Partition functions

1. Definitions

The Physics partition function is related to the probability to find a system in certain state. It is defined as follows: Z = exp{-βE} , where β = 1/kT (k is the Boltzmann constant and T is the temperature of the system). The factor "E" is the energy associated to the system.
If the number of particles of the system is fixed equal to N; and this system has "j" levels, with n_j particles occupaying each level "j", we can write:
N = ∑n_j. Plus, if the system has a surrounding as a reservoir that imposes to it its energy E (or temperature T), we can write also E = ∑n_j&epsilon._j = ∑E_j, where ε_j is the energy of each particle in the level "j", and E_j is the energy of the level "j", that is the energy of n_j particles in the level "j" carrying each the energy ε_j.

The probability to have n_j individual paricles in the level "j" is P(n_j) = n_j/N = constant x exp{- βε_j}. We normalize this probability to 1 to find the constant:
∑P(n_j) = constant x ∑ exp{ - βE_j} = 1.
Then constant = 1/Z ; where Z = ∑ exp{ - βE_j}, called the partition function for the canonical description.

In addition to exchange energy, the system exchanges particles with the reservoir; the expression of the partition function is Z = ∑ exp{ - β(E_j - μN)}, where μ is the chemical potential to feed the system with particles, and exp{ βμ} is the corresponding fugacity.

Generally, e write:
Ξ = ∑ exp{ - β(E_j - μN)} when N is fixed, and
Ξ = ∑ exp{ - β ∑(ε_j - μ)n_j}
when N fluctuates, in the case of grand canonical description


If &x;_1j = exp{ -β (ε_j -μ)} is the partition function for one particle of energy ε_j
in the level "j", then for n_j particles in the elevel "j", we have ξ_nj = Π &x;_1j [the product is n_j times].
ξ_nj = Π &x;_1j = Π exp{ -β (ε_j -μ)} = ∑ exp{ - β ∑(ε_j - μ)n_j}.
But n_j goes from 0 to N. within each level "j", we have a block {n_j} of n_j particles to consider. Hence,
the partition function of the level "j" is then Ξ_j = ∑&x;_nj [the sum is over {n_j}].

Forethermore, for the whole system, that is for all levels, we have:
Ξ = Π ξ_j = Π ∑ exp{ - β ∑(ε_j - μ)n_j}, [the sum is over n_j and the product is over "j"].

2. Grand canonical: different cases

1.N is fixed and the particles are distiguishables:
First, if we choose n1 particles for the level "1" (nj goes from 1 to N), we have then C(n1,N) possiblities to take this choice. The partition function for n1 particles is ξ_n1 = C(n1,N) x exp{ - β (ε₁ - μ)n₁} C(nx,N) is the number of combinations of nx among N, that is: N!/n1!(N - n1)!

Next: n2 = N - n1
If we choose n2 particles for the level "2", we have then C(n2,N) possiblities to take this choice among (n2 = N - n1) particles. The partition function for n2 particles is ξ₂ = C(n2, N - n1) x exp{ - β (ε₂ - μ)n₂}
... etc ...

The total of possible choices is the multiplicity Ω(j), that is the product Π [N!/n_j!(N - n_j)!].
It akes the form: Π [N!/n_j!(N - n_j)!] = N!/n1!(N - n1)! x N!/n1!(N - n1)! x ... = Π (N!/n_j!)

N!/Πn_j!. The total partition function for this configuration is the product, over "j", of the ones correponding to each choice n_j. We can write then: Ξ {n_j} = N!/Πn_j! x Π ξ_j [product over the levels "j"].
Or:
Ξ {n_j} = N!/Πn_j! x Π exp{ - β (ε_j - μ)n_j} [product over the levels "j"]
Or:
Ξ {n_j} = N!/Πn_j! x exp{ - β ∑(ε_j - μ)n_j} [sum over the levels "j"]

Now, for the first level, if we had chosen n1. But we could choose n2, or n3, ... or N within each level "j". We have then to add all this possibilities within each level "j". It follows that:
Ξ = ∑ Ξ {n_j} = ∑ Ω(j) x Π exp{ - β (ε_j - μ)n_j} = [product over the levels "j" and sum over {n_j}]