TITLE: About the $6N$ dimension when describing the Boltzmann equation
QUESTION [1 upvotes]: I have a question related with the formulation of the Boltzmann equation. In all the documents I read, it appears that the system is made of $N$ molecules, and so the phase space has $6N$ dimensions (in 3D). But later, all the treatment is with a $6$ dimension space. Why is the space reduced from $6N$ to $6$? In one of the descriptions, I found that in fact, the distribution $f(r,p,t)$ is the integration over all particles except one. Is this related to non-distinguibility of particles? Any physical clue on that?

REPLY [2 votes]: "...Is this related to non-distinguibility of particles?..."
Yes. But it is more subtle than that. We can integrate the full probability distribution function (for $N$ particles) , $f_N(x_1, p_1 ; x_2 , p_2 ; ... x_N , p_N ; t)$ over the variables $x_i, p_i$ for $i = 2...N$ , to generate the $1$ particle distribution $f_1 (x_1, p_1, t)$, because we are dealing with $N$ non-distinguishable particles .
Why we would be interested in $f_1$ is a different matter. What we are trying to generate, ultimately, are hydrodynamic equations for macroscopically conserved quantities (in the most simple case, mass/number , energy and momentum). All of these can be expressed as moments of $f_1$. Number density $N(x,t)$ = $\int_p f_1(x,p,t)$, energy density $E(x,t) = \int_p \frac{p^2}{2m} f_1(x,p,t)$ etc.
So, from the point of view of understanding macroscopic non-equilibrium phenomena, knowledge of $f_1$ is enough. This is why we want an equation for just $f_1$ (the Boltzmann equation).
Note that just because we would like an equation for $f_1$ doesn't mean that we can always write one down. The Liouville equation obeyed by the full distribution $f_N$ is equivalent to a chain of integro-differential equations, relating $f_1$ to $f_2$, $f_2$ to $f_3$ etc. This chain is called the BBGKY hierarchy. We can only write down the Boltzmann equation (an equation for $f_1$ that involves only $f_1$) , if we can truncate this chain after the first equation , by assuming that $f_2(1,2) \approx f_1(1) f_1(2)$. In other words, writing down the Boltzmann equation requires us to assume that motions of particles participating in "collisions" are uncorrelated before and after the collisions (basically, at any time except during the collisions) (also known as the assumption of "Molecular Chaos")
For example, see chapter $2$ of David Tong's notes on Kinetic theory (https://www.damtp.cam.ac.uk/user/tong/kinetic.html), or $\S$16, Physical Kinetics, Lifshitz and Pitaevskii
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