Statistical Physics, Second Revised and Enlarged Edition

124 Phase transitions

T < TTTc

T = 0

• 1 F^ + 1
  m

T = TTTc

T > TTTTc

Fig. 11.4Variation ofFwithmat various temperatures. The vertical arrows indicate the values ofmfor
whichFis a minimum at each temperature (compare Fig. 11.3).

M=(P 1 −P 2 )Nμ=Nμm,as required.] Hence

S=−NkkkB(P 1 lnP 1 +P 2 lnP 2 )

=(NkkkB/ 2 )[2ln2−( 1 +m)ln( 1 +m)−( 1 −m)ln( 1 −m)] (11.8)

The free energy due to the spinsF=U−TSis now obtained by combining (11.8)
and(11.7), together withthedefinition ofTTTC,(11.5), togive

F=(−NkkkB/ 2 ){TTTCm^2 +T[2ln 2 −( 1 +m)ln( 1 +m)

−( 1 −m)ln( 1 −m)]} (11.9)

ThestationaryvaluesofFarefoundbysettingdF/dm=0. Itisamatterofelementary
mathematics (try it!) to show that this leads to precisely (11.4), as it should. But we
havelearnedmuchmorefrom this exercise,inthat we nowknow thefull form of
F(m),as plottedinFig. 11.4. We confirm that atT>TTTCthereisjust the one minimum
atm= 0. WhenT<TTTC,there are the two minima rapidly moving towardsm=+ 1
and− 1 ;andthe stationary point atm=0is a maximum, an unstable position. Itis
alsointerestingto note the veryshallow minimum atT=TTTC,whichindicates a very
insecure equilibrium value ofm;this gives an explanation of the large fluctuations
often observedinthe properties ofasubstance close to a transition point.

1 1.2.3 The thermalproperties

Theordering ofthespinsis notjust evidentinthe magnetization,butitwillalso give
contributions to thethermalproperties,for exampletothe entropyandto theheat
capacity.
Since thevariation ofmwithTisknown (Fig. 11.3), the variation ofSwithTmay
be workedoutfrom (11.8) andthence theheat capacityCfromC=TdS/dT.The