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

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

where $\Omega (E)$ 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

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

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

$\left.Z(T)\right.=N\exp\{N(s(e_{0})-e_{0}/k_{B}T)\}$,

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

${\frac {1}{k_{B}T}}(TS(E_{0})-E_{0})=-{\frac {A}{k_{B}T}}$

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

$\left.A\right.=-k_{B}T\log Z(T).$

Connection with thermodynamics

We have the aforementioned Helmholtz energy function,

$\left.A\right.=-k_{B}T\log Z(T)$

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