===== Classical canonical partition function =====
==== Set ====
| @#55CCEE: context     | @#55CCEE: $ \langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ ... classical canonical ensemble |
| @#55CCEE: context     | @#55CCEE: $ \mathrm{dim}(\mathcal M) = 3N $ |
| @#55CCEE: context     | @#55CCEE: $ \hbar$ ... Reduced Planck's constant | 
| @#FFBB00: definiendum | @#FFBB00: $Z(\beta):=\frac{1}{h^{3N}N!}\int_{\Gamma_{\mathcal M}}\ \hat\rho(\beta;{\bf q},{\bf p}) \ \mathrm d\Gamma $ |

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=== Discussion ===
This mirrors the [[classical microcanonical phase volume]].

We have

$Z(\beta):=\frac{1}{h^{3N}N!}\int_{\Gamma_{\mathcal M}} \mathrm{e}^{-\beta\ H({\bf q},{\bf p})}\ \mathrm d\Gamma $

$U\equiv\langle H\rangle=-\dfrac{\partial}{\partial\beta}\log\,Z(\beta)$

== Usage as moment generating function for $H$ ==

The usage of $Z(\beta)$ is very similar to that of a characteristic function/moment generating function for a probability density $f(x)$:

$E[e^{-ikX}]:=\int f(x)\ \mathrm e^{-ikx}\ \mathrm dx$

in probability theory, where 

$E[X^n]=\left(i\frac{\partial}{\partial k}\right)_{k=0}^nE[e^{-ikX}].$

In fact, to compute expectation values with the partition function is even a little more straight forward, since in that case the probability function itself has exponential form and the temperature acts like the otherwise auxiliary parameter $k$. Notice a second difference: The parameter $\beta$ is multiplied by the energy $H(q,p)$, not directly by the variables of integration $q,p$.

=== In terms of the density of states ===
See [[Classical density of states|density of states]]

$ Z(\beta)=\int{\mathrm e}^{-\beta\,E}{\mathrm d}\varphi(E)=\int D(E)\ \mathrm{e}^{-\beta\ E}\ \mathrm d E  $

$U\equiv\langle H\rangle=-\dfrac{\partial}{\partial\beta}\log\,Z(\beta)=\int E\,\dfrac{{\mathrm e}^{-\beta\,E}}{Z(\beta)}\,D(E)\,{\mathrm d}E$

Note that in QM, there is no phase spaces to smoothly integrate over. 

>Experimental observation says that certain objects (the ones involved in experiments measuring the energy spectrum from a box, the sun, etc.) the itegrand $E\,\dfrac{{\mathrm e}^{-\beta\,E}}{Z(\beta)}\,D(E)$ ought to be replaces by $\propto E\,\dfrac{1}{\mathrm e^{\beta\hbar\omega}-1}\,E^2$. 
>For a single particle and $E\propto \omega\propto |k|$, it can indeed by argued that $D(E)\propto E^2$, see [[Classical density of states|density of states]]. The temperature dependency of the energy distribution however isn't observed to follow a Boltzmann Distribution but instead a statistics known from the [[Grand canonical partition function]].
>
>Planck's ad hoc step is to say that any radiation ray of energy $E$ is actually partitioned into $n$ parts of $(\hbar\omega)$. Here $n$ isn't fixed but partioning is instead also thermailzed (that's basically the Definition of a black body) and thus follows a Boltzmann distribution in $(n\hbar\omega)$. His derivation amounts to raplacing certain functions of energy (they are energy densities, if we introduce spatial variables into the model)
>
>If we compare the classical expression with the empirical result, the replacement must be
>
>$E\,\dfrac{{\mathrm e}^{-\beta\,E}}{Z(\beta)}\mapsto\sum_{n=0}^\infty\dfrac{(n(\hbar\omega))\,{\mathrm e}^{-\beta\,n\,(\hbar\omega)}}{\sum_{n=0}^\infty{\mathrm e}^{-\beta\,n\,(\hbar\omega)}}=(\hbar\omega)\dfrac{1}{\mathrm e^{\beta\hbar\omega}-1}$

The common derivation of **Plancks law** makes the above step and then neglects spatial inhomogenies and just introduces a characteristic length $L$. Then we can use the characteristic speed (of light) $c$ to get a characteristic frequency $c/L$. Thus we can intorduce powers of $\omega$ via the unitless expression $\omega\left/\right.\dfrac{c}{L}$. If $D$ is quadratic (the case of photon gas), we get

^ $U=\int_0^\infty \dfrac{1}{\pi^2}\dfrac{1}{\mathrm e^{\beta\hbar\omega}-1} \left(\omega\left/\right.\dfrac{c}{L}\right)^3 \mathrm d(\hbar\omega)$ ^

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=== Context ===
[[Classical canonical ensemble]]
=== Related ===
[[Classical density of states]]