Distributions 5Actuallyanythermodynamicsystem musthave someinteractionbetweenits parti-
cles, otherwise it would never reach equilibrium. The requirement rather is for the
interaction tobe smallenoughfor (1.1) tobevalid,hence ‘weaklyinteracting’ rather
than ‘non-interacting’ particles.
Of course this restriction of approach is extremely limiting, although less so than
one mightfirst suspect. Clearly, since the restrictionisalso one ofthe assumptions
ofsimplekinetictheory, our treatment will be usefulfor perfectgases. However,
it means that for a real fluid having strong interactions between molecules, i.e. an
imperfectgas or aliquid,the methodcannotbe applied.Weshallreturnbrieflyto
this point in Chapter 14, but a full treatment of interactingparticles is well outside
the scope of this book. At first sight it might seem that a description of solids is
also outsidethisframework,sinceinteractionsbetween atoms are obviouslystrong
in a solid. However, we shall see that manyof the thermal properties of solids are
nevertheless to be understood from a model based on an assembly ofNweakly inter-
actingparticles, when one recognizes that these particles neednotbethe atoms,but
other appropriate entities. For example theparticles can bephonons for a discussion
of lattice vibrations (Chapter 9); localized spins for a discussion of magnetic prop-
erties (Chapters 2 and11); or conduction electronsfor adescription ofmetalsand
semiconductors (Chapter 8).
Adistributionthen relates to the energies of a single particle. For each microstate
ofthe assemblyofNidenticalweakly interactingparticles, eachparticleisinan
identifiable one-particle state. In the distribution specification, intermediate in detail
between the macrostate andamicrostate, we choose not toinvestigatewhichpar-
ticles areinwhichstates,but onlyto specifythe totalnumberofparticlesinthe
states.
Weshalluse twoalternativedefinitions ofadistribution.
Definition1 – Distribution in states Thisis a set ofnumbers(n 1 ,n 2 ,...,nnj,...)
where thetypicaldistribution numbernnjisdefinedas the number ofparticlesin state
j,which has energyεεj.Often, but not always, this distribution will be an infinite set;
the labeljmust run over all the possible states for one particle. A useful shorthand
for thewhole set ofdistribution numbers(n 1 ,n 2 ,...,nnj,...)issimply{nnj}.
The above definition is the one we shall adopt until we specificallydiscussgases
(Chapter 4 onwards), at which stage an alternative, and somewhat less detailed,
definitionbecomes useful.Definition2 – Distribution in levels This is a set of numbers(n 1 ,n 2 ,...,ni,...)for
whichthetypicalnumberniis nowdefinedas the number ofparticlesinleveli,which
has energyεiand degeneracygi,the degeneracy being defined as the number of states
belonging to thatlevel.Theshorthand{ni}will beadoptedfor thisdistribution.
Itis worthpointingout that thedefinition tobeadoptedis a matter ofone’s choice.
The first definition is the more detailed, and is perfectly capable of handling the case
ofdegeneratelevels–degeneracy simply means that not alltheεεjs aredifferent.
Weshallreserve thelabeljfor the statesdescription andthelabelifor thelevels