Statistical Physics, Second Revised and Enlarged Edition

38 Two examples

T

 2 

1

2

U

2 NkkkB

NkkkB

UUU(0) = Nk^12 kkB

UU=NkkkBT

0

Fig. 3.9The variation ofinternalenergy withtemperaturefor an assemblyofharmonic oscillators. The
characteristic temperatureθdependsonthefrequency ofthe oscillators (see text). Note thehighandlow
temperaturelimits.

T

2 

Classical limit

Extreme quantum limit

0 

C

NkkkB

Fig. 3.1 0 The variation of heat capacity with temperature for an assembly of harmonic oscillators.

The entropySis also readily derived, the easiest route again being to evaluateF,
(2.28), andtodifferentiateFwithrespect toT.The resultis

S=NkkkB

[

ln

(

exp(θ/T)

exp(θ/T)− 1

)

+

(θ/T)

(exp(θ/T)− 1 )

]

(3.16)

The reader maywishto checkthat thelow temperaturelimitisS =0 (no sur-
prise), and that at high temperatures the expression becomesS=NkkkBln(T/θ)=
NkkkBln(kkkBT/hν).