Boltzmann-Grad limit ($N{a}^{d-1}\sim 1,N{a}^d\to 0$)

For $N$ hard particles in a fixed macroscopic volume, shrink their diameter $a$ while keeping $N{a}^{d-1}$ of order one. The occupied volume fraction tends to zero, but the mean free path stays macroscopic. Deriving the Boltzmann equation in this dilute limit requires control of recollisions and propagation of an incoming independence hypothesis related to molecular chaos for the Boltzmann equation. Its validity is not an automatic all-time consequence of small particle size.