Microcanonical Ensemble

It is curious that one of the fundamental ensembles of Statistical Physics, the Microcanonical, has a very simple but tricky expression for the entropy

$S = k_Bln\Omega(E;\Delta E)$

$k_B$is the Boltzmann constant and $\Omega(E;\Delta E)$is the number of states within the compatible energies$\Delta E$. In a formal manner one can write that as
$$\Omega(E;\Delta E) = \int_{E \le \mathcal H (\alpha) \le E +\Delta E}d\alpha$$where$d\alpha = d\vec q_1d\vec q_2\ldots d\vec q_Nd\vec p_1d\vec p_2\ldots d\vec p_N$.

This integral is not easy to compute even in the most simple cases because one should guess which is they way you count your states. But, searching for a more systematic calculation of the latter, the another day I founded in a book of Schwabl a easier way to handle this. Because we are "just" counting states, in the end one can use another "representation" for the microcanonical ensemble

$$\Omega(E;\Delta E) =\sum^{\infty}_{n_1 = 0}\ldots \sum^{\infty}_{n_N = 0}\delta(E-\mathcal H(q,p))=\sum^{\infty}_{n_1 = 0}\ldots \sum^{\infty}_{n_N = 0}\int \frac{dk}{2\pi}e^{ik(E-\mathcal H(q,p))}$$.

And from there is much easier to compute the solution. I will try to do some examples next time.