Statistical Physics, Second Revised and Enlarged Edition

20 Distinguishable particles

2 .3.2 Temperature and entropy

We now come to the form of this relationβandT. The simplest approach is to know
the answer(!), andwe shallchoose todefineastatisticaltemperaturein terms ofβ
from theequation

β=− 1 /kkkBT (2.20)

What will eventually become clear (notably when we discuss ideal gases in Chapter 6)
isthat thisdefinition ofTdoesindeedagree withtheabsolute thermodynamic scale
of temperature. Meanwhile we shall adopt (2.20) knowing that its justification will
follow.
Thereis muchsimilaritywithour early definition ofentropyasS =kkkBln,
introduced in section 1.7. And in fact the connection between these two results is
something wellworthexploring at this stage, particularlysinceit can give us a
microscopicpicture ofheat andworkin reversibleprocesses.
Consider how the internal energyUof a system can be changed. From a macro-
scopicviewpoint, this canbedonebyaddingheat and/or work,i.e. changeinU=
heatinput + workinput. Thelaws ofthermodynamicsfor adifferentialchangeina
simpleP−Vsystem tell us that

dU=TdS−PdV (2.21)

wherefor reversible processes (only)thefirst (TdS) term canbeidentifiedas theheat
input, and the second term(−PdV)as the work input.
Nowlet us consider themicroscopicpicture. Theinternalenergyissimplythe
sum ofenergies ofallthe particles ofthesystem,i.e.U=

∑

nnjεεj.Takingagaina
differential change inU,we obtain

dU=

∑

j

εεjdnnj+

∑

j

nnjdεεj (2.22)

where thefirst term allowsfor changesinthe occupation numbersnnj, and the second
termfor changesinthe energy levelsεεj.
It is not hard to convince oneself that the respective first and second terms of (2.21)
and(2.22) matchup. The energylevels aredependent onlyonV,so that−PdV
workinput can onlyaddress the secondterm of(2.22). And,bearinginmindthe
correlation betweenSand(and hencet∗,and hence{nnj∗}),it is equally clear that
occupation number changes aredirectlyrelatedto entropy changes. Hence the match-
ingup ofthefirst terms. Theseideas turn out tobebothinterestinganduseful.The
relation−PdV=

∑

nnjdεεjgives a direct and physical way of calculating the pressure
fromamicroscopicmodel.Andtheother relationbearsdirectlyonthe topicofthis
section.