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In this paper the problem is posed of the prescription of the so-calledBoltzmann-Grad (BG) limit ($\mathcal{L}_{BG}$) for the $N-$body system ofsmooth hard-spheres which undergo unary, binary as well as multiple elasticinstantaneous collisions. The statistical description is couched in terms ofthe Master kinetic equation, i.e., the kinetic equation which realizes theaxiomatic "\textit{ab initio}" approach to the classical statistical mechanicsof finite hard-sphere systems recently developed (Tessarotto \textit{et al.},2013-2017). The issue addressed here concerns the prescription of the BG-limitoperator and specifically the non-commutative property of $\mathcal{L}_{BG}$with the free-streaming operator which enters the same kinetic equation.\ It isshown\ that the form of the resulting limit equation remains in principlenon-unique, its precise realization depending critically on the way the actionof the same operator is prescribed. Implications for the global prescription ofthe Boltzmann equation are pointed out.

On the Boltzmann-Grad limit of the Master Kinetic Equation