The averagingpostulate 3wouldneedto specifythe (vector) position andmomentum ofeachoftheNgas par-
ticles,a total of 6Nco-ordinates. (Actually even this is assuming that each particle
is a structureless point, withnointernaldegrees offreedomlike rotation, vibration,
etc.) Ofcourse, thismicrostate contains a totally indigestible amount ofinforma-
tion, far too much for one to store even one microstate in the largest available
computer. But, worse still,the system changesits microstate very rapidlyindeed–
forinstance one moleofatypicalgas willchangeits microstate roughly 1032 times
a second.
Clearlysome sort ofaveragingover microstatesis needed.Andhereis one ofthose
happyoccasions where quantum mechanics turns out to be a lot easier than classical
mechanics.
The conceptualproblemfor classicalmicrostates, as outlinedabovefor agas,isthat
theyare infinite in number. The triumph of Boltzmann in the late 19th century–had
he lived to see the full justification of it – and of Gibbs around the turn of the century,
was to seehow todothe averagingnevertheless. Theyobservedthatasystem spends
equal times in equal volumes of ‘phase-space’ (a combinedposition and momentum
space; we shall develop these ideas much later in the book, in section 14.4). Hence the
volumeinphase-space canbe usedas a statisticalweightfor microstates withinthat
volume. Splitting the whole of phase-space into small volume elements, therefore,
leadstoafeasible procedurefor averaging over allmicrostates as required. However,
we can nowadaysadopt a muchsimpler approach.
In quantum mechanics a microstate by definition isaquantum state of the whole
assembly.It canbedescribedbyasingleN-particle wavefunction, containing all
theinformation possibleabout the state ofthesystem. Thepoint to appreciateisthat
quantum states are discrete in principle. Hence although the macrostate(N,U,V)
has an enormous number ofpossiblemicrostates consistent withit, the numberis
none thelessdefinite andfinite.Weshallcallthis number,andit turns out to play
acentral role in the statistical treatment.
1 .3 The averaging postulate
We now come to the assumption which is the whole basis of statistical physics:
Allaccessible microstates are equallyprobable.This averaging postulate is to be treated as an assumption, but it is of interest to
observe thatitis nevertheless a reasonable one. Two types ofsupporting argument
canbeproduced.
The first argument is to talk about time-averages. Making any physical measure-
ment (say, ofthe pressure ofa gas on thewallofits container) takes a non-zero time;
andinthetime ofthe measurement thesystem will have passedthroughaverylarge
number of microstates. In fact this is why we get a reproducible value ofP;observ-
ablefluctuations are smallover the appropriate time scale. Henceitis reasonablethat
we should be averagingeffectivelyover allaccessiblemicrostates. The qualification