114 Entropyin other situations
The secondtermsintheFDandBE expressions areintriguing. Equation (10.4)
displays that in an FD gas, one not only considers the disorder of the particles filling
the states (thefirst term),but also ofthe empty states (the secondterm). Bothterms
vanishiffffi=0or1,asisthe casefor eachstate atT= 0. Intuition concerning(10.5)
is not so clearcut, except to note that some such term is a clear necessity to ensure
thatS>0whenwenowallowfffitobe> 1.
1 0.2 An assembly at fixed temperature
The statistical method adopted earlier in the book has been based on an assembly
ofparticlesin a macrostate offixednumberN,volumeVandinternalenergyU.
Of course for a large assembly, the specification of anymacrostate in practice deter-
mines all the other thermodynamic quantities, such as temperatureT, pressurePand
chemicalpotentialμ.So wehave cheerfullyappliedour methodto assemblies witha
macrostate specified by(N,V,T),with little fear of anythinggoingwrong. However,
when we look back at the statistical method and its relation to entropy in particular, we
shall bepointedtowards a moregeneralstatisticalapproach, one that canbe applied
to new types of assembly.
Our old method is based on the properties of anisolated system(mechanically
isolatedsinceVis constant, thermally isolatedsinceinadditionUis constant, and
of fixedparticle numberN). One thing we really know about an isolated system is
that anyinternalrearrangementsitmakes, any spontaneous changes, willalwaysbe
suchas toincrease the entropyofthesystem. Thatisthe content ofthe secondlaw
of thermodynamics. The entropy of the equilibrium state of the isolated system isa
maximum.
Now our statisticalmethodcanbeviewedin exactlythisway,asatheorist mak-
ing changes until he finds the state of maximum entropy! We have seen that the
equilibriumdistributionisthat withthelargest number ofmicrostates,i.e. withthe
maximumt({nnj}). Our statistical mechanician fiddles with the distribution numbers
until he finds the maximumt(=t∗). But since in the equilibrium stateS=kkkBlnt∗,
thisisjust the same as the maximum entropyidea ofthe secondlaw. The exampleof
section 1.6.2 is of direct relevance here. A system prepared with a non-equilibrium
distribution,(i)will increase itstby adjusting the distribution, and (ii) will increase its
entropy.
Having reminded ourselves of how the thermodynamic view (together withS=
kkkBln)isrelatedto the statisticalmethodfor anisolatedsystem, we now turn
to considerasystem atfixedtemperature. Consider an assembly describedbyan
(N,V,T)macrostate.
The conditionfor thermodynamic equilibrium ofthe(N,V,T)systemis equally
secure.The free energy F mustbe a minimum in equilibrium.This condition (e.g.
Thermal Physics,by Finn, section 6.4) is again a statement of the second law. The
entropy oftheuniverse(i.e. ofthe system together withalargeheatbathat temper-
atureT)must notdecrease whenheatflowsin or out ofthesystem to maintainits