2.6 First results from the calculation of the state sum

The canonical partition function for a Hamiltonian which completely separates into subspaces can be written in the form

ZC(T,V,N) = igi exp(-βUi) = [nα]N exp-β αnαϵα . (2.40)

The equation on the right side must fulfill the restriction

N = αnα. (2.41)

The grand canonical partition function of a Hamiltonian which completely separates into subspaces can be written in the form

ZGC(T,V,μ) = N=0 [nα]N exp-β αnα(ϵα - μ) . (2.42)

The big advantage of this partition function is the identity

N=0 [nα]N... = n1 n2 n3... (2.43)

i.e. states can be occupied independently and within the partition function an independent sum over independent energy subspaces is performed, leading to a product over subspace sums.
Finally we get

ZGC(T,V,μ) = α nα exp(-βnα(ϵα - μ)) = α 1 - exp(-β(ϵα - μ) -1for Bosons = α 1 + exp(-β(ϵα - μ) +1for Fermions (2.44)

Just a reminder:

For the grand canonical potential we find

Ω(T,V,μ) = kT α ln1 exp-ϵα - μ kT (2.45)

and

N = -Ω μ = kT α 1 expϵα-μ kT 1 1 kT = α 1 expϵα-μ kT 1. (2.46)

If exp-ϵα-μ kT << 1 holds we get

ln1 exp-ϵα - μ kT exp-ϵα - μ kT . (2.47)

In this case we find for Fermions as well as for Bosons

N = -Ω μ = kT α exp-ϵα - μ kT 1 kT = α exp-ϵα - μ kT (2.48)

This is the so called Boltzmann statistics. As we will see later the above assumption is fulfilled for extremely deluded systems and at high temperatures (classical particles).


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© J. Carstensen (Stat. Meth.)