### 2.4 Calculation of the microcanonical ensemble

In the microcanonical ensemble we only sum up micro states ${p}_{i}$ with energies ${U}_{i}=U$. The remaining ${p}_{i}$ are zero.
We find

 $1=\sum _{i=1}^{W}{p}_{i}=\frac{1}{Z}exp\left(-\frac{U}{kT}\right)\sum _{i=1}^{W}1=\frac{1}{Z}exp\left(-\frac{U}{kT}\right)W\phantom{\rule{2em}{0ex}}.$ (2.23)

and

 $S=kln\left(Z\right)+\frac{1}{T}U=kln\left(W\right)\phantom{\rule{2em}{0ex}}.$ (2.24)

$W$: Number of the micro states of a system with energies $U$.
The isolated system is therefor characterized by $U=const.$ and $S=const.$