Statistical Physics, Second Revised and Enlarged Edition

26 Two examples

Thethermalequilibriumdistribution numbers (or occupation numbers)n 0 andn 1
may now be evaluated from (2.23) to give

n 0 =N/[ 1 +exp(−ε/kkkBT)]=N/Z(th)

and

n 1 =Nexp(−ε/kkkBT)/[ 1 +exp(−ε/kkkBT)]

=Nexp(−ε/kkkBT)/Z(th) (3.2)

These numbers are sketchedinFig. 3.1 asfunctions ofT. We can note thefollowing
points

1.The numbersn 0 ,1do not depend at all on theground state energyε 0 .The first
factorZ( 0 )of(3.1)does not enter into(3.2). Instead all the relevant information is
containedinZ(th). Thisshouldcause no surprise since all is containedinthesimple
statement (2.2^6 ) of the Boltzmann distribution, namelyn 1 /n 0 =exp(−ε/kkkBT),
together withn 0 +n 1 =N.Hence also:

2. The numbersn 0 ,1arefunctions onlyofε/kkkBT.One can thinkofthisasthe ratioof
   two ‘energyscales’. One scaleεrepresents the separation between the two energy
   levels of the particles, and is determined by the applied conditions, e.g. of volume
   V(orfor our spin-^12 solidofappliedmagneticfield– seelater). The secondenergy
   scaleiskkkBT,which should be thought of asathermal energyscale.Equivalently,
   the variable maybewritten asθ/T,the ratiooftwo temperatures, whereθ=ε/kkkB
   is a temperature characteristicofthe energy-levelspacing.Thisidea oftemperature
   or energy scales turns out to be valuable in many other situations also.
   3 .Atlow temperatures (meaningTθ, or equivalentlykkkBTε), equation (3.2)
   givesn 0 =N,n 1 = 0. To employusefulpicturelanguage, allthe particles are

0 3 

Upper state, n 1

Lower state, n 0

NNN/2

N

n 0 , n 1

Fig.3. 1 Occupation numbers for the two states of a spin-^12 solid in thermal equilibrium at temperature
T. The characteristic temperatureθdepends only on the energy difference between the two states. The
particles are all in the lower state whenTθ, but the occupation of the two states becomes equal when
Tθ.