Statistical Physics, Second Revised and Enlarged Edition

184 Appendix B

z

0 12 3 4 5 6 7^356 X

z

X–– 22

XX– 1

Fig. B. 1 Stirling’s approximation.

physics; evenfor a verymodestX= 1 000, the errorinusingequation (B.1)isless

than 0.1%, and the fractional error goes roughly as 1/X.

2 APROBLEM WITH PENNIES

An unbiasedpennyis tossedNtimes. (i) How many possible arrangements ofheads
(H)andtails(T) are there? (ii)Whatisthe mostprobable number ofHandTafter
theNtosses? (iii) What is its probability? In particular what is this probability when
Nislarge?
This problembears muchsimilarityto the statisticalphysics athightemperatures of
a spin-^12 solid (section 3.1), and we shall apply the usual nomenclature in discussing
it. The answer to the first problem is simply 2 N(=, say, by analogy), since each toss
has two possibilities. The answer to problem (ii)isalso readilysolvedsince thecoin
is unbiased.This means that each of the 2 Narrangements (or microstates) has equal
probability; hence the most probable distribution ofHandTis thatwhich includes
the most microstates. Andthatisthedistribution withequalnumbers ofheadsand
tails(we assumeNto be even for convenience!). This is readily proved from problem
2ofAppendixA.Adistribution ofnHandm(=N−n)Tcan occurint=N!/(n!m!)
ways, andthisis maximizedwhenn=m=N/2.
Problem (iii) is then also solved. The required probabilityPis given byt∗/,i.e.
P=N!/{(N/ 2 )!^22 N}. However, theinterest andtherelevance to statisticalphysics
comewhenweevaluatePwhenNislarge. Hereisthecalculation:

lnP=lnN!− 2 ln(N/ 2 )!−Nln2 from above

=NlnN−N− 2 (N/ 2 )ln(N/ 2 ) from (B.1)

+ 2 (N/ 2 )−Nln2

= 0 since everything cancels!