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:''For other topics related to 'Einstein' see Einstein (disambiguation).''
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.
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:
:$$
n_i = rac {g_i} {e^{(arepsilon_i-mu)/kT} - 1}

with 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''.

Contents

History

A derivation of the Bose–Einstein distribution

See also

History

In the early 1920s Satyendra Nath Bose 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, because 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 he used his influence to get it published.

A derivation of the Bose–Einstein distribution

Suppose we have a number of energy levels, labelled by index ''i'', each level
having energy ''ε_i''  and containing a total of ''n_i''  particles. Suppose each level contains ''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 ''g_i''  associated with level ''i'' is called the "degeneracy" of that energy level. Any number of bosons can occupy the same sublevel.
Let ''w(n,g)'' be the number of ways of distributing ''n'' particles among
the ''g'' sublevels of an energy level. There is only one way of distributing
''n'' particles with one sublevel, therefore ''w''(''n'',1) = 1. It's easy to see that
there are ''n'' + 1 ways of distributing ''n'' particles in two sublevels which we will write as:
:$$
w(n,2)=rac{(n+1)!}{n!1!}.

With a little thought it can be seen that the number of ways of distributing
''n'' particles in three sublevels is ''w''(''n'',3) = ''w''(''n'',2) + ''w''(''n''−1,2) + ... + ''w''(0,2) so that
:$$
w(n,3)=sum_{k=0}^n w(n-k,2) = sum_{k=0}^nrac{(n-k+1)!}{(n-k)!1!}=rac{(n+2)!}{n!2!}

where we have used the following theorem involving binomial coefficients:
:$$
sum_{k=0}^nrac{(k+a)!}{k!a!}=rac{(n+a+1)!}{n!(a+1)!}.

Continuing this process, we can see that ''w(n,g)'' is just a binomial coefficient
:$$
w(n,g)=rac{(n+g-1)!}{n!(g-1)!}.

The number of ways that a set of occupation numbers ''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 rac{(n_i+g_i-1)!}{n_i!(g_i-1)!}
pproxprod_i rac{(n_i+g_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 ''n_i''  for which $W$ is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of $W$ and $ln(W)$ occur at the value of $N\_i$ 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)+lpha(N-sum n_i)+eta(E-sum n_i arepsilon_i)

Using the $g\_i\; gg\; 1$ approximation and using Stirling's approximation for the factorials
ight) gives
:
ight)+etaleft(E-sum n_i arepsilon_i
ight).

Taking the derivative with respect to ''n_i'', and setting the result to zero and solving for ''n_i'' yields the Bose–Einstein population numbers:
:$$
n_i = rac{g_i}{e^{lpha+eta arepsilon_i}-1}.

It can be shown thermodynamically that β = 1/''kT'' where ''k''  is Boltzmann's constant and ''T'' is the temperature, and that α = −μ/''kT'' where μ is the chemical potential, so that finally:
:$$
n_i = rac{g_i}{e^{(arepsilon_i-mu)/kT}-1}.

Note that the above formula is sometimes written:
:$$
n_i = rac{g_i}{e^{arepsilon_i/kT}/z-1},

where $z=exp(mu/kT)$ is the absolute activity.

See also

★ Maxwell-Boltzmann statistics

★ Fermi-Dirac statistics

★ Parastatistics

★ Planck's law of black body radiation