Statistical Physics, Second Revised and Enlarged Edition

Summary 61

5. 4 .4 αandβrevisited

After the lengthy discussion concerningαandβin Chapter 2, brevity is now in order.
The parameterαagain is related to the restriction ( 5 .7) which spawned it. In the
MB case thisinvolves simplya normalizingconstant. In thefullFD andBE cases,
the general idea is the same (αis adjusted until the distribution contains the correct
numberNofparticles)but the mathematicsis not so simple!
For theparameterβ,thearguments ofsection 2.3.1 remainvalid.They didnot
depend on the two assemblies in thermal contact having localized particles. Therefore
βmustbe a common parameterbetweenanytwo assembliesinthermalequilibrium,
andfollowingsection 2.3.2 we continue to use theidentity:β=− 1 /kkkBT.
Thefinaloutcomeofthischapteristowritetheresultsofourhardworkallinasingle
composite equation. Thedistributionfunction,definedbythe average number of
particles per state of energyεi,isgiven for agaseous assemblyin thermal equilibrium
at temperatureTby

fffi=

1

+ 1 (FD)

Bexp(εi/kkkBT) 0 (MB)

− 1 (BE)

(5.13)

In(5.13)the choice of+ 1 ,0or− 1 is governedbywhichofthethree types ofstatistics

isrelevant. TheparameterB(equivalentlyαsinceB=exp(−α))isadjustedtogive

the correct number of gas particles. How the three distributions are used in practice

willemergefrom thefollowingfour chapters.

5.5 Summary

This chapter builds on previous ideas to derive the distribution functions for gases

inthermalequilibrium. We use the statisticalapproachfirst outlinedinChapter 1

together withthe countingofstates ofChapter 4

1. Groupingtogethergiofthe numerous one-particle statesis worthwhile (anddoes

no violence to the physics) since the simplifyingapproximations of large numbers

can then be used.

2. Thequantum mechanics ofidenticalparticles spawns two classes,bosons and
   fermions.
   3 .Half-integralspin particles arefermions; zero orintegralspin particles arebosons.


3. Occupation ofstatesforfermionsis restrictedbythe Exclusion Principle–notwo
   fermions can occupy the same state. There is no such restriction for the (friendly!)
   bosons.


4. Counting microstates in eachgroup reduces to a simple binomial problem
   (Appendix A).


5. When occupation of states is small (a ‘dilute gas’), fermion gases and boson gases
   bothtendto the samelimit,the Maxwell–Boltzmann or classicallimit.