Editing Partition function

The [[canonical ensemble]] '''partition function''' of a system in contact with a thermal bath
at temperature <math>T</math> is the normalization constant of the [[Boltzmann distribution]]
function, and therefore its expression is given by

:<math> \left. Z \right.= {\mathrm {Tr}} \{ e^{-\beta H} \}</math>

where ''H'' is the [[Hamiltonian]], or as

:<math>Z(T)=\int \Omega(E)\exp(-E/k_BT)\,dE</math>,

where <math>\Omega(E)</math> is the [[density of states]] with energy <math>E</math> and <math>k_B</math>
the [[Boltzmann constant]]. The symbol ''Z'' is from the German ''Zustandssumme'' meaning "sum over states".

In classical statistical mechanics, there is a close connection 
between the partition function and the 
[http://clesm.mae.ufl.edu/wiki.pub/index.php/Configuration_integral_%28statistical_mechanics%29 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

:<math>\left.A\right.=-k_BT\log Z.</math>

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

:<math>\left.k_B\right. \log\Omega(E)=Ns(e),</math>

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

:<math>\left.Z(T)\right.=N\int \exp\{N(s(e)-e/T)/k_B\}\,de.</math>

Since <math>N</math> is large, this integral can be performed through [[steepest descent]],
and we obtain

:<math>\left.Z(T)\right.=N\exp\{N(s(e_0)-e_0/k_BT)\}</math>,

where <math>e_0</math> is the value that maximizes the argument in the exponential; in other
words, the solution to

:<math>\left.s'(e_0)\right.=1/T.</math>

This is the thermodynamic formula for the inverse temperature provided <math>e_0</math> is
the mean energy per particle of the system. On the other hand, the argument in the exponential
is

:<math>\frac{1}{k_BT}(TS(E_0)-E_0)=-\frac{A}{k_BT}</math>

the thermodynamic definition of the [[Helmholtz energy function]]. Thus, when <math>N</math> is large,

:<math>\left.A\right.=-k_BT\log Z(T).</math>
==Connection with thermodynamics==
We have the aforementioned [[Helmholtz energy function]],

:<math>\left.A\right.=-k_BT\log Z(T)</math>

we also have the [[internal energy]], which is given by

:<math>U=k_B  T^{2} \left. \frac{\partial \log Z(T)}{\partial T} \right\vert_{N,V}</math>

and the pressure, which is given by

:<math>p=k_B  T \left. \frac{\partial \log Z(T)}{\partial V} \right\vert_{N,T}</math>.

These equations provide a link between [[Classical thermodynamics | classical thermodynamics]] and 
[[Statistical mechanics | statistical mechanics]]
==See also==
*[[Ideal gas partition function]]
[[category:classical thermodynamics]]
[[category:statistical mechanics]]