# Why does the Boltzmann equation deal with single-particle phase space density?

Why does the Boltzmann equation deal with single-particle phase space density $$\rho_{1}(\textbf{r}_1,\textbf{p}_1,t)$$ rather than the N-particle phase space density $$\rho(\{\textbf{r}_i,\textbf{p}_i,t\})$$? How'll deal with Boltzmann equation for a system of $$N$$ interacting particles?

That's the whole point. Rather than dealing with the complicated N-particle distribution we try to find an equation of motion for the 1-body distribution $$f_1(r_1,p_1,t)=\int d\Gamma_2\ldots d\Gamma_n\, f_N(r_1,p_1,r_2,p_2,\ldots).$$ Boltzmann brilliantly guessed an equation for $$f_1$$, and standard textbooks try to at least outline a proof that starts from the Liouville (classical) or von-Neumann (quantum) equation, but obviously there are some subtleties (how do you get the non-reversible Boltzmann equation from reversible microscopic dynamics), and much ink has been spilled discussing these subtleties.