2 Distinguishable particles
The next stepistoapplythe statisticalmethodoutlinedinChapter 1 to realistic
thermodynamic systems. This means addressing the properties of an assembly which
consists ofalarge numberNofweaklyinteractingidenticalparticles. There are two
types ofassemblywhichfulfilthe requirements.
One type is a gaseous assembly, in which the identical particles are the gas
molecules themselves. In quantum mechanics one recognizes that themolecules are
not only identical,but theyare also (inprincipleaswellasin practice)indistinguish-
able. It is not possible to ‘put a blob of red paint’ on one particular molecule and
tofollowitshistory. Hence themicrostatedescription must takefullaccount ofthe
indistinguishabilityofthe particles. Gaseous assemblies will beintroducedlaterin
Chapter 4.
Inthischapter we shalltreat theother type ofassembly,inwhichthe particles
are distinguishable. The physical example is that of a solid rather than that of agas.
Consider a simple solid which is made up ofNidentical atoms. It remains true that the
atoms themselves areindistinguishable. However, agooddescription ofour assembly
is to thinkabout thesolid as a set ofNlattice sites,in which each lattice site contains
an atom. A ‘particle’ of the assembly then becomes ‘the atom at lattice site 43 57
(or whatever)’. (Whichofthe atomsisatthissiteis not specified.) The particleis
distinguished not bythe identityof the atom, but bythe distinct location of each
lattice site. A solid is an assembly of localized particles, and it is this locality which
makes the particlesdistinguishable.
We shall now developthe statistical description of an ideal solid, in which the
atoms are weakly interacting. How far the behaviour of real solids can be explained
inthiswaywill become clearerinlater chapters. Themain results ofthischapter will
be the derivation of the thermal equilibrium distribution (the Boltzmann distribution)
together withmethodsfor thecalculation ofthermodynamic quantities. Two physical
examples aregiveninChapter 3.
13