Statistical Physics, Second Revised and Enlarged Edition

Astatistical definition oftemperature 19

and

∑

j

nnjεεj+

∑

k

n′kε′k=U (2.18)

whereUis the total energy(i.e.UUUP+UUUQ)of the two systems together.
The counting of microstates is easy when we recall (section 1.6.2) that we may
write

=P×Q or t=tttP×tQ (2.19)

withtheexpressionsfortttPandtQbeinganalogous to (2.3).
Finding the most probable distribution may again be achieved by the Lagrange
method as in section 2.1.5. The problem is to maximize lntwithtasgivenin (2.19),
subject now to the three conditions (2.1 6 ), (2.17) and (2.18). UsingmultipliersαP,
αQandβrespectively for the three conditions, the result on differentiation (compare
(2.12))is

∑

j

(−lnnn∗j+αP+βεεj)dnnj+

∑

k

(−lnnn′j∗+αQ+βεk′)dnk= 0

The Lagrange method then enables one to see that, for the appropriate values of the
three multipliers, eachterminthe two sumsis separately equalto zero, so that the
finalresultisfor system P

nnj∗=exp(αP+βεεj)

and for system Q

n′∗k =exp(αQ+βε′k)

Wheredoes thisleave us? Whatitshowsisthatbothsystem P andsystem Qhave
theirthermalequilibriumdistributions oftheBoltzmann type (compare (2.12)). The
distributions have their ownprivate values ofα,and we can see from the derivation
that thisfollowedfrom theintroduction ofthe separate conditionsfor particle conser-
vation ((2.1 6 ) and (2.17)). However, the two distributions have the same value ofβ.
This arose in the derivation from the single energy condition (2.18), in other words
from thethermalcontact or energyinterchangebetween the systems. So theimpor-
tant conclusionisthat two systemsin mutualthermalequilibrium anddistributions
with the sameβ.From thermodynamics we know that they necessarily have the same
empiric temperature, andthus the same thermodynamic temperatureT.Thereforeit
follows thatβisafunction ofT only.