I want to calculate the time evolution of the density of phase points for an ensemble of N harmonic oscillators.
However, I intended to do so without using the Liouville equation. Sure, I want to check my result is consistent with the Liouville equation.
I am not an expert in the topic; however, at this point I have deduced the following equation for the density:
$$\rho (p,q;t)=\frac{1}{\Omega}\delta(H(p,q)-E)$$
Which I obtained from solving the classical equation for phase space density:
$$\frac{1}{h^{3N}}\int d^{3N}p\,d^{3N}q\, \rho(p_{i},q_{i})=1$$
And remember:
$$f(n) = \begin{cases} \frac{1}{\Omega} & E \leq H(q_{i},p_{i})\leq E+\Delta E \\ 0 & \textrm{otherwise} \\ \end{cases}$$
Also, I got:
$$\Omega=\left(\frac{1}{\hbar \omega} \right)^{N}\frac{E}{\Gamma(N)}$$
Where $(p_{i}, q_{i})$ are the generalized coordinates of each harmonic oscillator.
However, from the results already obtained have not been able to calculate the time courses, I could make a suggestion. Or if my behavior is bad, they could help me to correct it. For your help and attention, thank you very much.
