= Boltzmann-Grad limit
{c}
{title2=$N a^{d-1}\sim1,\qquad N a^d\to0$}
For $N$ hard particles in a fixed macroscopic volume, shrink their diameter $a$ while keeping $Na^{d-1}$ 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.
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