> ： ボルツマン定数 <math> \mu </math> ： 化学ポテンシャル（ケミカルポテンシャル） <math>\mu\leq0</math> である。<math>\mu=0\,</math> となるのは生成および消滅が起こる光子やフォノンなどの粒子系か、 ボース＝アインシュタイン凝縮を起こしている粒子系である。
- 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.
- Statystyka Bosego-Einsteina – statystyka dotycząca bozonów traktowanych jako gaz bozonowy, cząstek o spinie całkowitym, których nie obowiązuje zakaz Pauliego. Zgodnie z rozkładem Bosego-Einsteina średnia ilość cząstek w danym stanie kwantowym jest równa <math>\langle n_i \rangle=\frac n Z \frac{g_i}{e^{\beta (E_i-\mu)}-1}</math> gdzie: <math>n_i</math> – średnia liczba cząsteczek w i-tym stanie, <math>E_i</math> – energia i-tego stanu, <math>g_i</math> – degeneracja i-tego stanu, <math>n</math> – całkowita liczba cząstek, <math>\mu </math> – potencjał chemiczny, <math>\beta = \frac1{k_BT}</math>, gdzie <math>k_B</math> jest stałą Boltzmanna, T – temperatura w skali Kelvina, <math>Z=\sum_i \frac{g_i}{e^{\beta (E_i-\mu)}-1}</math> suma statystyczna. Potencjał chemiczny w tym rozkładzie jest zawsze ujemny lub równy zeru. Gdy temperatura jest wysoka, można zaniedbać czynnik –1 i rozkład przechodzi w rozkład fizyki klasycznej, klasyczny rozkład Boltzmanna <math>\langle n_i\rangle =\frac n Z g_i e^{-\beta (E_i-\mu)}</math> Rozkładowi Bosego-Einsteina podlegają oczywiście fotony (o spinie 1) – nosi on wtedy nazwę rozkładu Plancka, który tłumaczy promieniowanie ciała doskonale czarnego. Jego wprowadzenie przez Plancka zapoczątkowało mechanikę kwantową. Zakaz Pauliego nie dotyczy bozonów, umożliwia to ich kondensację.
- Em mecânica estatística, a estatística de Bose–Einstein (ou mais coloquialmente estatística B-E) determina a distribuição estatística de bósons idênticos indistinguíveis sobre os estados de energia em equilíbrio térmico.
- Стати́стика Бо́зе — Эйнште́йна — квантовая статистика, применяемая к системам частиц с нулевым или целочисленным спином; предложена в 1924 году индийским физиком Ш. Бозе для квантов света; использована А. Эйнштейном для молекул идеальных газов. Характерная особенность — в одном и том же состоянии может находиться любое число одинаковых частиц (в противоположность статистике Ферми — Дирака для частиц с полуцелым спином, согласно которой каждое состояние может быть занято не более чем одной частицей). Для сильно разрежённых газов (как и статистика Ферми — Дирака) переходит в статистику Максвелла — Больцмана.
- Bose-Einstein-statistik, (B–E-statistik) är inom statistisk mekanik, liksom Fermi-Dirac-statistiken, fördelningar inom kvantstatistiken.
- Стати́стика Бозе—Ейнштейна — це особливий вид розподілу за енергією часток, які належать до бозонів. Згідно з розподілом Бозе-Ейнштейна ймовірність, що в квантовомеханічній багаточастинковій системі існує бозон у одночастинковому квантовому стані <math> |n\rangle </math> із енергією <math> \varepsilon_n </math> визначається формулою <math> f(\varepsilon_n) = \frac{1}{e^{(\varepsilon_n - \mu)/k_B T} -1} </math>, де <math> \mu </math> — хімічний потенціал, <math> k_B </math> — стала Больцмана, T — температура. Оскільки ймовірність повинна бути додатним числом, значення хімічного потенціалу завжди менше за енергію основного стану бозонів. Якщо кількість бозонів строго визначена (N), то хімічний потенціал визначається із умови нормування розподілу. <math> N = \sum_n \frac{1}{e^{(\varepsilon_n - \mu)/k_B T} -1}. </math>
- 玻色-爱因斯坦统计是玻色子所依从的统计规律。 根据量子力学，玻色子是自旋为整数的粒子，其本征波函数对称，在玻色子的某一个能级上，可以容纳无限个粒子。因而符合玻色-爱因斯坦统计分布的粒子，当他们处于某一分布<math>\left\{ n_j \right\}（“某一分布”指这样一种状态：即在能量为<math>\left\{ \epsilon_j \right\}的能级上同时有<math>n_j个粒子存在着，不难想象，当从宏观观察体系能量一定的时候，从微观角度观察体系可能有很多种不同的分布状态，而且在这些不同的分布状态中，总有一些状态出现的几率特别的大，而其中出现几率最大的分布状态被称为最可几分布）时，体系总状态数为： \Omega_j=\frac{(g_j+n_j-1)!}{n_j!(g_j-1)!} 对这一公式的理解是这样的：把g_j个简并能级看作一个拥有g_j个隔室的大盒子，把n_j个粒子看作准备放入盒子中的n_j个不可区分的小球，则可以把这个向盒子里面放小球的过程看作n_j个小球和盒子中g_j-1个隔室壁的随机排列过程，则这样的排列一共有(g_j+n_j-1)!种可能出现的状态；另一方面，小球和小球是不可区分的，隔室壁和隔室壁也是不可区分的，因此对小球和隔室壁的计数都有重复，需要除以这种重复计数(g_j-1)!和(n_j)!，最终得到的结果就是上述结果。 \Omega_j=\frac{(g_j+n_j-1)!}{n_j!(g_j-1)!} g_j=3;n_j=2;\Omega_j=6 玻色-爱因斯坦统计的最可几分布的数学表达式为： \left\{ n_j^{BE} \right\}=\frac{g_j e^\alpha e^{\beta\epsilon_j}}{1 - e^\alpha e^{\beta\epsilon_j}} 由于量子统计在数学处理上非常困难，因此在处理实际问题时经常引入一些近似条件，使费米-狄拉克统计和玻色-爱因斯坦统计退化成为经典的麦克斯韦-玻尔兹曼统计。
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