Bose–Einstein statistics

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.

Fermi-Dirac and Bose-Einstein statistics apply when quantum effects have to be taken into account and the particles are considered "indistinguishable". The quantum effects appear if the concentration of particles (N/V) ≥ n_q (where n_q is the quantum concentration). The quantum concentration is when the interparticle distance is equal to the thermal de Broglie wavelength i.e. when the wavefunctions of the particles are touching but not overlapping. As the quantum concentration depends on temperature; high temperatures will put most systems in the classical limit unless they have a very high density e.g. a White dwarf. Fermi-Dirac statistics apply to fermions (particles that obey the Pauli exclusion principle), Bose-Einstein statistics apply to bosons. Both Fermi-Dirac and Bose-Einstein become Maxwell-Boltzmann statistics at high temperatures or low concentrations.

Maxwell-Boltzmann statistics are often described as the statistics of "distinguishable" classical particles. In other words the configuration of particle A in state 1 and particle B in state 2 is different from the case where particle B is in state 1 and particle A is in state 2. When this idea is carried out fully, it yields the proper (Boltzmann) distribution of particles in the energy states, but yields non-physical results for the entropy, as embodied in Gibbs paradox. These problems disappear when it is realized that all particles are in fact indistinguishable. Both of these distributions approach the Maxwell-Boltzmann distribution in the limit of high temperature and low density, without the need for any ad hoc assumptions. Maxwell-Boltzmann statistics are particularly useful for studying gases. Fermi-Dirac statistics are most often used for the study of electrons in solids. As such, they form the basis of semiconductor device theory and electronics.

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 1924.

The expected number of particles in an energy state i  for B-E statistics is:

Failed to parse (Missing texvc executable; please see math/README to configure.): n_i = \frac {g_i} {e^{(\varepsilon_i-\mu)/kT} - 1}

with Failed to parse (Missing texvc executable; please see math/README to configure.): \varepsilon_i > \mu

and where:

n_i  is the number of particles in state i

g_i  is the degeneracy of state i

ε_i  is the energy of the i-th state

μ is the chemical potential

k is Boltzmann's constant

T is absolute temperature

This reduces to M-B statistics for energies ( ε_i − μ ) >> 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle i , each level having energy Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \varepsilon_i

and containing a total of 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i

particles.  Suppose each level contains 

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g_i

associated with level Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle i
is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.

Let Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) be the number of ways of distributing Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

particles among the 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g

sublevels of an energy level. There is only one way of distributing

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

particles with one sublevel, therefore 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,1)=1 . It is easy to see that there are Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (n+1)

ways of distributing

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

particles in two sublevels which we will write as: 

Failed to parse (Missing texvc executable; please see math/README to configure.): w(n,2)=\frac{(n+1)!}{n!1!}.

With a little thought (See Notes below) it can be seen that the number of ways of distributing Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

particles in three sublevels is  

Failed to parse (Missing texvc executable; please see math/README to configure.): w(n,3) = w(n,2) + w(n-1,2) + \cdots + w(1,2) + w(0,2)

so that

Failed to parse (Missing texvc executable; please see math/README to configure.): w(n,3)=\sum_{k=0}^n w(n-k,2) = \sum_{k=0}^n\frac{(n-k+1)!}{(n-k)!1!}=\frac{(n+2)!}{n!2!}

where we have used the following theorem involving binomial coefficients:

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.

Continuing this process, we can see that Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) is just a binomial coefficient (See Notes below)

Failed to parse (Missing texvc executable; please see math/README to configure.): w(n,g)=\frac{(n+g-1)!}{n!(g-1)!}.

The number of ways that a set of occupation numbers Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i

can be realized is the product of the ways that each individual energy level can be populated:

Failed to parse (Missing texvc executable; please see math/README to configure.): W = \prod_i w(n_i,g_i) = \prod_i \frac{(n_i+g_i-1)!}{n_i!(g_i-1)!} \approx\prod_i \frac{(n_i+g_i)!}{n_i!(g_i)!}

where the approximation assumes that Failed to parse (Missing texvc executable; please see math/README to configure.): g_i \gg 1 . Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i

for which Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle W

is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle W

and Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \ln(W)
occur at the value of 

Failed to parse (Missing texvc executable; please see math/README to configure.): \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:

Failed to parse (Missing texvc executable; please see math/README to configure.): f(n_i)=\ln(W)+\alpha(N-\sum n_i)+\beta(E-\sum n_i \varepsilon_i)

Using the Failed to parse (Missing texvc executable; please see math/README to configure.): g_i \gg 1

approximation and using Stirling's approximation for the factorials 

Failed to parse (Missing texvc executable; please see math/README to configure.): \left(\ln(x!)\approx x\ln(x)-x\right)

gives

Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i , and setting the result to zero and solving for Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i , yields the Bose–Einstein population numbers:

Failed to parse (Missing texvc executable; please see math/README to configure.): n_i = \frac{g_i}{e^{\alpha+\beta \varepsilon_i}-1}.

It can be shown thermodynamically that Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \beta = \frac{1}{kT} , where Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle k

is Boltzmann's constant and Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle T

is the temperature.

It can also be shown that Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \alpha = - \frac{\mu}{kT} , where Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \mu

is the chemical potential, so that finally:

Failed to parse (Missing texvc executable; please see math/README to configure.): n_i = \frac{g_i}{e^{(\varepsilon_i-\mu)/kT}-1}.

Note that the above formula is sometimes written:

Failed to parse (Missing texvc executable; please see math/README to configure.): n_i = \frac{g_i}{e^{\varepsilon_i/kT}/z-1},

where Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

dice, 

with each dice taking values in the set Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left\{ 1, \cdots, g \right\} , for Failed to parse (Missing texvc executable; please see math/README to configure.): g \ge 1 . The constraints of the game is that the value of a dice Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle i , denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n_i , has to be greater or equal to the value of dice Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (i-1) , denoted by Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_{i-1} , in the previous throw, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): m_i \ge m_{i-1} . Thus a valid sequence of dice throws can be described by an Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n -tuple Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left( m_1 , m_2 , \cdots , m_n \right) , such that Failed to parse (Missing texvc executable; please see math/README to configure.): m_i \ge m_{i-1} . Let Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g)

denote the set of these valid Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

-tuples:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(n,g) = \Big\{ \left( m_1 , m_2 , \cdots , m_n \right) \Big| \Big. m_i \ge m_{i-1} , m_i \in \left\{ 1 \cdots, g \right\} , \forall i = 1, \cdots , n \Big\}

(1)

Then the quantity Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g)

(defined above as the number of ways to distribute 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

particles among the 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g

sublevels of an energy level) is the cardinality of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g)

, i.e., the number of elements (or valid Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n -tuples) in Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g) . Thus the problem of finding and expression for Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g)

becomes the problem of counting the elements in Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g) .

Example n = 4, g = 3:

Failed to parse (Missing texvc executable; please see math/README to configure.): S(4,3) = \left\{ \underbrace{(1111), (1112), (1113)}_{(a)}, \underbrace{(1122), (1123), (1133)}_{(b)}, \underbrace{(1222), (1223), (1233), (1333)}_{(c)}, \right.

Failed to parse (Missing texvc executable; please see math/README to configure.): \left. \underbrace{(2222), (2223), (2233), (2333), (3333)}_{(d)} \right\}

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(4,3) = 15

(there are Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle 15
elements in Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3)

)

Subset Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (a)

is obtained by fixing all indices Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_i

to 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle 1 , except for the last index, Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_n , which is incremented from Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle 1

to

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g=3 . Subset Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (b)

is obtained by fixing Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_1 = m_2 = 1 , and increment Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_3

from 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle 2

to

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g=3

due to the constraint

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_i \ge m_{i-1}

on the indices in Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g) , the index Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_4

must 

automatically take values in Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left\{ 2, 3 \right\} . The construction of subsets Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (c)

and 

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (d)

follows in the same manner.

Each element of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3)

can be thought of as a 

multiset of cardinality Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n=4

the elements of such multiset are taken from the set Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left\{ 1, 2, 3 \right\}

of cardinality Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g=3 , and the number of such multisets is the multiset coefficient

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left\langle \begin{matrix} 3 \\ 4 \end{matrix} \right\rangle = {3 + 4 - 1 \choose 3-1} = {3 + 4 - 1 \choose 4} = \frac {6!} {4! 2!} = 15

More generally, each element of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g)

is a multiset of cardinality Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n

(number of dice) with elements taken from the set Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \left\{ 1, \cdots, g \right\}

of cardinality Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g

(number of possible values of each dice), and the number of such multisets, i.e., Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g)

is the multiset coefficient

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) = \left\langle \begin{matrix} g \\ n \end{matrix} \right\rangle = {g + n - 1 \choose g-1} = {g + n - 1 \choose n} = \frac{(g + n - 1)!} {n! (g-1)!}

(2)

which is exactly the same as the formula for Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) , as derived above with the aid of a theorem involving binomial coefficients, namely

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum_{k=0}^n\frac{(k+a)!}{k!a!}=\frac{(n+a+1)!}{n!(a+1)!}.

(3)

To understand the decomposition

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) = \sum_{k=0}^{n} w(n-k, g-1) = w(n, g-1) + w(n-1, g-1) + \cdots + w(1, g-1) + w(0, g-1)

(4)

or for example, Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle n=4

and Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle g=3

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(4,3) = w(4,2) + w(3,2) + w(2,2) + w(1,2) + w(0,2)

To this end, let's rearrange the elements of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3)

as follows

Failed to parse (Missing texvc executable; please see math/README to configure.): S(4,3) = \left\{ \underbrace{ (1111), (1112), (1122), (1222), (2222) }_{(\alpha)}, \underbrace{ (111{\color{Red}\underset{=}{3}}), (112{\color{Red}\underset{=}{3}}), (122{\color{Red}\underset{=}{3}}), (222{\color{Red}\underset{=}{3}}) }_{(\beta)}, \right.

Failed to parse (Missing texvc executable; please see math/README to configure.): \left. \underbrace{ (11{\color{Red}\underset{==}{33}}), (12{\color{Red}\underset{==}{33}}), (22{\color{Red}\underset{==}{33}}) }_{(\gamma)}, \underbrace{ (1{\color{Red}\underset{===}{333}}), (2{\color{Red}\underset{===}{333}}) }_{(\delta)} \underbrace{ ({\color{Red}\underset{====}{3333}}) }_{(\omega)} \right\}

Clearly, the subset Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\alpha)

of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3)

is the same as the set

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,2) = \left\{ (1111), (1112), (1122), (1222), (2222) \right\}

By deleting the index Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle m_4=3

(shown in red with double underline) in the subset Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\beta)

of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3) , one obtain the set

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(3,2) = \left\{ (111), (112), (122), (222) \right\}

In other words, there is a one-to-one correspondence between the subset Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\beta)

of Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3)

and the set Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(3,2) . We write

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\beta) \longleftrightarrow S(3,2)

Similarly, it is easy to see that

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\gamma) \longleftrightarrow S(2,2) = \left\{ (11), (12), (22) \right\}

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\delta) \longleftrightarrow S(1,2) = \left\{ (1), (2) \right\}

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle (\omega) \longleftrightarrow S(0,2) = \phi

(empty set)

Thus we can write

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(4,3) = \bigcup_{k=0}^{4} S(4-k,2)

or more generally,

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(n,g) = \bigcup_{k=0}^{n} S(n-k,g-1)

(5)

and since the sets

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle S(i,g-1) \ , \ {\rm for} \ i = 0, \cdots , n

are non-intersecting, we thus have

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) = \sum_{k=0}^{n} w(n-k,g-1)

(6)

with the convention that

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(0,g) = 1 \ , \forall g \ , {\rm and} \ w(n,0) = 1 \ , \forall n

(7)

Continue the process, we arrive at the following formula

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} w(n - k_1 - k_2, g-2) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} \cdots \sum_{k_g=0}^{n-\sum_{j=1}^{g-1} k_j} w(n - \sum_{i=1}^{g} k_i, 0)

Using the convention (7)₂ above, we obtain the formula

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(n,g) = \sum_{k_1=0}^{n} \sum_{k_2=0}^{n-k_1} \cdots \sum_{k_g=0}^{n-\sum_{j=1}^{g-1} k_j} 1

(8)

keeping in mind that for Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle q

and Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle p

being constants, we have

Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle \sum_{k=0}^{q} p = q p

(9)

It can then be verified that (8) and (2) give the same result for Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(4,3) , Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(3,3) , Failed to parse (Missing texvc executable; please see math/README to configure.): \displaystyle w(3,2) , etc.

References

Annett, James F., "Superconductivity, Superfluids and Condensates", Oxford University Press, 2004, New York.

Carter, Ashley H., "Classical ans Statistical Thermodynamics", Prentice-Hall, Inc., 2001, New Jersey.

Griffiths, David J., "Introduction to Quantum Mechanics", 2nd ed. Pearson Education, Inc., 2005.

See also

• Maxwell-Boltzmann statistics
• Fermi-Dirac statistics
• Parastatistics
• Planck's law of black body radiationbs:Bose-Einstein statistika

ca:Estadística de Bose-Einstein cs:Boseho-Einsteinovo rozdělení de:Bose-Einstein-Statistik et:Bose-Einsteini statistika es:Estadística de Bose-Einstein fr:Statistique de Bose-Einstein hr:Bose-Einstein statistika it:Statistica di Bose-Einstein he:התפלגות בוז-איינשטיין nl:Bose-Einsteinstatistiek ja:ボース分布関数 pl:Statystyka Bosego-Einsteina pt:Estatística de Bose-Einstein ru:Статистика Бозе — Эйнштейна sk:Boseho-Einsteinove rozdelenie sl:Bose-Einsteinova statistika uk:Статистика Бозе-Ейнштейна