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:
with and where:
This reduces to Maxwell–Boltzmann statistics for energies .
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.
Let be the number of ways of distributing particles among the sublevels of an energy level. There is only one way of distributing particles with one sublevel, therefore . It is easy to see that there are ways of distributing particles in two sublevels which we will write as:
With a little thought (See Notes below) it can be seen that the number of ways of distributing particles in three sublevels is
where we have used the following theorem involving binomial coefficients:
Continuing this process, we can see that is just a binomial coefficient (See Notes below)
The number of ways that a set of occupation numbers can be realized is the product of the ways that each individual energy level can be populated:
where the approximation assumes that . Following the same procedure used in deriving the Maxwell–Boltzmann statistics, we wish to find the set of for which is maximised, subject to the constraint that there be a fixed number of particles, and a fixed energy. The maxima of and occur at the value of 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:
Using the approximation and using Stirling's approximation for the factorials gives
Taking the derivative with respect to , and setting the result to zero and solving for , yields the Bose–Einstein population numbers:
It can be shown thermodynamically that , where is Boltzmann's constant and is the temperature.
It can also be shown that , where is the chemical potential, so that finally:
Note that the above formula is sometimes written:
where is the absolute activity.
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 dice, with each dice taking values in the set , for . The constraints of the game is that the value of a dice , denoted by , has to be greater or equal to the value of dice , denoted by , in the previous throw, i.e., . Thus a valid sequence of dice throws can be described by an -tuple , such that . Let denote the set of these valid -tuples:
| (1) |
Then the quantity (defined above as the number of ways to distribute particles among the sublevels of an energy level) is the cardinality of , i.e., the number of elements (or valid -tuples) in . Thus the problem of finding and expression for becomes the problem of counting the elements in .
Example n = 4, g = 3:
S(4,3) =
left{
underbrace{(1111), (1112), (1113)}_{(a)},
underbrace{(1122), (1123), (1133)}_{(b)},
underbrace{(1222), (1223), (1233), (1333)}_{(c)},
right.
left.underbrace{(2222), (2223), (2233), (2333), (3333)}_{(d)} right}
Subset is obtained by fixing all indices to , except for the last index, , which is incremented from to . Subset is obtained by fixing , and increment from to ; due to the constraint on the indices in , the index must automatically take values in . The construction of subsets and follows in the same manner.
Each element of can be thought of as a multiset of cardinality ; the elements of such multiset are taken from the set of cardinality , and the number of such multisets is the multiset coefficient
displaystyle
leftlanglebegin{matrix} 3 4 end{matrix}
rightrangle= {3 + 4 - 1 choose 3-1} = {3 + 4 - 1 choose 4}
=
frac{6!} {4! 2!}
= 15
More generally, each element of is a multiset of cardinality (number of dice) with elements taken from the set of cardinality (number of possible values of each dice), and the number of such multisets, i.e., is the multiset coefficient
| (2) |
| (3) |
To understand the decomposition
| (4) |
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 as follows
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.
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 of is the same as the set
displaystyle
S(4,2)
=
left{
(1111),
(1112),
(1122),
(1222),
(2222)
right}
By deleting the index (shown in red with double underline) in the subset of , one obtain the set
displaystyle
S(3,2)
=
left{
(111),
(112),
(122),
(222)
right}
In other words, there is a one-to-one correspondence between the subset
of
and the set
. We write
displaystyle
(beta)
longleftrightarrow
S(3,2)
Similarly, it is easy to see that
displaystyle
(gamma)
longleftrightarrow
S(2,2)
=
left{
(11),
(12),
(22)
right}
displaystyle
(delta)
longleftrightarrow
S(1,2)
=
left{
(1),
(2)
right}
displaystyle
(omega)
longleftrightarrow
S(0,2)
=
phi(empty set)
Thus we can write
displaystyle
S(4,3)
=bigcup_{k=0}^{4}
S(4-k,2)
or more generally,
| (5) |
displaystyleS(i,g-1) , {rm for} i = 0, cdots , n are non-intersecting, we thus have
| (6) |
displaystyle w(0,g) = 1 , forall g ,{rm and} w(n,0) = 1 , forall n | (7) |
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}
cdotssum_{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
| (8) |
keeping in mind that for and being constants, we have
| (9) |
It can then be verified that (8) and (2) give the same result for , , , etc.
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 Terrier project at the University of Glasgow.
Annett, James F., "Superconductivity, Superfluids and Condensates", Oxford University Press, 2004, New York.
Carter, Ashley H., "Classical and Statistical Thermodynamics", Prentice-Hall, Inc., 2001, New Jersey.
Griffiths, David J., "Introduction to Quantum Mechanics", 2nd ed. Pearson Education, Inc., 2005.