Statistical Physics, Second Revised and Enlarged Edition

18 Distinguishable particles

or

U/N=

∑

j

εεjexp(βεεj)

/∑

j

exp(βεεj) (2.15)

The appropriate value ofβis then that one which, when put into this equation,
gives preciselytheinternalenergy per particle(U/N)specifiedbythe macrostate.
Unfortunatelythisis not a verytidyresult,butitisasfar as we cango explicitly,
since one cannot in general invert (2.1 5 ) to give an explicit formula forβas afunction
of(U,V,N). Neverthelessfor agiven(U,V,N)macrostate,βisfullyspecifiedby
(2.1 5 ), and one can indeed describe it as a ‘potential for energy’, in that the equation
gives a very direct correlation between(U/N)andβ.
However, thatis not theendofthe story. It turns out that this untidy(but absolutely
specific) functionβhas a veryclear physical meaningin terms of thermodynamic
functions other than(U,V,N). In fact we shall see that it must be related to
temperatureonly,asufficiently important point that thefollowingsection will be
devoted to it.

2 .3 A statisticaldefinition oftemperature

2. 3 .1 βand temperature

To show that thereis a necessaryrelationshipbetweenβandtemperatureT,we
consider the thermodynamic and statistical treatments of two systems in thermal
equilibrium.
Thethermodynamic treatmentisobvious. Two systemsinthermalequilibrium
have, effectively by definition, the same temperature. This statement is based on the
‘zerothlaw ofthermodynamics’, whichstates that thereis some commonfunction of
state shared byall systems in mutual thermal equilibrium – and this function of state
is what is meant by (empiric) temperature.
The statisticaltreatment canfollowdirectlythelines ofsection 2.1. The problem
can be set up as follows. Consider two systems P and Q which are in thermal contact
with each other, but together are isolated from the rest of the universe. The statistical
methodapplies to this composite assembly in muchthe same wayasinthesimple
example of section 1. 6 .2. We suppose that system P consists of a fixed numberNNNPof
localizedparticles, whicheachhave states ofenergyεεj,asbefore. The system Q need
notbeofthe same type, so we takeit as containingNNNQparticles whose energystates
areεk′. The corresponding distributions are{nnj}for system P and{n′k}for system Q.
Clearlythe restrictions on thedistributions are
∑

j

nnj=NNNP (2.16)

∑

k

n′k=NNNQ (2.17)