The thermal equilibrium distribution 15(equation (2.3)) as statisticalweights. This taskcanbe performed,butfortunatelywe
are saved from the necessity of having to do anything so complicated by the large
numbersinvolved. Some ofthesimplifications oflarge numbers are exploredbriefly
inAppendixB.
The vital point is that it turns out that one distribution, say{nn∗j}, is overwhelmingly
more probablethan any oftheothers. In other words, thefunctiont({nnj})is very
sharplypeakedindeedaround{nnj∗}. Hence, rather than averagingover allpossible
distributions, one can obtain essentially the same result by picking out the most prob-
abledistribution alone. Thisthen reduces to the mathematicalproblem ofmaximizing
t({nnj})from (2.3) subject to the conditions (2.1) and (2.2).
Another even stronger way of looking at the sharp peaking oftis to consider the
relationshipbetweenandt.Sinceisdefinedas the totalnumber ofmicrostates
contained bythe macrostate, it follows that
=
∑
t({nnj})where the sum goes over all distributions. Whatis now suggested(andcompare the
pennies problem ofAppendixB)isthat this sum canin practicebe replacedbyits
maximum term,i.e.
≈t({nn∗j})=t∗(forshort) (2.4)2 .1.5 The mostprobable distribution
Tofindthethermalequilibriumdistribution, we needtofindthe maximumt∗andto
identify the distribution{nn∗j}at this maximum. Actually it is a lot simpler to work
withlnt, rather thantitself.
Sincelnxis a monotonically increasingfunction ofx,thisdoes not changethe
problem; it just makes the solution a lot more straightforward. Taking logarithms of
(2.3) we obtain
lnt=lnN!−∑
jlnnnj! (2.5)Here the large numbers come to our aid. Assumingthat all thensare large enough
for Stirling’s approximation (Appendix B) to be used, we can eliminate the factorials
to obtain
lnt=(NlnN−N)−∑
j(nnjlnnnj−nnj) (2. 6 )Tofindthe maximumvalue oflntfrom (2. 6 ) we express changes in the distribution
numbers as differentials (large numbers again!) so that the maximum will be obtained