Partition function

The 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

$Z(T)=\int \mathrm{\Omega }(E)\exp (-E/k_{B}T)dE$,

where $\mathrm{\Omega }(E)$ is the density of states with energy $E$ and $k_B$ the Boltzmann constant.

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

$A=-k_{B}T\log Z.$

This connection can be derived from the fact that $k_B\log \mathrm{\Omega }(E)$ 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 $N\to \mathrm{\infty }$ or the volume $V\to \mathrm{\infty }$), it is proportional to $N$ or $V$. In other words, if we assume $N$ large, then

$k_B\log \mathrm{\Omega }(E)=Ns(e),$

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

$Z(T)=N\int \exp \{N(s(e)-e/T)/k_B\}de.$

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

$Z(T)=N\exp \{N(s(e_0)-e_0/k_{B}T)\}$,

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

$s^{\prime }(e_0)=1/T.$

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

$\frac{1}{k_{B}T}(TS(E_0)-E_0)=-\frac{A}{k_{B}T}$

the thermodynamic definition of the Helmholtz energy function. Thus, when $N$ is large,

$A=-k_{B}T\log Z.$