136 Twonewideas
8 .Newhorizons. Finally, we can point out that thedegree ofabstraction neednot
end here.In the canonical ensemble the restriction thatUis fixed is removed,
in comparison to themicrocanonicalensemble. But the assemblies are stillcon-
strainedtohave agiven numberNofparticles. For somepurposes thisisquite
unphysical. For example consider the properties of a litre of air in the middle
ofthe room. The properties ofthis ‘open assembly’ are thermodynamicallywell
definedandcalculable. We candescribeits propertiesin statisticalphysics with
agrand canonical ensemblewhich consists of an array of these open assemblies.
Now thereis not onlyenergyexchangebetween assemblies,but thereisalso par-
ticle exchange. What happens now is that the whole ensemble defines a chemical
potentialμ, determined by the total number of particles in the ensemble, just as
Tinthe canonicalensemble wasdeterminedbythe totalenergy (NNAU)ofthe
ensemble. In thegrand canonical ensemble, an assemblymayhave anynumber
of particles, but, as one might anticipate, there is again a very sharp probability
functionP(E,N)whichensures thatEandNareinpractice well defined.Aswith
the canonical ensemble, thegrand canonical ensemble solves all our old problems
equally well. But in addition it opens new techniques, essential, for example, for
discussingchemicalreactions andequilibrium andfor a studyofsolutions. We
develop these ideas further in the following chapter.