Statistical Physics, Second Revised and Enlarged Edition

Counting microstatesfor gases 55

5 .3 Counting microstates for gases

Now we are equipped to attack step III of the statistical method, that of counting

upthe number ofmicrostates consistent withagiven valid distribution. Since we

postulate that allsuchmicrostates are equallyprobable, this stepisthe essential

prerequisite to finding the most probable distribution, i.e. that which best describes

thermalequilibrium,but thatfinalstep (step IV)followsinthe next section.

The countingproblem maybe set up asfollows. Theindistinguishable nature of

the particles is accounted for by the inclusion of all permutations of the particle co-

ordinates in the generalizations of (5.2) and (5.3). The microstate (effectively the

appropriate wavefunction eitherψSorψψψA)can therefore be labelledjust bythe one-

particle state labelsa,b, etc. As pointed out earlier, this is simply a recognition that

we cannotknow whichparticles areinwhichstates; eveninprinciple we can only

know which states are occupied. Therefore the countingof microstates is to specify

the occupation of each one-particle state.

Thedistribution whose number ofmicrostatesistobe countedisthegrouped

distribution defined in section 5 .1. The required numbert({ni})can be written as

the product of contributions from each group of states, but the form of these factors

will differforbosons andfermions. The treatments offermions (withthe exclusion

principle) and of bosons (without) are quite different in principle, which goes two-

thirdsofthe way to explaining whythis sectionhas three sub-sections. The reason

for thethirdwillemergebelow.

5. 3 .1 Fermions

As we have seen in section 5.2, the exclusionprinciple operates for fermions. There-
fore, the one-particle states can onlyhave occupation numbers of 0 or 1. Incidentally,
this implies that in theith group, the number of particlesnicannot exceedthe number
of statesgi.
The counting is now straightforward. The group ofgistates is divisible into two
subgroups:niofthe states are to contain one particle, andtherefore theother(gi−ni)
mustbe unoccupied.Thisisthe‘binomialtheorem’ countingproblem ofAppendix
A, problem 2. The number of different ways the states can be so divided is

gi!

ni!(gi−ni)!

This is the contribution to the number of microstates from theith group. The total

number ofmicrostates correspondingto an allowabledistribution{ni}istherefore

given by

tttFD({ni})=

∏

i

gi!

ni!(gi−ni)!

(5.4)