Classical microcanonical phase volume

Set

 context $\langle \mathcal M, \mathcal H,\pi,\pi_0,{\hat\rho},{\hat\rho}_0\rangle$ … classical microcanonical ensemble context $\mathrm{dim}(\mathcal M) = 3N$ context $\hbar$ … Reduced Planck's constant definiendum $\Gamma(E):=\frac{1}{h^{3N} N!}\int_{\{\langle{\bf q},{\bf p}\rangle\in \Gamma_{\mathcal M}\ |\ E\le H({\bf q},{\bf p})\le E+\Delta\}} \mathrm d\Gamma$

Alternative definitions

$\Gamma(E):=\frac{1}{h^{3N}N!}\int_{\Gamma_{\mathcal M}} \hat\rho(E;{\bf q},{\bf p}) \ \mathrm d\Gamma$

And here we see that this is the analog of Classical canonical partition function for the Classical canonical ensemble.