Partition function

The partition function of an ensemble is the normalization factor of the Probability density function ρ\rho for the states. It includes the statistical properties of a system in thermal equilibrium and most thermodynamic variables, like temperature and entropy, can be derived from it directly or from its derivatives. As such, it is a generating function. It is the sum of the density function over all possible states:

Z=∫ρ(q,p) d3Nq d3Np,Z=∑nρ^(q^,p^)=Tr(ρ^)Z=\int\rho(\mathbf{q},\mathbf{p})\ d^{3N}q\ d^{3N}p,\qquad Z=\sum_{n}\hat{\rho}(\hat{q},\hat{p})=\text{Tr}(\hat{\rho})

The first case is the classical one, where integration happens over all phase space. The second is the quantum case, which instead uses the density matrix and its trace.

It is usually denoted using ZZ, which comes from German "Zustandsumme", or QNQ_{N}. The most common partition functions are the canonical partition function of the canonical ensemble QNQ_{N} and the grand canonical partition function of the grand canonical ensemble Z\mathcal{Z}. Their expressions change depending on whether we are consider the classical or quantum ensemble and whether particles are distinguishable or not.

Properties

The probability of a state in an ensemble is given by the density function normalized by the partition function

p(n)=ρ(n)Zp(n)=\frac{\rho(n)}{Z}

For independent and classical systems, canonical partition functions multiply. That is, the partition function of two systems that do not interact is

Z=Z1Z2Z=Z_{1}Z_{2}

This is because the energy of the combined system is just the sum of the energy of each system:

Z=∑n∑me−β(En(1)+Em(2))=∑n∑me−βEn(1)e−βEm(n)=∑ne−βEn(1)∑me−βEm(2)=Z1Z2\begin{align} Z&=\sum_{n}\sum_{m}e^{-\beta(E_{n}^{(1)}+E_{m}^{(2)})}=\sum_{n}\sum_{m}e^{-\beta E_{n}^{(1)}}e^{-\beta E_{m}^{(n)}} =\sum_{n}e^{-\beta E_{n}^{(1)}}\sum_{m}e^{-\beta E_{m}^{(2)}}=Z_{1}Z_{2} \end{align}

More generally, for NN independent components equipped with the same partition function Z1Z_{1}, we have

Z=Z1NZ=Z_{1}^{N}

In quantum systems however, this is no longer true due to the wavefunction constraints of fermions and bosons and, more specifically, their indistinguishability. For bosons, every combination of states is permitted, but we don't want to overcount states that only differ by a permutation. To express this mathematically, nested sums now begin at the previous index:

Z=∑n∑m≥ne−β(En(1)+Em(2))(Bosons)Z=\sum_{n}\sum_{m\geq n}e^{-\beta(E_{n}^{(1)}+E_{m}^{(2)})}\quad\text{(Bosons)}

The sums are no longer independent, so we cannot solve them separately like in the classical case. The solution now depends on the shape of EnE_{n}. For fermions, particles also cannot simultaneously occupy the same state due to the Pauli exclusion principle, so the greater-or-equals becomes a simple greater:

Z=∑n∑m>ne−β(En(1)+Em(2))(Fermions)Z=\sum_{n}\sum_{m>n}e^{-\beta(E_{n}^{(1)}+E_{m}^{(2)})}\quad\text{(Fermions)}

Because of these constraints, solving quantum partition functions can be much more involved than classical ones. In practice, it shows that quantum systems are effectively never independent from each other, even if there are no formal interactions between components. For an abstract exercise on quantum state counting, see Occupation number > Average occupation.

To compute these sums, it is typically useful to consider the geometric series and some simple generalizations:

∑n=1∞xn=11−x,∑n≥m∞xn=xm1−x,∑x>m∞xm+11−x\sum_{n=1}^{\infty} x^{n}=\frac{1}{1-x},\qquad \sum_{n\geq m}^{\infty} x^{n}=\frac{x^{m}}{1-x},\qquad \sum_{x>m}^{\infty} \frac{x^{m+1}}{1-x}

Derived quantities

In general, the ensemble average of a quantity OO can be derived as

⟨O⟩=1Z∑Oρ(O)\langle O \rangle=\frac{1}{Z}\sum O\rho(O)

where summation (or integration, if continuous) happens over all states. If there exists an xx for which

Oρ(O)=∂∂xρ(O)O\rho(O)=\frac{ \partial }{ \partial x } \rho(O)

is true, then the ensemble average can be rewritten as

⟨O⟩=∂∂xln⁡Z\langle O \rangle =\frac{ \partial }{ \partial x } \ln Z

This can be seen, for instance, in the internal energy of a system (where x=−βx=-\beta):

U=−∂∂βln⁡ZU=-\frac{ \partial }{ \partial \beta } \ln Z

where β=1/kBT\beta=1/k_{B}T with kBk_{B} the Boltzmann constant and TT the temperature.

The canonical partition function is related to the Helmholtz free energy by

QN=e−βA⇒A=−1βln⁡QNQ_{N}=e^{-\beta A}\quad\Rightarrow \quad A=- \frac{1}{\beta} \ln Q_{N}

Using Maxwell relations, this leads to an expression for entropy:

S=kB∂∂T(Tln⁡QN)S=k_{B}\frac{ \partial }{ \partial T }(T\ln Q_{N})