Ensembles–alarger view 135(ii)Wemayalso calculateZZAusing(12.5a) together with the multinomial theorem
(Appendix A). Because the coefficientst({nnj})are identical (for an assembly
of distinguishable particles) to themultinomialcoefficients,itis not toohard
to prove that the sum in (12.5a) can be performed explicitly; and that the result
again isZN.
Method(ii)is worthexamininginaddition to method(i), since when we
come togases method (i) is not available. Thegas particles are competingfor
the same states,soZZAcannot factorize into a factor from each particle. How-
ever, method(ii)gives us the answerfor adilute (MB)gas. (Thecalculation of
ZZAfor FD and BEgases requiresyet further work, which we shall not pursue
here.)The values oft({nnj})for the MB gas simply differ by that factorN!from
thosefor thedistinguishableBoltzmann particles. Therefore the summation
goes through unscathed, except for this constant factor throughout, togive
ZZA=ZN/N!
for the assembly partition function. This expression, when put into equation
(12.7), givesF=−NkkkBTlnZ+kkkBTlnN!, identical to the result of Chapter 6
(6.1 6 ).
6 .Awayinto new problems. The importance of the new method is that a wider
range ofassemblies can nowbe studied, at any rateinprinciple. Thethree
equations (12.5), (12.6) and (12.7) outline a programme which should work for
an ensemble of any assemblies. The equations make no restrictions on ‘what is
inside eachbox’, theonly requirementbeing that wehaveinmindalarge number
of identicalboxes. So the assemblycouldcontaininteractingparticles, andstill
the method would be the same. Naturally it becomes mathematically difficult, but
atleast thephysicsisclear. The computationalproblemisimmense, as soon as the
energyEdependsoninteractionsbetween the10^23 particles. Nobodycanyet fully
work out the properties of a liquid from first principles. However, some progress
canbemade. For example, the small deviationsfrom theidealgaslawsfor real
gases (which are onlyslightlynon-ideal) can be related to the interaction poten-
tial between two gas particles, using expansion methods effectively in powers of
theinteraction. We shalluse theseideasfurtherinChapter 14. Another tractable
problem is that of the Isingmodel, mentioned as a (more physical?) alternative to
the mean field approximation of Chapter 11. Here we have a solid in which only
nearest neighboursinteract, so that the expressionforEis not toodesperate. In
fact it is exactlysoluble in two dimensions, and satisfactorynumerical results can
be obtained in three dimensions.
7 .Nomenclature.Youknow the methods, so whynotlearn their names? (Atleast
it helps to impress your friends!) The ensemble of this chapter, with a probability
functionforEgiven by (12.6), is called thecanonicalensemble.The earlier method
ofthebookcan alsobethoughtofasinvolvingan ensemble (see section 1.3);but
now each assembly of the ensemble had the same energyU. In the language of
(12.6),P(E)for the ensembleisadelta-function at energyU.Thisiscalledthe
micro-canonicalensemble.