##### Contents

Combinatorics

Probability & Statistics

# 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 nj particles occupaying each level "j", we can write:
N = ∑nj. 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 = ∑nj&epsilon.j = ∑Ej, where εj is the energy of each particle in the level "j", and Ej is the energy of the level "j", that is the energy of nj particles in the level "j" carrying each the energy εj.

The probability to have nj individual paricles in the level "j" is P(nj) = nj/N = constant x exp{- βεj}. We normalize this probability to 1 to find the constant:
∑P(nj) = constant x ∑ exp{ - βEj} = 1.
Then constant = 1/Z ; where Z = ∑ exp{ - βEj}, 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(Ej - 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(Ej - μN)} when N is fixed, and
Ξ = ∑ exp{ - β ∑(εj - μ)nj}
when N fluctuates, in the case of grand canonical description

If &x;1j = exp{ -β (εj -μ)} is the partition function for one particle of energy εj
in the level "j", then for nj particles in the elevel "j", we have ξnj = Π &x;1j [the product is nj times].
ξnj = Π &x;1j = Π exp{ -β (εj -μ)} = ∑ exp{ - β ∑(εj - μ)nj}.
But nj goes from 0 to N. within each level "j", we have a block {nj} of nj particles to consider. Hence,
the partition function of the level "j" is then Ξj = ∑&x;nj [the sum is over {nj}].

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

For all levels "j" of the system; occupied by nj particles:
Ξ = Π ξj = Π ∑ exp{ - β ∑(εj - μ)nj},
[the sum is over {nj} 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 ξn1 = 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!/nj!(N - nj)!].
It akes the form: Π [N!/nj!(N - nj)!] = N!/n1!(N - n1)! x N!/n1!(N - n1)! x ... = Π (N!/nj!)

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

The total partition function for a systen with "j" levels occupied by nj distinguishable particles carryiny each one the energy εj, along with the constaraint ∑nj = N, is:
Ξ = ∑ N!/Πnj! x Π exp{ - β (εj - μ)nj}
Ξ = ∑ N!/Πnj! x exp{ - β ∑ (εj - μ)nj}
[product over the levels "j" and sum over {nj}]

2.N is not fixed and the particles are distiguishables:
In this cas: ∑nj = N is released; then if we choose nj for the first level, we can choose it for any level as well. The multiplicity becomes:
Ω(j) = Π C(nj,N) [product over the levels "j"]. The total partition function becomes:
Ξ = ∑ Π C(nj,N) x Π exp{ - β (εj - μ)nj} = ∑ Π [N!/nj!(N - nj)!] x exp{ - β ∑ (εj - μ)nj} [product over the levels "j" and sum over {nj}]

The total partition function for a systen with "j" levels occupied by nj distinguishable particles carryiny each one the energy εj, without the constaraint ∑nj = N, is:
Ξ = ∑ Π [N!/nj!(N - nj)!] x exp{ - β ∑ (εj - μ)nj}
= ∑ Π [N!/nj!] x exp{ - β ∑ (εj - μ)nj}
[product over the levels "j" and sum over {nj}]

3. N is fixed and the particles are indistiguishables: In this case, we consider the particles are distinguishable and divide by the their multiplicity by the number of permutations, that is N! which is called Gibbs' correction. The multiplicity becomes: Ω(j) = Π [1/nj!]. The total number of the particles nj within each level are different; nj depends on "j" (since, for eaxample, n2 goes from 0 to N - n1). We have:

The total partition function for a systen with "j" levels occupied by nj indistinguishable particles carryiny each one the energy εj, along with the constaraint ∑nj = N, is:
Ξ = ∑ Π [1/nj!] x Π exp{ - β (εj - μ)nj} = ∑ Π [1/nj!] x exp{ - β ∑ (εj - μ)nj} [product over the levels "j" and sum over {nj}]

4.N is not fixed and the particles are indistiguishables:
In this case, the system is fed with particles directly from th reservoir.; the sum over the configuration (or block){j} for each level "j" becomes over the numbers j .
The multiplicity remains Ω(j) = Π [1/nj!]; but the total number of the particles nj within each level are not different; nj does not depend on "j" (since, for eaxample, n2 goes from 0 to N). We have:

The total partition function for a systen with "j" levels occupied by nj indistinguishable particles carryiny each one the energy εj, without the constaraint ∑nj = N, is:
Ξ = ∑ Π [1/nj!] x Π exp{ - β (εj - μ)nj}
= ∑ Π [1/nj!] x Π [exp{ - β (εj - μ)}]nj
= ∑ Π [1/nj!] x [exp{ - β (εj - μ)}]nj
[product over the levels "j" and sum over nj]

Once the constraint is released, we can invert the sum and the produst and obtain:
Ξ = Π ∑ [1/nj!] x [exp{ - β (εj - μ)}]nj [product over the levels "j" and sum over nj]
= Π ∑ [exp{ - β (εj - μ)}]nj/nj!
According to the formula: ∑ xn/n! = ex, we have:
Ξ = Π exp{ - β (εj - μ)}
[product over the levels "j"]

We cal also write:
The partition function of a system is written as:
Ξ = Π ξλ, [product over the levels λ]
where ξλ is the partition function for the level λ.
ξλ = ∑ exp{ - β (ελ - μ)nλ}
[sum over the numbers nλ within the state λ]

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