Statistical Physics, Second Revised and Enlarged Edition

14 Distinguishable particles

2.1 TheThermalEquilibrium Distribution

We follow the method outlined in section 1. 5. For the macrostate we consider an

assemblyofNidenticaldistinguishable(localized) particles containedinafixed

volumeVandhavingafixedinternalenergyU.Thesystemis mechanicallyand

thermally isolated, but we shall be considering sufficiently large assemblies that the

other thermodynamic quantities (T,S, etc.) are well defined.

2 .1.1 The one-particle states

The one-particle states will be specified by a state labelj(=0, 1, 2...).The corre-

sponding energiesεεjmay or may notbeall different. These states will bedependent

upon thevolumeperparticle(V///N)for ourlocalizedassembly.

2 .1.2 Possible distributions

We use thedistributionin states{nnj}definedin section 1.4. Thedistribution numbers

must satisfythe two conditions (1.2) and(1.3)imposedbythe macrostate

∑

j

nnj=N (1.2)and(2.1)

∑

j

nnjεεj=U (1.3) and(2.2)

2 .1.3 Countingmicrostates

Itishere that thefact that wehavedistinguishable particles shows up. It means that

the particles can be countedjust as macroscopic objects. Amicrostate will specifythe

state(i.e. the labelj) for each distinct particle. We wish to count up how many such

micro-states there are to an allowabledistribution{nnj}.The problemis essentiallythe

same as that discussed inAppendixA, namelythe possible arrangements ofNobjects

into piles withnnjin a typical pile. The answer is

t({nnj})=N!

/∏

j

nnj! (1.4)and(2.3)

2 .1.4 The average distribution

According to our postulate, the thermal distribution should now be obtained by eval-
uating the averagedistribution{nnj}av.This taskinvolvesaweightedaverage of
all the possible distributions (as allowed by(2.1) and (2.2)) usingthe values oft