Statistical Physics, Second Revised and Enlarged Edition

Amodel example 9

Step IV. The averagevalue ofeverydistribution number can nowbeobtainedbyan

equal averaging over every microstate, readily computed as a weighted mean over

thefive possibledistributions using thetvalues as theweight. Forinstance

(n 0 )av=(n( 01 )t(^1 )+n( 02 )t(^2 )+···)/

=( 3 × 4 + 2 × 12 + 2 × 6 + 1 × 12 + 0 × 1 )/ 35

= 60 / 35 =1.7 1

Similarlyforn 1 ,n 2 etc. givefinally

{nnj}av=( 1 .71, 1.14, 0. 6 9, 0.34, 0.11, 0, 0...)

The resultis not unlikeafallingexponentialcurve, thegeneralresultfor alarge

assembly which we shall derive in the next chapter.

1 .6.2 A composite assembly

We now briefly reconsider the four-particle model assembly as a composite two-part
assembly. We takeasaninitialmacrostate a more restrictive situation thanbefore. Let
us suppose that the two particlesAandBform one sub-assembly,whichisthermally
isolated from a second sub-assembly consisting of particles C and D; and that the
values ofinternalenergies areUUAB= 4 ε,UUUCD= 0 .Thisinitialmacrostateis more
fullyspecified than the situation in section 1. 6 .1, since the division of energybetween
AB and CD is fixed.
The total number of microstates is now only five rather than 35. This arises as
follows. For sub-assemblyCD we musthavej=0forbothparticles, theonlyway
to achieveUUUCD=0. HenceCD = 1 and{nnj}=(2, 0, 0, 0, 0,...).(This result
correspondstoalow-temperaturedistribution, as we shallseelater,inthat allthe
particlesareintheloweststate). Forsub-assemblyABtherearefivemicrostatestogive
UUAB= 4 ε. In a notation [[[j(A),j(B)] they are[ 4 , 0 ],[ 0 ,4],[ 3 , 1 ],[1, 3],[ 2 , 2 ].Hence
AB= 5 and{nnj}=( 0. 4 , 0. 4 , 0. 4 , 0. 4 , 0. 4 , 0 ,0,...).(Thisis now ahigh-temperature
distribution, with the states becomingmore evenlypopulated.)
In order to find the total number of microstates of any composite two-part assembly,
thevalues offor the two sub-assemblies mustbemultipliedtogether. Thisisbecause
for every microstate of the one sub-assembly, the other sub-assembly can be in any
one ofits microstates. Thereforeinthis casewehave=AB·CD= 5 , as stated
earlier.
Now let usplace the two sub-assemblies in thermal contact with each other, while
stillremainingisolatedfrom the rest oftheuniverse. This removes the restriction of
the 4:0 energy divisionbetweenAB andCD, andthe macrostate reverts to exactlythe
same as in section 1.6.1. When equilibrium is reached, this then implies a distribution
intermediateinshape (andin temperature). But ofparticularimportanceisthenew
value of(namely35). One wayof understandingthegreat increase (and indeed