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# 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

### Contributors

07:49, 28 Oct 2013

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This material is based upon work supported by the National Science Foundation under Grant Number 1246120