**Definition 0.0.1 **  A **classical ideal gas in the microcanonical ensemble** is the system with hamiltonian $\mathcal{H}$ defined by
\[\mathcal{H} = \sum_{i=1}^N\frac{\mathbf{p}_i^2}{2m}\]
\[\Omega \approx \frac{2\pi^{3N/2}}{(3N/2 - 1)!} \frac{2m^{3N/2}}{N!(2\pi\hbar)^{3N}3N} \frac{3N\Delta}{2E}V^NE^{3N/2}\]