Statistical Physics, Second Revised and Enlarged Edition

56 Gases: the distributions

ThesymbolFDisshortfor Fermi–Dirac, since these two physicists were responsible
for the first discussion of the statistics of what we now call fermions.

5.3.2 Bosons

The statistics for bosons is called Bose–Einstein(BE)statistics, also named after its
inventors. Counting theboson microstatesisalittle more trickythanforfermions
since now anynumber ofparticles are allowedto occupyanyone-particle state.
Adirect, if slightly abstract, way of calculating the contribution from theith group
ofthedistributionisasfollows. In thegroup there aregistates containingniidentical
particles with no restrictions on occupation numbers. A typical microstate can be
represented as in Fig. 5 .2 by(gi− 1 )lines andnicrosses. The lines represent divisions
betweenthegistates, andthecrossesrepresentanoccupyingparticle. Newmicrostates
representing the same distribution, i.e. the same value ofnifor the group, can be
obtainedbyshuffling thelines andcrosses on thepicture. Infact the contribution to
the number ofmicrostatesis preciselyafurther simplebinomialproblem:how many
ways can the(ni+gi− 1 )symbols be arranged intonicrosses and(gi− 1 )lines?
The answeristhebinomialcoefficient (see again AppendixA)

(ni+gi− 1 )!

ni!(gi− 1 )!

This is the correct answer to the problem. But in fact, bearing in mind that in statistical
physics we are alwaysdealingwithlarge numbersgiofstates, the−1additionis
negligible. It is adequate to usegiratherthan(gi− 1 ),and we shall do this forthwith.
(Actually,ifthe− 1 is retainedhere, theninthe next section the mathematics would
give adistributionniproportionalto(gi− 1 )rather than togiviolatingthegrouping
requirement of section 5 .1. The− 1 would have then to be omitted for consistency,
so we omititfrom the outset!)

××××× ×× × × ×××× × ×

groupi

gistates

niparticles nicrosses

(gi– 1) lines

Fig. 5. 2 Countingbosons. In eachgroup thegistates are represented by(gi− 1 )lines andtheniparticles
bynicrosses. In the illustration a micro-state is represented for the casegi=8,ni= 9.