# Partition Functions

The **partition function** of a system is given by

\[ \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}\]

where *H* is the Hamiltonian. The symbol *Z* is from the German *Zustandssumme* meaning "sum over states". The canonical ensemble partition function of a system in contact with a thermal bath at temperature \(T\) is the normalization constant of the Boltzmann distribution function, and therefore its expression is given by

\[Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE\],

where \(\Omega(E)\) is the density of states with energy \(E\) and \(k_B\) the Boltzmann constant. In classical statistical mechanics, there is a close connection between the partition function and the configuration integral, which has played an important role in many applications (e.g., drug design).

## Helmholtz energy function

The partition function of a system is related to the Helmholtz energy function through the formula

\[\left.A\right.=-k_BT\log Z.\]

This connection can be derived from the fact that \(k_B\log\Omega(E)\) is the entropy of a system with total energy \(E\). This is an extensive magnitude in the sense that, for large systems (i.e. in the thermodynamic limit, when the number of particles \(N\to\infty\) or the volume \(V\to\infty\)), it is proportional to \(N\) or \(V\). In other words, if we assume \(N\) large, then

\[\left.k_B\right. \log\Omega(E)=Ns(e),\]

where \(s(e)\) is the entropy per particle in the thermodynamic limit, which is a function of the energy per particle \(e=E/N\). We can therefore write

\[\left.Z(T)\right.=N\int \exp\{N(s(e)-e/T)/k_B\}\,de.\]

Since \(N\) is large, this integral can be performed through steepest descent, and we obtain

\[\left.Z(T)\right.=N\exp\{N(s(e_0)-e_0/k_BT)\}\],

where \(e_0\) is the value that maximizes the argument in the exponential; in other words, the solution to

\[\left.s'(e_0)\right.=1/T.\]

This is the thermodynamic formula for the inverse temperature provided \(e_0\) is the mean energy per particle of the system. On the other hand, the argument in the exponential is

\[\frac{1}{k_BT}(TS(E_0)-E_0)=-\frac{A}{k_BT}\]

the thermodynamic definition of the Helmholtz energy function. Thus, when \(N\) is large,

\[\left.A\right.=-k_BT\log Z(T).\]

## Connection with thermodynamics

We have the aforementioned Helmholtz energy function,

\[\left.A\right.=-k_BT\log Z(T)\]

we also have the internal energy, which is given by

\[U=k_B T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}\]

and the pressure, which is given by

\[p=k_B T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}\].

These equations provide a link between classical thermodynamics and statistical mechanics