Statistical Physics, Second Revised and Enlarged Edition

Appendix BSome problems with large numbers

1 STIRLING’SAPPROXIMATION

Stirling’s approximationgives a veryuseful method for dealingwith factorials of large
numbers. The form in which it should be known and used in statistical physics is:

lnX!=XlnX−X (B.1)

for anylargeintegerX.Andanother usefulresultmaybeobtained by differentiating
(B.1) to give:

d(lnX!)/dy=(dX/dy){lnX+ 1 − 1 }

=(dX/dy)lnX (B.2)

where one assumes thatXmay be considered a continuous variable.
Equation (B.1)is notdifficultfor theinterestedpartyto prove. Consider thedefinite
integralI =

∫X

1

∫∫

lnzdz. Integration by parts expresses this as[zlnz−z]X 1 ,which
when evaluatedbetween thelimits gives:

I=XlnX−X+ 1

However, a quick inspection of agraph of this integral (Fig. B.1) shows that the
integral (i.e. the area under the curve) lies almost midway between the areas of the
two staircases on thediagram. Andthe upper staircasehas an area of(ln1+ln 2 +
···+lnX)=lnX!.
Thereforewe see thatI=lnX!– an error term of approximately^12 lnX.
Thelower one’s areais

[ln 1 +ln2+···+ln(X− 1 )]=ln(X− 1 )!

=lnx!−lnX

Hence:lnX!=XlnX−X[+^12 lnX+term oforder unity].ThisisStirling’s approx-
imation. It really is remarkablyaccuratefor the sort ofnumbers we usein statistical

183