# Density of states for a system of $N$ particles (both interacting and non-interacting case)

The density of states $\rho(\epsilon)$ for a single particle (with $g$ number of internal states such as spin) confined in a volume $V$ can be calculated as $$\rho(\epsilon)=\frac{4\pi gV}{h^3}p^2\frac{dp}{d\epsilon}$$ and using the appropriate dispersion relation $\epsilon=\epsilon(p)$ to compute $\frac{dp}{d\epsilon}$. How does one calculate the density of states for a system of $N$ particles (interacting and non-interacting) confined in a box? Does it remain $\rho(\epsilon)$? If yes, why?

Update According to @physshyp's comment (if I understand it correctly) for a system of $N$ non-interacting particles confined in a $V$, the DOS of the full system will be equal to the one-particle DOS. But for $N$ interacting particles, the DOS of the full system will be different from the one-particle DOS, and you need many-body Green's function technique to calculate that. Why is this the case?

• Can the $N$ particles exchange energy with each other? – Michael Seifert Apr 3 '18 at 19:50
• I would like to know both cases when the particles are interacting and when they are not. @MichaelSeifert – SRS Apr 3 '18 at 19:51
• For non interacting case i suppose DOS is independent of number of particles, it just tells you the number of avaliable states per energy. you can use it to calculate the number of particles in your system $N=\int d\epsilon \rho (\epsilon) n_{F}(\epsilon)$. for interacting case you should use Greens functions to compute it. – physshyp Apr 4 '18 at 10:30
• @physshyp Is there a formula that relates the density of states of an interacting N-body system and the Green's function? And which Green's function are you talking about? Is the computed from the many-body Schrodinger equation? – SRS Apr 4 '18 at 15:33
• yes there is a formula. I am talking about a many body greens function. it hasa a precise definition you can check this. also i think in both cases the number of particles is not related to the DOS. DOS is related to available states that your system allows. – physshyp Apr 4 '18 at 15:41