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\sect{Boltzmann approximation from the canonical ensemble}

Given ${N}$ particle gas with uniparticle state density function ${g\left(E\right)}$.

In the grand canonical ensemble, in Boltzman approximation, the results
we get for the state functions ${N\left(\beta\mu\right),
E\left(\beta \mu\right)}$ are
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\[N\left(\beta\mu\right)=\int_{0}^{\infty}g\left(E\right)dE\,\,\,f\left(E-\mu\right)\]
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\[E\left(\beta\mu\right)=\int g\left(E\right)dE\,\,\, E\cdot
f\left(E-\mu\right)\]
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Where ${f\left(E-\mu\right)=e^{-\beta\left(E-\mu\right)}}$ is called
the Boltzman occupation function.

In this exercise you need to show that you get those equations in
the framework of the approximation ${Z_{N}\approx \frac{1}{N!}Z_{1}^{N}}$.

For that, calculate ${Z}$, that you get from this proximity for
${Z_{N}}$ and derive the expressions for ${N\left(\beta\mu\right),
E\left(\beta\mu\right)}$.



