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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?

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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.

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