Partition function

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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 $Ω(E)$ 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).

[edit] 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 $k B logΩ(E)$ 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

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