# Coarse graining in the derivation of Boltzmann equation

I am reading chapter 3 of Kardar's statistical physics of particles. I have a question about the coarse grainig. What is the origin of this coarse graining?

Is it from the integration of small relative distance region or from the approximation that we only consider the small relative distance limit?

From the explanation just above the highlight, it seems that this coarse graining is from the small relative distance approximation, i.e. , we only consider small relative distance. But the effective theory tells us that the high energy physics affects low energy only through some parameters while the details of high energy is not important. What we do is just integrating the high energy out. So for me integrate out small distance physics is some kind of coarse graining. What is the relation here?

• If recall the material right, it seems the author is suggesting that $f_2$ is essentially constant along $a$ a few $d$ from the collision meaning that the only contributions is the $\Delta f_2$ from before/after the collision, hence the two terms. This would be a coarse graining averaging the value of $f_2$ on each $d$ intervals. – G. Bergeron Dec 9 '16 at 11:04