132 Twonewideas
NNN 2 /N 1 =exp(−/kkkBT),itfollowsfrom (12.4) that thedistributionfunctionfor the
electronsmust be oftheform
[ 1 −f(ε)]///f(ε)=Bexp(ε/kkkBT)withBaconstant (independent ofε). This rearranges togive
f(ε)= 1 /[Bexp(ε/kkkBT)+ 1 ]whichis preciselytheFDdistribution (8.2) asindeeditshould be. The argument
ofthis section canbe seen either as an alternativederivation oftheFDdistribution,
or as a confirmation of the principle of detailed balance. We may further note that
similar arguments canbemade(withequally satisfactory results)for transitions ofthe
atoms which involve,instead of electrons,either massive bosons or other Boltzmann
particles.
1 2.2 Ensembles – a larger view
Statisticalphysicshas many possible entry points. In thisbook,wehave concentrated
on one oftheleast abstract routes, that whichconcentrates on apiece ofmatter
modelled by the(N,U,V)macro-state. The assembly under consideration consists
ofNweaklyinteracting particles, andthe totalenergyUisfixed. Bothofthese
limitations are essentialfor the methodofChapter 1 onwardstomake sense.
But here is the new idea. Let us raise the scale of our thoughts, and apply the
identicalmethodstoalarger universe. Previously, wehadan assemblyofNidentical
particles withafixedtotalenergyU.Nowweshallconsider as ‘ensemble’ ofNNA
identical assemblies with a fixed total energyNNAU.
Why?Who wants to consider 10^23 orsoblocksofcopper, when we are really
interested in onlyone of them? The answer, of course, concerns averaging. We are
interested in the average properties of one assembly, and this ensemble is one way of
lettingnaturedothe averagingfor us! We can thinkofthe ensemble as consistingof
one assembly(whose thermodynamic properties we wish to know) in a heat reservoir
provided by all the others, so that this is a way of modelling an assembly at fixed
temperature,rather than theoldmethodoffixedenergy.
Furthermore, our statistical method applies immediatelyand preciselyto this
ensemble. The assemblies are separate macroscopic entities, so they are distinguish-
able ‘particles’ without adoubt. Theycanbeplacedinasweakcontact as necessary,
so the ‘weakly interacting’ limitation is unimportant. And we can avoid any doubts
about numericalapproximationsbymakingNNAeffectivelyinfinite–after all,the
ensembleisafigment oftheimagination, not ablueprintfor a machinist. Hence, the
Boltzmann distribution (Chapter 2) applies.
For the assemblyofparticles, wederivedtheBoltzmanndistribution
nnj=(N/Z)exp(−εεj/kkkBT) (2.23)